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""" This is a shim file to provide backwards compatibility (ccode.py was renamed to c.py in SymPy 1.7). """ from sympy.utilities.exceptions import SymPyDeprecationWarning SymPyDeprecationWarning( feature="importing from sympy.printing.ccode", useinstead="Import from sympy.printing.c", issue=20256, deprecated_since_version="1.7").warn() from .c import (ccode, print_ccode, known_functions_C89, known_functions_C99, # noqa:F401 reserved_words, reserved_words_c99, get_math_macros, C89CodePrinter, C99CodePrinter, C11CodePrinter, c_code_printers)
0a6f58d30d400d3fe5f40d9c4c4ae1f07c90fe0572355102e58ac18305d789f3
""" A MathML printer. """ from typing import Any, Dict from sympy import sympify, S, Mul from sympy.core.compatibility import default_sort_key from sympy.core.function import _coeff_isneg from sympy.printing.conventions import split_super_sub, requires_partial from sympy.printing.precedence import \ precedence_traditional, PRECEDENCE, PRECEDENCE_TRADITIONAL from sympy.printing.pretty.pretty_symbology import greek_unicode from sympy.printing.printer import Printer, print_function import mpmath.libmp as mlib from mpmath.libmp import prec_to_dps, repr_dps, to_str as mlib_to_str class MathMLPrinterBase(Printer): """Contains common code required for MathMLContentPrinter and MathMLPresentationPrinter. """ _default_settings = { "order": None, "encoding": "utf-8", "fold_frac_powers": False, "fold_func_brackets": False, "fold_short_frac": None, "inv_trig_style": "abbreviated", "ln_notation": False, "long_frac_ratio": None, "mat_delim": "[", "mat_symbol_style": "plain", "mul_symbol": None, "root_notation": True, "symbol_names": {}, "mul_symbol_mathml_numbers": '&#xB7;', } # type: Dict[str, Any] def __init__(self, settings=None): Printer.__init__(self, settings) from xml.dom.minidom import Document, Text self.dom = Document() # Workaround to allow strings to remain unescaped # Based on # https://stackoverflow.com/questions/38015864/python-xml-dom-minidom-\ # please-dont-escape-my-strings/38041194 class RawText(Text): def writexml(self, writer, indent='', addindent='', newl=''): if self.data: writer.write('{}{}{}'.format(indent, self.data, newl)) def createRawTextNode(data): r = RawText() r.data = data r.ownerDocument = self.dom return r self.dom.createTextNode = createRawTextNode def doprint(self, expr): """ Prints the expression as MathML. """ mathML = Printer._print(self, expr) unistr = mathML.toxml() xmlbstr = unistr.encode('ascii', 'xmlcharrefreplace') res = xmlbstr.decode() return res def apply_patch(self): # Applying the patch of xml.dom.minidom bug # Date: 2011-11-18 # Description: http://ronrothman.com/public/leftbraned/xml-dom-minidom\ # -toprettyxml-and-silly-whitespace/#best-solution # Issue: http://bugs.python.org/issue4147 # Patch: http://hg.python.org/cpython/rev/7262f8f276ff/ from xml.dom.minidom import Element, Text, Node, _write_data def writexml(self, writer, indent="", addindent="", newl=""): # indent = current indentation # addindent = indentation to add to higher levels # newl = newline string writer.write(indent + "<" + self.tagName) attrs = self._get_attributes() a_names = list(attrs.keys()) a_names.sort() for a_name in a_names: writer.write(" %s=\"" % a_name) _write_data(writer, attrs[a_name].value) writer.write("\"") if self.childNodes: writer.write(">") if (len(self.childNodes) == 1 and self.childNodes[0].nodeType == Node.TEXT_NODE): self.childNodes[0].writexml(writer, '', '', '') else: writer.write(newl) for node in self.childNodes: node.writexml( writer, indent + addindent, addindent, newl) writer.write(indent) writer.write("</%s>%s" % (self.tagName, newl)) else: writer.write("/>%s" % (newl)) self._Element_writexml_old = Element.writexml Element.writexml = writexml def writexml(self, writer, indent="", addindent="", newl=""): _write_data(writer, "%s%s%s" % (indent, self.data, newl)) self._Text_writexml_old = Text.writexml Text.writexml = writexml def restore_patch(self): from xml.dom.minidom import Element, Text Element.writexml = self._Element_writexml_old Text.writexml = self._Text_writexml_old class MathMLContentPrinter(MathMLPrinterBase): """Prints an expression to the Content MathML markup language. References: https://www.w3.org/TR/MathML2/chapter4.html """ printmethod = "_mathml_content" def mathml_tag(self, e): """Returns the MathML tag for an expression.""" translate = { 'Add': 'plus', 'Mul': 'times', 'Derivative': 'diff', 'Number': 'cn', 'int': 'cn', 'Pow': 'power', 'Max': 'max', 'Min': 'min', 'Abs': 'abs', 'And': 'and', 'Or': 'or', 'Xor': 'xor', 'Not': 'not', 'Implies': 'implies', 'Symbol': 'ci', 'MatrixSymbol': 'ci', 'RandomSymbol': 'ci', 'Integral': 'int', 'Sum': 'sum', 'sin': 'sin', 'cos': 'cos', 'tan': 'tan', 'cot': 'cot', 'csc': 'csc', 'sec': 'sec', 'sinh': 'sinh', 'cosh': 'cosh', 'tanh': 'tanh', 'coth': 'coth', 'csch': 'csch', 'sech': 'sech', 'asin': 'arcsin', 'asinh': 'arcsinh', 'acos': 'arccos', 'acosh': 'arccosh', 'atan': 'arctan', 'atanh': 'arctanh', 'atan2': 'arctan', 'acot': 'arccot', 'acoth': 'arccoth', 'asec': 'arcsec', 'asech': 'arcsech', 'acsc': 'arccsc', 'acsch': 'arccsch', 'log': 'ln', 'Equality': 'eq', 'Unequality': 'neq', 'GreaterThan': 'geq', 'LessThan': 'leq', 'StrictGreaterThan': 'gt', 'StrictLessThan': 'lt', 'Union': 'union', 'Intersection': 'intersect', } for cls in e.__class__.__mro__: n = cls.__name__ if n in translate: return translate[n] # Not found in the MRO set n = e.__class__.__name__ return n.lower() def _print_Mul(self, expr): if _coeff_isneg(expr): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('minus')) x.appendChild(self._print_Mul(-expr)) return x from sympy.simplify import fraction numer, denom = fraction(expr) if denom is not S.One: x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('divide')) x.appendChild(self._print(numer)) x.appendChild(self._print(denom)) return x coeff, terms = expr.as_coeff_mul() if coeff is S.One and len(terms) == 1: # XXX since the negative coefficient has been handled, I don't # think a coeff of 1 can remain return self._print(terms[0]) if self.order != 'old': terms = Mul._from_args(terms).as_ordered_factors() x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('times')) if coeff != 1: x.appendChild(self._print(coeff)) for term in terms: x.appendChild(self._print(term)) return x def _print_Add(self, expr, order=None): args = self._as_ordered_terms(expr, order=order) lastProcessed = self._print(args[0]) plusNodes = [] for arg in args[1:]: if _coeff_isneg(arg): # use minus x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('minus')) x.appendChild(lastProcessed) x.appendChild(self._print(-arg)) # invert expression since this is now minused lastProcessed = x if arg == args[-1]: plusNodes.append(lastProcessed) else: plusNodes.append(lastProcessed) lastProcessed = self._print(arg) if arg == args[-1]: plusNodes.append(self._print(arg)) if len(plusNodes) == 1: return lastProcessed x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('plus')) while plusNodes: x.appendChild(plusNodes.pop(0)) return x def _print_Piecewise(self, expr): if expr.args[-1].cond != True: # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") root = self.dom.createElement('piecewise') for i, (e, c) in enumerate(expr.args): if i == len(expr.args) - 1 and c == True: piece = self.dom.createElement('otherwise') piece.appendChild(self._print(e)) else: piece = self.dom.createElement('piece') piece.appendChild(self._print(e)) piece.appendChild(self._print(c)) root.appendChild(piece) return root def _print_MatrixBase(self, m): x = self.dom.createElement('matrix') for i in range(m.rows): x_r = self.dom.createElement('matrixrow') for j in range(m.cols): x_r.appendChild(self._print(m[i, j])) x.appendChild(x_r) return x def _print_Rational(self, e): if e.q == 1: # don't divide x = self.dom.createElement('cn') x.appendChild(self.dom.createTextNode(str(e.p))) return x x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('divide')) # numerator xnum = self.dom.createElement('cn') xnum.appendChild(self.dom.createTextNode(str(e.p))) # denominator xdenom = self.dom.createElement('cn') xdenom.appendChild(self.dom.createTextNode(str(e.q))) x.appendChild(xnum) x.appendChild(xdenom) return x def _print_Limit(self, e): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement(self.mathml_tag(e))) x_1 = self.dom.createElement('bvar') x_2 = self.dom.createElement('lowlimit') x_1.appendChild(self._print(e.args[1])) x_2.appendChild(self._print(e.args[2])) x.appendChild(x_1) x.appendChild(x_2) x.appendChild(self._print(e.args[0])) return x def _print_ImaginaryUnit(self, e): return self.dom.createElement('imaginaryi') def _print_EulerGamma(self, e): return self.dom.createElement('eulergamma') def _print_GoldenRatio(self, e): """We use unicode #x3c6 for Greek letter phi as defined here http://www.w3.org/2003/entities/2007doc/isogrk1.html""" x = self.dom.createElement('cn') x.appendChild(self.dom.createTextNode("\N{GREEK SMALL LETTER PHI}")) return x def _print_Exp1(self, e): return self.dom.createElement('exponentiale') def _print_Pi(self, e): return self.dom.createElement('pi') def _print_Infinity(self, e): return self.dom.createElement('infinity') def _print_NaN(self, e): return self.dom.createElement('notanumber') def _print_EmptySet(self, e): return self.dom.createElement('emptyset') def _print_BooleanTrue(self, e): return self.dom.createElement('true') def _print_BooleanFalse(self, e): return self.dom.createElement('false') def _print_NegativeInfinity(self, e): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('minus')) x.appendChild(self.dom.createElement('infinity')) return x def _print_Integral(self, e): def lime_recur(limits): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement(self.mathml_tag(e))) bvar_elem = self.dom.createElement('bvar') bvar_elem.appendChild(self._print(limits[0][0])) x.appendChild(bvar_elem) if len(limits[0]) == 3: low_elem = self.dom.createElement('lowlimit') low_elem.appendChild(self._print(limits[0][1])) x.appendChild(low_elem) up_elem = self.dom.createElement('uplimit') up_elem.appendChild(self._print(limits[0][2])) x.appendChild(up_elem) if len(limits[0]) == 2: up_elem = self.dom.createElement('uplimit') up_elem.appendChild(self._print(limits[0][1])) x.appendChild(up_elem) if len(limits) == 1: x.appendChild(self._print(e.function)) else: x.appendChild(lime_recur(limits[1:])) return x limits = list(e.limits) limits.reverse() return lime_recur(limits) def _print_Sum(self, e): # Printer can be shared because Sum and Integral have the # same internal representation. return self._print_Integral(e) def _print_Symbol(self, sym): ci = self.dom.createElement(self.mathml_tag(sym)) def join(items): if len(items) > 1: mrow = self.dom.createElement('mml:mrow') for i, item in enumerate(items): if i > 0: mo = self.dom.createElement('mml:mo') mo.appendChild(self.dom.createTextNode(" ")) mrow.appendChild(mo) mi = self.dom.createElement('mml:mi') mi.appendChild(self.dom.createTextNode(item)) mrow.appendChild(mi) return mrow else: mi = self.dom.createElement('mml:mi') mi.appendChild(self.dom.createTextNode(items[0])) return mi # translate name, supers and subs to unicode characters def translate(s): if s in greek_unicode: return greek_unicode.get(s) else: return s name, supers, subs = split_super_sub(sym.name) name = translate(name) supers = [translate(sup) for sup in supers] subs = [translate(sub) for sub in subs] mname = self.dom.createElement('mml:mi') mname.appendChild(self.dom.createTextNode(name)) if not supers: if not subs: ci.appendChild(self.dom.createTextNode(name)) else: msub = self.dom.createElement('mml:msub') msub.appendChild(mname) msub.appendChild(join(subs)) ci.appendChild(msub) else: if not subs: msup = self.dom.createElement('mml:msup') msup.appendChild(mname) msup.appendChild(join(supers)) ci.appendChild(msup) else: msubsup = self.dom.createElement('mml:msubsup') msubsup.appendChild(mname) msubsup.appendChild(join(subs)) msubsup.appendChild(join(supers)) ci.appendChild(msubsup) return ci _print_MatrixSymbol = _print_Symbol _print_RandomSymbol = _print_Symbol def _print_Pow(self, e): # Here we use root instead of power if the exponent is the reciprocal # of an integer if (self._settings['root_notation'] and e.exp.is_Rational and e.exp.p == 1): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('root')) if e.exp.q != 2: xmldeg = self.dom.createElement('degree') xmlci = self.dom.createElement('ci') xmlci.appendChild(self.dom.createTextNode(str(e.exp.q))) xmldeg.appendChild(xmlci) x.appendChild(xmldeg) x.appendChild(self._print(e.base)) return x x = self.dom.createElement('apply') x_1 = self.dom.createElement(self.mathml_tag(e)) x.appendChild(x_1) x.appendChild(self._print(e.base)) x.appendChild(self._print(e.exp)) return x def _print_Number(self, e): x = self.dom.createElement(self.mathml_tag(e)) x.appendChild(self.dom.createTextNode(str(e))) return x def _print_Float(self, e): x = self.dom.createElement(self.mathml_tag(e)) repr_e = mlib_to_str(e._mpf_, repr_dps(e._prec)) x.appendChild(self.dom.createTextNode(repr_e)) return x def _print_Derivative(self, e): x = self.dom.createElement('apply') diff_symbol = self.mathml_tag(e) if requires_partial(e.expr): diff_symbol = 'partialdiff' x.appendChild(self.dom.createElement(diff_symbol)) x_1 = self.dom.createElement('bvar') for sym, times in reversed(e.variable_count): x_1.appendChild(self._print(sym)) if times > 1: degree = self.dom.createElement('degree') degree.appendChild(self._print(sympify(times))) x_1.appendChild(degree) x.appendChild(x_1) x.appendChild(self._print(e.expr)) return x def _print_Function(self, e): x = self.dom.createElement("apply") x.appendChild(self.dom.createElement(self.mathml_tag(e))) for arg in e.args: x.appendChild(self._print(arg)) return x def _print_Basic(self, e): x = self.dom.createElement(self.mathml_tag(e)) for arg in e.args: x.appendChild(self._print(arg)) return x def _print_AssocOp(self, e): x = self.dom.createElement('apply') x_1 = self.dom.createElement(self.mathml_tag(e)) x.appendChild(x_1) for arg in e.args: x.appendChild(self._print(arg)) return x def _print_Relational(self, e): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement(self.mathml_tag(e))) x.appendChild(self._print(e.lhs)) x.appendChild(self._print(e.rhs)) return x def _print_list(self, seq): """MathML reference for the <list> element: http://www.w3.org/TR/MathML2/chapter4.html#contm.list""" dom_element = self.dom.createElement('list') for item in seq: dom_element.appendChild(self._print(item)) return dom_element def _print_int(self, p): dom_element = self.dom.createElement(self.mathml_tag(p)) dom_element.appendChild(self.dom.createTextNode(str(p))) return dom_element _print_Implies = _print_AssocOp _print_Not = _print_AssocOp _print_Xor = _print_AssocOp def _print_FiniteSet(self, e): x = self.dom.createElement('set') for arg in e.args: x.appendChild(self._print(arg)) return x def _print_Complement(self, e): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('setdiff')) for arg in e.args: x.appendChild(self._print(arg)) return x def _print_ProductSet(self, e): x = self.dom.createElement('apply') x.appendChild(self.dom.createElement('cartesianproduct')) for arg in e.args: x.appendChild(self._print(arg)) return x # XXX Symmetric difference is not supported for MathML content printers. class MathMLPresentationPrinter(MathMLPrinterBase): """Prints an expression to the Presentation MathML markup language. References: https://www.w3.org/TR/MathML2/chapter3.html """ printmethod = "_mathml_presentation" def mathml_tag(self, e): """Returns the MathML tag for an expression.""" translate = { 'Number': 'mn', 'Limit': '&#x2192;', 'Derivative': '&dd;', 'int': 'mn', 'Symbol': 'mi', 'Integral': '&int;', 'Sum': '&#x2211;', 'sin': 'sin', 'cos': 'cos', 'tan': 'tan', 'cot': 'cot', 'asin': 'arcsin', 'asinh': 'arcsinh', 'acos': 'arccos', 'acosh': 'arccosh', 'atan': 'arctan', 'atanh': 'arctanh', 'acot': 'arccot', 'atan2': 'arctan', 'Equality': '=', 'Unequality': '&#x2260;', 'GreaterThan': '&#x2265;', 'LessThan': '&#x2264;', 'StrictGreaterThan': '>', 'StrictLessThan': '<', 'lerchphi': '&#x3A6;', 'zeta': '&#x3B6;', 'dirichlet_eta': '&#x3B7;', 'elliptic_k': '&#x39A;', 'lowergamma': '&#x3B3;', 'uppergamma': '&#x393;', 'gamma': '&#x393;', 'totient': '&#x3D5;', 'reduced_totient': '&#x3BB;', 'primenu': '&#x3BD;', 'primeomega': '&#x3A9;', 'fresnels': 'S', 'fresnelc': 'C', 'LambertW': 'W', 'Heaviside': '&#x398;', 'BooleanTrue': 'True', 'BooleanFalse': 'False', 'NoneType': 'None', 'mathieus': 'S', 'mathieuc': 'C', 'mathieusprime': 'S&#x2032;', 'mathieucprime': 'C&#x2032;', } def mul_symbol_selection(): if (self._settings["mul_symbol"] is None or self._settings["mul_symbol"] == 'None'): return '&InvisibleTimes;' elif self._settings["mul_symbol"] == 'times': return '&#xD7;' elif self._settings["mul_symbol"] == 'dot': return '&#xB7;' elif self._settings["mul_symbol"] == 'ldot': return '&#x2024;' elif not isinstance(self._settings["mul_symbol"], str): raise TypeError else: return self._settings["mul_symbol"] for cls in e.__class__.__mro__: n = cls.__name__ if n in translate: return translate[n] # Not found in the MRO set if e.__class__.__name__ == "Mul": return mul_symbol_selection() n = e.__class__.__name__ return n.lower() def parenthesize(self, item, level, strict=False): prec_val = precedence_traditional(item) if (prec_val < level) or ((not strict) and prec_val <= level): brac = self.dom.createElement('mfenced') brac.appendChild(self._print(item)) return brac else: return self._print(item) def _print_Mul(self, expr): def multiply(expr, mrow): from sympy.simplify import fraction numer, denom = fraction(expr) if denom is not S.One: frac = self.dom.createElement('mfrac') if self._settings["fold_short_frac"] and len(str(expr)) < 7: frac.setAttribute('bevelled', 'true') xnum = self._print(numer) xden = self._print(denom) frac.appendChild(xnum) frac.appendChild(xden) mrow.appendChild(frac) return mrow coeff, terms = expr.as_coeff_mul() if coeff is S.One and len(terms) == 1: mrow.appendChild(self._print(terms[0])) return mrow if self.order != 'old': terms = Mul._from_args(terms).as_ordered_factors() if coeff != 1: x = self._print(coeff) y = self.dom.createElement('mo') y.appendChild(self.dom.createTextNode(self.mathml_tag(expr))) mrow.appendChild(x) mrow.appendChild(y) for term in terms: mrow.appendChild(self.parenthesize(term, PRECEDENCE['Mul'])) if not term == terms[-1]: y = self.dom.createElement('mo') y.appendChild(self.dom.createTextNode(self.mathml_tag(expr))) mrow.appendChild(y) return mrow mrow = self.dom.createElement('mrow') if _coeff_isneg(expr): x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode('-')) mrow.appendChild(x) mrow = multiply(-expr, mrow) else: mrow = multiply(expr, mrow) return mrow def _print_Add(self, expr, order=None): mrow = self.dom.createElement('mrow') args = self._as_ordered_terms(expr, order=order) mrow.appendChild(self._print(args[0])) for arg in args[1:]: if _coeff_isneg(arg): # use minus x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode('-')) y = self._print(-arg) # invert expression since this is now minused else: x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode('+')) y = self._print(arg) mrow.appendChild(x) mrow.appendChild(y) return mrow def _print_MatrixBase(self, m): table = self.dom.createElement('mtable') for i in range(m.rows): x = self.dom.createElement('mtr') for j in range(m.cols): y = self.dom.createElement('mtd') y.appendChild(self._print(m[i, j])) x.appendChild(y) table.appendChild(x) if self._settings["mat_delim"] == '': return table brac = self.dom.createElement('mfenced') if self._settings["mat_delim"] == "[": brac.setAttribute('close', ']') brac.setAttribute('open', '[') brac.appendChild(table) return brac def _get_printed_Rational(self, e, folded=None): if e.p < 0: p = -e.p else: p = e.p x = self.dom.createElement('mfrac') if folded or self._settings["fold_short_frac"]: x.setAttribute('bevelled', 'true') x.appendChild(self._print(p)) x.appendChild(self._print(e.q)) if e.p < 0: mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('-')) mrow.appendChild(mo) mrow.appendChild(x) return mrow else: return x def _print_Rational(self, e): if e.q == 1: # don't divide return self._print(e.p) return self._get_printed_Rational(e, self._settings["fold_short_frac"]) def _print_Limit(self, e): mrow = self.dom.createElement('mrow') munder = self.dom.createElement('munder') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('lim')) x = self.dom.createElement('mrow') x_1 = self._print(e.args[1]) arrow = self.dom.createElement('mo') arrow.appendChild(self.dom.createTextNode(self.mathml_tag(e))) x_2 = self._print(e.args[2]) x.appendChild(x_1) x.appendChild(arrow) x.appendChild(x_2) munder.appendChild(mi) munder.appendChild(x) mrow.appendChild(munder) mrow.appendChild(self._print(e.args[0])) return mrow def _print_ImaginaryUnit(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&ImaginaryI;')) return x def _print_GoldenRatio(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x3A6;')) return x def _print_Exp1(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&ExponentialE;')) return x def _print_Pi(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&pi;')) return x def _print_Infinity(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x221E;')) return x def _print_NegativeInfinity(self, e): mrow = self.dom.createElement('mrow') y = self.dom.createElement('mo') y.appendChild(self.dom.createTextNode('-')) x = self._print_Infinity(e) mrow.appendChild(y) mrow.appendChild(x) return mrow def _print_HBar(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x210F;')) return x def _print_EulerGamma(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x3B3;')) return x def _print_TribonacciConstant(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('TribonacciConstant')) return x def _print_Dagger(self, e): msup = self.dom.createElement('msup') msup.appendChild(self._print(e.args[0])) msup.appendChild(self.dom.createTextNode('&#x2020;')) return msup def _print_Contains(self, e): mrow = self.dom.createElement('mrow') mrow.appendChild(self._print(e.args[0])) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2208;')) mrow.appendChild(mo) mrow.appendChild(self._print(e.args[1])) return mrow def _print_HilbertSpace(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x210B;')) return x def _print_ComplexSpace(self, e): msup = self.dom.createElement('msup') msup.appendChild(self.dom.createTextNode('&#x1D49E;')) msup.appendChild(self._print(e.args[0])) return msup def _print_FockSpace(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x2131;')) return x def _print_Integral(self, expr): intsymbols = {1: "&#x222B;", 2: "&#x222C;", 3: "&#x222D;"} mrow = self.dom.createElement('mrow') if len(expr.limits) <= 3 and all(len(lim) == 1 for lim in expr.limits): # Only up to three-integral signs exists mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode(intsymbols[len(expr.limits)])) mrow.appendChild(mo) else: # Either more than three or limits provided for lim in reversed(expr.limits): mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode(intsymbols[1])) if len(lim) == 1: mrow.appendChild(mo) if len(lim) == 2: msup = self.dom.createElement('msup') msup.appendChild(mo) msup.appendChild(self._print(lim[1])) mrow.appendChild(msup) if len(lim) == 3: msubsup = self.dom.createElement('msubsup') msubsup.appendChild(mo) msubsup.appendChild(self._print(lim[1])) msubsup.appendChild(self._print(lim[2])) mrow.appendChild(msubsup) # print function mrow.appendChild(self.parenthesize(expr.function, PRECEDENCE["Mul"], strict=True)) # print integration variables for lim in reversed(expr.limits): d = self.dom.createElement('mo') d.appendChild(self.dom.createTextNode('&dd;')) mrow.appendChild(d) mrow.appendChild(self._print(lim[0])) return mrow def _print_Sum(self, e): limits = list(e.limits) subsup = self.dom.createElement('munderover') low_elem = self._print(limits[0][1]) up_elem = self._print(limits[0][2]) summand = self.dom.createElement('mo') summand.appendChild(self.dom.createTextNode(self.mathml_tag(e))) low = self.dom.createElement('mrow') var = self._print(limits[0][0]) equal = self.dom.createElement('mo') equal.appendChild(self.dom.createTextNode('=')) low.appendChild(var) low.appendChild(equal) low.appendChild(low_elem) subsup.appendChild(summand) subsup.appendChild(low) subsup.appendChild(up_elem) mrow = self.dom.createElement('mrow') mrow.appendChild(subsup) if len(str(e.function)) == 1: mrow.appendChild(self._print(e.function)) else: fence = self.dom.createElement('mfenced') fence.appendChild(self._print(e.function)) mrow.appendChild(fence) return mrow def _print_Symbol(self, sym, style='plain'): def join(items): if len(items) > 1: mrow = self.dom.createElement('mrow') for i, item in enumerate(items): if i > 0: mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode(" ")) mrow.appendChild(mo) mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(item)) mrow.appendChild(mi) return mrow else: mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(items[0])) return mi # translate name, supers and subs to unicode characters def translate(s): if s in greek_unicode: return greek_unicode.get(s) else: return s name, supers, subs = split_super_sub(sym.name) name = translate(name) supers = [translate(sup) for sup in supers] subs = [translate(sub) for sub in subs] mname = self.dom.createElement('mi') mname.appendChild(self.dom.createTextNode(name)) if len(supers) == 0: if len(subs) == 0: x = mname else: x = self.dom.createElement('msub') x.appendChild(mname) x.appendChild(join(subs)) else: if len(subs) == 0: x = self.dom.createElement('msup') x.appendChild(mname) x.appendChild(join(supers)) else: x = self.dom.createElement('msubsup') x.appendChild(mname) x.appendChild(join(subs)) x.appendChild(join(supers)) # Set bold font? if style == 'bold': x.setAttribute('mathvariant', 'bold') return x def _print_MatrixSymbol(self, sym): return self._print_Symbol(sym, style=self._settings['mat_symbol_style']) _print_RandomSymbol = _print_Symbol def _print_conjugate(self, expr): enc = self.dom.createElement('menclose') enc.setAttribute('notation', 'top') enc.appendChild(self._print(expr.args[0])) return enc def _print_operator_after(self, op, expr): row = self.dom.createElement('mrow') row.appendChild(self.parenthesize(expr, PRECEDENCE["Func"])) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode(op)) row.appendChild(mo) return row def _print_factorial(self, expr): return self._print_operator_after('!', expr.args[0]) def _print_factorial2(self, expr): return self._print_operator_after('!!', expr.args[0]) def _print_binomial(self, expr): brac = self.dom.createElement('mfenced') frac = self.dom.createElement('mfrac') frac.setAttribute('linethickness', '0') frac.appendChild(self._print(expr.args[0])) frac.appendChild(self._print(expr.args[1])) brac.appendChild(frac) return brac def _print_Pow(self, e): # Here we use root instead of power if the exponent is the # reciprocal of an integer if (e.exp.is_Rational and abs(e.exp.p) == 1 and e.exp.q != 1 and self._settings['root_notation']): if e.exp.q == 2: x = self.dom.createElement('msqrt') x.appendChild(self._print(e.base)) if e.exp.q != 2: x = self.dom.createElement('mroot') x.appendChild(self._print(e.base)) x.appendChild(self._print(e.exp.q)) if e.exp.p == -1: frac = self.dom.createElement('mfrac') frac.appendChild(self._print(1)) frac.appendChild(x) return frac else: return x if e.exp.is_Rational and e.exp.q != 1: if e.exp.is_negative: top = self.dom.createElement('mfrac') top.appendChild(self._print(1)) x = self.dom.createElement('msup') x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) x.appendChild(self._get_printed_Rational(-e.exp, self._settings['fold_frac_powers'])) top.appendChild(x) return top else: x = self.dom.createElement('msup') x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) x.appendChild(self._get_printed_Rational(e.exp, self._settings['fold_frac_powers'])) return x if e.exp.is_negative: top = self.dom.createElement('mfrac') top.appendChild(self._print(1)) if e.exp == -1: top.appendChild(self._print(e.base)) else: x = self.dom.createElement('msup') x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) x.appendChild(self._print(-e.exp)) top.appendChild(x) return top x = self.dom.createElement('msup') x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) x.appendChild(self._print(e.exp)) return x def _print_Number(self, e): x = self.dom.createElement(self.mathml_tag(e)) x.appendChild(self.dom.createTextNode(str(e))) return x def _print_AccumulationBounds(self, i): brac = self.dom.createElement('mfenced') brac.setAttribute('close', '\u27e9') brac.setAttribute('open', '\u27e8') brac.appendChild(self._print(i.min)) brac.appendChild(self._print(i.max)) return brac def _print_Derivative(self, e): if requires_partial(e.expr): d = '&#x2202;' else: d = self.mathml_tag(e) # Determine denominator m = self.dom.createElement('mrow') dim = 0 # Total diff dimension, for numerator for sym, num in reversed(e.variable_count): dim += num if num >= 2: x = self.dom.createElement('msup') xx = self.dom.createElement('mo') xx.appendChild(self.dom.createTextNode(d)) x.appendChild(xx) x.appendChild(self._print(num)) else: x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode(d)) m.appendChild(x) y = self._print(sym) m.appendChild(y) mnum = self.dom.createElement('mrow') if dim >= 2: x = self.dom.createElement('msup') xx = self.dom.createElement('mo') xx.appendChild(self.dom.createTextNode(d)) x.appendChild(xx) x.appendChild(self._print(dim)) else: x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode(d)) mnum.appendChild(x) mrow = self.dom.createElement('mrow') frac = self.dom.createElement('mfrac') frac.appendChild(mnum) frac.appendChild(m) mrow.appendChild(frac) # Print function mrow.appendChild(self._print(e.expr)) return mrow def _print_Function(self, e): mrow = self.dom.createElement('mrow') x = self.dom.createElement('mi') if self.mathml_tag(e) == 'log' and self._settings["ln_notation"]: x.appendChild(self.dom.createTextNode('ln')) else: x.appendChild(self.dom.createTextNode(self.mathml_tag(e))) y = self.dom.createElement('mfenced') for arg in e.args: y.appendChild(self._print(arg)) mrow.appendChild(x) mrow.appendChild(y) return mrow def _print_Float(self, expr): # Based off of that in StrPrinter dps = prec_to_dps(expr._prec) str_real = mlib.to_str(expr._mpf_, dps, strip_zeros=True) # Must always have a mul symbol (as 2.5 10^{20} just looks odd) # thus we use the number separator separator = self._settings['mul_symbol_mathml_numbers'] mrow = self.dom.createElement('mrow') if 'e' in str_real: (mant, exp) = str_real.split('e') if exp[0] == '+': exp = exp[1:] mn = self.dom.createElement('mn') mn.appendChild(self.dom.createTextNode(mant)) mrow.appendChild(mn) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode(separator)) mrow.appendChild(mo) msup = self.dom.createElement('msup') mn = self.dom.createElement('mn') mn.appendChild(self.dom.createTextNode("10")) msup.appendChild(mn) mn = self.dom.createElement('mn') mn.appendChild(self.dom.createTextNode(exp)) msup.appendChild(mn) mrow.appendChild(msup) return mrow elif str_real == "+inf": return self._print_Infinity(None) elif str_real == "-inf": return self._print_NegativeInfinity(None) else: mn = self.dom.createElement('mn') mn.appendChild(self.dom.createTextNode(str_real)) return mn def _print_polylog(self, expr): mrow = self.dom.createElement('mrow') m = self.dom.createElement('msub') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('Li')) m.appendChild(mi) m.appendChild(self._print(expr.args[0])) mrow.appendChild(m) brac = self.dom.createElement('mfenced') brac.appendChild(self._print(expr.args[1])) mrow.appendChild(brac) return mrow def _print_Basic(self, e): mrow = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) mrow.appendChild(mi) brac = self.dom.createElement('mfenced') for arg in e.args: brac.appendChild(self._print(arg)) mrow.appendChild(brac) return mrow def _print_Tuple(self, e): mrow = self.dom.createElement('mrow') x = self.dom.createElement('mfenced') for arg in e.args: x.appendChild(self._print(arg)) mrow.appendChild(x) return mrow def _print_Interval(self, i): mrow = self.dom.createElement('mrow') brac = self.dom.createElement('mfenced') if i.start == i.end: # Most often, this type of Interval is converted to a FiniteSet brac.setAttribute('close', '}') brac.setAttribute('open', '{') brac.appendChild(self._print(i.start)) else: if i.right_open: brac.setAttribute('close', ')') else: brac.setAttribute('close', ']') if i.left_open: brac.setAttribute('open', '(') else: brac.setAttribute('open', '[') brac.appendChild(self._print(i.start)) brac.appendChild(self._print(i.end)) mrow.appendChild(brac) return mrow def _print_Abs(self, expr, exp=None): mrow = self.dom.createElement('mrow') x = self.dom.createElement('mfenced') x.setAttribute('close', '|') x.setAttribute('open', '|') x.appendChild(self._print(expr.args[0])) mrow.appendChild(x) return mrow _print_Determinant = _print_Abs def _print_re_im(self, c, expr): mrow = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'fraktur') mi.appendChild(self.dom.createTextNode(c)) mrow.appendChild(mi) brac = self.dom.createElement('mfenced') brac.appendChild(self._print(expr)) mrow.appendChild(brac) return mrow def _print_re(self, expr, exp=None): return self._print_re_im('R', expr.args[0]) def _print_im(self, expr, exp=None): return self._print_re_im('I', expr.args[0]) def _print_AssocOp(self, e): mrow = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) mrow.appendChild(mi) for arg in e.args: mrow.appendChild(self._print(arg)) return mrow def _print_SetOp(self, expr, symbol, prec): mrow = self.dom.createElement('mrow') mrow.appendChild(self.parenthesize(expr.args[0], prec)) for arg in expr.args[1:]: x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode(symbol)) y = self.parenthesize(arg, prec) mrow.appendChild(x) mrow.appendChild(y) return mrow def _print_Union(self, expr): prec = PRECEDENCE_TRADITIONAL['Union'] return self._print_SetOp(expr, '&#x222A;', prec) def _print_Intersection(self, expr): prec = PRECEDENCE_TRADITIONAL['Intersection'] return self._print_SetOp(expr, '&#x2229;', prec) def _print_Complement(self, expr): prec = PRECEDENCE_TRADITIONAL['Complement'] return self._print_SetOp(expr, '&#x2216;', prec) def _print_SymmetricDifference(self, expr): prec = PRECEDENCE_TRADITIONAL['SymmetricDifference'] return self._print_SetOp(expr, '&#x2206;', prec) def _print_ProductSet(self, expr): prec = PRECEDENCE_TRADITIONAL['ProductSet'] return self._print_SetOp(expr, '&#x00d7;', prec) def _print_FiniteSet(self, s): return self._print_set(s.args) def _print_set(self, s): items = sorted(s, key=default_sort_key) brac = self.dom.createElement('mfenced') brac.setAttribute('close', '}') brac.setAttribute('open', '{') for item in items: brac.appendChild(self._print(item)) return brac _print_frozenset = _print_set def _print_LogOp(self, args, symbol): mrow = self.dom.createElement('mrow') if args[0].is_Boolean and not args[0].is_Not: brac = self.dom.createElement('mfenced') brac.appendChild(self._print(args[0])) mrow.appendChild(brac) else: mrow.appendChild(self._print(args[0])) for arg in args[1:]: x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode(symbol)) if arg.is_Boolean and not arg.is_Not: y = self.dom.createElement('mfenced') y.appendChild(self._print(arg)) else: y = self._print(arg) mrow.appendChild(x) mrow.appendChild(y) return mrow def _print_BasisDependent(self, expr): from sympy.vector import Vector if expr == expr.zero: # Not clear if this is ever called return self._print(expr.zero) if isinstance(expr, Vector): items = expr.separate().items() else: items = [(0, expr)] mrow = self.dom.createElement('mrow') for system, vect in items: inneritems = list(vect.components.items()) inneritems.sort(key = lambda x:x[0].__str__()) for i, (k, v) in enumerate(inneritems): if v == 1: if i: # No + for first item mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('+')) mrow.appendChild(mo) mrow.appendChild(self._print(k)) elif v == -1: mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('-')) mrow.appendChild(mo) mrow.appendChild(self._print(k)) else: if i: # No + for first item mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('+')) mrow.appendChild(mo) mbrac = self.dom.createElement('mfenced') mbrac.appendChild(self._print(v)) mrow.appendChild(mbrac) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&InvisibleTimes;')) mrow.appendChild(mo) mrow.appendChild(self._print(k)) return mrow def _print_And(self, expr): args = sorted(expr.args, key=default_sort_key) return self._print_LogOp(args, '&#x2227;') def _print_Or(self, expr): args = sorted(expr.args, key=default_sort_key) return self._print_LogOp(args, '&#x2228;') def _print_Xor(self, expr): args = sorted(expr.args, key=default_sort_key) return self._print_LogOp(args, '&#x22BB;') def _print_Implies(self, expr): return self._print_LogOp(expr.args, '&#x21D2;') def _print_Equivalent(self, expr): args = sorted(expr.args, key=default_sort_key) return self._print_LogOp(args, '&#x21D4;') def _print_Not(self, e): mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#xAC;')) mrow.appendChild(mo) if (e.args[0].is_Boolean): x = self.dom.createElement('mfenced') x.appendChild(self._print(e.args[0])) else: x = self._print(e.args[0]) mrow.appendChild(x) return mrow def _print_bool(self, e): mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) return mi _print_BooleanTrue = _print_bool _print_BooleanFalse = _print_bool def _print_NoneType(self, e): mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) return mi def _print_Range(self, s): dots = "\u2026" brac = self.dom.createElement('mfenced') brac.setAttribute('close', '}') brac.setAttribute('open', '{') if s.start.is_infinite and s.stop.is_infinite: if s.step.is_positive: printset = dots, -1, 0, 1, dots else: printset = dots, 1, 0, -1, dots elif s.start.is_infinite: printset = dots, s[-1] - s.step, s[-1] elif s.stop.is_infinite: it = iter(s) printset = next(it), next(it), dots elif len(s) > 4: it = iter(s) printset = next(it), next(it), dots, s[-1] else: printset = tuple(s) for el in printset: if el == dots: mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(dots)) brac.appendChild(mi) else: brac.appendChild(self._print(el)) return brac def _hprint_variadic_function(self, expr): args = sorted(expr.args, key=default_sort_key) mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode((str(expr.func)).lower())) mrow.appendChild(mo) brac = self.dom.createElement('mfenced') for symbol in args: brac.appendChild(self._print(symbol)) mrow.appendChild(brac) return mrow _print_Min = _print_Max = _hprint_variadic_function def _print_exp(self, expr): msup = self.dom.createElement('msup') msup.appendChild(self._print_Exp1(None)) msup.appendChild(self._print(expr.args[0])) return msup def _print_Relational(self, e): mrow = self.dom.createElement('mrow') mrow.appendChild(self._print(e.lhs)) x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode(self.mathml_tag(e))) mrow.appendChild(x) mrow.appendChild(self._print(e.rhs)) return mrow def _print_int(self, p): dom_element = self.dom.createElement(self.mathml_tag(p)) dom_element.appendChild(self.dom.createTextNode(str(p))) return dom_element def _print_BaseScalar(self, e): msub = self.dom.createElement('msub') index, system = e._id mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'bold') mi.appendChild(self.dom.createTextNode(system._variable_names[index])) msub.appendChild(mi) mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'bold') mi.appendChild(self.dom.createTextNode(system._name)) msub.appendChild(mi) return msub def _print_BaseVector(self, e): msub = self.dom.createElement('msub') index, system = e._id mover = self.dom.createElement('mover') mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'bold') mi.appendChild(self.dom.createTextNode(system._vector_names[index])) mover.appendChild(mi) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('^')) mover.appendChild(mo) msub.appendChild(mover) mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'bold') mi.appendChild(self.dom.createTextNode(system._name)) msub.appendChild(mi) return msub def _print_VectorZero(self, e): mover = self.dom.createElement('mover') mi = self.dom.createElement('mi') mi.setAttribute('mathvariant', 'bold') mi.appendChild(self.dom.createTextNode("0")) mover.appendChild(mi) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('^')) mover.appendChild(mo) return mover def _print_Cross(self, expr): mrow = self.dom.createElement('mrow') vec1 = expr._expr1 vec2 = expr._expr2 mrow.appendChild(self.parenthesize(vec1, PRECEDENCE['Mul'])) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#xD7;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(vec2, PRECEDENCE['Mul'])) return mrow def _print_Curl(self, expr): mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2207;')) mrow.appendChild(mo) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#xD7;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) return mrow def _print_Divergence(self, expr): mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2207;')) mrow.appendChild(mo) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#xB7;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) return mrow def _print_Dot(self, expr): mrow = self.dom.createElement('mrow') vec1 = expr._expr1 vec2 = expr._expr2 mrow.appendChild(self.parenthesize(vec1, PRECEDENCE['Mul'])) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#xB7;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(vec2, PRECEDENCE['Mul'])) return mrow def _print_Gradient(self, expr): mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2207;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) return mrow def _print_Laplacian(self, expr): mrow = self.dom.createElement('mrow') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2206;')) mrow.appendChild(mo) mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) return mrow def _print_Integers(self, e): x = self.dom.createElement('mi') x.setAttribute('mathvariant', 'normal') x.appendChild(self.dom.createTextNode('&#x2124;')) return x def _print_Complexes(self, e): x = self.dom.createElement('mi') x.setAttribute('mathvariant', 'normal') x.appendChild(self.dom.createTextNode('&#x2102;')) return x def _print_Reals(self, e): x = self.dom.createElement('mi') x.setAttribute('mathvariant', 'normal') x.appendChild(self.dom.createTextNode('&#x211D;')) return x def _print_Naturals(self, e): x = self.dom.createElement('mi') x.setAttribute('mathvariant', 'normal') x.appendChild(self.dom.createTextNode('&#x2115;')) return x def _print_Naturals0(self, e): sub = self.dom.createElement('msub') x = self.dom.createElement('mi') x.setAttribute('mathvariant', 'normal') x.appendChild(self.dom.createTextNode('&#x2115;')) sub.appendChild(x) sub.appendChild(self._print(S.Zero)) return sub def _print_SingularityFunction(self, expr): shift = expr.args[0] - expr.args[1] power = expr.args[2] sup = self.dom.createElement('msup') brac = self.dom.createElement('mfenced') brac.setAttribute('close', '\u27e9') brac.setAttribute('open', '\u27e8') brac.appendChild(self._print(shift)) sup.appendChild(brac) sup.appendChild(self._print(power)) return sup def _print_NaN(self, e): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('NaN')) return x def _print_number_function(self, e, name): # Print name_arg[0] for one argument or name_arg[0](arg[1]) # for more than one argument sub = self.dom.createElement('msub') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode(name)) sub.appendChild(mi) sub.appendChild(self._print(e.args[0])) if len(e.args) == 1: return sub # TODO: copy-pasted from _print_Function: can we do better? mrow = self.dom.createElement('mrow') y = self.dom.createElement('mfenced') for arg in e.args[1:]: y.appendChild(self._print(arg)) mrow.appendChild(sub) mrow.appendChild(y) return mrow def _print_bernoulli(self, e): return self._print_number_function(e, 'B') _print_bell = _print_bernoulli def _print_catalan(self, e): return self._print_number_function(e, 'C') def _print_euler(self, e): return self._print_number_function(e, 'E') def _print_fibonacci(self, e): return self._print_number_function(e, 'F') def _print_lucas(self, e): return self._print_number_function(e, 'L') def _print_stieltjes(self, e): return self._print_number_function(e, '&#x03B3;') def _print_tribonacci(self, e): return self._print_number_function(e, 'T') def _print_ComplexInfinity(self, e): x = self.dom.createElement('mover') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x221E;')) x.appendChild(mo) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('~')) x.appendChild(mo) return x def _print_EmptySet(self, e): x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode('&#x2205;')) return x def _print_UniversalSet(self, e): x = self.dom.createElement('mo') x.appendChild(self.dom.createTextNode('&#x1D54C;')) return x def _print_Adjoint(self, expr): from sympy.matrices import MatrixSymbol mat = expr.arg sup = self.dom.createElement('msup') if not isinstance(mat, MatrixSymbol): brac = self.dom.createElement('mfenced') brac.appendChild(self._print(mat)) sup.appendChild(brac) else: sup.appendChild(self._print(mat)) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2020;')) sup.appendChild(mo) return sup def _print_Transpose(self, expr): from sympy.matrices import MatrixSymbol mat = expr.arg sup = self.dom.createElement('msup') if not isinstance(mat, MatrixSymbol): brac = self.dom.createElement('mfenced') brac.appendChild(self._print(mat)) sup.appendChild(brac) else: sup.appendChild(self._print(mat)) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('T')) sup.appendChild(mo) return sup def _print_Inverse(self, expr): from sympy.matrices import MatrixSymbol mat = expr.arg sup = self.dom.createElement('msup') if not isinstance(mat, MatrixSymbol): brac = self.dom.createElement('mfenced') brac.appendChild(self._print(mat)) sup.appendChild(brac) else: sup.appendChild(self._print(mat)) sup.appendChild(self._print(-1)) return sup def _print_MatMul(self, expr): from sympy import MatMul x = self.dom.createElement('mrow') args = expr.args if isinstance(args[0], Mul): args = args[0].as_ordered_factors() + list(args[1:]) else: args = list(args) if isinstance(expr, MatMul) and _coeff_isneg(expr): if args[0] == -1: args = args[1:] else: args[0] = -args[0] mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('-')) x.appendChild(mo) for arg in args[:-1]: x.appendChild(self.parenthesize(arg, precedence_traditional(expr), False)) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&InvisibleTimes;')) x.appendChild(mo) x.appendChild(self.parenthesize(args[-1], precedence_traditional(expr), False)) return x def _print_MatPow(self, expr): from sympy.matrices import MatrixSymbol base, exp = expr.base, expr.exp sup = self.dom.createElement('msup') if not isinstance(base, MatrixSymbol): brac = self.dom.createElement('mfenced') brac.appendChild(self._print(base)) sup.appendChild(brac) else: sup.appendChild(self._print(base)) sup.appendChild(self._print(exp)) return sup def _print_HadamardProduct(self, expr): x = self.dom.createElement('mrow') args = expr.args for arg in args[:-1]: x.appendChild( self.parenthesize(arg, precedence_traditional(expr), False)) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x2218;')) x.appendChild(mo) x.appendChild( self.parenthesize(args[-1], precedence_traditional(expr), False)) return x def _print_ZeroMatrix(self, Z): x = self.dom.createElement('mn') x.appendChild(self.dom.createTextNode('&#x1D7D8')) return x def _print_OneMatrix(self, Z): x = self.dom.createElement('mn') x.appendChild(self.dom.createTextNode('&#x1D7D9')) return x def _print_Identity(self, I): x = self.dom.createElement('mi') x.appendChild(self.dom.createTextNode('&#x1D540;')) return x def _print_floor(self, e): mrow = self.dom.createElement('mrow') x = self.dom.createElement('mfenced') x.setAttribute('close', '\u230B') x.setAttribute('open', '\u230A') x.appendChild(self._print(e.args[0])) mrow.appendChild(x) return mrow def _print_ceiling(self, e): mrow = self.dom.createElement('mrow') x = self.dom.createElement('mfenced') x.setAttribute('close', '\u2309') x.setAttribute('open', '\u2308') x.appendChild(self._print(e.args[0])) mrow.appendChild(x) return mrow def _print_Lambda(self, e): x = self.dom.createElement('mfenced') mrow = self.dom.createElement('mrow') symbols = e.args[0] if len(symbols) == 1: symbols = self._print(symbols[0]) else: symbols = self._print(symbols) mrow.appendChild(symbols) mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('&#x21A6;')) mrow.appendChild(mo) mrow.appendChild(self._print(e.args[1])) x.appendChild(mrow) return x def _print_tuple(self, e): x = self.dom.createElement('mfenced') for i in e: x.appendChild(self._print(i)) return x def _print_IndexedBase(self, e): return self._print(e.label) def _print_Indexed(self, e): x = self.dom.createElement('msub') x.appendChild(self._print(e.base)) if len(e.indices) == 1: x.appendChild(self._print(e.indices[0])) return x x.appendChild(self._print(e.indices)) return x def _print_MatrixElement(self, e): x = self.dom.createElement('msub') x.appendChild(self.parenthesize(e.parent, PRECEDENCE["Atom"], strict = True)) brac = self.dom.createElement('mfenced') brac.setAttribute("close", "") brac.setAttribute("open", "") for i in e.indices: brac.appendChild(self._print(i)) x.appendChild(brac) return x def _print_elliptic_f(self, e): x = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('&#x1d5a5;')) x.appendChild(mi) y = self.dom.createElement('mfenced') y.setAttribute("separators", "|") for i in e.args: y.appendChild(self._print(i)) x.appendChild(y) return x def _print_elliptic_e(self, e): x = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('&#x1d5a4;')) x.appendChild(mi) y = self.dom.createElement('mfenced') y.setAttribute("separators", "|") for i in e.args: y.appendChild(self._print(i)) x.appendChild(y) return x def _print_elliptic_pi(self, e): x = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('&#x1d6f1;')) x.appendChild(mi) y = self.dom.createElement('mfenced') if len(e.args) == 2: y.setAttribute("separators", "|") else: y.setAttribute("separators", ";|") for i in e.args: y.appendChild(self._print(i)) x.appendChild(y) return x def _print_Ei(self, e): x = self.dom.createElement('mrow') mi = self.dom.createElement('mi') mi.appendChild(self.dom.createTextNode('Ei')) x.appendChild(mi) x.appendChild(self._print(e.args)) return x def _print_expint(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('E')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x def _print_jacobi(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msubsup') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('P')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) y.appendChild(self._print(e.args[1:3])) x.appendChild(y) x.appendChild(self._print(e.args[3:])) return x def _print_gegenbauer(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msubsup') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('C')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) y.appendChild(self._print(e.args[1:2])) x.appendChild(y) x.appendChild(self._print(e.args[2:])) return x def _print_chebyshevt(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('T')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x def _print_chebyshevu(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('U')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x def _print_legendre(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('P')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x def _print_assoc_legendre(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msubsup') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('P')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) y.appendChild(self._print(e.args[1:2])) x.appendChild(y) x.appendChild(self._print(e.args[2:])) return x def _print_laguerre(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('L')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x def _print_assoc_laguerre(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msubsup') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('L')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) y.appendChild(self._print(e.args[1:2])) x.appendChild(y) x.appendChild(self._print(e.args[2:])) return x def _print_hermite(self, e): x = self.dom.createElement('mrow') y = self.dom.createElement('msub') mo = self.dom.createElement('mo') mo.appendChild(self.dom.createTextNode('H')) y.appendChild(mo) y.appendChild(self._print(e.args[0])) x.appendChild(y) x.appendChild(self._print(e.args[1:])) return x @print_function(MathMLPrinterBase) def mathml(expr, printer='content', **settings): """Returns the MathML representation of expr. If printer is presentation then prints Presentation MathML else prints content MathML. """ if printer == 'presentation': return MathMLPresentationPrinter(settings).doprint(expr) else: return MathMLContentPrinter(settings).doprint(expr) def print_mathml(expr, printer='content', **settings): """ Prints a pretty representation of the MathML code for expr. If printer is presentation then prints Presentation MathML else prints content MathML. Examples ======== >>> ## >>> from sympy.printing.mathml import print_mathml >>> from sympy.abc import x >>> print_mathml(x+1) #doctest: +NORMALIZE_WHITESPACE <apply> <plus/> <ci>x</ci> <cn>1</cn> </apply> >>> print_mathml(x+1, printer='presentation') <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> """ if printer == 'presentation': s = MathMLPresentationPrinter(settings) else: s = MathMLContentPrinter(settings) xml = s._print(sympify(expr)) s.apply_patch() pretty_xml = xml.toprettyxml() s.restore_patch() print(pretty_xml) # For backward compatibility MathMLPrinter = MathMLContentPrinter
0e6e70033972a7899bddcc0d503b351d57dd25dc791f1044679ffb423e32b3b0
""" R code printer The RCodePrinter converts single sympy expressions into single R expressions, using the functions defined in math.h where possible. """ from typing import Any, Dict from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence, PRECEDENCE from sympy.sets.fancysets import Range # dictionary mapping sympy function to (argument_conditions, C_function). # Used in RCodePrinter._print_Function(self) known_functions = { #"Abs": [(lambda x: not x.is_integer, "fabs")], "Abs": "abs", "sin": "sin", "cos": "cos", "tan": "tan", "asin": "asin", "acos": "acos", "atan": "atan", "atan2": "atan2", "exp": "exp", "log": "log", "erf": "erf", "sinh": "sinh", "cosh": "cosh", "tanh": "tanh", "asinh": "asinh", "acosh": "acosh", "atanh": "atanh", "floor": "floor", "ceiling": "ceiling", "sign": "sign", "Max": "max", "Min": "min", "factorial": "factorial", "gamma": "gamma", "digamma": "digamma", "trigamma": "trigamma", "beta": "beta", } # These are the core reserved words in the R language. Taken from: # https://cran.r-project.org/doc/manuals/r-release/R-lang.html#Reserved-words reserved_words = ['if', 'else', 'repeat', 'while', 'function', 'for', 'in', 'next', 'break', 'TRUE', 'FALSE', 'NULL', 'Inf', 'NaN', 'NA', 'NA_integer_', 'NA_real_', 'NA_complex_', 'NA_character_', 'volatile'] class RCodePrinter(CodePrinter): """A printer to convert python expressions to strings of R code""" printmethod = "_rcode" language = "R" _default_settings = { 'order': None, 'full_prec': 'auto', 'precision': 15, 'user_functions': {}, 'human': True, 'contract': True, 'dereference': set(), 'error_on_reserved': False, 'reserved_word_suffix': '_', } # type: Dict[str, Any] _operators = { 'and': '&', 'or': '|', 'not': '!', } _relationals = { } # type: Dict[str, str] def __init__(self, settings={}): CodePrinter.__init__(self, settings) self.known_functions = dict(known_functions) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) self._dereference = set(settings.get('dereference', [])) self.reserved_words = set(reserved_words) def _rate_index_position(self, p): return p*5 def _get_statement(self, codestring): return "%s;" % codestring def _get_comment(self, text): return "// {}".format(text) def _declare_number_const(self, name, value): return "{} = {};".format(name, value) def _format_code(self, lines): return self.indent_code(lines) def _traverse_matrix_indices(self, mat): rows, cols = mat.shape return ((i, j) for i in range(rows) for j in range(cols)) def _get_loop_opening_ending(self, indices): """Returns a tuple (open_lines, close_lines) containing lists of codelines """ open_lines = [] close_lines = [] loopstart = "for (%(var)s in %(start)s:%(end)s){" for i in indices: # R arrays start at 1 and end at dimension open_lines.append(loopstart % { 'var': self._print(i.label), 'start': self._print(i.lower+1), 'end': self._print(i.upper + 1)}) close_lines.append("}") return open_lines, close_lines def _print_Pow(self, expr): if "Pow" in self.known_functions: return self._print_Function(expr) PREC = precedence(expr) if expr.exp == -1: return '1.0/%s' % (self.parenthesize(expr.base, PREC)) elif expr.exp == 0.5: return 'sqrt(%s)' % self._print(expr.base) else: return '%s^%s' % (self.parenthesize(expr.base, PREC), self.parenthesize(expr.exp, PREC)) def _print_Rational(self, expr): p, q = int(expr.p), int(expr.q) return '%d.0/%d.0' % (p, q) def _print_Indexed(self, expr): inds = [ self._print(i) for i in expr.indices ] return "%s[%s]" % (self._print(expr.base.label), ", ".join(inds)) def _print_Idx(self, expr): return self._print(expr.label) def _print_Exp1(self, expr): return "exp(1)" def _print_Pi(self, expr): return 'pi' def _print_Infinity(self, expr): return 'Inf' def _print_NegativeInfinity(self, expr): return '-Inf' def _print_Assignment(self, expr): from sympy.codegen.ast import Assignment from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.tensor.indexed import IndexedBase lhs = expr.lhs rhs = expr.rhs # We special case assignments that take multiple lines #if isinstance(expr.rhs, Piecewise): # from sympy.functions.elementary.piecewise import Piecewise # # Here we modify Piecewise so each expression is now # # an Assignment, and then continue on the print. # expressions = [] # conditions = [] # for (e, c) in rhs.args: # expressions.append(Assignment(lhs, e)) # conditions.append(c) # temp = Piecewise(*zip(expressions, conditions)) # return self._print(temp) #elif isinstance(lhs, MatrixSymbol): if isinstance(lhs, MatrixSymbol): # Here we form an Assignment for each element in the array, # printing each one. lines = [] for (i, j) in self._traverse_matrix_indices(lhs): temp = Assignment(lhs[i, j], rhs[i, j]) code0 = self._print(temp) lines.append(code0) return "\n".join(lines) elif self._settings["contract"] and (lhs.has(IndexedBase) or rhs.has(IndexedBase)): # Here we check if there is looping to be done, and if so # print the required loops. return self._doprint_loops(rhs, lhs) else: lhs_code = self._print(lhs) rhs_code = self._print(rhs) return self._get_statement("%s = %s" % (lhs_code, rhs_code)) def _print_Piecewise(self, expr): # This method is called only for inline if constructs # Top level piecewise is handled in doprint() if expr.args[-1].cond == True: last_line = "%s" % self._print(expr.args[-1].expr) else: last_line = "ifelse(%s,%s,NA)" % (self._print(expr.args[-1].cond), self._print(expr.args[-1].expr)) code=last_line for e, c in reversed(expr.args[:-1]): code= "ifelse(%s,%s," % (self._print(c), self._print(e))+code+")" return(code) def _print_ITE(self, expr): from sympy.functions import Piecewise _piecewise = Piecewise((expr.args[1], expr.args[0]), (expr.args[2], True)) return self._print(_piecewise) def _print_MatrixElement(self, expr): return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True), expr.j + expr.i*expr.parent.shape[1]) def _print_Symbol(self, expr): name = super()._print_Symbol(expr) if expr in self._dereference: return '(*{})'.format(name) else: return name def _print_Relational(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) op = expr.rel_op return "{} {} {}".format(lhs_code, op, rhs_code) def _print_sinc(self, expr): from sympy.functions.elementary.trigonometric import sin from sympy.core.relational import Ne from sympy.functions import Piecewise _piecewise = Piecewise( (sin(expr.args[0]) / expr.args[0], Ne(expr.args[0], 0)), (1, True)) return self._print(_piecewise) def _print_AugmentedAssignment(self, expr): lhs_code = self._print(expr.lhs) op = expr.op rhs_code = self._print(expr.rhs) return "{} {} {};".format(lhs_code, op, rhs_code) def _print_For(self, expr): target = self._print(expr.target) if isinstance(expr.iterable, Range): start, stop, step = expr.iterable.args else: raise NotImplementedError("Only iterable currently supported is Range") body = self._print(expr.body) return ('for ({target} = {start}; {target} < {stop}; {target} += ' '{step}) {{\n{body}\n}}').format(target=target, start=start, stop=stop, step=step, body=body) def indent_code(self, code): """Accepts a string of code or a list of code lines""" if isinstance(code, str): code_lines = self.indent_code(code.splitlines(True)) return ''.join(code_lines) tab = " " inc_token = ('{', '(', '{\n', '(\n') dec_token = ('}', ')') code = [ line.lstrip(' \t') for line in code ] increase = [ int(any(map(line.endswith, inc_token))) for line in code ] decrease = [ int(any(map(line.startswith, dec_token))) for line in code ] pretty = [] level = 0 for n, line in enumerate(code): if line == '' or line == '\n': pretty.append(line) continue level -= decrease[n] pretty.append("%s%s" % (tab*level, line)) level += increase[n] return pretty def rcode(expr, assign_to=None, **settings): """Converts an expr to a string of r code Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of line-wrapping, or for expressions that generate multi-line statements. precision : integer, optional The precision for numbers such as pi [default=15]. user_functions : dict, optional A dictionary where the keys are string representations of either ``FunctionClass`` or ``UndefinedFunction`` instances and the values are their desired R string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, rfunction_string)] or [(argument_test, rfunction_formater)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. Examples ======== >>> from sympy import rcode, symbols, Rational, sin, ceiling, Abs, Function >>> x, tau = symbols("x, tau") >>> rcode((2*tau)**Rational(7, 2)) '8*sqrt(2)*tau^(7.0/2.0)' >>> rcode(sin(x), assign_to="s") 's = sin(x);' Simple custom printing can be defined for certain types by passing a dictionary of {"type" : "function"} to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. >>> custom_functions = { ... "ceiling": "CEIL", ... "Abs": [(lambda x: not x.is_integer, "fabs"), ... (lambda x: x.is_integer, "ABS")], ... "func": "f" ... } >>> func = Function('func') >>> rcode(func(Abs(x) + ceiling(x)), user_functions=custom_functions) 'f(fabs(x) + CEIL(x))' or if the R-function takes a subset of the original arguments: >>> rcode(2**x + 3**x, user_functions={'Pow': [ ... (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e), ... (lambda b, e: b != 2, 'pow')]}) 'exp2(x) + pow(3, x)' ``Piecewise`` expressions are converted into conditionals. If an ``assign_to`` variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> expr = Piecewise((x + 1, x > 0), (x, True)) >>> print(rcode(expr, assign_to=tau)) tau = ifelse(x > 0,x + 1,x); Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> rcode(e.rhs, assign_to=e.lhs, contract=False) 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions must be provided to ``assign_to``. Note that any expression that can be generated normally can also exist inside a Matrix: >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) >>> A = MatrixSymbol('A', 3, 1) >>> print(rcode(mat, A)) A[0] = x^2; A[1] = ifelse(x > 0,x + 1,x); A[2] = sin(x); """ return RCodePrinter(settings).doprint(expr, assign_to) def print_rcode(expr, **settings): """Prints R representation of the given expression.""" print(rcode(expr, **settings))
4bb580cbda07667946185148f7cfc8ba0ae8e4d6c8ae74dc9b2bbcfb50236034
""" Octave (and Matlab) code printer The `OctaveCodePrinter` converts SymPy expressions into Octave expressions. It uses a subset of the Octave language for Matlab compatibility. A complete code generator, which uses `octave_code` extensively, can be found in `sympy.utilities.codegen`. The `codegen` module can be used to generate complete source code files. """ from typing import Any, Dict from sympy.core import Mul, Pow, S, Rational from sympy.core.mul import _keep_coeff from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence, PRECEDENCE from re import search # List of known functions. First, those that have the same name in # SymPy and Octave. This is almost certainly incomplete! known_fcns_src1 = ["sin", "cos", "tan", "cot", "sec", "csc", "asin", "acos", "acot", "atan", "atan2", "asec", "acsc", "sinh", "cosh", "tanh", "coth", "csch", "sech", "asinh", "acosh", "atanh", "acoth", "asech", "acsch", "erfc", "erfi", "erf", "erfinv", "erfcinv", "besseli", "besselj", "besselk", "bessely", "bernoulli", "beta", "euler", "exp", "factorial", "floor", "fresnelc", "fresnels", "gamma", "harmonic", "log", "polylog", "sign", "zeta", "legendre"] # These functions have different names ("Sympy": "Octave"), more # generally a mapping to (argument_conditions, octave_function). known_fcns_src2 = { "Abs": "abs", "arg": "angle", # arg/angle ok in Octave but only angle in Matlab "binomial": "bincoeff", "ceiling": "ceil", "chebyshevu": "chebyshevU", "chebyshevt": "chebyshevT", "Chi": "coshint", "Ci": "cosint", "conjugate": "conj", "DiracDelta": "dirac", "Heaviside": "heaviside", "im": "imag", "laguerre": "laguerreL", "LambertW": "lambertw", "li": "logint", "loggamma": "gammaln", "Max": "max", "Min": "min", "Mod": "mod", "polygamma": "psi", "re": "real", "RisingFactorial": "pochhammer", "Shi": "sinhint", "Si": "sinint", } class OctaveCodePrinter(CodePrinter): """ A printer to convert expressions to strings of Octave/Matlab code. """ printmethod = "_octave" language = "Octave" _operators = { 'and': '&', 'or': '|', 'not': '~', } _default_settings = { 'order': None, 'full_prec': 'auto', 'precision': 17, 'user_functions': {}, 'human': True, 'allow_unknown_functions': False, 'contract': True, 'inline': True, } # type: Dict[str, Any] # Note: contract is for expressing tensors as loops (if True), or just # assignment (if False). FIXME: this should be looked a more carefully # for Octave. def __init__(self, settings={}): super().__init__(settings) self.known_functions = dict(zip(known_fcns_src1, known_fcns_src1)) self.known_functions.update(dict(known_fcns_src2)) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) def _rate_index_position(self, p): return p*5 def _get_statement(self, codestring): return "%s;" % codestring def _get_comment(self, text): return "% {}".format(text) def _declare_number_const(self, name, value): return "{} = {};".format(name, value) def _format_code(self, lines): return self.indent_code(lines) def _traverse_matrix_indices(self, mat): # Octave uses Fortran order (column-major) rows, cols = mat.shape return ((i, j) for j in range(cols) for i in range(rows)) def _get_loop_opening_ending(self, indices): open_lines = [] close_lines = [] for i in indices: # Octave arrays start at 1 and end at dimension var, start, stop = map(self._print, [i.label, i.lower + 1, i.upper + 1]) open_lines.append("for %s = %s:%s" % (var, start, stop)) close_lines.append("end") return open_lines, close_lines def _print_Mul(self, expr): # print complex numbers nicely in Octave if (expr.is_number and expr.is_imaginary and (S.ImaginaryUnit*expr).is_Integer): return "%si" % self._print(-S.ImaginaryUnit*expr) # cribbed from str.py prec = precedence(expr) c, e = expr.as_coeff_Mul() if c < 0: expr = _keep_coeff(-c, e) sign = "-" else: sign = "" a = [] # items in the numerator b = [] # items that are in the denominator (if any) pow_paren = [] # Will collect all pow with more than one base element and exp = -1 if self.order not in ('old', 'none'): args = expr.as_ordered_factors() else: # use make_args in case expr was something like -x -> x args = Mul.make_args(expr) # Gather args for numerator/denominator for item in args: if (item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative): if item.exp != -1: b.append(Pow(item.base, -item.exp, evaluate=False)) else: if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 pow_paren.append(item) b.append(Pow(item.base, -item.exp)) elif item.is_Rational and item is not S.Infinity: if item.p != 1: a.append(Rational(item.p)) if item.q != 1: b.append(Rational(item.q)) else: a.append(item) a = a or [S.One] a_str = [self.parenthesize(x, prec) for x in a] b_str = [self.parenthesize(x, prec) for x in b] # To parenthesize Pow with exp = -1 and having more than one Symbol for item in pow_paren: if item.base in b: b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] # from here it differs from str.py to deal with "*" and ".*" def multjoin(a, a_str): # here we probably are assuming the constants will come first r = a_str[0] for i in range(1, len(a)): mulsym = '*' if a[i-1].is_number else '.*' r = r + mulsym + a_str[i] return r if not b: return sign + multjoin(a, a_str) elif len(b) == 1: divsym = '/' if b[0].is_number else './' return sign + multjoin(a, a_str) + divsym + b_str[0] else: divsym = '/' if all([bi.is_number for bi in b]) else './' return (sign + multjoin(a, a_str) + divsym + "(%s)" % multjoin(b, b_str)) def _print_Relational(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) op = expr.rel_op return "{} {} {}".format(lhs_code, op, rhs_code) def _print_Pow(self, expr): powsymbol = '^' if all([x.is_number for x in expr.args]) else '.^' PREC = precedence(expr) if expr.exp == S.Half: return "sqrt(%s)" % self._print(expr.base) if expr.is_commutative: if expr.exp == -S.Half: sym = '/' if expr.base.is_number else './' return "1" + sym + "sqrt(%s)" % self._print(expr.base) if expr.exp == -S.One: sym = '/' if expr.base.is_number else './' return "1" + sym + "%s" % self.parenthesize(expr.base, PREC) return '%s%s%s' % (self.parenthesize(expr.base, PREC), powsymbol, self.parenthesize(expr.exp, PREC)) def _print_MatPow(self, expr): PREC = precedence(expr) return '%s^%s' % (self.parenthesize(expr.base, PREC), self.parenthesize(expr.exp, PREC)) def _print_MatrixSolve(self, expr): PREC = precedence(expr) return "%s \\ %s" % (self.parenthesize(expr.matrix, PREC), self.parenthesize(expr.vector, PREC)) def _print_Pi(self, expr): return 'pi' def _print_ImaginaryUnit(self, expr): return "1i" def _print_Exp1(self, expr): return "exp(1)" def _print_GoldenRatio(self, expr): # FIXME: how to do better, e.g., for octave_code(2*GoldenRatio)? #return self._print((1+sqrt(S(5)))/2) return "(1+sqrt(5))/2" def _print_Assignment(self, expr): from sympy.codegen.ast import Assignment from sympy.functions.elementary.piecewise import Piecewise from sympy.tensor.indexed import IndexedBase # Copied from codeprinter, but remove special MatrixSymbol treatment lhs = expr.lhs rhs = expr.rhs # We special case assignments that take multiple lines if not self._settings["inline"] and isinstance(expr.rhs, Piecewise): # Here we modify Piecewise so each expression is now # an Assignment, and then continue on the print. expressions = [] conditions = [] for (e, c) in rhs.args: expressions.append(Assignment(lhs, e)) conditions.append(c) temp = Piecewise(*zip(expressions, conditions)) return self._print(temp) if self._settings["contract"] and (lhs.has(IndexedBase) or rhs.has(IndexedBase)): # Here we check if there is looping to be done, and if so # print the required loops. return self._doprint_loops(rhs, lhs) else: lhs_code = self._print(lhs) rhs_code = self._print(rhs) return self._get_statement("%s = %s" % (lhs_code, rhs_code)) def _print_Infinity(self, expr): return 'inf' def _print_NegativeInfinity(self, expr): return '-inf' def _print_NaN(self, expr): return 'NaN' def _print_list(self, expr): return '{' + ', '.join(self._print(a) for a in expr) + '}' _print_tuple = _print_list _print_Tuple = _print_list def _print_BooleanTrue(self, expr): return "true" def _print_BooleanFalse(self, expr): return "false" def _print_bool(self, expr): return str(expr).lower() # Could generate quadrature code for definite Integrals? #_print_Integral = _print_not_supported def _print_MatrixBase(self, A): # Handle zero dimensions: if (A.rows, A.cols) == (0, 0): return '[]' elif A.rows == 0 or A.cols == 0: return 'zeros(%s, %s)' % (A.rows, A.cols) elif (A.rows, A.cols) == (1, 1): # Octave does not distinguish between scalars and 1x1 matrices return self._print(A[0, 0]) return "[%s]" % "; ".join(" ".join([self._print(a) for a in A[r, :]]) for r in range(A.rows)) def _print_SparseMatrix(self, A): from sympy.matrices import Matrix L = A.col_list(); # make row vectors of the indices and entries I = Matrix([[k[0] + 1 for k in L]]) J = Matrix([[k[1] + 1 for k in L]]) AIJ = Matrix([[k[2] for k in L]]) return "sparse(%s, %s, %s, %s, %s)" % (self._print(I), self._print(J), self._print(AIJ), A.rows, A.cols) def _print_MatrixElement(self, expr): return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \ + '(%s, %s)' % (expr.i + 1, expr.j + 1) def _print_MatrixSlice(self, expr): def strslice(x, lim): l = x[0] + 1 h = x[1] step = x[2] lstr = self._print(l) hstr = 'end' if h == lim else self._print(h) if step == 1: if l == 1 and h == lim: return ':' if l == h: return lstr else: return lstr + ':' + hstr else: return ':'.join((lstr, self._print(step), hstr)) return (self._print(expr.parent) + '(' + strslice(expr.rowslice, expr.parent.shape[0]) + ', ' + strslice(expr.colslice, expr.parent.shape[1]) + ')') def _print_Indexed(self, expr): inds = [ self._print(i) for i in expr.indices ] return "%s(%s)" % (self._print(expr.base.label), ", ".join(inds)) def _print_Idx(self, expr): return self._print(expr.label) def _print_KroneckerDelta(self, expr): prec = PRECEDENCE["Pow"] return "double(%s == %s)" % tuple(self.parenthesize(x, prec) for x in expr.args) def _print_HadamardProduct(self, expr): return '.*'.join([self.parenthesize(arg, precedence(expr)) for arg in expr.args]) def _print_HadamardPower(self, expr): PREC = precedence(expr) return '.**'.join([ self.parenthesize(expr.base, PREC), self.parenthesize(expr.exp, PREC) ]) def _print_Identity(self, expr): shape = expr.shape if len(shape) == 2 and shape[0] == shape[1]: shape = [shape[0]] s = ", ".join(self._print(n) for n in shape) return "eye(" + s + ")" def _print_lowergamma(self, expr): # Octave implements regularized incomplete gamma function return "(gammainc({1}, {0}).*gamma({0}))".format( self._print(expr.args[0]), self._print(expr.args[1])) def _print_uppergamma(self, expr): return "(gammainc({1}, {0}, 'upper').*gamma({0}))".format( self._print(expr.args[0]), self._print(expr.args[1])) def _print_sinc(self, expr): #Note: Divide by pi because Octave implements normalized sinc function. return "sinc(%s)" % self._print(expr.args[0]/S.Pi) def _print_hankel1(self, expr): return "besselh(%s, 1, %s)" % (self._print(expr.order), self._print(expr.argument)) def _print_hankel2(self, expr): return "besselh(%s, 2, %s)" % (self._print(expr.order), self._print(expr.argument)) # Note: as of 2015, Octave doesn't have spherical Bessel functions def _print_jn(self, expr): from sympy.functions import sqrt, besselj x = expr.argument expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x) return self._print(expr2) def _print_yn(self, expr): from sympy.functions import sqrt, bessely x = expr.argument expr2 = sqrt(S.Pi/(2*x))*bessely(expr.order + S.Half, x) return self._print(expr2) def _print_airyai(self, expr): return "airy(0, %s)" % self._print(expr.args[0]) def _print_airyaiprime(self, expr): return "airy(1, %s)" % self._print(expr.args[0]) def _print_airybi(self, expr): return "airy(2, %s)" % self._print(expr.args[0]) def _print_airybiprime(self, expr): return "airy(3, %s)" % self._print(expr.args[0]) def _print_expint(self, expr): mu, x = expr.args if mu != 1: return self._print_not_supported(expr) return "expint(%s)" % self._print(x) def _one_or_two_reversed_args(self, expr): assert len(expr.args) <= 2 return '{name}({args})'.format( name=self.known_functions[expr.__class__.__name__], args=", ".join([self._print(x) for x in reversed(expr.args)]) ) _print_DiracDelta = _print_LambertW = _one_or_two_reversed_args def _nested_binary_math_func(self, expr): return '{name}({arg1}, {arg2})'.format( name=self.known_functions[expr.__class__.__name__], arg1=self._print(expr.args[0]), arg2=self._print(expr.func(*expr.args[1:])) ) _print_Max = _print_Min = _nested_binary_math_func def _print_Piecewise(self, expr): if expr.args[-1].cond != True: # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") lines = [] if self._settings["inline"]: # Express each (cond, expr) pair in a nested Horner form: # (condition) .* (expr) + (not cond) .* (<others>) # Expressions that result in multiple statements won't work here. ecpairs = ["({0}).*({1}) + (~({0})).*(".format (self._print(c), self._print(e)) for e, c in expr.args[:-1]] elast = "%s" % self._print(expr.args[-1].expr) pw = " ...\n".join(ecpairs) + elast + ")"*len(ecpairs) # Note: current need these outer brackets for 2*pw. Would be # nicer to teach parenthesize() to do this for us when needed! return "(" + pw + ")" else: for i, (e, c) in enumerate(expr.args): if i == 0: lines.append("if (%s)" % self._print(c)) elif i == len(expr.args) - 1 and c == True: lines.append("else") else: lines.append("elseif (%s)" % self._print(c)) code0 = self._print(e) lines.append(code0) if i == len(expr.args) - 1: lines.append("end") return "\n".join(lines) def _print_zeta(self, expr): if len(expr.args) == 1: return "zeta(%s)" % self._print(expr.args[0]) else: # Matlab two argument zeta is not equivalent to SymPy's return self._print_not_supported(expr) def indent_code(self, code): """Accepts a string of code or a list of code lines""" # code mostly copied from ccode if isinstance(code, str): code_lines = self.indent_code(code.splitlines(True)) return ''.join(code_lines) tab = " " inc_regex = ('^function ', '^if ', '^elseif ', '^else$', '^for ') dec_regex = ('^end$', '^elseif ', '^else$') # pre-strip left-space from the code code = [ line.lstrip(' \t') for line in code ] increase = [ int(any([search(re, line) for re in inc_regex])) for line in code ] decrease = [ int(any([search(re, line) for re in dec_regex])) for line in code ] pretty = [] level = 0 for n, line in enumerate(code): if line == '' or line == '\n': pretty.append(line) continue level -= decrease[n] pretty.append("%s%s" % (tab*level, line)) level += increase[n] return pretty def octave_code(expr, assign_to=None, **settings): r"""Converts `expr` to a string of Octave (or Matlab) code. The string uses a subset of the Octave language for Matlab compatibility. Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for expressions that generate multi-line statements. precision : integer, optional The precision for numbers such as pi [default=16]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. inline: bool, optional If True, we try to create single-statement code instead of multiple statements. [default=True]. Examples ======== >>> from sympy import octave_code, symbols, sin, pi >>> x = symbols('x') >>> octave_code(sin(x).series(x).removeO()) 'x.^5/120 - x.^3/6 + x' >>> from sympy import Rational, ceiling >>> x, y, tau = symbols("x, y, tau") >>> octave_code((2*tau)**Rational(7, 2)) '8*sqrt(2)*tau.^(7/2)' Note that element-wise (Hadamard) operations are used by default between symbols. This is because its very common in Octave to write "vectorized" code. It is harmless if the values are scalars. >>> octave_code(sin(pi*x*y), assign_to="s") 's = sin(pi*x.*y);' If you need a matrix product "*" or matrix power "^", you can specify the symbol as a ``MatrixSymbol``. >>> from sympy import Symbol, MatrixSymbol >>> n = Symbol('n', integer=True, positive=True) >>> A = MatrixSymbol('A', n, n) >>> octave_code(3*pi*A**3) '(3*pi)*A^3' This class uses several rules to decide which symbol to use a product. Pure numbers use "*", Symbols use ".*" and MatrixSymbols use "*". A HadamardProduct can be used to specify componentwise multiplication ".*" of two MatrixSymbols. There is currently there is no easy way to specify scalar symbols, so sometimes the code might have some minor cosmetic issues. For example, suppose x and y are scalars and A is a Matrix, then while a human programmer might write "(x^2*y)*A^3", we generate: >>> octave_code(x**2*y*A**3) '(x.^2.*y)*A^3' Matrices are supported using Octave inline notation. When using ``assign_to`` with matrices, the name can be specified either as a string or as a ``MatrixSymbol``. The dimensions must align in the latter case. >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([[x**2, sin(x), ceiling(x)]]) >>> octave_code(mat, assign_to='A') 'A = [x.^2 sin(x) ceil(x)];' ``Piecewise`` expressions are implemented with logical masking by default. Alternatively, you can pass "inline=False" to use if-else conditionals. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> pw = Piecewise((x + 1, x > 0), (x, True)) >>> octave_code(pw, assign_to=tau) 'tau = ((x > 0).*(x + 1) + (~(x > 0)).*(x));' Note that any expression that can be generated normally can also exist inside a Matrix: >>> mat = Matrix([[x**2, pw, sin(x)]]) >>> octave_code(mat, assign_to='A') 'A = [x.^2 ((x > 0).*(x + 1) + (~(x > 0)).*(x)) sin(x)];' Custom printing can be defined for certain types by passing a dictionary of "type" : "function" to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e., [(argument_test, cfunction_string)]. This can be used to call a custom Octave function. >>> from sympy import Function >>> f = Function('f') >>> g = Function('g') >>> custom_functions = { ... "f": "existing_octave_fcn", ... "g": [(lambda x: x.is_Matrix, "my_mat_fcn"), ... (lambda x: not x.is_Matrix, "my_fcn")] ... } >>> mat = Matrix([[1, x]]) >>> octave_code(f(x) + g(x) + g(mat), user_functions=custom_functions) 'existing_octave_fcn(x) + my_fcn(x) + my_mat_fcn([1 x])' Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e = Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> octave_code(e.rhs, assign_to=e.lhs, contract=False) 'Dy(i) = (y(i + 1) - y(i))./(t(i + 1) - t(i));' """ return OctaveCodePrinter(settings).doprint(expr, assign_to) def print_octave_code(expr, **settings): """Prints the Octave (or Matlab) representation of the given expression. See `octave_code` for the meaning of the optional arguments. """ print(octave_code(expr, **settings))
edcd917453f98c1e2ae9c66b21893906516d8a8147e9393bf0bc1672da7b82dc
""" This is a shim file to provide backwards compatibility (fcode.py was renamed to fortran.py in SymPy 1.7). """ from sympy.utilities.exceptions import SymPyDeprecationWarning SymPyDeprecationWarning( feature="importing from sympy.printing.fcode", useinstead="Import from sympy.printing.fortran", issue=20256, deprecated_since_version="1.7").warn() from .fortran import fcode, print_fcode, known_functions, FCodePrinter # noqa:F401
0901172461697b28c8d780a4b41e896644ff211495ca4344d52fa64b7abaa1a9
from typing import Any, Dict, Set, Tuple from functools import wraps from sympy.core import Add, Expr, Mul, Pow, S, sympify, Float from sympy.core.basic import Basic from sympy.core.compatibility import default_sort_key from sympy.core.function import Lambda from sympy.core.mul import _keep_coeff from sympy.core.symbol import Symbol from sympy.printing.str import StrPrinter from sympy.printing.precedence import precedence class requires: """ Decorator for registering requirements on print methods. """ def __init__(self, **kwargs): self._req = kwargs def __call__(self, method): def _method_wrapper(self_, *args, **kwargs): for k, v in self._req.items(): getattr(self_, k).update(v) return method(self_, *args, **kwargs) return wraps(method)(_method_wrapper) class AssignmentError(Exception): """ Raised if an assignment variable for a loop is missing. """ pass class CodePrinter(StrPrinter): """ The base class for code-printing subclasses. """ _operators = { 'and': '&&', 'or': '||', 'not': '!', } _default_settings = { 'order': None, 'full_prec': 'auto', 'error_on_reserved': False, 'reserved_word_suffix': '_', 'human': True, 'inline': False, 'allow_unknown_functions': False, } # type: Dict[str, Any] # Functions which are "simple" to rewrite to other functions that # may be supported _rewriteable_functions = { 'erf2': 'erf', 'Li': 'li', 'beta': 'gamma' } def __init__(self, settings=None): super().__init__(settings=settings) if not hasattr(self, 'reserved_words'): self.reserved_words = set() def doprint(self, expr, assign_to=None): """ Print the expression as code. Parameters ---------- expr : Expression The expression to be printed. assign_to : Symbol, MatrixSymbol, or string (optional) If provided, the printed code will set the expression to a variable with name ``assign_to``. """ from sympy.codegen.ast import Assignment from sympy.matrices.expressions.matexpr import MatrixSymbol if isinstance(assign_to, str): if expr.is_Matrix: assign_to = MatrixSymbol(assign_to, *expr.shape) else: assign_to = Symbol(assign_to) elif not isinstance(assign_to, (Basic, type(None))): raise TypeError("{} cannot assign to object of type {}".format( type(self).__name__, type(assign_to))) if assign_to: expr = Assignment(assign_to, expr) else: # _sympify is not enough b/c it errors on iterables expr = sympify(expr) # keep a set of expressions that are not strictly translatable to Code # and number constants that must be declared and initialized self._not_supported = set() self._number_symbols = set() # type: Set[Tuple[Expr, Float]] lines = self._print(expr).splitlines() # format the output if self._settings["human"]: frontlines = [] if self._not_supported: frontlines.append(self._get_comment( "Not supported in {}:".format(self.language))) for expr in sorted(self._not_supported, key=str): frontlines.append(self._get_comment(type(expr).__name__)) for name, value in sorted(self._number_symbols, key=str): frontlines.append(self._declare_number_const(name, value)) lines = frontlines + lines lines = self._format_code(lines) result = "\n".join(lines) else: lines = self._format_code(lines) num_syms = {(k, self._print(v)) for k, v in self._number_symbols} result = (num_syms, self._not_supported, "\n".join(lines)) self._not_supported = set() self._number_symbols = set() return result def _doprint_loops(self, expr, assign_to=None): # Here we print an expression that contains Indexed objects, they # correspond to arrays in the generated code. The low-level implementation # involves looping over array elements and possibly storing results in temporary # variables or accumulate it in the assign_to object. if self._settings.get('contract', True): from sympy.tensor import get_contraction_structure # Setup loops over non-dummy indices -- all terms need these indices = self._get_expression_indices(expr, assign_to) # Setup loops over dummy indices -- each term needs separate treatment dummies = get_contraction_structure(expr) else: indices = [] dummies = {None: (expr,)} openloop, closeloop = self._get_loop_opening_ending(indices) # terms with no summations first if None in dummies: text = StrPrinter.doprint(self, Add(*dummies[None])) else: # If all terms have summations we must initialize array to Zero text = StrPrinter.doprint(self, 0) # skip redundant assignments (where lhs == rhs) lhs_printed = self._print(assign_to) lines = [] if text != lhs_printed: lines.extend(openloop) if assign_to is not None: text = self._get_statement("%s = %s" % (lhs_printed, text)) lines.append(text) lines.extend(closeloop) # then terms with summations for d in dummies: if isinstance(d, tuple): indices = self._sort_optimized(d, expr) openloop_d, closeloop_d = self._get_loop_opening_ending( indices) for term in dummies[d]: if term in dummies and not ([list(f.keys()) for f in dummies[term]] == [[None] for f in dummies[term]]): # If one factor in the term has it's own internal # contractions, those must be computed first. # (temporary variables?) raise NotImplementedError( "FIXME: no support for contractions in factor yet") else: # We need the lhs expression as an accumulator for # the loops, i.e # # for (int d=0; d < dim; d++){ # lhs[] = lhs[] + term[][d] # } ^.................. the accumulator # # We check if the expression already contains the # lhs, and raise an exception if it does, as that # syntax is currently undefined. FIXME: What would be # a good interpretation? if assign_to is None: raise AssignmentError( "need assignment variable for loops") if term.has(assign_to): raise ValueError("FIXME: lhs present in rhs,\ this is undefined in CodePrinter") lines.extend(openloop) lines.extend(openloop_d) text = "%s = %s" % (lhs_printed, StrPrinter.doprint( self, assign_to + term)) lines.append(self._get_statement(text)) lines.extend(closeloop_d) lines.extend(closeloop) return "\n".join(lines) def _get_expression_indices(self, expr, assign_to): from sympy.tensor import get_indices rinds, junk = get_indices(expr) linds, junk = get_indices(assign_to) # support broadcast of scalar if linds and not rinds: rinds = linds if rinds != linds: raise ValueError("lhs indices must match non-dummy" " rhs indices in %s" % expr) return self._sort_optimized(rinds, assign_to) def _sort_optimized(self, indices, expr): from sympy.tensor.indexed import Indexed if not indices: return [] # determine optimized loop order by giving a score to each index # the index with the highest score are put in the innermost loop. score_table = {} for i in indices: score_table[i] = 0 arrays = expr.atoms(Indexed) for arr in arrays: for p, ind in enumerate(arr.indices): try: score_table[ind] += self._rate_index_position(p) except KeyError: pass return sorted(indices, key=lambda x: score_table[x]) def _rate_index_position(self, p): """function to calculate score based on position among indices This method is used to sort loops in an optimized order, see CodePrinter._sort_optimized() """ raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _get_statement(self, codestring): """Formats a codestring with the proper line ending.""" raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _get_comment(self, text): """Formats a text string as a comment.""" raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _declare_number_const(self, name, value): """Declare a numeric constant at the top of a function""" raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _format_code(self, lines): """Take in a list of lines of code, and format them accordingly. This may include indenting, wrapping long lines, etc...""" raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _get_loop_opening_ending(self, indices): """Returns a tuple (open_lines, close_lines) containing lists of codelines""" raise NotImplementedError("This function must be implemented by " "subclass of CodePrinter.") def _print_Dummy(self, expr): if expr.name.startswith('Dummy_'): return '_' + expr.name else: return '%s_%d' % (expr.name, expr.dummy_index) def _print_CodeBlock(self, expr): return '\n'.join([self._print(i) for i in expr.args]) def _print_String(self, string): return str(string) def _print_QuotedString(self, arg): return '"%s"' % arg.text def _print_Comment(self, string): return self._get_comment(str(string)) def _print_Assignment(self, expr): from sympy.codegen.ast import Assignment from sympy.functions.elementary.piecewise import Piecewise from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.tensor.indexed import IndexedBase lhs = expr.lhs rhs = expr.rhs # We special case assignments that take multiple lines if isinstance(expr.rhs, Piecewise): # Here we modify Piecewise so each expression is now # an Assignment, and then continue on the print. expressions = [] conditions = [] for (e, c) in rhs.args: expressions.append(Assignment(lhs, e)) conditions.append(c) temp = Piecewise(*zip(expressions, conditions)) return self._print(temp) elif isinstance(lhs, MatrixSymbol): # Here we form an Assignment for each element in the array, # printing each one. lines = [] for (i, j) in self._traverse_matrix_indices(lhs): temp = Assignment(lhs[i, j], rhs[i, j]) code0 = self._print(temp) lines.append(code0) return "\n".join(lines) elif self._settings.get("contract", False) and (lhs.has(IndexedBase) or rhs.has(IndexedBase)): # Here we check if there is looping to be done, and if so # print the required loops. return self._doprint_loops(rhs, lhs) else: lhs_code = self._print(lhs) rhs_code = self._print(rhs) return self._get_statement("%s = %s" % (lhs_code, rhs_code)) def _print_AugmentedAssignment(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) return self._get_statement("{} {} {}".format( *map(lambda arg: self._print(arg), [lhs_code, expr.op, rhs_code]))) def _print_FunctionCall(self, expr): return '%s(%s)' % ( expr.name, ', '.join(map(lambda arg: self._print(arg), expr.function_args))) def _print_Variable(self, expr): return self._print(expr.symbol) def _print_Statement(self, expr): arg, = expr.args return self._get_statement(self._print(arg)) def _print_Symbol(self, expr): name = super()._print_Symbol(expr) if name in self.reserved_words: if self._settings['error_on_reserved']: msg = ('This expression includes the symbol "{}" which is a ' 'reserved keyword in this language.') raise ValueError(msg.format(name)) return name + self._settings['reserved_word_suffix'] else: return name def _print_Function(self, expr): if expr.func.__name__ in self.known_functions: cond_func = self.known_functions[expr.func.__name__] func = None if isinstance(cond_func, str): func = cond_func else: for cond, func in cond_func: if cond(*expr.args): break if func is not None: try: return func(*[self.parenthesize(item, 0) for item in expr.args]) except TypeError: return "%s(%s)" % (func, self.stringify(expr.args, ", ")) elif hasattr(expr, '_imp_') and isinstance(expr._imp_, Lambda): # inlined function return self._print(expr._imp_(*expr.args)) elif (expr.func.__name__ in self._rewriteable_functions and self._rewriteable_functions[expr.func.__name__] in self.known_functions): # Simple rewrite to supported function possible return self._print(expr.rewrite(self._rewriteable_functions[expr.func.__name__])) elif expr.is_Function and self._settings.get('allow_unknown_functions', False): return '%s(%s)' % (self._print(expr.func), ', '.join(map(self._print, expr.args))) else: return self._print_not_supported(expr) _print_Expr = _print_Function def _print_NumberSymbol(self, expr): if self._settings.get("inline", False): return self._print(Float(expr.evalf(self._settings["precision"]))) else: # A Number symbol that is not implemented here or with _printmethod # is registered and evaluated self._number_symbols.add((expr, Float(expr.evalf(self._settings["precision"])))) return str(expr) def _print_Catalan(self, expr): return self._print_NumberSymbol(expr) def _print_EulerGamma(self, expr): return self._print_NumberSymbol(expr) def _print_GoldenRatio(self, expr): return self._print_NumberSymbol(expr) def _print_TribonacciConstant(self, expr): return self._print_NumberSymbol(expr) def _print_Exp1(self, expr): return self._print_NumberSymbol(expr) def _print_Pi(self, expr): return self._print_NumberSymbol(expr) def _print_And(self, expr): PREC = precedence(expr) return (" %s " % self._operators['and']).join(self.parenthesize(a, PREC) for a in sorted(expr.args, key=default_sort_key)) def _print_Or(self, expr): PREC = precedence(expr) return (" %s " % self._operators['or']).join(self.parenthesize(a, PREC) for a in sorted(expr.args, key=default_sort_key)) def _print_Xor(self, expr): if self._operators.get('xor') is None: return self._print_not_supported(expr) PREC = precedence(expr) return (" %s " % self._operators['xor']).join(self.parenthesize(a, PREC) for a in expr.args) def _print_Equivalent(self, expr): if self._operators.get('equivalent') is None: return self._print_not_supported(expr) PREC = precedence(expr) return (" %s " % self._operators['equivalent']).join(self.parenthesize(a, PREC) for a in expr.args) def _print_Not(self, expr): PREC = precedence(expr) return self._operators['not'] + self.parenthesize(expr.args[0], PREC) def _print_Mul(self, expr): prec = precedence(expr) c, e = expr.as_coeff_Mul() if c < 0: expr = _keep_coeff(-c, e) sign = "-" else: sign = "" a = [] # items in the numerator b = [] # items that are in the denominator (if any) pow_paren = [] # Will collect all pow with more than one base element and exp = -1 if self.order not in ('old', 'none'): args = expr.as_ordered_factors() else: # use make_args in case expr was something like -x -> x args = Mul.make_args(expr) # Gather args for numerator/denominator for item in args: if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative: if item.exp != -1: b.append(Pow(item.base, -item.exp, evaluate=False)) else: if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 pow_paren.append(item) b.append(Pow(item.base, -item.exp)) else: a.append(item) a = a or [S.One] a_str = [self.parenthesize(x, prec) for x in a] b_str = [self.parenthesize(x, prec) for x in b] # To parenthesize Pow with exp = -1 and having more than one Symbol for item in pow_paren: if item.base in b: b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] if not b: return sign + '*'.join(a_str) elif len(b) == 1: return sign + '*'.join(a_str) + "/" + b_str[0] else: return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str) def _print_not_supported(self, expr): try: self._not_supported.add(expr) except TypeError: # not hashable pass return self.emptyPrinter(expr) # The following can not be simply translated into C or Fortran _print_Basic = _print_not_supported _print_ComplexInfinity = _print_not_supported _print_Derivative = _print_not_supported _print_ExprCondPair = _print_not_supported _print_GeometryEntity = _print_not_supported _print_Infinity = _print_not_supported _print_Integral = _print_not_supported _print_Interval = _print_not_supported _print_AccumulationBounds = _print_not_supported _print_Limit = _print_not_supported _print_MatrixBase = _print_not_supported _print_DeferredVector = _print_not_supported _print_NaN = _print_not_supported _print_NegativeInfinity = _print_not_supported _print_Order = _print_not_supported _print_RootOf = _print_not_supported _print_RootsOf = _print_not_supported _print_RootSum = _print_not_supported _print_Uniform = _print_not_supported _print_Unit = _print_not_supported _print_Wild = _print_not_supported _print_WildFunction = _print_not_supported _print_Relational = _print_not_supported # Code printer functions. These are included in this file so that they can be # imported in the top-level __init__.py without importing the sympy.codegen # module. def ccode(expr, assign_to=None, standard='c99', **settings): """Converts an expr to a string of c code Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of line-wrapping, or for expressions that generate multi-line statements. standard : str, optional String specifying the standard. If your compiler supports a more modern standard you may set this to 'c99' to allow the printer to use more math functions. [default='c89']. precision : integer, optional The precision for numbers such as pi [default=17]. user_functions : dict, optional A dictionary where the keys are string representations of either ``FunctionClass`` or ``UndefinedFunction`` instances and the values are their desired C string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)] or [(argument_test, cfunction_formater)]. See below for examples. dereference : iterable, optional An iterable of symbols that should be dereferenced in the printed code expression. These would be values passed by address to the function. For example, if ``dereference=[a]``, the resulting code would print ``(*a)`` instead of ``a``. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. Examples ======== >>> from sympy import ccode, symbols, Rational, sin, ceiling, Abs, Function >>> x, tau = symbols("x, tau") >>> expr = (2*tau)**Rational(7, 2) >>> ccode(expr) '8*M_SQRT2*pow(tau, 7.0/2.0)' >>> ccode(expr, math_macros={}) '8*sqrt(2)*pow(tau, 7.0/2.0)' >>> ccode(sin(x), assign_to="s") 's = sin(x);' >>> from sympy.codegen.ast import real, float80 >>> ccode(expr, type_aliases={real: float80}) '8*M_SQRT2l*powl(tau, 7.0L/2.0L)' Simple custom printing can be defined for certain types by passing a dictionary of {"type" : "function"} to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. >>> custom_functions = { ... "ceiling": "CEIL", ... "Abs": [(lambda x: not x.is_integer, "fabs"), ... (lambda x: x.is_integer, "ABS")], ... "func": "f" ... } >>> func = Function('func') >>> ccode(func(Abs(x) + ceiling(x)), standard='C89', user_functions=custom_functions) 'f(fabs(x) + CEIL(x))' or if the C-function takes a subset of the original arguments: >>> ccode(2**x + 3**x, standard='C99', user_functions={'Pow': [ ... (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e), ... (lambda b, e: b != 2, 'pow')]}) 'exp2(x) + pow(3, x)' ``Piecewise`` expressions are converted into conditionals. If an ``assign_to`` variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> expr = Piecewise((x + 1, x > 0), (x, True)) >>> print(ccode(expr, tau, standard='C89')) if (x > 0) { tau = x + 1; } else { tau = x; } Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> ccode(e.rhs, assign_to=e.lhs, contract=False, standard='C89') 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions must be provided to ``assign_to``. Note that any expression that can be generated normally can also exist inside a Matrix: >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) >>> A = MatrixSymbol('A', 3, 1) >>> print(ccode(mat, A, standard='C89')) A[0] = pow(x, 2); if (x > 0) { A[1] = x + 1; } else { A[1] = x; } A[2] = sin(x); """ from sympy.printing.c import c_code_printers return c_code_printers[standard.lower()](settings).doprint(expr, assign_to) def print_ccode(expr, **settings): """Prints C representation of the given expression.""" print(ccode(expr, **settings)) def fcode(expr, assign_to=None, **settings): """Converts an expr to a string of fortran code Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of line-wrapping, or for expressions that generate multi-line statements. precision : integer, optional DEPRECATED. Use type_mappings instead. The precision for numbers such as pi [default=17]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. source_format : optional The source format can be either 'fixed' or 'free'. [default='fixed'] standard : integer, optional The Fortran standard to be followed. This is specified as an integer. Acceptable standards are 66, 77, 90, 95, 2003, and 2008. Default is 77. Note that currently the only distinction internally is between standards before 95, and those 95 and after. This may change later as more features are added. name_mangling : bool, optional If True, then the variables that would become identical in case-insensitive Fortran are mangled by appending different number of ``_`` at the end. If False, SymPy won't interfere with naming of variables. [default=True] Examples ======== >>> from sympy import fcode, symbols, Rational, sin, ceiling, floor >>> x, tau = symbols("x, tau") >>> fcode((2*tau)**Rational(7, 2)) ' 8*sqrt(2.0d0)*tau**(7.0d0/2.0d0)' >>> fcode(sin(x), assign_to="s") ' s = sin(x)' Custom printing can be defined for certain types by passing a dictionary of "type" : "function" to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. >>> custom_functions = { ... "ceiling": "CEIL", ... "floor": [(lambda x: not x.is_integer, "FLOOR1"), ... (lambda x: x.is_integer, "FLOOR2")] ... } >>> fcode(floor(x) + ceiling(x), user_functions=custom_functions) ' CEIL(x) + FLOOR1(x)' ``Piecewise`` expressions are converted into conditionals. If an ``assign_to`` variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> expr = Piecewise((x + 1, x > 0), (x, True)) >>> print(fcode(expr, tau)) if (x > 0) then tau = x + 1 else tau = x end if Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> fcode(e.rhs, assign_to=e.lhs, contract=False) ' Dy(i) = (y(i + 1) - y(i))/(t(i + 1) - t(i))' Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions must be provided to ``assign_to``. Note that any expression that can be generated normally can also exist inside a Matrix: >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) >>> A = MatrixSymbol('A', 3, 1) >>> print(fcode(mat, A)) A(1, 1) = x**2 if (x > 0) then A(2, 1) = x + 1 else A(2, 1) = x end if A(3, 1) = sin(x) """ from sympy.printing.fortran import FCodePrinter return FCodePrinter(settings).doprint(expr, assign_to) def print_fcode(expr, **settings): """Prints the Fortran representation of the given expression. See fcode for the meaning of the optional arguments. """ print(fcode(expr, **settings)) def cxxcode(expr, assign_to=None, standard='c++11', **settings): """ C++ equivalent of :func:`~.ccode`. """ from sympy.printing.cxx import cxx_code_printers return cxx_code_printers[standard.lower()](settings).doprint(expr, assign_to)
10e2c0c6e85ea0d87b744a74c77f37ebaa55107852f2bff44e9df7b916cc9a0f
import keyword as kw import sympy from .repr import ReprPrinter from .str import StrPrinter # A list of classes that should be printed using StrPrinter STRPRINT = ("Add", "Infinity", "Integer", "Mul", "NegativeInfinity", "Pow", "Zero") class PythonPrinter(ReprPrinter, StrPrinter): """A printer which converts an expression into its Python interpretation.""" def __init__(self, settings=None): super().__init__(settings) self.symbols = [] self.functions = [] # Create print methods for classes that should use StrPrinter instead # of ReprPrinter. for name in STRPRINT: f_name = "_print_%s" % name f = getattr(StrPrinter, f_name) setattr(PythonPrinter, f_name, f) def _print_Function(self, expr): func = expr.func.__name__ if not hasattr(sympy, func) and not func in self.functions: self.functions.append(func) return StrPrinter._print_Function(self, expr) # procedure (!) for defining symbols which have be defined in print_python() def _print_Symbol(self, expr): symbol = self._str(expr) if symbol not in self.symbols: self.symbols.append(symbol) return StrPrinter._print_Symbol(self, expr) def _print_module(self, expr): raise ValueError('Modules in the expression are unacceptable') def python(expr, **settings): """Return Python interpretation of passed expression (can be passed to the exec() function without any modifications)""" printer = PythonPrinter(settings) exprp = printer.doprint(expr) result = '' # Returning found symbols and functions renamings = {} for symbolname in printer.symbols: newsymbolname = symbolname # Escape symbol names that are reserved python keywords if kw.iskeyword(newsymbolname): while True: newsymbolname += "_" if (newsymbolname not in printer.symbols and newsymbolname not in printer.functions): renamings[sympy.Symbol( symbolname)] = sympy.Symbol(newsymbolname) break result += newsymbolname + ' = Symbol(\'' + symbolname + '\')\n' for functionname in printer.functions: newfunctionname = functionname # Escape function names that are reserved python keywords if kw.iskeyword(newfunctionname): while True: newfunctionname += "_" if (newfunctionname not in printer.symbols and newfunctionname not in printer.functions): renamings[sympy.Function( functionname)] = sympy.Function(newfunctionname) break result += newfunctionname + ' = Function(\'' + functionname + '\')\n' if renamings: exprp = expr.subs(renamings) result += 'e = ' + printer._str(exprp) return result def print_python(expr, **settings): """Print output of python() function""" print(python(expr, **settings))
086157bd4f4714a92ec721072e920d88691fe2ae03105768dbc04e1fc4477c77
""" Maple code printer The MapleCodePrinter converts single sympy expressions into single Maple expressions, using the functions defined in the Maple objects where possible. FIXME: This module is still under actively developed. Some functions may be not completed. """ from sympy.core import S from sympy.core.numbers import Integer, IntegerConstant from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence, PRECEDENCE import sympy _known_func_same_name = [ 'sin', 'cos', 'tan', 'sec', 'csc', 'cot', 'sinh', 'cosh', 'tanh', 'sech', 'csch', 'coth', 'exp', 'floor', 'factorial' ] known_functions = { # Sympy -> Maple 'Abs': 'abs', 'log': 'ln', 'asin': 'arcsin', 'acos': 'arccos', 'atan': 'arctan', 'asec': 'arcsec', 'acsc': 'arccsc', 'acot': 'arccot', 'asinh': 'arcsinh', 'acosh': 'arccosh', 'atanh': 'arctanh', 'asech': 'arcsech', 'acsch': 'arccsch', 'acoth': 'arccoth', 'ceiling': 'ceil', 'besseli': 'BesselI', 'besselj': 'BesselJ', 'besselk': 'BesselK', 'bessely': 'BesselY', 'hankelh1': 'HankelH1', 'hankelh2': 'HankelH2', 'airyai': 'AiryAi', 'airybi': 'AiryBi' } for _func in _known_func_same_name: known_functions[_func] = _func number_symbols = { # Sympy -> Maple S.Pi: 'Pi', S.Exp1: 'exp(1)', S.Catalan: 'Catalan', S.EulerGamma: 'gamma', S.GoldenRatio: '(1/2 + (1/2)*sqrt(5))' } spec_relational_ops = { # Sympy -> Maple '==': '=', '!=': '<>' } not_supported_symbol = [ S.ComplexInfinity ] class MapleCodePrinter(CodePrinter): """ Printer which converts a sympy expression into a maple code. """ printmethod = "_maple" language = "maple" _default_settings = { 'order': None, 'full_prec': 'auto', 'human': True, 'inline': True, 'allow_unknown_functions': True, } def __init__(self, settings=None): if settings is None: settings = dict() super().__init__(settings) self.known_functions = dict(known_functions) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) def _get_statement(self, codestring): return "%s;" % codestring def _get_comment(self, text): return "# {}".format(text) def _declare_number_const(self, name, value): return "{} := {};".format(name, value.evalf(self._settings['precision'])) def _format_code(self, lines): return lines def _print_tuple(self, expr): return self._print(list(expr)) def _print_Tuple(self, expr): return self._print(list(expr)) def _print_Assignment(self, expr): lhs = self._print(expr.lhs) rhs = self._print(expr.rhs) return "{lhs} := {rhs}".format(lhs=lhs, rhs=rhs) def _print_Pow(self, expr, **kwargs): PREC = precedence(expr) if expr.exp == -1: return '1/%s' % (self.parenthesize(expr.base, PREC)) elif expr.exp == 0.5 or expr.exp == S(1) / 2: return 'sqrt(%s)' % self._print(expr.base) elif expr.exp == -0.5 or expr.exp == -S(1) / 2: return '1/sqrt(%s)' % self._print(expr.base) else: return '{base}^{exp}'.format( base=self.parenthesize(expr.base, PREC), exp=self.parenthesize(expr.exp, PREC)) def _print_Piecewise(self, expr): if (expr.args[-1].cond is not True) and (expr.args[-1].cond != S.BooleanTrue): # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") _coup_list = [ ("{c}, {e}".format(c=self._print(c), e=self._print(e)) if c is not True and c is not S.BooleanTrue else "{e}".format( e=self._print(e))) for e, c in expr.args] _inbrace = ', '.join(_coup_list) return 'piecewise({_inbrace})'.format(_inbrace=_inbrace) def _print_Rational(self, expr): p, q = int(expr.p), int(expr.q) return "{p}/{q}".format(p=str(p), q=str(q)) def _print_Relational(self, expr): PREC=precedence(expr) lhs_code = self.parenthesize(expr.lhs, PREC) rhs_code = self.parenthesize(expr.rhs, PREC) op = expr.rel_op if op in spec_relational_ops: op = spec_relational_ops[op] return "{lhs} {rel_op} {rhs}".format(lhs=lhs_code, rel_op=op, rhs=rhs_code) def _print_NumberSymbol(self, expr): return number_symbols[expr] def _print_NegativeInfinity(self, expr): return '-infinity' def _print_Infinity(self, expr): return 'infinity' def _print_Idx(self, expr): return self._print(expr.label) def _print_BooleanTrue(self, expr): return "true" def _print_BooleanFalse(self, expr): return "false" def _print_bool(self, expr): return 'true' if expr else 'false' def _print_NaN(self, expr): return 'undefined' def _get_matrix(self, expr, sparse=False): if expr.cols == 0 or expr.rows == 0: _strM = 'Matrix([], storage = {storage})'.format( storage='sparse' if sparse else 'rectangular') else: _strM = 'Matrix({list}, storage = {storage})'.format( list=self._print(expr.tolist()), storage='sparse' if sparse else 'rectangular') return _strM def _print_MatrixElement(self, expr): return "{parent}[{i_maple}, {j_maple}]".format( parent=self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True), i_maple=self._print(expr.i + 1), j_maple=self._print(expr.j + 1)) def _print_MatrixBase(self, expr): return self._get_matrix(expr, sparse=False) def _print_SparseMatrix(self, expr): return self._get_matrix(expr, sparse=True) def _print_Identity(self, expr): if isinstance(expr.rows, Integer) or isinstance(expr.rows, IntegerConstant): return self._print(sympy.SparseMatrix(expr)) else: return "Matrix({var_size}, shape = identity)".format(var_size=self._print(expr.rows)) def _print_MatMul(self, expr): PREC=precedence(expr) _fact_list = list(expr.args) _const = None if not ( isinstance(_fact_list[0], sympy.MatrixBase) or isinstance( _fact_list[0], sympy.MatrixExpr) or isinstance( _fact_list[0], sympy.MatrixSlice) or isinstance( _fact_list[0], sympy.MatrixSymbol)): _const, _fact_list = _fact_list[0], _fact_list[1:] if _const is None or _const == 1: return '.'.join(self.parenthesize(_m, PREC) for _m in _fact_list) else: return '{c}*{m}'.format(c=_const, m='.'.join(self.parenthesize(_m, PREC) for _m in _fact_list)) def _print_MatPow(self, expr): # This function requires LinearAlgebra Function in Maple return 'MatrixPower({A}, {n})'.format(A=self._print(expr.base), n=self._print(expr.exp)) def _print_HadamardProduct(self, expr): PREC = precedence(expr) _fact_list = list(expr.args) return '*'.join(self.parenthesize(_m, PREC) for _m in _fact_list) def _print_Derivative(self, expr): _f, (_var, _order) = expr.args if _order != 1: _second_arg = '{var}${order}'.format(var=self._print(_var), order=self._print(_order)) else: _second_arg = '{var}'.format(var=self._print(_var)) return 'diff({func_expr}, {sec_arg})'.format(func_expr=self._print(_f), sec_arg=_second_arg) def maple_code(expr, assign_to=None, **settings): r"""Converts ``expr`` to a string of Maple code. Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for expressions that generate multi-line statements. precision : integer, optional The precision for numbers such as pi [default=16]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. inline: bool, optional If True, we try to create single-statement code instead of multiple statements. [default=True]. """ return MapleCodePrinter(settings).doprint(expr, assign_to) def print_maple_code(expr, **settings): """Prints the Maple representation of the given expression. See :func:`maple_code` for the meaning of the optional arguments. Examples ======== >>> from sympy.printing.maple import print_maple_code >>> from sympy import symbols >>> x, y = symbols('x y') >>> print_maple_code(x, assign_to=y) y := x """ print(maple_code(expr, **settings))
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from distutils.version import LooseVersion as V from sympy import Mul, S from sympy.codegen.cfunctions import Sqrt from sympy.core.compatibility import Iterable from sympy.external import import_module from sympy.printing.precedence import PRECEDENCE from sympy.printing.pycode import AbstractPythonCodePrinter import sympy tensorflow = import_module('tensorflow') class TensorflowPrinter(AbstractPythonCodePrinter): """ Tensorflow printer which handles vectorized piecewise functions, logical operators, max/min, and relational operators. """ printmethod = "_tensorflowcode" mapping = { sympy.Abs: "tensorflow.math.abs", sympy.sign: "tensorflow.math.sign", # XXX May raise error for ints. sympy.ceiling: "tensorflow.math.ceil", sympy.floor: "tensorflow.math.floor", sympy.log: "tensorflow.math.log", sympy.exp: "tensorflow.math.exp", Sqrt: "tensorflow.math.sqrt", sympy.cos: "tensorflow.math.cos", sympy.acos: "tensorflow.math.acos", sympy.sin: "tensorflow.math.sin", sympy.asin: "tensorflow.math.asin", sympy.tan: "tensorflow.math.tan", sympy.atan: "tensorflow.math.atan", sympy.atan2: "tensorflow.math.atan2", # XXX Also may give NaN for complex results. sympy.cosh: "tensorflow.math.cosh", sympy.acosh: "tensorflow.math.acosh", sympy.sinh: "tensorflow.math.sinh", sympy.asinh: "tensorflow.math.asinh", sympy.tanh: "tensorflow.math.tanh", sympy.atanh: "tensorflow.math.atanh", sympy.re: "tensorflow.math.real", sympy.im: "tensorflow.math.imag", sympy.arg: "tensorflow.math.angle", # XXX May raise error for ints and complexes sympy.erf: "tensorflow.math.erf", sympy.loggamma: "tensorflow.math.lgamma", sympy.Eq: "tensorflow.math.equal", sympy.Ne: "tensorflow.math.not_equal", sympy.StrictGreaterThan: "tensorflow.math.greater", sympy.StrictLessThan: "tensorflow.math.less", sympy.LessThan: "tensorflow.math.less_equal", sympy.GreaterThan: "tensorflow.math.greater_equal", sympy.And: "tensorflow.math.logical_and", sympy.Or: "tensorflow.math.logical_or", sympy.Not: "tensorflow.math.logical_not", sympy.Max: "tensorflow.math.maximum", sympy.Min: "tensorflow.math.minimum", # Matrices sympy.MatAdd: "tensorflow.math.add", sympy.HadamardProduct: "tensorflow.math.multiply", sympy.Trace: "tensorflow.linalg.trace", # XXX May raise error for integer matrices. sympy.Determinant : "tensorflow.linalg.det", } _default_settings = dict( AbstractPythonCodePrinter._default_settings, tensorflow_version=None ) def __init__(self, settings=None): super().__init__(settings) version = self._settings['tensorflow_version'] if version is None and tensorflow: version = tensorflow.__version__ self.tensorflow_version = version def _print_Function(self, expr): op = self.mapping.get(type(expr), None) if op is None: return super()._print_Basic(expr) children = [self._print(arg) for arg in expr.args] if len(children) == 1: return "%s(%s)" % ( self._module_format(op), children[0] ) else: return self._expand_fold_binary_op(op, children) _print_Expr = _print_Function _print_Application = _print_Function _print_MatrixExpr = _print_Function # TODO: a better class structure would avoid this mess: _print_Relational = _print_Function _print_Not = _print_Function _print_And = _print_Function _print_Or = _print_Function _print_HadamardProduct = _print_Function _print_Trace = _print_Function _print_Determinant = _print_Function def _print_Inverse(self, expr): op = self._module_format('tensorflow.linalg.inv') return "{}({})".format(op, self._print(expr.arg)) def _print_Transpose(self, expr): version = self.tensorflow_version if version and V(version) < V('1.14'): op = self._module_format('tensorflow.matrix_transpose') else: op = self._module_format('tensorflow.linalg.matrix_transpose') return "{}({})".format(op, self._print(expr.arg)) def _print_Derivative(self, expr): variables = expr.variables if any(isinstance(i, Iterable) for i in variables): raise NotImplementedError("derivation by multiple variables is not supported") def unfold(expr, args): if not args: return self._print(expr) return "%s(%s, %s)[0]" % ( self._module_format("tensorflow.gradients"), unfold(expr, args[:-1]), self._print(args[-1]), ) return unfold(expr.expr, variables) def _print_Piecewise(self, expr): version = self.tensorflow_version if version and V(version) < V('1.0'): tensorflow_piecewise = "tensorflow.select" else: tensorflow_piecewise = "tensorflow.where" from sympy import Piecewise e, cond = expr.args[0].args if len(expr.args) == 1: return '{}({}, {}, {})'.format( self._module_format(tensorflow_piecewise), self._print(cond), self._print(e), 0) return '{}({}, {}, {})'.format( self._module_format(tensorflow_piecewise), self._print(cond), self._print(e), self._print(Piecewise(*expr.args[1:]))) def _print_Pow(self, expr): # XXX May raise error for # int**float or int**complex or float**complex base, exp = expr.args if expr.exp == S.Half: return "{}({})".format( self._module_format("tensorflow.math.sqrt"), self._print(base)) return "{}({}, {})".format( self._module_format("tensorflow.math.pow"), self._print(base), self._print(exp)) def _print_MatrixBase(self, expr): tensorflow_f = "tensorflow.Variable" if expr.free_symbols else "tensorflow.constant" data = "["+", ".join(["["+", ".join([self._print(j) for j in i])+"]" for i in expr.tolist()])+"]" return "%s(%s)" % ( self._module_format(tensorflow_f), data, ) def _print_MatMul(self, expr): from sympy.matrices.expressions import MatrixExpr mat_args = [arg for arg in expr.args if isinstance(arg, MatrixExpr)] args = [arg for arg in expr.args if arg not in mat_args] if args: return "%s*%s" % ( self.parenthesize(Mul.fromiter(args), PRECEDENCE["Mul"]), self._expand_fold_binary_op( "tensorflow.linalg.matmul", mat_args) ) else: return self._expand_fold_binary_op( "tensorflow.linalg.matmul", mat_args) def _print_MatPow(self, expr): return self._expand_fold_binary_op( "tensorflow.linalg.matmul", [expr.base]*expr.exp) def _print_Assignment(self, expr): # TODO: is this necessary? return "%s = %s" % ( self._print(expr.lhs), self._print(expr.rhs), ) def _print_CodeBlock(self, expr): # TODO: is this necessary? ret = [] for subexpr in expr.args: ret.append(self._print(subexpr)) return "\n".join(ret) def _get_letter_generator_for_einsum(self): for i in range(97, 123): yield chr(i) for i in range(65, 91): yield chr(i) raise ValueError("out of letters") def _print_CodegenArrayTensorProduct(self, expr): letters = self._get_letter_generator_for_einsum() contraction_string = ",".join(["".join([next(letters) for j in range(i)]) for i in expr.subranks]) return '%s("%s", %s)' % ( self._module_format('tensorflow.linalg.einsum'), contraction_string, ", ".join([self._print(arg) for arg in expr.args]) ) def _print_CodegenArrayContraction(self, expr): from sympy.codegen.array_utils import CodegenArrayTensorProduct base = expr.expr contraction_indices = expr.contraction_indices contraction_string, letters_free, letters_dum = self._get_einsum_string(base.subranks, contraction_indices) if not contraction_indices: return self._print(base) if isinstance(base, CodegenArrayTensorProduct): elems = ["%s" % (self._print(arg)) for arg in base.args] return "%s(\"%s\", %s)" % ( self._module_format("tensorflow.linalg.einsum"), contraction_string, ", ".join(elems) ) raise NotImplementedError() def _print_CodegenArrayDiagonal(self, expr): from sympy.codegen.array_utils import CodegenArrayTensorProduct diagonal_indices = list(expr.diagonal_indices) if len(diagonal_indices) > 1: # TODO: this should be handled in sympy.codegen.array_utils, # possibly by creating the possibility of unfolding the # CodegenArrayDiagonal object into nested ones. Same reasoning for # the array contraction. raise NotImplementedError if len(diagonal_indices[0]) != 2: raise NotImplementedError if isinstance(expr.expr, CodegenArrayTensorProduct): subranks = expr.expr.subranks elems = expr.expr.args else: subranks = expr.subranks elems = [expr.expr] diagonal_string, letters_free, letters_dum = self._get_einsum_string(subranks, diagonal_indices) elems = [self._print(i) for i in elems] return '%s("%s", %s)' % ( self._module_format("tensorflow.linalg.einsum"), "{}->{}{}".format(diagonal_string, "".join(letters_free), "".join(letters_dum)), ", ".join(elems) ) def _print_CodegenArrayPermuteDims(self, expr): return "%s(%s, %s)" % ( self._module_format("tensorflow.transpose"), self._print(expr.expr), self._print(expr.permutation.array_form), ) def _print_CodegenArrayElementwiseAdd(self, expr): return self._expand_fold_binary_op('tensorflow.math.add', expr.args) def tensorflow_code(expr, **settings): printer = TensorflowPrinter(settings) return printer.doprint(expr)
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from sympy.core.basic import Basic from sympy.core.expr import Expr from sympy.core.symbol import Symbol from sympy.core.numbers import Integer, Rational, Float from sympy.printing.repr import srepr __all__ = ['dotprint'] default_styles = ( (Basic, {'color': 'blue', 'shape': 'ellipse'}), (Expr, {'color': 'black'}) ) slotClasses = (Symbol, Integer, Rational, Float) def purestr(x, with_args=False): """A string that follows ```obj = type(obj)(*obj.args)``` exactly. Parameters ========== with_args : boolean, optional If ``True``, there will be a second argument for the return value, which is a tuple containing ``purestr`` applied to each of the subnodes. If ``False``, there will not be a second argument for the return. Default is ``False`` Examples ======== >>> from sympy import Float, Symbol, MatrixSymbol >>> from sympy import Integer # noqa: F401 >>> from sympy.core.symbol import Str # noqa: F401 >>> from sympy.printing.dot import purestr Applying ``purestr`` for basic symbolic object: >>> code = purestr(Symbol('x')) >>> code "Symbol('x')" >>> eval(code) == Symbol('x') True For basic numeric object: >>> purestr(Float(2)) "Float('2.0', precision=53)" For matrix symbol: >>> code = purestr(MatrixSymbol('x', 2, 2)) >>> code "MatrixSymbol(Str('x'), Integer(2), Integer(2))" >>> eval(code) == MatrixSymbol('x', 2, 2) True With ``with_args=True``: >>> purestr(Float(2), with_args=True) ("Float('2.0', precision=53)", ()) >>> purestr(MatrixSymbol('x', 2, 2), with_args=True) ("MatrixSymbol(Str('x'), Integer(2), Integer(2))", ("Str('x')", 'Integer(2)', 'Integer(2)')) """ sargs = () if not isinstance(x, Basic): rv = str(x) elif not x.args: rv = srepr(x) else: args = x.args sargs = tuple(map(purestr, args)) rv = "%s(%s)"%(type(x).__name__, ', '.join(sargs)) if with_args: rv = rv, sargs return rv def styleof(expr, styles=default_styles): """ Merge style dictionaries in order Examples ======== >>> from sympy import Symbol, Basic, Expr >>> from sympy.printing.dot import styleof >>> styles = [(Basic, {'color': 'blue', 'shape': 'ellipse'}), ... (Expr, {'color': 'black'})] >>> styleof(Basic(1), styles) {'color': 'blue', 'shape': 'ellipse'} >>> x = Symbol('x') >>> styleof(x + 1, styles) # this is an Expr {'color': 'black', 'shape': 'ellipse'} """ style = dict() for typ, sty in styles: if isinstance(expr, typ): style.update(sty) return style def attrprint(d, delimiter=', '): """ Print a dictionary of attributes Examples ======== >>> from sympy.printing.dot import attrprint >>> print(attrprint({'color': 'blue', 'shape': 'ellipse'})) "color"="blue", "shape"="ellipse" """ return delimiter.join('"%s"="%s"'%item for item in sorted(d.items())) def dotnode(expr, styles=default_styles, labelfunc=str, pos=(), repeat=True): """ String defining a node Examples ======== >>> from sympy.printing.dot import dotnode >>> from sympy.abc import x >>> print(dotnode(x)) "Symbol('x')_()" ["color"="black", "label"="x", "shape"="ellipse"]; """ style = styleof(expr, styles) if isinstance(expr, Basic) and not expr.is_Atom: label = str(expr.__class__.__name__) else: label = labelfunc(expr) style['label'] = label expr_str = purestr(expr) if repeat: expr_str += '_%s' % str(pos) return '"%s" [%s];' % (expr_str, attrprint(style)) def dotedges(expr, atom=lambda x: not isinstance(x, Basic), pos=(), repeat=True): """ List of strings for all expr->expr.arg pairs See the docstring of dotprint for explanations of the options. Examples ======== >>> from sympy.printing.dot import dotedges >>> from sympy.abc import x >>> for e in dotedges(x+2): ... print(e) "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)"; "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)"; """ if atom(expr): return [] else: expr_str, arg_strs = purestr(expr, with_args=True) if repeat: expr_str += '_%s' % str(pos) arg_strs = ['%s_%s' % (a, str(pos + (i,))) for i, a in enumerate(arg_strs)] return ['"%s" -> "%s";' % (expr_str, a) for a in arg_strs] template = \ """digraph{ # Graph style %(graphstyle)s ######### # Nodes # ######### %(nodes)s ######### # Edges # ######### %(edges)s }""" _graphstyle = {'rankdir': 'TD', 'ordering': 'out'} def dotprint(expr, styles=default_styles, atom=lambda x: not isinstance(x, Basic), maxdepth=None, repeat=True, labelfunc=str, **kwargs): """DOT description of a SymPy expression tree Parameters ========== styles : list of lists composed of (Class, mapping), optional Styles for different classes. The default is .. code-block:: python ( (Basic, {'color': 'blue', 'shape': 'ellipse'}), (Expr, {'color': 'black'}) ) atom : function, optional Function used to determine if an arg is an atom. A good choice is ``lambda x: not x.args``. The default is ``lambda x: not isinstance(x, Basic)``. maxdepth : integer, optional The maximum depth. The default is ``None``, meaning no limit. repeat : boolean, optional Whether to use different nodes for common subexpressions. The default is ``True``. For example, for ``x + x*y`` with ``repeat=True``, it will have two nodes for ``x``; with ``repeat=False``, it will have one node. .. warning:: Even if a node appears twice in the same object like ``x`` in ``Pow(x, x)``, it will still only appear once. Hence, with ``repeat=False``, the number of arrows out of an object might not equal the number of args it has. labelfunc : function, optional A function to create a label for a given leaf node. The default is ``str``. Another good option is ``srepr``. For example with ``str``, the leaf nodes of ``x + 1`` are labeled, ``x`` and ``1``. With ``srepr``, they are labeled ``Symbol('x')`` and ``Integer(1)``. **kwargs : optional Additional keyword arguments are included as styles for the graph. Examples ======== >>> from sympy.printing.dot import dotprint >>> from sympy.abc import x >>> print(dotprint(x+2)) # doctest: +NORMALIZE_WHITESPACE digraph{ <BLANKLINE> # Graph style "ordering"="out" "rankdir"="TD" <BLANKLINE> ######### # Nodes # ######### <BLANKLINE> "Add(Integer(2), Symbol('x'))_()" ["color"="black", "label"="Add", "shape"="ellipse"]; "Integer(2)_(0,)" ["color"="black", "label"="2", "shape"="ellipse"]; "Symbol('x')_(1,)" ["color"="black", "label"="x", "shape"="ellipse"]; <BLANKLINE> ######### # Edges # ######### <BLANKLINE> "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)"; "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)"; } """ # repeat works by adding a signature tuple to the end of each node for its # position in the graph. For example, for expr = Add(x, Pow(x, 2)), the x in the # Pow will have the tuple (1, 0), meaning it is expr.args[1].args[0]. graphstyle = _graphstyle.copy() graphstyle.update(kwargs) nodes = [] edges = [] def traverse(e, depth, pos=()): nodes.append(dotnode(e, styles, labelfunc=labelfunc, pos=pos, repeat=repeat)) if maxdepth and depth >= maxdepth: return edges.extend(dotedges(e, atom=atom, pos=pos, repeat=repeat)) [traverse(arg, depth+1, pos + (i,)) for i, arg in enumerate(e.args) if not atom(arg)] traverse(expr, 0) return template%{'graphstyle': attrprint(graphstyle, delimiter='\n'), 'nodes': '\n'.join(nodes), 'edges': '\n'.join(edges)}
fa56866e432bfc868c08d5cbb684a7574d3c583f163be8eaeb807a6dce636496
import os from os.path import join import shutil import tempfile try: from subprocess import STDOUT, CalledProcessError, check_output except ImportError: pass from sympy.utilities.decorator import doctest_depends_on from .latex import latex __doctest_requires__ = {('preview',): ['pyglet']} def _check_output_no_window(*args, **kwargs): # Avoid showing a cmd.exe window when running this # on Windows if os.name == 'nt': creation_flag = 0x08000000 # CREATE_NO_WINDOW else: creation_flag = 0 # Default value return check_output(*args, creationflags=creation_flag, **kwargs) def _run_pyglet(fname, fmt): from pyglet import window, image, gl from pyglet.window import key if fmt == "png": from pyglet.image.codecs.png import PNGImageDecoder img = image.load(fname, decoder=PNGImageDecoder()) else: raise ValueError("pyglet preview works only for 'png' files.") offset = 25 config = gl.Config(double_buffer=False) win = window.Window( width=img.width + 2*offset, height=img.height + 2*offset, caption="sympy", resizable=False, config=config ) win.set_vsync(False) try: def on_close(): win.has_exit = True win.on_close = on_close def on_key_press(symbol, modifiers): if symbol in [key.Q, key.ESCAPE]: on_close() win.on_key_press = on_key_press def on_expose(): gl.glClearColor(1.0, 1.0, 1.0, 1.0) gl.glClear(gl.GL_COLOR_BUFFER_BIT) img.blit( (win.width - img.width) / 2, (win.height - img.height) / 2 ) win.on_expose = on_expose while not win.has_exit: win.dispatch_events() win.flip() except KeyboardInterrupt: pass win.close() @doctest_depends_on(exe=('latex', 'dvipng'), modules=('pyglet',), disable_viewers=('evince', 'gimp', 'superior-dvi-viewer')) def preview(expr, output='png', viewer=None, euler=True, packages=(), filename=None, outputbuffer=None, preamble=None, dvioptions=None, outputTexFile=None, **latex_settings): r""" View expression or LaTeX markup in PNG, DVI, PostScript or PDF form. If the expr argument is an expression, it will be exported to LaTeX and then compiled using the available TeX distribution. The first argument, 'expr', may also be a LaTeX string. The function will then run the appropriate viewer for the given output format or use the user defined one. By default png output is generated. By default pretty Euler fonts are used for typesetting (they were used to typeset the well known "Concrete Mathematics" book). For that to work, you need the 'eulervm.sty' LaTeX style (in Debian/Ubuntu, install the texlive-fonts-extra package). If you prefer default AMS fonts or your system lacks 'eulervm' LaTeX package then unset the 'euler' keyword argument. To use viewer auto-detection, lets say for 'png' output, issue >>> from sympy import symbols, preview, Symbol >>> x, y = symbols("x,y") >>> preview(x + y, output='png') This will choose 'pyglet' by default. To select a different one, do >>> preview(x + y, output='png', viewer='gimp') The 'png' format is considered special. For all other formats the rules are slightly different. As an example we will take 'dvi' output format. If you would run >>> preview(x + y, output='dvi') then 'view' will look for available 'dvi' viewers on your system (predefined in the function, so it will try evince, first, then kdvi and xdvi). If nothing is found you will need to set the viewer explicitly. >>> preview(x + y, output='dvi', viewer='superior-dvi-viewer') This will skip auto-detection and will run user specified 'superior-dvi-viewer'. If 'view' fails to find it on your system it will gracefully raise an exception. You may also enter 'file' for the viewer argument. Doing so will cause this function to return a file object in read-only mode, if 'filename' is unset. However, if it was set, then 'preview' writes the genereted file to this filename instead. There is also support for writing to a BytesIO like object, which needs to be passed to the 'outputbuffer' argument. >>> from io import BytesIO >>> obj = BytesIO() >>> preview(x + y, output='png', viewer='BytesIO', ... outputbuffer=obj) The LaTeX preamble can be customized by setting the 'preamble' keyword argument. This can be used, e.g., to set a different font size, use a custom documentclass or import certain set of LaTeX packages. >>> preamble = "\\documentclass[10pt]{article}\n" \ ... "\\usepackage{amsmath,amsfonts}\\begin{document}" >>> preview(x + y, output='png', preamble=preamble) If the value of 'output' is different from 'dvi' then command line options can be set ('dvioptions' argument) for the execution of the 'dvi'+output conversion tool. These options have to be in the form of a list of strings (see subprocess.Popen). Additional keyword args will be passed to the latex call, e.g., the symbol_names flag. >>> phidd = Symbol('phidd') >>> preview(phidd, symbol_names={phidd:r'\ddot{\varphi}'}) For post-processing the generated TeX File can be written to a file by passing the desired filename to the 'outputTexFile' keyword argument. To write the TeX code to a file named "sample.tex" and run the default png viewer to display the resulting bitmap, do >>> preview(x + y, outputTexFile="sample.tex") """ special = [ 'pyglet' ] if viewer is None: if output == "png": viewer = "pyglet" else: # sorted in order from most pretty to most ugly # very discussable, but indeed 'gv' looks awful :) # TODO add candidates for windows to list candidates = { "dvi": [ "evince", "okular", "kdvi", "xdvi" ], "ps": [ "evince", "okular", "gsview", "gv" ], "pdf": [ "evince", "okular", "kpdf", "acroread", "xpdf", "gv" ], } try: candidate_viewers = candidates[output] except KeyError: raise ValueError("Invalid output format: %s" % output) from None for candidate in candidate_viewers: path = shutil.which(candidate) if path is not None: viewer = path break else: raise OSError( "No viewers found for '%s' output format." % output) else: if viewer == "file": if filename is None: raise ValueError("filename has to be specified if viewer=\"file\"") elif viewer == "BytesIO": if outputbuffer is None: raise ValueError("outputbuffer has to be a BytesIO " "compatible object if viewer=\"BytesIO\"") elif viewer not in special and not shutil.which(viewer): raise OSError("Unrecognized viewer: %s" % viewer) if preamble is None: actual_packages = packages + ("amsmath", "amsfonts") if euler: actual_packages += ("euler",) package_includes = "\n" + "\n".join(["\\usepackage{%s}" % p for p in actual_packages]) preamble = r"""\documentclass[varwidth,12pt]{standalone} %s \begin{document} """ % (package_includes) else: if packages: raise ValueError("The \"packages\" keyword must not be set if a " "custom LaTeX preamble was specified") if isinstance(expr, str): latex_string = expr else: latex_string = ('$\\displaystyle ' + latex(expr, mode='plain', **latex_settings) + '$') latex_main = preamble + '\n' + latex_string + '\n\n' + r"\end{document}" with tempfile.TemporaryDirectory() as workdir: with open(join(workdir, 'texput.tex'), 'w', encoding='utf-8') as fh: fh.write(latex_main) if outputTexFile is not None: shutil.copyfile(join(workdir, 'texput.tex'), outputTexFile) if not shutil.which('latex'): raise RuntimeError("latex program is not installed") try: _check_output_no_window( ['latex', '-halt-on-error', '-interaction=nonstopmode', 'texput.tex'], cwd=workdir, stderr=STDOUT) except CalledProcessError as e: raise RuntimeError( "'latex' exited abnormally with the following output:\n%s" % e.output) src = "texput.%s" % (output) if output != "dvi": # in order of preference commandnames = { "ps": ["dvips"], "pdf": ["dvipdfmx", "dvipdfm", "dvipdf"], "png": ["dvipng"], "svg": ["dvisvgm"], } try: cmd_variants = commandnames[output] except KeyError: raise ValueError("Invalid output format: %s" % output) from None # find an appropriate command for cmd_variant in cmd_variants: cmd_path = shutil.which(cmd_variant) if cmd_path: cmd = [cmd_path] break else: if len(cmd_variants) > 1: raise RuntimeError("None of %s are installed" % ", ".join(cmd_variants)) else: raise RuntimeError("%s is not installed" % cmd_variants[0]) defaultoptions = { "dvipng": ["-T", "tight", "-z", "9", "--truecolor"], "dvisvgm": ["--no-fonts"], } commandend = { "dvips": ["-o", src, "texput.dvi"], "dvipdf": ["texput.dvi", src], "dvipdfm": ["-o", src, "texput.dvi"], "dvipdfmx": ["-o", src, "texput.dvi"], "dvipng": ["-o", src, "texput.dvi"], "dvisvgm": ["-o", src, "texput.dvi"], } if dvioptions is not None: cmd.extend(dvioptions) else: cmd.extend(defaultoptions.get(cmd_variant, [])) cmd.extend(commandend[cmd_variant]) try: _check_output_no_window(cmd, cwd=workdir, stderr=STDOUT) except CalledProcessError as e: raise RuntimeError( "'%s' exited abnormally with the following output:\n%s" % (' '.join(cmd), e.output)) if viewer == "file": shutil.move(join(workdir, src), filename) elif viewer == "BytesIO": with open(join(workdir, src), 'rb') as fh: outputbuffer.write(fh.read()) elif viewer == "pyglet": try: import pyglet # noqa: F401 except ImportError: raise ImportError("pyglet is required for preview.\n visit http://www.pyglet.org/") return _run_pyglet(join(workdir, src), fmt=output) else: try: _check_output_no_window( [viewer, src], cwd=workdir, stderr=STDOUT) except CalledProcessError as e: raise RuntimeError( "'%s %s' exited abnormally with the following output:\n%s" % (viewer, src, e.output))
bbc08e3e1f9bff1710174065bd562a0b7dbe293509dcfa02e67d15bf2fc8ba96
""" Javascript code printer The JavascriptCodePrinter converts single sympy expressions into single Javascript expressions, using the functions defined in the Javascript Math object where possible. """ from typing import Any, Dict from sympy.core import S from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence, PRECEDENCE # dictionary mapping sympy function to (argument_conditions, Javascript_function). # Used in JavascriptCodePrinter._print_Function(self) known_functions = { 'Abs': 'Math.abs', 'acos': 'Math.acos', 'acosh': 'Math.acosh', 'asin': 'Math.asin', 'asinh': 'Math.asinh', 'atan': 'Math.atan', 'atan2': 'Math.atan2', 'atanh': 'Math.atanh', 'ceiling': 'Math.ceil', 'cos': 'Math.cos', 'cosh': 'Math.cosh', 'exp': 'Math.exp', 'floor': 'Math.floor', 'log': 'Math.log', 'Max': 'Math.max', 'Min': 'Math.min', 'sign': 'Math.sign', 'sin': 'Math.sin', 'sinh': 'Math.sinh', 'tan': 'Math.tan', 'tanh': 'Math.tanh', } class JavascriptCodePrinter(CodePrinter): """"A Printer to convert python expressions to strings of javascript code """ printmethod = '_javascript' language = 'Javascript' _default_settings = { 'order': None, 'full_prec': 'auto', 'precision': 17, 'user_functions': {}, 'human': True, 'allow_unknown_functions': False, 'contract': True, } # type: Dict[str, Any] def __init__(self, settings={}): CodePrinter.__init__(self, settings) self.known_functions = dict(known_functions) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) def _rate_index_position(self, p): return p*5 def _get_statement(self, codestring): return "%s;" % codestring def _get_comment(self, text): return "// {}".format(text) def _declare_number_const(self, name, value): return "var {} = {};".format(name, value.evalf(self._settings['precision'])) def _format_code(self, lines): return self.indent_code(lines) def _traverse_matrix_indices(self, mat): rows, cols = mat.shape return ((i, j) for i in range(rows) for j in range(cols)) def _get_loop_opening_ending(self, indices): open_lines = [] close_lines = [] loopstart = "for (var %(varble)s=%(start)s; %(varble)s<%(end)s; %(varble)s++){" for i in indices: # Javascript arrays start at 0 and end at dimension-1 open_lines.append(loopstart % { 'varble': self._print(i.label), 'start': self._print(i.lower), 'end': self._print(i.upper + 1)}) close_lines.append("}") return open_lines, close_lines def _print_Pow(self, expr): PREC = precedence(expr) if expr.exp == -1: return '1/%s' % (self.parenthesize(expr.base, PREC)) elif expr.exp == 0.5: return 'Math.sqrt(%s)' % self._print(expr.base) elif expr.exp == S.One/3: return 'Math.cbrt(%s)' % self._print(expr.base) else: return 'Math.pow(%s, %s)' % (self._print(expr.base), self._print(expr.exp)) def _print_Rational(self, expr): p, q = int(expr.p), int(expr.q) return '%d/%d' % (p, q) def _print_Relational(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) op = expr.rel_op return "{} {} {}".format(lhs_code, op, rhs_code) def _print_Indexed(self, expr): # calculate index for 1d array dims = expr.shape elem = S.Zero offset = S.One for i in reversed(range(expr.rank)): elem += expr.indices[i]*offset offset *= dims[i] return "%s[%s]" % (self._print(expr.base.label), self._print(elem)) def _print_Idx(self, expr): return self._print(expr.label) def _print_Exp1(self, expr): return "Math.E" def _print_Pi(self, expr): return 'Math.PI' def _print_Infinity(self, expr): return 'Number.POSITIVE_INFINITY' def _print_NegativeInfinity(self, expr): return 'Number.NEGATIVE_INFINITY' def _print_Piecewise(self, expr): from sympy.codegen.ast import Assignment if expr.args[-1].cond != True: # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") lines = [] if expr.has(Assignment): for i, (e, c) in enumerate(expr.args): if i == 0: lines.append("if (%s) {" % self._print(c)) elif i == len(expr.args) - 1 and c == True: lines.append("else {") else: lines.append("else if (%s) {" % self._print(c)) code0 = self._print(e) lines.append(code0) lines.append("}") return "\n".join(lines) else: # The piecewise was used in an expression, need to do inline # operators. This has the downside that inline operators will # not work for statements that span multiple lines (Matrix or # Indexed expressions). ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c), self._print(e)) for e, c in expr.args[:-1]] last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr) return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)]) def _print_MatrixElement(self, expr): return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True), expr.j + expr.i*expr.parent.shape[1]) def indent_code(self, code): """Accepts a string of code or a list of code lines""" if isinstance(code, str): code_lines = self.indent_code(code.splitlines(True)) return ''.join(code_lines) tab = " " inc_token = ('{', '(', '{\n', '(\n') dec_token = ('}', ')') code = [ line.lstrip(' \t') for line in code ] increase = [ int(any(map(line.endswith, inc_token))) for line in code ] decrease = [ int(any(map(line.startswith, dec_token))) for line in code ] pretty = [] level = 0 for n, line in enumerate(code): if line == '' or line == '\n': pretty.append(line) continue level -= decrease[n] pretty.append("%s%s" % (tab*level, line)) level += increase[n] return pretty def jscode(expr, assign_to=None, **settings): """Converts an expr to a string of javascript code Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of line-wrapping, or for expressions that generate multi-line statements. precision : integer, optional The precision for numbers such as pi [default=15]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, js_function_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. Examples ======== >>> from sympy import jscode, symbols, Rational, sin, ceiling, Abs >>> x, tau = symbols("x, tau") >>> jscode((2*tau)**Rational(7, 2)) '8*Math.sqrt(2)*Math.pow(tau, 7/2)' >>> jscode(sin(x), assign_to="s") 's = Math.sin(x);' Custom printing can be defined for certain types by passing a dictionary of "type" : "function" to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, js_function_string)]. >>> custom_functions = { ... "ceiling": "CEIL", ... "Abs": [(lambda x: not x.is_integer, "fabs"), ... (lambda x: x.is_integer, "ABS")] ... } >>> jscode(Abs(x) + ceiling(x), user_functions=custom_functions) 'fabs(x) + CEIL(x)' ``Piecewise`` expressions are converted into conditionals. If an ``assign_to`` variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> expr = Piecewise((x + 1, x > 0), (x, True)) >>> print(jscode(expr, tau)) if (x > 0) { tau = x + 1; } else { tau = x; } Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> jscode(e.rhs, assign_to=e.lhs, contract=False) 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions must be provided to ``assign_to``. Note that any expression that can be generated normally can also exist inside a Matrix: >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) >>> A = MatrixSymbol('A', 3, 1) >>> print(jscode(mat, A)) A[0] = Math.pow(x, 2); if (x > 0) { A[1] = x + 1; } else { A[1] = x; } A[2] = Math.sin(x); """ return JavascriptCodePrinter(settings).doprint(expr, assign_to) def print_jscode(expr, **settings): """Prints the Javascript representation of the given expression. See jscode for the meaning of the optional arguments. """ print(jscode(expr, **settings))
622b4b7c845f147f25b4d775f18e98b4182ae47dad7b4c052a51ee20ee1cbe8f
from sympy.core._print_helpers import Printable # alias for compatibility Printable.__module__ = __name__ DefaultPrinting = Printable
0252fb94f490ff17caa893ac69c988ba11e843ba5ec2531621beebda8a376cf0
""" Julia code printer The `JuliaCodePrinter` converts SymPy expressions into Julia expressions. A complete code generator, which uses `julia_code` extensively, can be found in `sympy.utilities.codegen`. The `codegen` module can be used to generate complete source code files. """ from typing import Any, Dict from sympy.core import Mul, Pow, S, Rational from sympy.core.mul import _keep_coeff from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence, PRECEDENCE from re import search # List of known functions. First, those that have the same name in # SymPy and Julia. This is almost certainly incomplete! known_fcns_src1 = ["sin", "cos", "tan", "cot", "sec", "csc", "asin", "acos", "atan", "acot", "asec", "acsc", "sinh", "cosh", "tanh", "coth", "sech", "csch", "asinh", "acosh", "atanh", "acoth", "asech", "acsch", "sinc", "atan2", "sign", "floor", "log", "exp", "cbrt", "sqrt", "erf", "erfc", "erfi", "factorial", "gamma", "digamma", "trigamma", "polygamma", "beta", "airyai", "airyaiprime", "airybi", "airybiprime", "besselj", "bessely", "besseli", "besselk", "erfinv", "erfcinv"] # These functions have different names ("Sympy": "Julia"), more # generally a mapping to (argument_conditions, julia_function). known_fcns_src2 = { "Abs": "abs", "ceiling": "ceil", "conjugate": "conj", "hankel1": "hankelh1", "hankel2": "hankelh2", "im": "imag", "re": "real" } class JuliaCodePrinter(CodePrinter): """ A printer to convert expressions to strings of Julia code. """ printmethod = "_julia" language = "Julia" _operators = { 'and': '&&', 'or': '||', 'not': '!', } _default_settings = { 'order': None, 'full_prec': 'auto', 'precision': 17, 'user_functions': {}, 'human': True, 'allow_unknown_functions': False, 'contract': True, 'inline': True, } # type: Dict[str, Any] # Note: contract is for expressing tensors as loops (if True), or just # assignment (if False). FIXME: this should be looked a more carefully # for Julia. def __init__(self, settings={}): super().__init__(settings) self.known_functions = dict(zip(known_fcns_src1, known_fcns_src1)) self.known_functions.update(dict(known_fcns_src2)) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) def _rate_index_position(self, p): return p*5 def _get_statement(self, codestring): return "%s" % codestring def _get_comment(self, text): return "# {}".format(text) def _declare_number_const(self, name, value): return "const {} = {}".format(name, value) def _format_code(self, lines): return self.indent_code(lines) def _traverse_matrix_indices(self, mat): # Julia uses Fortran order (column-major) rows, cols = mat.shape return ((i, j) for j in range(cols) for i in range(rows)) def _get_loop_opening_ending(self, indices): open_lines = [] close_lines = [] for i in indices: # Julia arrays start at 1 and end at dimension var, start, stop = map(self._print, [i.label, i.lower + 1, i.upper + 1]) open_lines.append("for %s = %s:%s" % (var, start, stop)) close_lines.append("end") return open_lines, close_lines def _print_Mul(self, expr): # print complex numbers nicely in Julia if (expr.is_number and expr.is_imaginary and expr.as_coeff_Mul()[0].is_integer): return "%sim" % self._print(-S.ImaginaryUnit*expr) # cribbed from str.py prec = precedence(expr) c, e = expr.as_coeff_Mul() if c < 0: expr = _keep_coeff(-c, e) sign = "-" else: sign = "" a = [] # items in the numerator b = [] # items that are in the denominator (if any) pow_paren = [] # Will collect all pow with more than one base element and exp = -1 if self.order not in ('old', 'none'): args = expr.as_ordered_factors() else: # use make_args in case expr was something like -x -> x args = Mul.make_args(expr) # Gather args for numerator/denominator for item in args: if (item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative): if item.exp != -1: b.append(Pow(item.base, -item.exp, evaluate=False)) else: if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 pow_paren.append(item) b.append(Pow(item.base, -item.exp)) elif item.is_Rational and item is not S.Infinity: if item.p != 1: a.append(Rational(item.p)) if item.q != 1: b.append(Rational(item.q)) else: a.append(item) a = a or [S.One] a_str = [self.parenthesize(x, prec) for x in a] b_str = [self.parenthesize(x, prec) for x in b] # To parenthesize Pow with exp = -1 and having more than one Symbol for item in pow_paren: if item.base in b: b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] # from here it differs from str.py to deal with "*" and ".*" def multjoin(a, a_str): # here we probably are assuming the constants will come first r = a_str[0] for i in range(1, len(a)): mulsym = '*' if a[i-1].is_number else '.*' r = r + mulsym + a_str[i] return r if not b: return sign + multjoin(a, a_str) elif len(b) == 1: divsym = '/' if b[0].is_number else './' return sign + multjoin(a, a_str) + divsym + b_str[0] else: divsym = '/' if all([bi.is_number for bi in b]) else './' return (sign + multjoin(a, a_str) + divsym + "(%s)" % multjoin(b, b_str)) def _print_Relational(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) op = expr.rel_op return "{} {} {}".format(lhs_code, op, rhs_code) def _print_Pow(self, expr): powsymbol = '^' if all([x.is_number for x in expr.args]) else '.^' PREC = precedence(expr) if expr.exp == S.Half: return "sqrt(%s)" % self._print(expr.base) if expr.is_commutative: if expr.exp == -S.Half: sym = '/' if expr.base.is_number else './' return "1" + sym + "sqrt(%s)" % self._print(expr.base) if expr.exp == -S.One: sym = '/' if expr.base.is_number else './' return "1" + sym + "%s" % self.parenthesize(expr.base, PREC) return '%s%s%s' % (self.parenthesize(expr.base, PREC), powsymbol, self.parenthesize(expr.exp, PREC)) def _print_MatPow(self, expr): PREC = precedence(expr) return '%s^%s' % (self.parenthesize(expr.base, PREC), self.parenthesize(expr.exp, PREC)) def _print_Pi(self, expr): if self._settings["inline"]: return "pi" else: return super()._print_NumberSymbol(expr) def _print_ImaginaryUnit(self, expr): return "im" def _print_Exp1(self, expr): if self._settings["inline"]: return "e" else: return super()._print_NumberSymbol(expr) def _print_EulerGamma(self, expr): if self._settings["inline"]: return "eulergamma" else: return super()._print_NumberSymbol(expr) def _print_Catalan(self, expr): if self._settings["inline"]: return "catalan" else: return super()._print_NumberSymbol(expr) def _print_GoldenRatio(self, expr): if self._settings["inline"]: return "golden" else: return super()._print_NumberSymbol(expr) def _print_Assignment(self, expr): from sympy.codegen.ast import Assignment from sympy.functions.elementary.piecewise import Piecewise from sympy.tensor.indexed import IndexedBase # Copied from codeprinter, but remove special MatrixSymbol treatment lhs = expr.lhs rhs = expr.rhs # We special case assignments that take multiple lines if not self._settings["inline"] and isinstance(expr.rhs, Piecewise): # Here we modify Piecewise so each expression is now # an Assignment, and then continue on the print. expressions = [] conditions = [] for (e, c) in rhs.args: expressions.append(Assignment(lhs, e)) conditions.append(c) temp = Piecewise(*zip(expressions, conditions)) return self._print(temp) if self._settings["contract"] and (lhs.has(IndexedBase) or rhs.has(IndexedBase)): # Here we check if there is looping to be done, and if so # print the required loops. return self._doprint_loops(rhs, lhs) else: lhs_code = self._print(lhs) rhs_code = self._print(rhs) return self._get_statement("%s = %s" % (lhs_code, rhs_code)) def _print_Infinity(self, expr): return 'Inf' def _print_NegativeInfinity(self, expr): return '-Inf' def _print_NaN(self, expr): return 'NaN' def _print_list(self, expr): return 'Any[' + ', '.join(self._print(a) for a in expr) + ']' def _print_tuple(self, expr): if len(expr) == 1: return "(%s,)" % self._print(expr[0]) else: return "(%s)" % self.stringify(expr, ", ") _print_Tuple = _print_tuple def _print_BooleanTrue(self, expr): return "true" def _print_BooleanFalse(self, expr): return "false" def _print_bool(self, expr): return str(expr).lower() # Could generate quadrature code for definite Integrals? #_print_Integral = _print_not_supported def _print_MatrixBase(self, A): # Handle zero dimensions: if A.rows == 0 or A.cols == 0: return 'zeros(%s, %s)' % (A.rows, A.cols) elif (A.rows, A.cols) == (1, 1): return "[%s]" % A[0, 0] elif A.rows == 1: return "[%s]" % A.table(self, rowstart='', rowend='', colsep=' ') elif A.cols == 1: # note .table would unnecessarily equispace the rows return "[%s]" % ", ".join([self._print(a) for a in A]) return "[%s]" % A.table(self, rowstart='', rowend='', rowsep=';\n', colsep=' ') def _print_SparseMatrix(self, A): from sympy.matrices import Matrix L = A.col_list(); # make row vectors of the indices and entries I = Matrix([k[0] + 1 for k in L]) J = Matrix([k[1] + 1 for k in L]) AIJ = Matrix([k[2] for k in L]) return "sparse(%s, %s, %s, %s, %s)" % (self._print(I), self._print(J), self._print(AIJ), A.rows, A.cols) def _print_MatrixElement(self, expr): return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \ + '[%s,%s]' % (expr.i + 1, expr.j + 1) def _print_MatrixSlice(self, expr): def strslice(x, lim): l = x[0] + 1 h = x[1] step = x[2] lstr = self._print(l) hstr = 'end' if h == lim else self._print(h) if step == 1: if l == 1 and h == lim: return ':' if l == h: return lstr else: return lstr + ':' + hstr else: return ':'.join((lstr, self._print(step), hstr)) return (self._print(expr.parent) + '[' + strslice(expr.rowslice, expr.parent.shape[0]) + ',' + strslice(expr.colslice, expr.parent.shape[1]) + ']') def _print_Indexed(self, expr): inds = [ self._print(i) for i in expr.indices ] return "%s[%s]" % (self._print(expr.base.label), ",".join(inds)) def _print_Idx(self, expr): return self._print(expr.label) def _print_Identity(self, expr): return "eye(%s)" % self._print(expr.shape[0]) def _print_HadamardProduct(self, expr): return '.*'.join([self.parenthesize(arg, precedence(expr)) for arg in expr.args]) def _print_HadamardPower(self, expr): PREC = precedence(expr) return '.**'.join([ self.parenthesize(expr.base, PREC), self.parenthesize(expr.exp, PREC) ]) # Note: as of 2015, Julia doesn't have spherical Bessel functions def _print_jn(self, expr): from sympy.functions import sqrt, besselj x = expr.argument expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x) return self._print(expr2) def _print_yn(self, expr): from sympy.functions import sqrt, bessely x = expr.argument expr2 = sqrt(S.Pi/(2*x))*bessely(expr.order + S.Half, x) return self._print(expr2) def _print_Piecewise(self, expr): if expr.args[-1].cond != True: # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") lines = [] if self._settings["inline"]: # Express each (cond, expr) pair in a nested Horner form: # (condition) .* (expr) + (not cond) .* (<others>) # Expressions that result in multiple statements won't work here. ecpairs = ["({}) ? ({}) :".format (self._print(c), self._print(e)) for e, c in expr.args[:-1]] elast = " (%s)" % self._print(expr.args[-1].expr) pw = "\n".join(ecpairs) + elast # Note: current need these outer brackets for 2*pw. Would be # nicer to teach parenthesize() to do this for us when needed! return "(" + pw + ")" else: for i, (e, c) in enumerate(expr.args): if i == 0: lines.append("if (%s)" % self._print(c)) elif i == len(expr.args) - 1 and c == True: lines.append("else") else: lines.append("elseif (%s)" % self._print(c)) code0 = self._print(e) lines.append(code0) if i == len(expr.args) - 1: lines.append("end") return "\n".join(lines) def indent_code(self, code): """Accepts a string of code or a list of code lines""" # code mostly copied from ccode if isinstance(code, str): code_lines = self.indent_code(code.splitlines(True)) return ''.join(code_lines) tab = " " inc_regex = ('^function ', '^if ', '^elseif ', '^else$', '^for ') dec_regex = ('^end$', '^elseif ', '^else$') # pre-strip left-space from the code code = [ line.lstrip(' \t') for line in code ] increase = [ int(any([search(re, line) for re in inc_regex])) for line in code ] decrease = [ int(any([search(re, line) for re in dec_regex])) for line in code ] pretty = [] level = 0 for n, line in enumerate(code): if line == '' or line == '\n': pretty.append(line) continue level -= decrease[n] pretty.append("%s%s" % (tab*level, line)) level += increase[n] return pretty def julia_code(expr, assign_to=None, **settings): r"""Converts `expr` to a string of Julia code. Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for expressions that generate multi-line statements. precision : integer, optional The precision for numbers such as pi [default=16]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, cfunction_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. inline: bool, optional If True, we try to create single-statement code instead of multiple statements. [default=True]. Examples ======== >>> from sympy import julia_code, symbols, sin, pi >>> x = symbols('x') >>> julia_code(sin(x).series(x).removeO()) 'x.^5/120 - x.^3/6 + x' >>> from sympy import Rational, ceiling >>> x, y, tau = symbols("x, y, tau") >>> julia_code((2*tau)**Rational(7, 2)) '8*sqrt(2)*tau.^(7/2)' Note that element-wise (Hadamard) operations are used by default between symbols. This is because its possible in Julia to write "vectorized" code. It is harmless if the values are scalars. >>> julia_code(sin(pi*x*y), assign_to="s") 's = sin(pi*x.*y)' If you need a matrix product "*" or matrix power "^", you can specify the symbol as a ``MatrixSymbol``. >>> from sympy import Symbol, MatrixSymbol >>> n = Symbol('n', integer=True, positive=True) >>> A = MatrixSymbol('A', n, n) >>> julia_code(3*pi*A**3) '(3*pi)*A^3' This class uses several rules to decide which symbol to use a product. Pure numbers use "*", Symbols use ".*" and MatrixSymbols use "*". A HadamardProduct can be used to specify componentwise multiplication ".*" of two MatrixSymbols. There is currently there is no easy way to specify scalar symbols, so sometimes the code might have some minor cosmetic issues. For example, suppose x and y are scalars and A is a Matrix, then while a human programmer might write "(x^2*y)*A^3", we generate: >>> julia_code(x**2*y*A**3) '(x.^2.*y)*A^3' Matrices are supported using Julia inline notation. When using ``assign_to`` with matrices, the name can be specified either as a string or as a ``MatrixSymbol``. The dimensions must align in the latter case. >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([[x**2, sin(x), ceiling(x)]]) >>> julia_code(mat, assign_to='A') 'A = [x.^2 sin(x) ceil(x)]' ``Piecewise`` expressions are implemented with logical masking by default. Alternatively, you can pass "inline=False" to use if-else conditionals. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> pw = Piecewise((x + 1, x > 0), (x, True)) >>> julia_code(pw, assign_to=tau) 'tau = ((x > 0) ? (x + 1) : (x))' Note that any expression that can be generated normally can also exist inside a Matrix: >>> mat = Matrix([[x**2, pw, sin(x)]]) >>> julia_code(mat, assign_to='A') 'A = [x.^2 ((x > 0) ? (x + 1) : (x)) sin(x)]' Custom printing can be defined for certain types by passing a dictionary of "type" : "function" to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e., [(argument_test, cfunction_string)]. This can be used to call a custom Julia function. >>> from sympy import Function >>> f = Function('f') >>> g = Function('g') >>> custom_functions = { ... "f": "existing_julia_fcn", ... "g": [(lambda x: x.is_Matrix, "my_mat_fcn"), ... (lambda x: not x.is_Matrix, "my_fcn")] ... } >>> mat = Matrix([[1, x]]) >>> julia_code(f(x) + g(x) + g(mat), user_functions=custom_functions) 'existing_julia_fcn(x) + my_fcn(x) + my_mat_fcn([1 x])' Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e = Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> julia_code(e.rhs, assign_to=e.lhs, contract=False) 'Dy[i] = (y[i + 1] - y[i])./(t[i + 1] - t[i])' """ return JuliaCodePrinter(settings).doprint(expr, assign_to) def print_julia_code(expr, **settings): """Prints the Julia representation of the given expression. See `julia_code` for the meaning of the optional arguments. """ print(julia_code(expr, **settings))
54cd693095a6b7fd30d720db88d9bf892c38a5dff8f4d0c64849e0f5847518fc
from typing import Set from sympy.core import Basic, S from sympy.core.function import _coeff_isneg, Lambda from sympy.printing.codeprinter import CodePrinter from sympy.printing.precedence import precedence from functools import reduce known_functions = { 'Abs': 'abs', 'sin': 'sin', 'cos': 'cos', 'tan': 'tan', 'acos': 'acos', 'asin': 'asin', 'atan': 'atan', 'atan2': 'atan', 'ceiling': 'ceil', 'floor': 'floor', 'sign': 'sign', 'exp': 'exp', 'log': 'log', 'add': 'add', 'sub': 'sub', 'mul': 'mul', 'pow': 'pow' } class GLSLPrinter(CodePrinter): """ Rudimentary, generic GLSL printing tools. Additional settings: 'use_operators': Boolean (should the printer use operators for +,-,*, or functions?) """ _not_supported = set() # type: Set[Basic] printmethod = "_glsl" language = "GLSL" _default_settings = { 'use_operators': True, 'mat_nested': False, 'mat_separator': ',\n', 'mat_transpose': False, 'glsl_types': True, 'order': None, 'full_prec': 'auto', 'precision': 9, 'user_functions': {}, 'human': True, 'allow_unknown_functions': False, 'contract': True, 'error_on_reserved': False, 'reserved_word_suffix': '_', } def __init__(self, settings={}): CodePrinter.__init__(self, settings) self.known_functions = dict(known_functions) userfuncs = settings.get('user_functions', {}) self.known_functions.update(userfuncs) def _rate_index_position(self, p): return p*5 def _get_statement(self, codestring): return "%s;" % codestring def _get_comment(self, text): return "// {}".format(text) def _declare_number_const(self, name, value): return "float {} = {};".format(name, value) def _format_code(self, lines): return self.indent_code(lines) def indent_code(self, code): """Accepts a string of code or a list of code lines""" if isinstance(code, str): code_lines = self.indent_code(code.splitlines(True)) return ''.join(code_lines) tab = " " inc_token = ('{', '(', '{\n', '(\n') dec_token = ('}', ')') code = [line.lstrip(' \t') for line in code] increase = [int(any(map(line.endswith, inc_token))) for line in code] decrease = [int(any(map(line.startswith, dec_token))) for line in code] pretty = [] level = 0 for n, line in enumerate(code): if line == '' or line == '\n': pretty.append(line) continue level -= decrease[n] pretty.append("%s%s" % (tab*level, line)) level += increase[n] return pretty def _print_MatrixBase(self, mat): mat_separator = self._settings['mat_separator'] mat_transpose = self._settings['mat_transpose'] glsl_types = self._settings['glsl_types'] column_vector = (mat.rows == 1) if mat_transpose else (mat.cols == 1) A = mat.transpose() if mat_transpose != column_vector else mat if A.cols == 1: return self._print(A[0]); if A.rows <= 4 and A.cols <= 4 and glsl_types: if A.rows == 1: return 'vec%s%s' % (A.cols, A.table(self,rowstart='(',rowend=')')) elif A.rows == A.cols: return 'mat%s(%s)' % (A.rows, A.table(self,rowsep=', ', rowstart='',rowend='')) else: return 'mat%sx%s(%s)' % (A.cols, A.rows, A.table(self,rowsep=', ', rowstart='',rowend='')) elif A.cols == 1 or A.rows == 1: return 'float[%s](%s)' % (A.cols*A.rows, A.table(self,rowsep=mat_separator,rowstart='',rowend='')) elif not self._settings['mat_nested']: return 'float[%s](\n%s\n) /* a %sx%s matrix */' % (A.cols*A.rows, A.table(self,rowsep=mat_separator,rowstart='',rowend=''), A.rows,A.cols) elif self._settings['mat_nested']: return 'float[%s][%s](\n%s\n)' % (A.rows,A.cols,A.table(self,rowsep=mat_separator,rowstart='float[](',rowend=')')) def _print_SparseMatrix(self, mat): # do not allow sparse matrices to be made dense return self._print_not_supported(mat) def _traverse_matrix_indices(self, mat): mat_transpose = self._settings['mat_transpose'] if mat_transpose: rows,cols = mat.shape else: cols,rows = mat.shape return ((i, j) for i in range(cols) for j in range(rows)) def _print_MatrixElement(self, expr): # print('begin _print_MatrixElement') nest = self._settings['mat_nested']; glsl_types = self._settings['glsl_types']; mat_transpose = self._settings['mat_transpose']; if mat_transpose: cols,rows = expr.parent.shape i,j = expr.j,expr.i else: rows,cols = expr.parent.shape i,j = expr.i,expr.j pnt = self._print(expr.parent) if glsl_types and ((rows <= 4 and cols <=4) or nest): # print('end _print_MatrixElement case A',nest,glsl_types) return "%s[%s][%s]" % (pnt, i, j) else: # print('end _print_MatrixElement case B',nest,glsl_types) return "{}[{}]".format(pnt, i + j*rows) def _print_list(self, expr): l = ', '.join(self._print(item) for item in expr) glsl_types = self._settings['glsl_types'] if len(expr) <= 4 and glsl_types: return 'vec%s(%s)' % (len(expr),l) else: return 'float[%s](%s)' % (len(expr),l) _print_tuple = _print_list _print_Tuple = _print_list def _get_loop_opening_ending(self, indices): open_lines = [] close_lines = [] loopstart = "for (int %(varble)s=%(start)s; %(varble)s<%(end)s; %(varble)s++){" for i in indices: # GLSL arrays start at 0 and end at dimension-1 open_lines.append(loopstart % { 'varble': self._print(i.label), 'start': self._print(i.lower), 'end': self._print(i.upper + 1)}) close_lines.append("}") return open_lines, close_lines def _print_Function_with_args(self, func, func_args): if func in self.known_functions: cond_func = self.known_functions[func] func = None if isinstance(cond_func, str): func = cond_func else: for cond, func in cond_func: if cond(func_args): break if func is not None: try: return func(*[self.parenthesize(item, 0) for item in func_args]) except TypeError: return "%s(%s)" % (func, self.stringify(func_args, ", ")) elif isinstance(func, Lambda): # inlined function return self._print(func(*func_args)) else: return self._print_not_supported(func) def _print_Piecewise(self, expr): from sympy.codegen.ast import Assignment if expr.args[-1].cond != True: # We need the last conditional to be a True, otherwise the resulting # function may not return a result. raise ValueError("All Piecewise expressions must contain an " "(expr, True) statement to be used as a default " "condition. Without one, the generated " "expression may not evaluate to anything under " "some condition.") lines = [] if expr.has(Assignment): for i, (e, c) in enumerate(expr.args): if i == 0: lines.append("if (%s) {" % self._print(c)) elif i == len(expr.args) - 1 and c == True: lines.append("else {") else: lines.append("else if (%s) {" % self._print(c)) code0 = self._print(e) lines.append(code0) lines.append("}") return "\n".join(lines) else: # The piecewise was used in an expression, need to do inline # operators. This has the downside that inline operators will # not work for statements that span multiple lines (Matrix or # Indexed expressions). ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c), self._print(e)) for e, c in expr.args[:-1]] last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr) return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)]) def _print_Idx(self, expr): return self._print(expr.label) def _print_Indexed(self, expr): # calculate index for 1d array dims = expr.shape elem = S.Zero offset = S.One for i in reversed(range(expr.rank)): elem += expr.indices[i]*offset offset *= dims[i] return "%s[%s]" % (self._print(expr.base.label), self._print(elem)) def _print_Pow(self, expr): PREC = precedence(expr) if expr.exp == -1: return '1.0/%s' % (self.parenthesize(expr.base, PREC)) elif expr.exp == 0.5: return 'sqrt(%s)' % self._print(expr.base) else: try: e = self._print(float(expr.exp)) except TypeError: e = self._print(expr.exp) # return self.known_functions['pow']+'(%s, %s)' % (self._print(expr.base),e) return self._print_Function_with_args('pow', ( self._print(expr.base), e )) def _print_int(self, expr): return str(float(expr)) def _print_Rational(self, expr): return "%s.0/%s.0" % (expr.p, expr.q) def _print_Relational(self, expr): lhs_code = self._print(expr.lhs) rhs_code = self._print(expr.rhs) op = expr.rel_op return "{} {} {}".format(lhs_code, op, rhs_code) def _print_Add(self, expr, order=None): if self._settings['use_operators']: return CodePrinter._print_Add(self, expr, order=order) terms = expr.as_ordered_terms() def partition(p,l): return reduce(lambda x, y: (x[0]+[y], x[1]) if p(y) else (x[0], x[1]+[y]), l, ([], [])) def add(a,b): return self._print_Function_with_args('add', (a, b)) # return self.known_functions['add']+'(%s, %s)' % (a,b) neg, pos = partition(lambda arg: _coeff_isneg(arg), terms) s = pos = reduce(lambda a,b: add(a,b), map(lambda t: self._print(t),pos)) if neg: # sum the absolute values of the negative terms neg = reduce(lambda a,b: add(a,b), map(lambda n: self._print(-n),neg)) # then subtract them from the positive terms s = self._print_Function_with_args('sub', (pos,neg)) # s = self.known_functions['sub']+'(%s, %s)' % (pos,neg) return s def _print_Mul(self, expr, **kwargs): if self._settings['use_operators']: return CodePrinter._print_Mul(self, expr, **kwargs) terms = expr.as_ordered_factors() def mul(a,b): # return self.known_functions['mul']+'(%s, %s)' % (a,b) return self._print_Function_with_args('mul', (a,b)) s = reduce(lambda a,b: mul(a,b), map(lambda t: self._print(t), terms)) return s def glsl_code(expr,assign_to=None,**settings): """Converts an expr to a string of GLSL code Parameters ========== expr : Expr A sympy expression to be converted. assign_to : optional When given, the argument is used as the name of the variable to which the expression is assigned. Can be a string, ``Symbol``, ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of line-wrapping, or for expressions that generate multi-line statements. use_operators: bool, optional If set to False, then *,/,+,- operators will be replaced with functions mul, add, and sub, which must be implemented by the user, e.g. for implementing non-standard rings or emulated quad/octal precision. [default=True] glsl_types: bool, optional Set this argument to ``False`` in order to avoid using the ``vec`` and ``mat`` types. The printer will instead use arrays (or nested arrays). [default=True] mat_nested: bool, optional GLSL version 4.3 and above support nested arrays (arrays of arrays). Set this to ``True`` to render matrices as nested arrays. [default=False] mat_separator: str, optional By default, matrices are rendered with newlines using this separator, making them easier to read, but less compact. By removing the newline this option can be used to make them more vertically compact. [default=',\n'] mat_transpose: bool, optional GLSL's matrix multiplication implementation assumes column-major indexing. By default, this printer ignores that convention. Setting this option to ``True`` transposes all matrix output. [default=False] precision : integer, optional The precision for numbers such as pi [default=15]. user_functions : dict, optional A dictionary where keys are ``FunctionClass`` instances and values are their string representations. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, js_function_string)]. See below for examples. human : bool, optional If True, the result is a single string that may contain some constant declarations for the number symbols. If False, the same information is returned in a tuple of (symbols_to_declare, not_supported_functions, code_text). [default=True]. contract: bool, optional If True, ``Indexed`` instances are assumed to obey tensor contraction rules and the corresponding nested loops over indices are generated. Setting contract=False will not generate loops, instead the user is responsible to provide values for the indices in the code. [default=True]. Examples ======== >>> from sympy import glsl_code, symbols, Rational, sin, ceiling, Abs >>> x, tau = symbols("x, tau") >>> glsl_code((2*tau)**Rational(7, 2)) '8*sqrt(2)*pow(tau, 3.5)' >>> glsl_code(sin(x), assign_to="float y") 'float y = sin(x);' Various GLSL types are supported: >>> from sympy import Matrix, glsl_code >>> glsl_code(Matrix([1,2,3])) 'vec3(1, 2, 3)' >>> glsl_code(Matrix([[1, 2],[3, 4]])) 'mat2(1, 2, 3, 4)' Pass ``mat_transpose = True`` to switch to column-major indexing: >>> glsl_code(Matrix([[1, 2],[3, 4]]), mat_transpose = True) 'mat2(1, 3, 2, 4)' By default, larger matrices get collapsed into float arrays: >>> print(glsl_code( Matrix([[1,2,3,4,5],[6,7,8,9,10]]) )) float[10]( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ) /* a 2x5 matrix */ Passing ``mat_nested = True`` instead prints out nested float arrays, which are supported in GLSL 4.3 and above. >>> mat = Matrix([ ... [ 0, 1, 2], ... [ 3, 4, 5], ... [ 6, 7, 8], ... [ 9, 10, 11], ... [12, 13, 14]]) >>> print(glsl_code( mat, mat_nested = True )) float[5][3]( float[]( 0, 1, 2), float[]( 3, 4, 5), float[]( 6, 7, 8), float[]( 9, 10, 11), float[](12, 13, 14) ) Custom printing can be defined for certain types by passing a dictionary of "type" : "function" to the ``user_functions`` kwarg. Alternatively, the dictionary value can be a list of tuples i.e. [(argument_test, js_function_string)]. >>> custom_functions = { ... "ceiling": "CEIL", ... "Abs": [(lambda x: not x.is_integer, "fabs"), ... (lambda x: x.is_integer, "ABS")] ... } >>> glsl_code(Abs(x) + ceiling(x), user_functions=custom_functions) 'fabs(x) + CEIL(x)' If further control is needed, addition, subtraction, multiplication and division operators can be replaced with ``add``, ``sub``, and ``mul`` functions. This is done by passing ``use_operators = False``: >>> x,y,z = symbols('x,y,z') >>> glsl_code(x*(y+z), use_operators = False) 'mul(x, add(y, z))' >>> glsl_code(x*(y+z*(x-y)**z), use_operators = False) 'mul(x, add(y, mul(z, pow(sub(x, y), z))))' ``Piecewise`` expressions are converted into conditionals. If an ``assign_to`` variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the ``Piecewise`` lacks a default term, represented by ``(expr, True)`` then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. >>> from sympy import Piecewise >>> expr = Piecewise((x + 1, x > 0), (x, True)) >>> print(glsl_code(expr, tau)) if (x > 0) { tau = x + 1; } else { tau = x; } Support for loops is provided through ``Indexed`` types. With ``contract=True`` these expressions will be turned into loops, whereas ``contract=False`` will just print the assignment expression that should be looped over: >>> from sympy import Eq, IndexedBase, Idx >>> len_y = 5 >>> y = IndexedBase('y', shape=(len_y,)) >>> t = IndexedBase('t', shape=(len_y,)) >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) >>> i = Idx('i', len_y-1) >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) >>> glsl_code(e.rhs, assign_to=e.lhs, contract=False) 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' >>> from sympy import Matrix, MatrixSymbol >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) >>> A = MatrixSymbol('A', 3, 1) >>> print(glsl_code(mat, A)) A[0][0] = pow(x, 2.0); if (x > 0) { A[1][0] = x + 1; } else { A[1][0] = x; } A[2][0] = sin(x); """ return GLSLPrinter(settings).doprint(expr,assign_to) def print_glsl(expr, **settings): """Prints the GLSL representation of the given expression. See GLSLPrinter init function for settings. """ print(glsl_code(expr, **settings))
e58cf4de9be93e85405d11e5fe2d768651b1d1a3a79fdd246fc93ffe31d6e54e
from typing import Any, Dict from sympy.core.compatibility import is_sequence from sympy.external import import_module from sympy.printing.printer import Printer import sympy from functools import partial theano = import_module('theano') if theano: ts = theano.scalar tt = theano.tensor from theano.sandbox import linalg as tlinalg mapping = { sympy.Add: tt.add, sympy.Mul: tt.mul, sympy.Abs: tt.abs_, sympy.sign: tt.sgn, sympy.ceiling: tt.ceil, sympy.floor: tt.floor, sympy.log: tt.log, sympy.exp: tt.exp, sympy.sqrt: tt.sqrt, sympy.cos: tt.cos, sympy.acos: tt.arccos, sympy.sin: tt.sin, sympy.asin: tt.arcsin, sympy.tan: tt.tan, sympy.atan: tt.arctan, sympy.atan2: tt.arctan2, sympy.cosh: tt.cosh, sympy.acosh: tt.arccosh, sympy.sinh: tt.sinh, sympy.asinh: tt.arcsinh, sympy.tanh: tt.tanh, sympy.atanh: tt.arctanh, sympy.re: tt.real, sympy.im: tt.imag, sympy.arg: tt.angle, sympy.erf: tt.erf, sympy.gamma: tt.gamma, sympy.loggamma: tt.gammaln, sympy.Pow: tt.pow, sympy.Eq: tt.eq, sympy.StrictGreaterThan: tt.gt, sympy.StrictLessThan: tt.lt, sympy.LessThan: tt.le, sympy.GreaterThan: tt.ge, sympy.And: tt.and_, sympy.Or: tt.or_, sympy.Max: tt.maximum, # Sympy accept >2 inputs, Theano only 2 sympy.Min: tt.minimum, # Sympy accept >2 inputs, Theano only 2 sympy.conjugate: tt.conj, sympy.core.numbers.ImaginaryUnit: lambda:tt.complex(0,1), # Matrices sympy.MatAdd: tt.Elemwise(ts.add), sympy.HadamardProduct: tt.Elemwise(ts.mul), sympy.Trace: tlinalg.trace, sympy.Determinant : tlinalg.det, sympy.Inverse: tlinalg.matrix_inverse, sympy.Transpose: tt.DimShuffle((False, False), [1, 0]), } class TheanoPrinter(Printer): """ Code printer which creates Theano symbolic expression graphs. Parameters ========== cache : dict Cache dictionary to use. If None (default) will use the global cache. To create a printer which does not depend on or alter global state pass an empty dictionary. Note: the dictionary is not copied on initialization of the printer and will be updated in-place, so using the same dict object when creating multiple printers or making multiple calls to :func:`.theano_code` or :func:`.theano_function` means the cache is shared between all these applications. Attributes ========== cache : dict A cache of Theano variables which have been created for Sympy symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or :class:`sympy.matrices.expressions.MatrixSymbol`). This is used to ensure that all references to a given symbol in an expression (or multiple expressions) are printed as the same Theano variable, which is created only once. Symbols are differentiated only by name and type. The format of the cache's contents should be considered opaque to the user. """ printmethod = "_theano" def __init__(self, *args, **kwargs): self.cache = kwargs.pop('cache', dict()) super().__init__(*args, **kwargs) def _get_key(self, s, name=None, dtype=None, broadcastable=None): """ Get the cache key for a Sympy object. Parameters ========== s : sympy.core.basic.Basic Sympy object to get key for. name : str Name of object, if it does not have a ``name`` attribute. """ if name is None: name = s.name return (name, type(s), s.args, dtype, broadcastable) def _get_or_create(self, s, name=None, dtype=None, broadcastable=None): """ Get the Theano variable for a Sympy symbol from the cache, or create it if it does not exist. """ # Defaults if name is None: name = s.name if dtype is None: dtype = 'floatX' if broadcastable is None: broadcastable = () key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable) if key in self.cache: return self.cache[key] value = tt.tensor(name=name, dtype=dtype, broadcastable=broadcastable) self.cache[key] = value return value def _print_Symbol(self, s, **kwargs): dtype = kwargs.get('dtypes', {}).get(s) bc = kwargs.get('broadcastables', {}).get(s) return self._get_or_create(s, dtype=dtype, broadcastable=bc) def _print_AppliedUndef(self, s, **kwargs): name = str(type(s)) + '_' + str(s.args[0]) dtype = kwargs.get('dtypes', {}).get(s) bc = kwargs.get('broadcastables', {}).get(s) return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc) def _print_Basic(self, expr, **kwargs): op = mapping[type(expr)] children = [self._print(arg, **kwargs) for arg in expr.args] return op(*children) def _print_Number(self, n, **kwargs): # Integers already taken care of below, interpret as float return float(n.evalf()) def _print_MatrixSymbol(self, X, **kwargs): dtype = kwargs.get('dtypes', {}).get(X) return self._get_or_create(X, dtype=dtype, broadcastable=(None, None)) def _print_DenseMatrix(self, X, **kwargs): if not hasattr(tt, 'stacklists'): raise NotImplementedError( "Matrix translation not yet supported in this version of Theano") return tt.stacklists([ [self._print(arg, **kwargs) for arg in L] for L in X.tolist() ]) _print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix def _print_MatMul(self, expr, **kwargs): children = [self._print(arg, **kwargs) for arg in expr.args] result = children[0] for child in children[1:]: result = tt.dot(result, child) return result def _print_MatPow(self, expr, **kwargs): children = [self._print(arg, **kwargs) for arg in expr.args] result = 1 if isinstance(children[1], int) and children[1] > 0: for i in range(children[1]): result = tt.dot(result, children[0]) else: raise NotImplementedError('''Only non-negative integer powers of matrices can be handled by Theano at the moment''') return result def _print_MatrixSlice(self, expr, **kwargs): parent = self._print(expr.parent, **kwargs) rowslice = self._print(slice(*expr.rowslice), **kwargs) colslice = self._print(slice(*expr.colslice), **kwargs) return parent[rowslice, colslice] def _print_BlockMatrix(self, expr, **kwargs): nrows, ncols = expr.blocks.shape blocks = [[self._print(expr.blocks[r, c], **kwargs) for c in range(ncols)] for r in range(nrows)] return tt.join(0, *[tt.join(1, *row) for row in blocks]) def _print_slice(self, expr, **kwargs): return slice(*[self._print(i, **kwargs) if isinstance(i, sympy.Basic) else i for i in (expr.start, expr.stop, expr.step)]) def _print_Pi(self, expr, **kwargs): return 3.141592653589793 def _print_Piecewise(self, expr, **kwargs): import numpy as np e, cond = expr.args[0].args # First condition and corresponding value # Print conditional expression and value for first condition p_cond = self._print(cond, **kwargs) p_e = self._print(e, **kwargs) # One condition only if len(expr.args) == 1: # Return value if condition else NaN return tt.switch(p_cond, p_e, np.nan) # Return value_1 if condition_1 else evaluate remaining conditions p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs) return tt.switch(p_cond, p_e, p_remaining) def _print_Rational(self, expr, **kwargs): return tt.true_div(self._print(expr.p, **kwargs), self._print(expr.q, **kwargs)) def _print_Integer(self, expr, **kwargs): return expr.p def _print_factorial(self, expr, **kwargs): return self._print(sympy.gamma(expr.args[0] + 1), **kwargs) def _print_Derivative(self, deriv, **kwargs): rv = self._print(deriv.expr, **kwargs) for var in deriv.variables: var = self._print(var, **kwargs) rv = tt.Rop(rv, var, tt.ones_like(var)) return rv def emptyPrinter(self, expr): return expr def doprint(self, expr, dtypes=None, broadcastables=None): """ Convert a Sympy expression to a Theano graph variable. The ``dtypes`` and ``broadcastables`` arguments are used to specify the data type, dimension, and broadcasting behavior of the Theano variables corresponding to the free symbols in ``expr``. Each is a mapping from Sympy symbols to the value of the corresponding argument to ``theano.tensor.Tensor``. See the corresponding `documentation page`__ for more information on broadcasting in Theano. .. __: http://deeplearning.net/software/theano/tutorial/broadcasting.html Parameters ========== expr : sympy.core.expr.Expr Sympy expression to print. dtypes : dict Mapping from Sympy symbols to Theano datatypes to use when creating new Theano variables for those symbols. Corresponds to the ``dtype`` argument to ``theano.tensor.Tensor``. Defaults to ``'floatX'`` for symbols not included in the mapping. broadcastables : dict Mapping from Sympy symbols to the value of the ``broadcastable`` argument to ``theano.tensor.Tensor`` to use when creating Theano variables for those symbols. Defaults to the empty tuple for symbols not included in the mapping (resulting in a scalar). Returns ======= theano.gof.graph.Variable A variable corresponding to the expression's value in a Theano symbolic expression graph. """ if dtypes is None: dtypes = {} if broadcastables is None: broadcastables = {} return self._print(expr, dtypes=dtypes, broadcastables=broadcastables) global_cache = {} # type: Dict[Any, Any] def theano_code(expr, cache=None, **kwargs): """ Convert a Sympy expression into a Theano graph variable. Parameters ========== expr : sympy.core.expr.Expr Sympy expression object to convert. cache : dict Cached Theano variables (see :class:`TheanoPrinter.cache <TheanoPrinter>`). Defaults to the module-level global cache. dtypes : dict Passed to :meth:`.TheanoPrinter.doprint`. broadcastables : dict Passed to :meth:`.TheanoPrinter.doprint`. Returns ======= theano.gof.graph.Variable A variable corresponding to the expression's value in a Theano symbolic expression graph. """ if not theano: raise ImportError("theano is required for theano_code") if cache is None: cache = global_cache return TheanoPrinter(cache=cache, settings={}).doprint(expr, **kwargs) def dim_handling(inputs, dim=None, dims=None, broadcastables=None): r""" Get value of ``broadcastables`` argument to :func:`.theano_code` from keyword arguments to :func:`.theano_function`. Included for backwards compatibility. Parameters ========== inputs Sequence of input symbols. dim : int Common number of dimensions for all inputs. Overrides other arguments if given. dims : dict Mapping from input symbols to number of dimensions. Overrides ``broadcastables`` argument if given. broadcastables : dict Explicit value of ``broadcastables`` argument to :meth:`.TheanoPrinter.doprint`. If not None function will return this value unchanged. Returns ======= dict Dictionary mapping elements of ``inputs`` to their "broadcastable" values (tuple of ``bool``\ s). """ if dim is not None: return {s: (False,) * dim for s in inputs} if dims is not None: maxdim = max(dims.values()) return { s: (False,) * d + (True,) * (maxdim - d) for s, d in dims.items() } if broadcastables is not None: return broadcastables return {} def theano_function(inputs, outputs, scalar=False, *, dim=None, dims=None, broadcastables=None, **kwargs): """ Create a Theano function from SymPy expressions. The inputs and outputs are converted to Theano variables using :func:`.theano_code` and then passed to ``theano.function``. Parameters ========== inputs Sequence of symbols which constitute the inputs of the function. outputs Sequence of expressions which constitute the outputs(s) of the function. The free symbols of each expression must be a subset of ``inputs``. scalar : bool Convert 0-dimensional arrays in output to scalars. This will return a Python wrapper function around the Theano function object. cache : dict Cached Theano variables (see :class:`TheanoPrinter.cache <TheanoPrinter>`). Defaults to the module-level global cache. dtypes : dict Passed to :meth:`.TheanoPrinter.doprint`. broadcastables : dict Passed to :meth:`.TheanoPrinter.doprint`. dims : dict Alternative to ``broadcastables`` argument. Mapping from elements of ``inputs`` to integers indicating the dimension of their associated arrays/tensors. Overrides ``broadcastables`` argument if given. dim : int Another alternative to the ``broadcastables`` argument. Common number of dimensions to use for all arrays/tensors. ``theano_function([x, y], [...], dim=2)`` is equivalent to using ``broadcastables={x: (False, False), y: (False, False)}``. Returns ======= callable A callable object which takes values of ``inputs`` as positional arguments and returns an output array for each of the expressions in ``outputs``. If ``outputs`` is a single expression the function will return a Numpy array, if it is a list of multiple expressions the function will return a list of arrays. See description of the ``squeeze`` argument above for the behavior when a single output is passed in a list. The returned object will either be an instance of ``theano.compile.function_module.Function`` or a Python wrapper function around one. In both cases, the returned value will have a ``theano_function`` attribute which points to the return value of ``theano.function``. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.printing.theanocode import theano_function A simple function with one input and one output: >>> f1 = theano_function([x], [x**2 - 1], scalar=True) >>> f1(3) 8.0 A function with multiple inputs and one output: >>> f2 = theano_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True) >>> f2(3, 4, 2) 5.0 A function with multiple inputs and multiple outputs: >>> f3 = theano_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True) >>> f3(2, 3) [13.0, -5.0] See also ======== dim_handling """ if not theano: raise ImportError("theano is required for theano_function") # Pop off non-theano keyword args cache = kwargs.pop('cache', {}) dtypes = kwargs.pop('dtypes', {}) broadcastables = dim_handling( inputs, dim=dim, dims=dims, broadcastables=broadcastables, ) # Print inputs/outputs code = partial(theano_code, cache=cache, dtypes=dtypes, broadcastables=broadcastables) tinputs = list(map(code, inputs)) toutputs = list(map(code, outputs)) #fix constant expressions as variables toutputs = [output if isinstance(output, theano.Variable) else tt.as_tensor_variable(output) for output in toutputs] if len(toutputs) == 1: toutputs = toutputs[0] # Compile theano func func = theano.function(tinputs, toutputs, **kwargs) is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs] # No wrapper required if not scalar or not any(is_0d): func.theano_function = func return func # Create wrapper to convert 0-dimensional outputs to scalars def wrapper(*args): out = func(*args) # out can be array(1.0) or [array(1.0), array(2.0)] if is_sequence(out): return [o[()] if is_0d[i] else o for i, o in enumerate(out)] else: return out[()] wrapper.__wrapped__ = func wrapper.__doc__ = func.__doc__ wrapper.theano_function = func return wrapper
889e040c00c266f699a093bb715f89678b508666a8b67de666f08fdec2ec668d
"""Integration method that emulates by-hand techniques. This module also provides functionality to get the steps used to evaluate a particular integral, in the ``integral_steps`` function. This will return nested namedtuples representing the integration rules used. The ``manualintegrate`` function computes the integral using those steps given an integrand; given the steps, ``_manualintegrate`` will evaluate them. The integrator can be extended with new heuristics and evaluation techniques. To do so, write a function that accepts an ``IntegralInfo`` object and returns either a namedtuple representing a rule or ``None``. Then, write another function that accepts the namedtuple's fields and returns the antiderivative, and decorate it with ``@evaluates(namedtuple_type)``. If the new technique requires a new match, add the key and call to the antiderivative function to integral_steps. To enable simple substitutions, add the match to find_substitutions. """ from typing import Dict as tDict, Optional from collections import namedtuple, defaultdict import sympy from sympy.core.compatibility import reduce, Mapping, iterable from sympy.core.containers import Dict from sympy.core.expr import Expr from sympy.core.logic import fuzzy_not from sympy.functions.elementary.trigonometric import TrigonometricFunction from sympy.functions.special.polynomials import OrthogonalPolynomial from sympy.functions.elementary.piecewise import Piecewise from sympy.strategies.core import switch, do_one, null_safe, condition from sympy.core.relational import Eq, Ne from sympy.polys.polytools import degree from sympy.ntheory.factor_ import divisors from sympy.utilities.misc import debug ZERO = sympy.S.Zero def Rule(name, props=""): # GOTCHA: namedtuple class name not considered! def __eq__(self, other): return self.__class__ == other.__class__ and tuple.__eq__(self, other) __neq__ = lambda self, other: not __eq__(self, other) cls = namedtuple(name, props + " context symbol") cls.__eq__ = __eq__ cls.__ne__ = __neq__ return cls ConstantRule = Rule("ConstantRule", "constant") ConstantTimesRule = Rule("ConstantTimesRule", "constant other substep") PowerRule = Rule("PowerRule", "base exp") AddRule = Rule("AddRule", "substeps") URule = Rule("URule", "u_var u_func constant substep") PartsRule = Rule("PartsRule", "u dv v_step second_step") CyclicPartsRule = Rule("CyclicPartsRule", "parts_rules coefficient") TrigRule = Rule("TrigRule", "func arg") ExpRule = Rule("ExpRule", "base exp") ReciprocalRule = Rule("ReciprocalRule", "func") ArcsinRule = Rule("ArcsinRule") InverseHyperbolicRule = Rule("InverseHyperbolicRule", "func") AlternativeRule = Rule("AlternativeRule", "alternatives") DontKnowRule = Rule("DontKnowRule") DerivativeRule = Rule("DerivativeRule") RewriteRule = Rule("RewriteRule", "rewritten substep") PiecewiseRule = Rule("PiecewiseRule", "subfunctions") HeavisideRule = Rule("HeavisideRule", "harg ibnd substep") TrigSubstitutionRule = Rule("TrigSubstitutionRule", "theta func rewritten substep restriction") ArctanRule = Rule("ArctanRule", "a b c") ArccothRule = Rule("ArccothRule", "a b c") ArctanhRule = Rule("ArctanhRule", "a b c") JacobiRule = Rule("JacobiRule", "n a b") GegenbauerRule = Rule("GegenbauerRule", "n a") ChebyshevTRule = Rule("ChebyshevTRule", "n") ChebyshevURule = Rule("ChebyshevURule", "n") LegendreRule = Rule("LegendreRule", "n") HermiteRule = Rule("HermiteRule", "n") LaguerreRule = Rule("LaguerreRule", "n") AssocLaguerreRule = Rule("AssocLaguerreRule", "n a") CiRule = Rule("CiRule", "a b") ChiRule = Rule("ChiRule", "a b") EiRule = Rule("EiRule", "a b") SiRule = Rule("SiRule", "a b") ShiRule = Rule("ShiRule", "a b") ErfRule = Rule("ErfRule", "a b c") FresnelCRule = Rule("FresnelCRule", "a b c") FresnelSRule = Rule("FresnelSRule", "a b c") LiRule = Rule("LiRule", "a b") PolylogRule = Rule("PolylogRule", "a b") UpperGammaRule = Rule("UpperGammaRule", "a e") EllipticFRule = Rule("EllipticFRule", "a d") EllipticERule = Rule("EllipticERule", "a d") IntegralInfo = namedtuple('IntegralInfo', 'integrand symbol') evaluators = {} def evaluates(rule): def _evaluates(func): func.rule = rule evaluators[rule] = func return func return _evaluates def contains_dont_know(rule): if isinstance(rule, DontKnowRule): return True else: for val in rule: if isinstance(val, tuple): if contains_dont_know(val): return True elif isinstance(val, list): if any(contains_dont_know(i) for i in val): return True return False def manual_diff(f, symbol): """Derivative of f in form expected by find_substitutions SymPy's derivatives for some trig functions (like cot) aren't in a form that works well with finding substitutions; this replaces the derivatives for those particular forms with something that works better. """ if f.args: arg = f.args[0] if isinstance(f, sympy.tan): return arg.diff(symbol) * sympy.sec(arg)**2 elif isinstance(f, sympy.cot): return -arg.diff(symbol) * sympy.csc(arg)**2 elif isinstance(f, sympy.sec): return arg.diff(symbol) * sympy.sec(arg) * sympy.tan(arg) elif isinstance(f, sympy.csc): return -arg.diff(symbol) * sympy.csc(arg) * sympy.cot(arg) elif isinstance(f, sympy.Add): return sum([manual_diff(arg, symbol) for arg in f.args]) elif isinstance(f, sympy.Mul): if len(f.args) == 2 and isinstance(f.args[0], sympy.Number): return f.args[0] * manual_diff(f.args[1], symbol) return f.diff(symbol) def manual_subs(expr, *args): """ A wrapper for `expr.subs(*args)` with additional logic for substitution of invertible functions. """ if len(args) == 1: sequence = args[0] if isinstance(sequence, (Dict, Mapping)): sequence = sequence.items() elif not iterable(sequence): raise ValueError("Expected an iterable of (old, new) pairs") elif len(args) == 2: sequence = [args] else: raise ValueError("subs accepts either 1 or 2 arguments") new_subs = [] for old, new in sequence: if isinstance(old, sympy.log): # If log(x) = y, then exp(a*log(x)) = exp(a*y) # that is, x**a = exp(a*y). Replace nontrivial powers of x # before subs turns them into `exp(y)**a`, but # do not replace x itself yet, to avoid `log(exp(y))`. x0 = old.args[0] expr = expr.replace(lambda x: x.is_Pow and x.base == x0, lambda x: sympy.exp(x.exp*new)) new_subs.append((x0, sympy.exp(new))) return expr.subs(list(sequence) + new_subs) # Method based on that on SIN, described in "Symbolic Integration: The # Stormy Decade" inverse_trig_functions = (sympy.atan, sympy.asin, sympy.acos, sympy.acot, sympy.acsc, sympy.asec) def find_substitutions(integrand, symbol, u_var): results = [] def test_subterm(u, u_diff): if u_diff == 0: return False substituted = integrand / u_diff if symbol not in substituted.free_symbols: # replaced everything already return False debug("substituted: {}, u: {}, u_var: {}".format(substituted, u, u_var)) substituted = manual_subs(substituted, u, u_var).cancel() if symbol not in substituted.free_symbols: # avoid increasing the degree of a rational function if integrand.is_rational_function(symbol) and substituted.is_rational_function(u_var): deg_before = max([degree(t, symbol) for t in integrand.as_numer_denom()]) deg_after = max([degree(t, u_var) for t in substituted.as_numer_denom()]) if deg_after > deg_before: return False return substituted.as_independent(u_var, as_Add=False) # special treatment for substitutions u = (a*x+b)**(1/n) if (isinstance(u, sympy.Pow) and (1/u.exp).is_Integer and sympy.Abs(u.exp) < 1): a = sympy.Wild('a', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) match = u.base.match(a*symbol + b) if match: a, b = [match.get(i, ZERO) for i in (a, b)] if a != 0 and b != 0: substituted = substituted.subs(symbol, (u_var**(1/u.exp) - b)/a) return substituted.as_independent(u_var, as_Add=False) return False def possible_subterms(term): if isinstance(term, (TrigonometricFunction, *inverse_trig_functions, sympy.exp, sympy.log, sympy.Heaviside)): return [term.args[0]] elif isinstance(term, (sympy.chebyshevt, sympy.chebyshevu, sympy.legendre, sympy.hermite, sympy.laguerre)): return [term.args[1]] elif isinstance(term, (sympy.gegenbauer, sympy.assoc_laguerre)): return [term.args[2]] elif isinstance(term, sympy.jacobi): return [term.args[3]] elif isinstance(term, sympy.Mul): r = [] for u in term.args: r.append(u) r.extend(possible_subterms(u)) return r elif isinstance(term, sympy.Pow): r = [] if term.args[1].is_constant(symbol): r.append(term.args[0]) elif term.args[0].is_constant(symbol): r.append(term.args[1]) if term.args[1].is_Integer: r.extend([term.args[0]**d for d in divisors(term.args[1]) if 1 < d < abs(term.args[1])]) if term.args[0].is_Add: r.extend([t for t in possible_subterms(term.args[0]) if t.is_Pow]) return r elif isinstance(term, sympy.Add): r = [] for arg in term.args: r.append(arg) r.extend(possible_subterms(arg)) return r return [] for u in possible_subterms(integrand): if u == symbol: continue u_diff = manual_diff(u, symbol) new_integrand = test_subterm(u, u_diff) if new_integrand is not False: constant, new_integrand = new_integrand if new_integrand == integrand.subs(symbol, u_var): continue substitution = (u, constant, new_integrand) if substitution not in results: results.append(substitution) return results def rewriter(condition, rewrite): """Strategy that rewrites an integrand.""" def _rewriter(integral): integrand, symbol = integral debug("Integral: {} is rewritten with {} on symbol: {}".format(integrand, rewrite, symbol)) if condition(*integral): rewritten = rewrite(*integral) if rewritten != integrand: substep = integral_steps(rewritten, symbol) if not isinstance(substep, DontKnowRule) and substep: return RewriteRule( rewritten, substep, integrand, symbol) return _rewriter def proxy_rewriter(condition, rewrite): """Strategy that rewrites an integrand based on some other criteria.""" def _proxy_rewriter(criteria): criteria, integral = criteria integrand, symbol = integral debug("Integral: {} is rewritten with {} on symbol: {} and criteria: {}".format(integrand, rewrite, symbol, criteria)) args = criteria + list(integral) if condition(*args): rewritten = rewrite(*args) if rewritten != integrand: return RewriteRule( rewritten, integral_steps(rewritten, symbol), integrand, symbol) return _proxy_rewriter def multiplexer(conditions): """Apply the rule that matches the condition, else None""" def multiplexer_rl(expr): for key, rule in conditions.items(): if key(expr): return rule(expr) return multiplexer_rl def alternatives(*rules): """Strategy that makes an AlternativeRule out of multiple possible results.""" def _alternatives(integral): alts = [] count = 0 debug("List of Alternative Rules") for rule in rules: count = count + 1 debug("Rule {}: {}".format(count, rule)) result = rule(integral) if (result and not isinstance(result, DontKnowRule) and result != integral and result not in alts): alts.append(result) if len(alts) == 1: return alts[0] elif alts: doable = [rule for rule in alts if not contains_dont_know(rule)] if doable: return AlternativeRule(doable, *integral) else: return AlternativeRule(alts, *integral) return _alternatives def constant_rule(integral): integrand, symbol = integral return ConstantRule(integral.integrand, *integral) def power_rule(integral): integrand, symbol = integral base, exp = integrand.as_base_exp() if symbol not in exp.free_symbols and isinstance(base, sympy.Symbol): if sympy.simplify(exp + 1) == 0: return ReciprocalRule(base, integrand, symbol) return PowerRule(base, exp, integrand, symbol) elif symbol not in base.free_symbols and isinstance(exp, sympy.Symbol): rule = ExpRule(base, exp, integrand, symbol) if fuzzy_not(sympy.log(base).is_zero): return rule elif sympy.log(base).is_zero: return ConstantRule(1, 1, symbol) return PiecewiseRule([ (rule, sympy.Ne(sympy.log(base), 0)), (ConstantRule(1, 1, symbol), True) ], integrand, symbol) def exp_rule(integral): integrand, symbol = integral if isinstance(integrand.args[0], sympy.Symbol): return ExpRule(sympy.E, integrand.args[0], integrand, symbol) def orthogonal_poly_rule(integral): orthogonal_poly_classes = { sympy.jacobi: JacobiRule, sympy.gegenbauer: GegenbauerRule, sympy.chebyshevt: ChebyshevTRule, sympy.chebyshevu: ChebyshevURule, sympy.legendre: LegendreRule, sympy.hermite: HermiteRule, sympy.laguerre: LaguerreRule, sympy.assoc_laguerre: AssocLaguerreRule } orthogonal_poly_var_index = { sympy.jacobi: 3, sympy.gegenbauer: 2, sympy.assoc_laguerre: 2 } integrand, symbol = integral for klass in orthogonal_poly_classes: if isinstance(integrand, klass): var_index = orthogonal_poly_var_index.get(klass, 1) if (integrand.args[var_index] is symbol and not any(v.has(symbol) for v in integrand.args[:var_index])): args = integrand.args[:var_index] + (integrand, symbol) return orthogonal_poly_classes[klass](*args) def special_function_rule(integral): integrand, symbol = integral a = sympy.Wild('a', exclude=[symbol], properties=[lambda x: not x.is_zero]) b = sympy.Wild('b', exclude=[symbol]) c = sympy.Wild('c', exclude=[symbol]) d = sympy.Wild('d', exclude=[symbol], properties=[lambda x: not x.is_zero]) e = sympy.Wild('e', exclude=[symbol], properties=[ lambda x: not (x.is_nonnegative and x.is_integer)]) wilds = (a, b, c, d, e) # patterns consist of a SymPy class, a wildcard expr, an optional # condition coded as a lambda (when Wild properties are not enough), # followed by an applicable rule patterns = ( (sympy.Mul, sympy.exp(a*symbol + b)/symbol, None, EiRule), (sympy.Mul, sympy.cos(a*symbol + b)/symbol, None, CiRule), (sympy.Mul, sympy.cosh(a*symbol + b)/symbol, None, ChiRule), (sympy.Mul, sympy.sin(a*symbol + b)/symbol, None, SiRule), (sympy.Mul, sympy.sinh(a*symbol + b)/symbol, None, ShiRule), (sympy.Pow, 1/sympy.log(a*symbol + b), None, LiRule), (sympy.exp, sympy.exp(a*symbol**2 + b*symbol + c), None, ErfRule), (sympy.sin, sympy.sin(a*symbol**2 + b*symbol + c), None, FresnelSRule), (sympy.cos, sympy.cos(a*symbol**2 + b*symbol + c), None, FresnelCRule), (sympy.Mul, symbol**e*sympy.exp(a*symbol), None, UpperGammaRule), (sympy.Mul, sympy.polylog(b, a*symbol)/symbol, None, PolylogRule), (sympy.Pow, 1/sympy.sqrt(a - d*sympy.sin(symbol)**2), lambda a, d: a != d, EllipticFRule), (sympy.Pow, sympy.sqrt(a - d*sympy.sin(symbol)**2), lambda a, d: a != d, EllipticERule), ) for p in patterns: if isinstance(integrand, p[0]): match = integrand.match(p[1]) if match: wild_vals = tuple(match.get(w) for w in wilds if match.get(w) is not None) if p[2] is None or p[2](*wild_vals): args = wild_vals + (integrand, symbol) return p[3](*args) def inverse_trig_rule(integral): integrand, symbol = integral base, exp = integrand.as_base_exp() a = sympy.Wild('a', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) match = base.match(a + b*symbol**2) if not match: return def negative(x): return x.is_negative or x.could_extract_minus_sign() def ArcsinhRule(integrand, symbol): return InverseHyperbolicRule(sympy.asinh, integrand, symbol) def ArccoshRule(integrand, symbol): return InverseHyperbolicRule(sympy.acosh, integrand, symbol) def make_inverse_trig(RuleClass, base_exp, a, sign_a, b, sign_b): u_var = sympy.Dummy("u") current_base = base current_symbol = symbol constant = u_func = u_constant = substep = None factored = integrand if a != 1: constant = a**base_exp current_base = sign_a + sign_b * (b/a) * current_symbol**2 factored = current_base ** base_exp if (b/a) != 1: u_func = sympy.sqrt(b/a) * symbol u_constant = sympy.sqrt(a/b) current_symbol = u_var current_base = sign_a + sign_b * current_symbol**2 substep = RuleClass(current_base ** base_exp, current_symbol) if u_func is not None: if u_constant != 1 and substep is not None: substep = ConstantTimesRule( u_constant, current_base ** base_exp, substep, u_constant * current_base ** base_exp, symbol) substep = URule(u_var, u_func, u_constant, substep, factored, symbol) if constant is not None and substep is not None: substep = ConstantTimesRule(constant, factored, substep, integrand, symbol) return substep a, b = [match.get(i, ZERO) for i in (a, b)] # list of (rule, base_exp, a, sign_a, b, sign_b, condition) possibilities = [] if sympy.simplify(2*exp + 1) == 0: possibilities.append((ArcsinRule, exp, a, 1, -b, -1, sympy.And(a > 0, b < 0))) possibilities.append((ArcsinhRule, exp, a, 1, b, 1, sympy.And(a > 0, b > 0))) possibilities.append((ArccoshRule, exp, -a, -1, b, 1, sympy.And(a < 0, b > 0))) possibilities = [p for p in possibilities if p[-1] is not sympy.false] if a.is_number and b.is_number: possibility = [p for p in possibilities if p[-1] is sympy.true] if len(possibility) == 1: return make_inverse_trig(*possibility[0][:-1]) elif possibilities: return PiecewiseRule( [(make_inverse_trig(*p[:-1]), p[-1]) for p in possibilities], integrand, symbol) def add_rule(integral): integrand, symbol = integral results = [integral_steps(g, symbol) for g in integrand.as_ordered_terms()] return None if None in results else AddRule(results, integrand, symbol) def mul_rule(integral): integrand, symbol = integral # Constant times function case coeff, f = integrand.as_independent(symbol) next_step = integral_steps(f, symbol) if coeff != 1 and next_step is not None: return ConstantTimesRule( coeff, f, next_step, integrand, symbol) def _parts_rule(integrand, symbol): # LIATE rule: # log, inverse trig, algebraic, trigonometric, exponential def pull_out_algebraic(integrand): integrand = integrand.cancel().together() # iterating over Piecewise args would not work here algebraic = ([] if isinstance(integrand, sympy.Piecewise) else [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)]) if algebraic: u = sympy.Mul(*algebraic) dv = (integrand / u).cancel() return u, dv def pull_out_u(*functions): def pull_out_u_rl(integrand): if any([integrand.has(f) for f in functions]): args = [arg for arg in integrand.args if any(isinstance(arg, cls) for cls in functions)] if args: u = reduce(lambda a,b: a*b, args) dv = integrand / u return u, dv return pull_out_u_rl liate_rules = [pull_out_u(sympy.log), pull_out_u(*inverse_trig_functions), pull_out_algebraic, pull_out_u(sympy.sin, sympy.cos), pull_out_u(sympy.exp)] dummy = sympy.Dummy("temporary") # we can integrate log(x) and atan(x) by setting dv = 1 if isinstance(integrand, (sympy.log, *inverse_trig_functions)): integrand = dummy * integrand for index, rule in enumerate(liate_rules): result = rule(integrand) if result: u, dv = result # Don't pick u to be a constant if possible if symbol not in u.free_symbols and not u.has(dummy): return u = u.subs(dummy, 1) dv = dv.subs(dummy, 1) # Don't pick a non-polynomial algebraic to be differentiated if rule == pull_out_algebraic and not u.is_polynomial(symbol): return # Don't trade one logarithm for another if isinstance(u, sympy.log): rec_dv = 1/dv if (rec_dv.is_polynomial(symbol) and degree(rec_dv, symbol) == 1): return # Can integrate a polynomial times OrthogonalPolynomial if rule == pull_out_algebraic and isinstance(dv, OrthogonalPolynomial): v_step = integral_steps(dv, symbol) if contains_dont_know(v_step): return else: du = u.diff(symbol) v = _manualintegrate(v_step) return u, dv, v, du, v_step # make sure dv is amenable to integration accept = False if index < 2: # log and inverse trig are usually worth trying accept = True elif (rule == pull_out_algebraic and dv.args and all(isinstance(a, (sympy.sin, sympy.cos, sympy.exp)) for a in dv.args)): accept = True else: for rule in liate_rules[index + 1:]: r = rule(integrand) if r and r[0].subs(dummy, 1).equals(dv): accept = True break if accept: du = u.diff(symbol) v_step = integral_steps(sympy.simplify(dv), symbol) if not contains_dont_know(v_step): v = _manualintegrate(v_step) return u, dv, v, du, v_step def parts_rule(integral): integrand, symbol = integral constant, integrand = integrand.as_coeff_Mul() result = _parts_rule(integrand, symbol) steps = [] if result: u, dv, v, du, v_step = result debug("u : {}, dv : {}, v : {}, du : {}, v_step: {}".format(u, dv, v, du, v_step)) steps.append(result) if isinstance(v, sympy.Integral): return # Set a limit on the number of times u can be used if isinstance(u, (sympy.sin, sympy.cos, sympy.exp, sympy.sinh, sympy.cosh)): cachekey = u.xreplace({symbol: _cache_dummy}) if _parts_u_cache[cachekey] > 2: return _parts_u_cache[cachekey] += 1 # Try cyclic integration by parts a few times for _ in range(4): debug("Cyclic integration {} with v: {}, du: {}, integrand: {}".format(_, v, du, integrand)) coefficient = ((v * du) / integrand).cancel() if coefficient == 1: break if symbol not in coefficient.free_symbols: rule = CyclicPartsRule( [PartsRule(u, dv, v_step, None, None, None) for (u, dv, v, du, v_step) in steps], (-1) ** len(steps) * coefficient, integrand, symbol ) if (constant != 1) and rule: rule = ConstantTimesRule(constant, integrand, rule, constant * integrand, symbol) return rule # _parts_rule is sensitive to constants, factor it out next_constant, next_integrand = (v * du).as_coeff_Mul() result = _parts_rule(next_integrand, symbol) if result: u, dv, v, du, v_step = result u *= next_constant du *= next_constant steps.append((u, dv, v, du, v_step)) else: break def make_second_step(steps, integrand): if steps: u, dv, v, du, v_step = steps[0] return PartsRule(u, dv, v_step, make_second_step(steps[1:], v * du), integrand, symbol) else: steps = integral_steps(integrand, symbol) if steps: return steps else: return DontKnowRule(integrand, symbol) if steps: u, dv, v, du, v_step = steps[0] rule = PartsRule(u, dv, v_step, make_second_step(steps[1:], v * du), integrand, symbol) if (constant != 1) and rule: rule = ConstantTimesRule(constant, integrand, rule, constant * integrand, symbol) return rule def trig_rule(integral): integrand, symbol = integral if isinstance(integrand, sympy.sin) or isinstance(integrand, sympy.cos): arg = integrand.args[0] if not isinstance(arg, sympy.Symbol): return # perhaps a substitution can deal with it if isinstance(integrand, sympy.sin): func = 'sin' else: func = 'cos' return TrigRule(func, arg, integrand, symbol) if integrand == sympy.sec(symbol)**2: return TrigRule('sec**2', symbol, integrand, symbol) elif integrand == sympy.csc(symbol)**2: return TrigRule('csc**2', symbol, integrand, symbol) if isinstance(integrand, sympy.tan): rewritten = sympy.sin(*integrand.args) / sympy.cos(*integrand.args) elif isinstance(integrand, sympy.cot): rewritten = sympy.cos(*integrand.args) / sympy.sin(*integrand.args) elif isinstance(integrand, sympy.sec): arg = integrand.args[0] rewritten = ((sympy.sec(arg)**2 + sympy.tan(arg) * sympy.sec(arg)) / (sympy.sec(arg) + sympy.tan(arg))) elif isinstance(integrand, sympy.csc): arg = integrand.args[0] rewritten = ((sympy.csc(arg)**2 + sympy.cot(arg) * sympy.csc(arg)) / (sympy.csc(arg) + sympy.cot(arg))) else: return return RewriteRule( rewritten, integral_steps(rewritten, symbol), integrand, symbol ) def trig_product_rule(integral): integrand, symbol = integral sectan = sympy.sec(symbol) * sympy.tan(symbol) q = integrand / sectan if symbol not in q.free_symbols: rule = TrigRule('sec*tan', symbol, sectan, symbol) if q != 1 and rule: rule = ConstantTimesRule(q, sectan, rule, integrand, symbol) return rule csccot = -sympy.csc(symbol) * sympy.cot(symbol) q = integrand / csccot if symbol not in q.free_symbols: rule = TrigRule('csc*cot', symbol, csccot, symbol) if q != 1 and rule: rule = ConstantTimesRule(q, csccot, rule, integrand, symbol) return rule def quadratic_denom_rule(integral): integrand, symbol = integral a = sympy.Wild('a', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) c = sympy.Wild('c', exclude=[symbol]) match = integrand.match(a / (b * symbol ** 2 + c)) if match: a, b, c = match[a], match[b], match[c] if b.is_extended_real and c.is_extended_real: return PiecewiseRule([(ArctanRule(a, b, c, integrand, symbol), sympy.Gt(c / b, 0)), (ArccothRule(a, b, c, integrand, symbol), sympy.And(sympy.Gt(symbol ** 2, -c / b), sympy.Lt(c / b, 0))), (ArctanhRule(a, b, c, integrand, symbol), sympy.And(sympy.Lt(symbol ** 2, -c / b), sympy.Lt(c / b, 0))), ], integrand, symbol) else: return ArctanRule(a, b, c, integrand, symbol) d = sympy.Wild('d', exclude=[symbol]) match2 = integrand.match(a / (b * symbol ** 2 + c * symbol + d)) if match2: b, c = match2[b], match2[c] if b.is_zero: return u = sympy.Dummy('u') u_func = symbol + c/(2*b) integrand2 = integrand.subs(symbol, u - c / (2*b)) next_step = integral_steps(integrand2, u) if next_step: return URule(u, u_func, None, next_step, integrand2, symbol) else: return e = sympy.Wild('e', exclude=[symbol]) match3 = integrand.match((a* symbol + b) / (c * symbol ** 2 + d * symbol + e)) if match3: a, b, c, d, e = match3[a], match3[b], match3[c], match3[d], match3[e] if c.is_zero: return denominator = c * symbol**2 + d * symbol + e const = a/(2*c) numer1 = (2*c*symbol+d) numer2 = - const*d + b u = sympy.Dummy('u') step1 = URule(u, denominator, const, integral_steps(u**(-1), u), integrand, symbol) if const != 1: step1 = ConstantTimesRule(const, numer1/denominator, step1, const*numer1/denominator, symbol) if numer2.is_zero: return step1 step2 = integral_steps(numer2/denominator, symbol) substeps = AddRule([step1, step2], integrand, symbol) rewriten = const*numer1/denominator+numer2/denominator return RewriteRule(rewriten, substeps, integrand, symbol) return def root_mul_rule(integral): integrand, symbol = integral a = sympy.Wild('a', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) c = sympy.Wild('c') match = integrand.match(sympy.sqrt(a * symbol + b) * c) if not match: return a, b, c = match[a], match[b], match[c] d = sympy.Wild('d', exclude=[symbol]) e = sympy.Wild('e', exclude=[symbol]) f = sympy.Wild('f') recursion_test = c.match(sympy.sqrt(d * symbol + e) * f) if recursion_test: return u = sympy.Dummy('u') u_func = sympy.sqrt(a * symbol + b) integrand = integrand.subs(u_func, u) integrand = integrand.subs(symbol, (u**2 - b) / a) integrand = integrand * 2 * u / a next_step = integral_steps(integrand, u) if next_step: return URule(u, u_func, None, next_step, integrand, symbol) @sympy.cacheit def make_wilds(symbol): a = sympy.Wild('a', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) m = sympy.Wild('m', exclude=[symbol], properties=[lambda n: isinstance(n, sympy.Integer)]) n = sympy.Wild('n', exclude=[symbol], properties=[lambda n: isinstance(n, sympy.Integer)]) return a, b, m, n @sympy.cacheit def sincos_pattern(symbol): a, b, m, n = make_wilds(symbol) pattern = sympy.sin(a*symbol)**m * sympy.cos(b*symbol)**n return pattern, a, b, m, n @sympy.cacheit def tansec_pattern(symbol): a, b, m, n = make_wilds(symbol) pattern = sympy.tan(a*symbol)**m * sympy.sec(b*symbol)**n return pattern, a, b, m, n @sympy.cacheit def cotcsc_pattern(symbol): a, b, m, n = make_wilds(symbol) pattern = sympy.cot(a*symbol)**m * sympy.csc(b*symbol)**n return pattern, a, b, m, n @sympy.cacheit def heaviside_pattern(symbol): m = sympy.Wild('m', exclude=[symbol]) b = sympy.Wild('b', exclude=[symbol]) g = sympy.Wild('g') pattern = sympy.Heaviside(m*symbol + b) * g return pattern, m, b, g def uncurry(func): def uncurry_rl(args): return func(*args) return uncurry_rl def trig_rewriter(rewrite): def trig_rewriter_rl(args): a, b, m, n, integrand, symbol = args rewritten = rewrite(a, b, m, n, integrand, symbol) if rewritten != integrand: return RewriteRule( rewritten, integral_steps(rewritten, symbol), integrand, symbol) return trig_rewriter_rl sincos_botheven_condition = uncurry( lambda a, b, m, n, i, s: m.is_even and n.is_even and m.is_nonnegative and n.is_nonnegative) sincos_botheven = trig_rewriter( lambda a, b, m, n, i, symbol: ( (((1 - sympy.cos(2*a*symbol)) / 2) ** (m / 2)) * (((1 + sympy.cos(2*b*symbol)) / 2) ** (n / 2)) )) sincos_sinodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd and m >= 3) sincos_sinodd = trig_rewriter( lambda a, b, m, n, i, symbol: ( (1 - sympy.cos(a*symbol)**2)**((m - 1) / 2) * sympy.sin(a*symbol) * sympy.cos(b*symbol) ** n)) sincos_cosodd_condition = uncurry(lambda a, b, m, n, i, s: n.is_odd and n >= 3) sincos_cosodd = trig_rewriter( lambda a, b, m, n, i, symbol: ( (1 - sympy.sin(b*symbol)**2)**((n - 1) / 2) * sympy.cos(b*symbol) * sympy.sin(a*symbol) ** m)) tansec_seceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4) tansec_seceven = trig_rewriter( lambda a, b, m, n, i, symbol: ( (1 + sympy.tan(b*symbol)**2) ** (n/2 - 1) * sympy.sec(b*symbol)**2 * sympy.tan(a*symbol) ** m )) tansec_tanodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd) tansec_tanodd = trig_rewriter( lambda a, b, m, n, i, symbol: ( (sympy.sec(a*symbol)**2 - 1) ** ((m - 1) / 2) * sympy.tan(a*symbol) * sympy.sec(b*symbol) ** n )) tan_tansquared_condition = uncurry(lambda a, b, m, n, i, s: m == 2 and n == 0) tan_tansquared = trig_rewriter( lambda a, b, m, n, i, symbol: ( sympy.sec(a*symbol)**2 - 1)) cotcsc_csceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4) cotcsc_csceven = trig_rewriter( lambda a, b, m, n, i, symbol: ( (1 + sympy.cot(b*symbol)**2) ** (n/2 - 1) * sympy.csc(b*symbol)**2 * sympy.cot(a*symbol) ** m )) cotcsc_cotodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd) cotcsc_cotodd = trig_rewriter( lambda a, b, m, n, i, symbol: ( (sympy.csc(a*symbol)**2 - 1) ** ((m - 1) / 2) * sympy.cot(a*symbol) * sympy.csc(b*symbol) ** n )) def trig_sincos_rule(integral): integrand, symbol = integral if any(integrand.has(f) for f in (sympy.sin, sympy.cos)): pattern, a, b, m, n = sincos_pattern(symbol) match = integrand.match(pattern) if not match: return return multiplexer({ sincos_botheven_condition: sincos_botheven, sincos_sinodd_condition: sincos_sinodd, sincos_cosodd_condition: sincos_cosodd })(tuple( [match.get(i, ZERO) for i in (a, b, m, n)] + [integrand, symbol])) def trig_tansec_rule(integral): integrand, symbol = integral integrand = integrand.subs({ 1 / sympy.cos(symbol): sympy.sec(symbol) }) if any(integrand.has(f) for f in (sympy.tan, sympy.sec)): pattern, a, b, m, n = tansec_pattern(symbol) match = integrand.match(pattern) if not match: return return multiplexer({ tansec_tanodd_condition: tansec_tanodd, tansec_seceven_condition: tansec_seceven, tan_tansquared_condition: tan_tansquared })(tuple( [match.get(i, ZERO) for i in (a, b, m, n)] + [integrand, symbol])) def trig_cotcsc_rule(integral): integrand, symbol = integral integrand = integrand.subs({ 1 / sympy.sin(symbol): sympy.csc(symbol), 1 / sympy.tan(symbol): sympy.cot(symbol), sympy.cos(symbol) / sympy.tan(symbol): sympy.cot(symbol) }) if any(integrand.has(f) for f in (sympy.cot, sympy.csc)): pattern, a, b, m, n = cotcsc_pattern(symbol) match = integrand.match(pattern) if not match: return return multiplexer({ cotcsc_cotodd_condition: cotcsc_cotodd, cotcsc_csceven_condition: cotcsc_csceven })(tuple( [match.get(i, ZERO) for i in (a, b, m, n)] + [integrand, symbol])) def trig_sindouble_rule(integral): integrand, symbol = integral a = sympy.Wild('a', exclude=[sympy.sin(2*symbol)]) match = integrand.match(sympy.sin(2*symbol)*a) if match: sin_double = 2*sympy.sin(symbol)*sympy.cos(symbol)/sympy.sin(2*symbol) return integral_steps(integrand * sin_double, symbol) def trig_powers_products_rule(integral): return do_one(null_safe(trig_sincos_rule), null_safe(trig_tansec_rule), null_safe(trig_cotcsc_rule), null_safe(trig_sindouble_rule))(integral) def trig_substitution_rule(integral): integrand, symbol = integral A = sympy.Wild('a', exclude=[0, symbol]) B = sympy.Wild('b', exclude=[0, symbol]) theta = sympy.Dummy("theta") target_pattern = A + B*symbol**2 matches = integrand.find(target_pattern) for expr in matches: match = expr.match(target_pattern) a = match.get(A, ZERO) b = match.get(B, ZERO) a_positive = ((a.is_number and a > 0) or a.is_positive) b_positive = ((b.is_number and b > 0) or b.is_positive) a_negative = ((a.is_number and a < 0) or a.is_negative) b_negative = ((b.is_number and b < 0) or b.is_negative) x_func = None if a_positive and b_positive: # a**2 + b*x**2. Assume sec(theta) > 0, -pi/2 < theta < pi/2 x_func = (sympy.sqrt(a)/sympy.sqrt(b)) * sympy.tan(theta) # Do not restrict the domain: tan(theta) takes on any real # value on the interval -pi/2 < theta < pi/2 so x takes on # any value restriction = True elif a_positive and b_negative: # a**2 - b*x**2. Assume cos(theta) > 0, -pi/2 < theta < pi/2 constant = sympy.sqrt(a)/sympy.sqrt(-b) x_func = constant * sympy.sin(theta) restriction = sympy.And(symbol > -constant, symbol < constant) elif a_negative and b_positive: # b*x**2 - a**2. Assume sin(theta) > 0, 0 < theta < pi constant = sympy.sqrt(-a)/sympy.sqrt(b) x_func = constant * sympy.sec(theta) restriction = sympy.And(symbol > -constant, symbol < constant) if x_func: # Manually simplify sqrt(trig(theta)**2) to trig(theta) # Valid due to assumed domain restriction substitutions = {} for f in [sympy.sin, sympy.cos, sympy.tan, sympy.sec, sympy.csc, sympy.cot]: substitutions[sympy.sqrt(f(theta)**2)] = f(theta) substitutions[sympy.sqrt(f(theta)**(-2))] = 1/f(theta) replaced = integrand.subs(symbol, x_func).trigsimp() replaced = manual_subs(replaced, substitutions) if not replaced.has(symbol): replaced *= manual_diff(x_func, theta) replaced = replaced.trigsimp() secants = replaced.find(1/sympy.cos(theta)) if secants: replaced = replaced.xreplace({ 1/sympy.cos(theta): sympy.sec(theta) }) substep = integral_steps(replaced, theta) if not contains_dont_know(substep): return TrigSubstitutionRule( theta, x_func, replaced, substep, restriction, integrand, symbol) def heaviside_rule(integral): integrand, symbol = integral pattern, m, b, g = heaviside_pattern(symbol) match = integrand.match(pattern) if match and 0 != match[g]: # f = Heaviside(m*x + b)*g v_step = integral_steps(match[g], symbol) result = _manualintegrate(v_step) m, b = match[m], match[b] return HeavisideRule(m*symbol + b, -b/m, result, integrand, symbol) def substitution_rule(integral): integrand, symbol = integral u_var = sympy.Dummy("u") substitutions = find_substitutions(integrand, symbol, u_var) count = 0 if substitutions: debug("List of Substitution Rules") ways = [] for u_func, c, substituted in substitutions: subrule = integral_steps(substituted, u_var) count = count + 1 debug("Rule {}: {}".format(count, subrule)) if contains_dont_know(subrule): continue if sympy.simplify(c - 1) != 0: _, denom = c.as_numer_denom() if subrule: subrule = ConstantTimesRule(c, substituted, subrule, substituted, u_var) if denom.free_symbols: piecewise = [] could_be_zero = [] if isinstance(denom, sympy.Mul): could_be_zero = denom.args else: could_be_zero.append(denom) for expr in could_be_zero: if not fuzzy_not(expr.is_zero): substep = integral_steps(manual_subs(integrand, expr, 0), symbol) if substep: piecewise.append(( substep, sympy.Eq(expr, 0) )) piecewise.append((subrule, True)) subrule = PiecewiseRule(piecewise, substituted, symbol) ways.append(URule(u_var, u_func, c, subrule, integrand, symbol)) if len(ways) > 1: return AlternativeRule(ways, integrand, symbol) elif ways: return ways[0] elif integrand.has(sympy.exp): u_func = sympy.exp(symbol) c = 1 substituted = integrand / u_func.diff(symbol) substituted = substituted.subs(u_func, u_var) if symbol not in substituted.free_symbols: return URule(u_var, u_func, c, integral_steps(substituted, u_var), integrand, symbol) partial_fractions_rule = rewriter( lambda integrand, symbol: integrand.is_rational_function(), lambda integrand, symbol: integrand.apart(symbol)) cancel_rule = rewriter( # lambda integrand, symbol: integrand.is_algebraic_expr(), # lambda integrand, symbol: isinstance(integrand, sympy.Mul), lambda integrand, symbol: True, lambda integrand, symbol: integrand.cancel()) distribute_expand_rule = rewriter( lambda integrand, symbol: ( all(arg.is_Pow or arg.is_polynomial(symbol) for arg in integrand.args) or isinstance(integrand, sympy.Pow) or isinstance(integrand, sympy.Mul)), lambda integrand, symbol: integrand.expand()) trig_expand_rule = rewriter( # If there are trig functions with different arguments, expand them lambda integrand, symbol: ( len({a.args[0] for a in integrand.atoms(TrigonometricFunction)}) > 1), lambda integrand, symbol: integrand.expand(trig=True)) def derivative_rule(integral): integrand = integral[0] diff_variables = integrand.variables undifferentiated_function = integrand.expr integrand_variables = undifferentiated_function.free_symbols if integral.symbol in integrand_variables: if integral.symbol in diff_variables: return DerivativeRule(*integral) else: return DontKnowRule(integrand, integral.symbol) else: return ConstantRule(integral.integrand, *integral) def rewrites_rule(integral): integrand, symbol = integral if integrand.match(1/sympy.cos(symbol)): rewritten = integrand.subs(1/sympy.cos(symbol), sympy.sec(symbol)) return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand, symbol) def fallback_rule(integral): return DontKnowRule(*integral) # Cache is used to break cyclic integrals. # Need to use the same dummy variable in cached expressions for them to match. # Also record "u" of integration by parts, to avoid infinite repetition. _integral_cache = {} # type: tDict[Expr, Optional[Expr]] _parts_u_cache = defaultdict(int) # type: tDict[Expr, int] _cache_dummy = sympy.Dummy("z") def integral_steps(integrand, symbol, **options): """Returns the steps needed to compute an integral. Explanation =========== This function attempts to mirror what a student would do by hand as closely as possible. SymPy Gamma uses this to provide a step-by-step explanation of an integral. The code it uses to format the results of this function can be found at https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py. Examples ======== >>> from sympy import exp, sin >>> from sympy.integrals.manualintegrate import integral_steps >>> from sympy.abc import x >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \ # doctest: +NORMALIZE_WHITESPACE URule(u_var=_u, u_func=exp(x), constant=1, substep=PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), True), (ArccothRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False), (ArctanhRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False)], context=1/(_u**2 + 1), symbol=_u), context=exp(x)/(exp(2*x) + 1), symbol=x) >>> print(repr(integral_steps(sin(x), x))) \ # doctest: +NORMALIZE_WHITESPACE TrigRule(func='sin', arg=x, context=sin(x), symbol=x) >>> print(repr(integral_steps((x**2 + 3)**2 , x))) \ # doctest: +NORMALIZE_WHITESPACE RewriteRule(rewritten=x**4 + 6*x**2 + 9, substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x), ConstantTimesRule(constant=6, other=x**2, substep=PowerRule(base=x, exp=2, context=x**2, symbol=x), context=6*x**2, symbol=x), ConstantRule(constant=9, context=9, symbol=x)], context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x) Returns ======= rule : namedtuple The first step; most rules have substeps that must also be considered. These substeps can be evaluated using ``manualintegrate`` to obtain a result. """ cachekey = integrand.xreplace({symbol: _cache_dummy}) if cachekey in _integral_cache: if _integral_cache[cachekey] is None: # Stop this attempt, because it leads around in a loop return DontKnowRule(integrand, symbol) else: # TODO: This is for future development, as currently # _integral_cache gets no values other than None return (_integral_cache[cachekey].xreplace(_cache_dummy, symbol), symbol) else: _integral_cache[cachekey] = None integral = IntegralInfo(integrand, symbol) def key(integral): integrand = integral.integrand if isinstance(integrand, TrigonometricFunction): return TrigonometricFunction elif isinstance(integrand, sympy.Derivative): return sympy.Derivative elif symbol not in integrand.free_symbols: return sympy.Number else: for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.log, sympy.Add, sympy.Mul, *inverse_trig_functions, sympy.Heaviside, OrthogonalPolynomial): if isinstance(integrand, cls): return cls def integral_is_subclass(*klasses): def _integral_is_subclass(integral): k = key(integral) return k and issubclass(k, klasses) return _integral_is_subclass result = do_one( null_safe(special_function_rule), null_safe(switch(key, { sympy.Pow: do_one(null_safe(power_rule), null_safe(inverse_trig_rule), \ null_safe(quadratic_denom_rule)), sympy.Symbol: power_rule, sympy.exp: exp_rule, sympy.Add: add_rule, sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule), \ null_safe(heaviside_rule), null_safe(quadratic_denom_rule), \ null_safe(root_mul_rule)), sympy.Derivative: derivative_rule, TrigonometricFunction: trig_rule, sympy.Heaviside: heaviside_rule, OrthogonalPolynomial: orthogonal_poly_rule, sympy.Number: constant_rule })), do_one( null_safe(trig_rule), null_safe(alternatives( rewrites_rule, substitution_rule, condition( integral_is_subclass(sympy.Mul, sympy.Pow), partial_fractions_rule), condition( integral_is_subclass(sympy.Mul, sympy.Pow), cancel_rule), condition( integral_is_subclass(sympy.Mul, sympy.log, *inverse_trig_functions), parts_rule), condition( integral_is_subclass(sympy.Mul, sympy.Pow), distribute_expand_rule), trig_powers_products_rule, trig_expand_rule )), null_safe(trig_substitution_rule) ), fallback_rule)(integral) del _integral_cache[cachekey] return result @evaluates(ConstantRule) def eval_constant(constant, integrand, symbol): return constant * symbol @evaluates(ConstantTimesRule) def eval_constanttimes(constant, other, substep, integrand, symbol): return constant * _manualintegrate(substep) @evaluates(PowerRule) def eval_power(base, exp, integrand, symbol): return sympy.Piecewise( ((base**(exp + 1))/(exp + 1), sympy.Ne(exp, -1)), (sympy.log(base), True), ) @evaluates(ExpRule) def eval_exp(base, exp, integrand, symbol): return integrand / sympy.ln(base) @evaluates(AddRule) def eval_add(substeps, integrand, symbol): return sum(map(_manualintegrate, substeps)) @evaluates(URule) def eval_u(u_var, u_func, constant, substep, integrand, symbol): result = _manualintegrate(substep) if u_func.is_Pow and u_func.exp == -1: # avoid needless -log(1/x) from substitution result = result.subs(sympy.log(u_var), -sympy.log(u_func.base)) return result.subs(u_var, u_func) @evaluates(PartsRule) def eval_parts(u, dv, v_step, second_step, integrand, symbol): v = _manualintegrate(v_step) return u * v - _manualintegrate(second_step) @evaluates(CyclicPartsRule) def eval_cyclicparts(parts_rules, coefficient, integrand, symbol): coefficient = 1 - coefficient result = [] sign = 1 for rule in parts_rules: result.append(sign * rule.u * _manualintegrate(rule.v_step)) sign *= -1 return sympy.Add(*result) / coefficient @evaluates(TrigRule) def eval_trig(func, arg, integrand, symbol): if func == 'sin': return -sympy.cos(arg) elif func == 'cos': return sympy.sin(arg) elif func == 'sec*tan': return sympy.sec(arg) elif func == 'csc*cot': return sympy.csc(arg) elif func == 'sec**2': return sympy.tan(arg) elif func == 'csc**2': return -sympy.cot(arg) @evaluates(ArctanRule) def eval_arctan(a, b, c, integrand, symbol): return a / b * 1 / sympy.sqrt(c / b) * sympy.atan(symbol / sympy.sqrt(c / b)) @evaluates(ArccothRule) def eval_arccoth(a, b, c, integrand, symbol): return - a / b * 1 / sympy.sqrt(-c / b) * sympy.acoth(symbol / sympy.sqrt(-c / b)) @evaluates(ArctanhRule) def eval_arctanh(a, b, c, integrand, symbol): return - a / b * 1 / sympy.sqrt(-c / b) * sympy.atanh(symbol / sympy.sqrt(-c / b)) @evaluates(ReciprocalRule) def eval_reciprocal(func, integrand, symbol): return sympy.ln(func) @evaluates(ArcsinRule) def eval_arcsin(integrand, symbol): return sympy.asin(symbol) @evaluates(InverseHyperbolicRule) def eval_inversehyperbolic(func, integrand, symbol): return func(symbol) @evaluates(AlternativeRule) def eval_alternative(alternatives, integrand, symbol): return _manualintegrate(alternatives[0]) @evaluates(RewriteRule) def eval_rewrite(rewritten, substep, integrand, symbol): return _manualintegrate(substep) @evaluates(PiecewiseRule) def eval_piecewise(substeps, integrand, symbol): return sympy.Piecewise(*[(_manualintegrate(substep), cond) for substep, cond in substeps]) @evaluates(TrigSubstitutionRule) def eval_trigsubstitution(theta, func, rewritten, substep, restriction, integrand, symbol): func = func.subs(sympy.sec(theta), 1/sympy.cos(theta)) func = func.subs(sympy.csc(theta), 1/sympy.sin(theta)) func = func.subs(sympy.cot(theta), 1/sympy.tan(theta)) trig_function = list(func.find(TrigonometricFunction)) assert len(trig_function) == 1 trig_function = trig_function[0] relation = sympy.solve(symbol - func, trig_function) assert len(relation) == 1 numer, denom = sympy.fraction(relation[0]) if isinstance(trig_function, sympy.sin): opposite = numer hypotenuse = denom adjacent = sympy.sqrt(denom**2 - numer**2) inverse = sympy.asin(relation[0]) elif isinstance(trig_function, sympy.cos): adjacent = numer hypotenuse = denom opposite = sympy.sqrt(denom**2 - numer**2) inverse = sympy.acos(relation[0]) elif isinstance(trig_function, sympy.tan): opposite = numer adjacent = denom hypotenuse = sympy.sqrt(denom**2 + numer**2) inverse = sympy.atan(relation[0]) substitution = [ (sympy.sin(theta), opposite/hypotenuse), (sympy.cos(theta), adjacent/hypotenuse), (sympy.tan(theta), opposite/adjacent), (theta, inverse) ] return sympy.Piecewise( (_manualintegrate(substep).subs(substitution).trigsimp(), restriction) ) @evaluates(DerivativeRule) def eval_derivativerule(integrand, symbol): # isinstance(integrand, Derivative) should be True variable_count = list(integrand.variable_count) for i, (var, count) in enumerate(variable_count): if var == symbol: variable_count[i] = (var, count-1) break return sympy.Derivative(integrand.expr, *variable_count) @evaluates(HeavisideRule) def eval_heaviside(harg, ibnd, substep, integrand, symbol): # If we are integrating over x and the integrand has the form # Heaviside(m*x+b)*g(x) == Heaviside(harg)*g(symbol) # then there needs to be continuity at -b/m == ibnd, # so we subtract the appropriate term. return sympy.Heaviside(harg)*(substep - substep.subs(symbol, ibnd)) @evaluates(JacobiRule) def eval_jacobi(n, a, b, integrand, symbol): return Piecewise( (2*sympy.jacobi(n + 1, a - 1, b - 1, symbol)/(n + a + b), Ne(n + a + b, 0)), (symbol, Eq(n, 0)), ((a + b + 2)*symbol**2/4 + (a - b)*symbol/2, Eq(n, 1))) @evaluates(GegenbauerRule) def eval_gegenbauer(n, a, integrand, symbol): return Piecewise( (sympy.gegenbauer(n + 1, a - 1, symbol)/(2*(a - 1)), Ne(a, 1)), (sympy.chebyshevt(n + 1, symbol)/(n + 1), Ne(n, -1)), (sympy.S.Zero, True)) @evaluates(ChebyshevTRule) def eval_chebyshevt(n, integrand, symbol): return Piecewise(((sympy.chebyshevt(n + 1, symbol)/(n + 1) - sympy.chebyshevt(n - 1, symbol)/(n - 1))/2, Ne(sympy.Abs(n), 1)), (symbol**2/2, True)) @evaluates(ChebyshevURule) def eval_chebyshevu(n, integrand, symbol): return Piecewise( (sympy.chebyshevt(n + 1, symbol)/(n + 1), Ne(n, -1)), (sympy.S.Zero, True)) @evaluates(LegendreRule) def eval_legendre(n, integrand, symbol): return (sympy.legendre(n + 1, symbol) - sympy.legendre(n - 1, symbol))/(2*n + 1) @evaluates(HermiteRule) def eval_hermite(n, integrand, symbol): return sympy.hermite(n + 1, symbol)/(2*(n + 1)) @evaluates(LaguerreRule) def eval_laguerre(n, integrand, symbol): return sympy.laguerre(n, symbol) - sympy.laguerre(n + 1, symbol) @evaluates(AssocLaguerreRule) def eval_assoclaguerre(n, a, integrand, symbol): return -sympy.assoc_laguerre(n + 1, a - 1, symbol) @evaluates(CiRule) def eval_ci(a, b, integrand, symbol): return sympy.cos(b)*sympy.Ci(a*symbol) - sympy.sin(b)*sympy.Si(a*symbol) @evaluates(ChiRule) def eval_chi(a, b, integrand, symbol): return sympy.cosh(b)*sympy.Chi(a*symbol) + sympy.sinh(b)*sympy.Shi(a*symbol) @evaluates(EiRule) def eval_ei(a, b, integrand, symbol): return sympy.exp(b)*sympy.Ei(a*symbol) @evaluates(SiRule) def eval_si(a, b, integrand, symbol): return sympy.sin(b)*sympy.Ci(a*symbol) + sympy.cos(b)*sympy.Si(a*symbol) @evaluates(ShiRule) def eval_shi(a, b, integrand, symbol): return sympy.sinh(b)*sympy.Chi(a*symbol) + sympy.cosh(b)*sympy.Shi(a*symbol) @evaluates(ErfRule) def eval_erf(a, b, c, integrand, symbol): if a.is_extended_real: return Piecewise( (sympy.sqrt(sympy.pi/(-a))/2 * sympy.exp(c - b**2/(4*a)) * sympy.erf((-2*a*symbol - b)/(2*sympy.sqrt(-a))), a < 0), (sympy.sqrt(sympy.pi/a)/2 * sympy.exp(c - b**2/(4*a)) * sympy.erfi((2*a*symbol + b)/(2*sympy.sqrt(a))), True)) else: return sympy.sqrt(sympy.pi/a)/2 * sympy.exp(c - b**2/(4*a)) * \ sympy.erfi((2*a*symbol + b)/(2*sympy.sqrt(a))) @evaluates(FresnelCRule) def eval_fresnelc(a, b, c, integrand, symbol): return sympy.sqrt(sympy.pi/(2*a)) * ( sympy.cos(b**2/(4*a) - c)*sympy.fresnelc((2*a*symbol + b)/sympy.sqrt(2*a*sympy.pi)) + sympy.sin(b**2/(4*a) - c)*sympy.fresnels((2*a*symbol + b)/sympy.sqrt(2*a*sympy.pi))) @evaluates(FresnelSRule) def eval_fresnels(a, b, c, integrand, symbol): return sympy.sqrt(sympy.pi/(2*a)) * ( sympy.cos(b**2/(4*a) - c)*sympy.fresnels((2*a*symbol + b)/sympy.sqrt(2*a*sympy.pi)) - sympy.sin(b**2/(4*a) - c)*sympy.fresnelc((2*a*symbol + b)/sympy.sqrt(2*a*sympy.pi))) @evaluates(LiRule) def eval_li(a, b, integrand, symbol): return sympy.li(a*symbol + b)/a @evaluates(PolylogRule) def eval_polylog(a, b, integrand, symbol): return sympy.polylog(b + 1, a*symbol) @evaluates(UpperGammaRule) def eval_uppergamma(a, e, integrand, symbol): return symbol**e * (-a*symbol)**(-e) * sympy.uppergamma(e + 1, -a*symbol)/a @evaluates(EllipticFRule) def eval_elliptic_f(a, d, integrand, symbol): return sympy.elliptic_f(symbol, d/a)/sympy.sqrt(a) @evaluates(EllipticERule) def eval_elliptic_e(a, d, integrand, symbol): return sympy.elliptic_e(symbol, d/a)*sympy.sqrt(a) @evaluates(DontKnowRule) def eval_dontknowrule(integrand, symbol): return sympy.Integral(integrand, symbol) def _manualintegrate(rule): evaluator = evaluators.get(rule.__class__) if not evaluator: raise ValueError("Cannot evaluate rule %s" % repr(rule)) return evaluator(*rule) def manualintegrate(f, var): """manualintegrate(f, var) Explanation =========== Compute indefinite integral of a single variable using an algorithm that resembles what a student would do by hand. Unlike :func:`~.integrate`, var can only be a single symbol. Examples ======== >>> from sympy import sin, cos, tan, exp, log, integrate >>> from sympy.integrals.manualintegrate import manualintegrate >>> from sympy.abc import x >>> manualintegrate(1 / x, x) log(x) >>> integrate(1/x) log(x) >>> manualintegrate(log(x), x) x*log(x) - x >>> integrate(log(x)) x*log(x) - x >>> manualintegrate(exp(x) / (1 + exp(2 * x)), x) atan(exp(x)) >>> integrate(exp(x) / (1 + exp(2 * x))) RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x)))) >>> manualintegrate(cos(x)**4 * sin(x), x) -cos(x)**5/5 >>> integrate(cos(x)**4 * sin(x), x) -cos(x)**5/5 >>> manualintegrate(cos(x)**4 * sin(x)**3, x) cos(x)**7/7 - cos(x)**5/5 >>> integrate(cos(x)**4 * sin(x)**3, x) cos(x)**7/7 - cos(x)**5/5 >>> manualintegrate(tan(x), x) -log(cos(x)) >>> integrate(tan(x), x) -log(cos(x)) See Also ======== sympy.integrals.integrals.integrate sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral """ result = _manualintegrate(integral_steps(f, var)) # Clear the cache of u-parts _parts_u_cache.clear() # If we got Piecewise with two parts, put generic first if isinstance(result, Piecewise) and len(result.args) == 2: cond = result.args[0][1] if isinstance(cond, Eq) and result.args[1][1] == True: result = result.func( (result.args[1][0], sympy.Ne(*cond.args)), (result.args[0][0], True)) return result
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from sympy.functions import SingularityFunction, DiracDelta from sympy.core import sympify from sympy.integrals import integrate def singularityintegrate(f, x): """ This function handles the indefinite integrations of Singularity functions. The ``integrate`` function calls this function internally whenever an instance of SingularityFunction is passed as argument. Explanation =========== The idea for integration is the following: - If we are dealing with a SingularityFunction expression, i.e. ``SingularityFunction(x, a, n)``, we just return ``SingularityFunction(x, a, n + 1)/(n + 1)`` if ``n >= 0`` and ``SingularityFunction(x, a, n + 1)`` if ``n < 0``. - If the node is a multiplication or power node having a SingularityFunction term we rewrite the whole expression in terms of Heaviside and DiracDelta and then integrate the output. Lastly, we rewrite the output of integration back in terms of SingularityFunction. - If none of the above case arises, we return None. Examples ======== >>> from sympy.integrals.singularityfunctions import singularityintegrate >>> from sympy import SingularityFunction, symbols, Function >>> x, a, n, y = symbols('x a n y') >>> f = Function('f') >>> singularityintegrate(SingularityFunction(x, a, 3), x) SingularityFunction(x, a, 4)/4 >>> singularityintegrate(5*SingularityFunction(x, 5, -2), x) 5*SingularityFunction(x, 5, -1) >>> singularityintegrate(6*SingularityFunction(x, 5, -1), x) 6*SingularityFunction(x, 5, 0) >>> singularityintegrate(x*SingularityFunction(x, 0, -1), x) 0 >>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x) f(1)*SingularityFunction(x, 1, 0) """ if not f.has(SingularityFunction): return None if f.func == SingularityFunction: x = sympify(f.args[0]) a = sympify(f.args[1]) n = sympify(f.args[2]) if n.is_positive or n.is_zero: return SingularityFunction(x, a, n + 1)/(n + 1) elif n == -1 or n == -2: return SingularityFunction(x, a, n + 1) if f.is_Mul or f.is_Pow: expr = f.rewrite(DiracDelta) expr = integrate(expr, x) return expr.rewrite(SingularityFunction) return None
1e47956b57f4c8f373b224dbaa3b62a418afd1a73d23ef15b300ca1b7b5b4689
""" Integral Transforms """ from sympy.core import S from sympy.core.compatibility import reduce, iterable from sympy.core.function import Function from sympy.core.relational import _canonical, Ge, Gt from sympy.core.numbers import oo from sympy.core.symbol import Dummy from sympy.integrals import integrate, Integral from sympy.integrals.meijerint import _dummy from sympy.logic.boolalg import to_cnf, conjuncts, disjuncts, Or, And from sympy.simplify import simplify from sympy.utilities import default_sort_key from sympy.matrices.matrices import MatrixBase ########################################################################## # Helpers / Utilities ########################################################################## class IntegralTransformError(NotImplementedError): """ Exception raised in relation to problems computing transforms. Explanation =========== This class is mostly used internally; if integrals cannot be computed objects representing unevaluated transforms are usually returned. The hint ``needeval=True`` can be used to disable returning transform objects, and instead raise this exception if an integral cannot be computed. """ def __init__(self, transform, function, msg): super().__init__( "%s Transform could not be computed: %s." % (transform, msg)) self.function = function class IntegralTransform(Function): """ Base class for integral transforms. Explanation =========== This class represents unevaluated transforms. To implement a concrete transform, derive from this class and implement the ``_compute_transform(f, x, s, **hints)`` and ``_as_integral(f, x, s)`` functions. If the transform cannot be computed, raise :obj:`IntegralTransformError`. Also set ``cls._name``. For instance, >>> from sympy.integrals.transforms import LaplaceTransform >>> LaplaceTransform._name 'Laplace' Implement ``self._collapse_extra`` if your function returns more than just a number and possibly a convergence condition. """ @property def function(self): """ The function to be transformed. """ return self.args[0] @property def function_variable(self): """ The dependent variable of the function to be transformed. """ return self.args[1] @property def transform_variable(self): """ The independent transform variable. """ return self.args[2] @property def free_symbols(self): """ This method returns the symbols that will exist when the transform is evaluated. """ return self.function.free_symbols.union({self.transform_variable}) \ - {self.function_variable} def _compute_transform(self, f, x, s, **hints): raise NotImplementedError def _as_integral(self, f, x, s): raise NotImplementedError def _collapse_extra(self, extra): cond = And(*extra) if cond == False: raise IntegralTransformError(self.__class__.name, None, '') return cond def doit(self, **hints): """ Try to evaluate the transform in closed form. Explanation =========== This general function handles linearity, but apart from that leaves pretty much everything to _compute_transform. Standard hints are the following: - ``simplify``: whether or not to simplify the result - ``noconds``: if True, don't return convergence conditions - ``needeval``: if True, raise IntegralTransformError instead of returning IntegralTransform objects The default values of these hints depend on the concrete transform, usually the default is ``(simplify, noconds, needeval) = (True, False, False)``. """ from sympy import Add, expand_mul, Mul from sympy.core.function import AppliedUndef needeval = hints.pop('needeval', False) try_directly = not any(func.has(self.function_variable) for func in self.function.atoms(AppliedUndef)) if try_directly: try: return self._compute_transform(self.function, self.function_variable, self.transform_variable, **hints) except IntegralTransformError: pass fn = self.function if not fn.is_Add: fn = expand_mul(fn) if fn.is_Add: hints['needeval'] = needeval res = [self.__class__(*([x] + list(self.args[1:]))).doit(**hints) for x in fn.args] extra = [] ress = [] for x in res: if not isinstance(x, tuple): x = [x] ress.append(x[0]) if len(x) == 2: # only a condition extra.append(x[1]) elif len(x) > 2: # some region parameters and a condition (Mellin, Laplace) extra += [x[1:]] res = Add(*ress) if not extra: return res try: extra = self._collapse_extra(extra) if iterable(extra): return tuple([res]) + tuple(extra) else: return (res, extra) except IntegralTransformError: pass if needeval: raise IntegralTransformError( self.__class__._name, self.function, 'needeval') # TODO handle derivatives etc # pull out constant coefficients coeff, rest = fn.as_coeff_mul(self.function_variable) return coeff*self.__class__(*([Mul(*rest)] + list(self.args[1:]))) @property def as_integral(self): return self._as_integral(self.function, self.function_variable, self.transform_variable) def _eval_rewrite_as_Integral(self, *args, **kwargs): return self.as_integral from sympy.solvers.inequalities import _solve_inequality def _simplify(expr, doit): from sympy import powdenest, piecewise_fold if doit: return simplify(powdenest(piecewise_fold(expr), polar=True)) return expr def _noconds_(default): """ This is a decorator generator for dropping convergence conditions. Explanation =========== Suppose you define a function ``transform(*args)`` which returns a tuple of the form ``(result, cond1, cond2, ...)``. Decorating it ``@_noconds_(default)`` will add a new keyword argument ``noconds`` to it. If ``noconds=True``, the return value will be altered to be only ``result``, whereas if ``noconds=False`` the return value will not be altered. The default value of the ``noconds`` keyword will be ``default`` (i.e. the argument of this function). """ def make_wrapper(func): from sympy.core.decorators import wraps @wraps(func) def wrapper(*args, noconds=default, **kwargs): res = func(*args, **kwargs) if noconds: return res[0] return res return wrapper return make_wrapper _noconds = _noconds_(False) ########################################################################## # Mellin Transform ########################################################################## def _default_integrator(f, x): return integrate(f, (x, 0, oo)) @_noconds def _mellin_transform(f, x, s_, integrator=_default_integrator, simplify=True): """ Backend function to compute Mellin transforms. """ from sympy import re, Max, Min, count_ops # We use a fresh dummy, because assumptions on s might drop conditions on # convergence of the integral. s = _dummy('s', 'mellin-transform', f) F = integrator(x**(s - 1) * f, x) if not F.has(Integral): return _simplify(F.subs(s, s_), simplify), (-oo, oo), S.true if not F.is_Piecewise: # XXX can this work if integration gives continuous result now? raise IntegralTransformError('Mellin', f, 'could not compute integral') F, cond = F.args[0] if F.has(Integral): raise IntegralTransformError( 'Mellin', f, 'integral in unexpected form') def process_conds(cond): """ Turn ``cond`` into a strip (a, b), and auxiliary conditions. """ a = -oo b = oo aux = S.true conds = conjuncts(to_cnf(cond)) t = Dummy('t', real=True) for c in conds: a_ = oo b_ = -oo aux_ = [] for d in disjuncts(c): d_ = d.replace( re, lambda x: x.as_real_imag()[0]).subs(re(s), t) if not d.is_Relational or \ d.rel_op in ('==', '!=') \ or d_.has(s) or not d_.has(t): aux_ += [d] continue soln = _solve_inequality(d_, t) if not soln.is_Relational or \ soln.rel_op in ('==', '!='): aux_ += [d] continue if soln.lts == t: b_ = Max(soln.gts, b_) else: a_ = Min(soln.lts, a_) if a_ != oo and a_ != b: a = Max(a_, a) elif b_ != -oo and b_ != a: b = Min(b_, b) else: aux = And(aux, Or(*aux_)) return a, b, aux conds = [process_conds(c) for c in disjuncts(cond)] conds = [x for x in conds if x[2] != False] conds.sort(key=lambda x: (x[0] - x[1], count_ops(x[2]))) if not conds: raise IntegralTransformError('Mellin', f, 'no convergence found') a, b, aux = conds[0] return _simplify(F.subs(s, s_), simplify), (a, b), aux class MellinTransform(IntegralTransform): """ Class representing unevaluated Mellin transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Mellin transforms, see the :func:`mellin_transform` docstring. """ _name = 'Mellin' def _compute_transform(self, f, x, s, **hints): return _mellin_transform(f, x, s, **hints) def _as_integral(self, f, x, s): return Integral(f*x**(s - 1), (x, 0, oo)) def _collapse_extra(self, extra): from sympy import Max, Min a = [] b = [] cond = [] for (sa, sb), c in extra: a += [sa] b += [sb] cond += [c] res = (Max(*a), Min(*b)), And(*cond) if (res[0][0] >= res[0][1]) == True or res[1] == False: raise IntegralTransformError( 'Mellin', None, 'no combined convergence.') return res def mellin_transform(f, x, s, **hints): r""" Compute the Mellin transform `F(s)` of `f(x)`, .. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x. For all "sensible" functions, this converges absolutely in a strip `a < \operatorname{Re}(s) < b`. Explanation =========== The Mellin transform is related via change of variables to the Fourier transform, and also to the (bilateral) Laplace transform. This function returns ``(F, (a, b), cond)`` where ``F`` is the Mellin transform of ``f``, ``(a, b)`` is the fundamental strip (as above), and ``cond`` are auxiliary convergence conditions. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`MellinTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``, then only `F` will be returned (i.e. not ``cond``, and also not the strip ``(a, b)``). Examples ======== >>> from sympy.integrals.transforms import mellin_transform >>> from sympy import exp >>> from sympy.abc import x, s >>> mellin_transform(exp(-x), x, s) (gamma(s), (0, oo), True) See Also ======== inverse_mellin_transform, laplace_transform, fourier_transform hankel_transform, inverse_hankel_transform """ return MellinTransform(f, x, s).doit(**hints) def _rewrite_sin(m_n, s, a, b): """ Re-write the sine function ``sin(m*s + n)`` as gamma functions, compatible with the strip (a, b). Return ``(gamma1, gamma2, fac)`` so that ``f == fac/(gamma1 * gamma2)``. Examples ======== >>> from sympy.integrals.transforms import _rewrite_sin >>> from sympy import pi, S >>> from sympy.abc import s >>> _rewrite_sin((pi, 0), s, 0, 1) (gamma(s), gamma(1 - s), pi) >>> _rewrite_sin((pi, 0), s, 1, 0) (gamma(s - 1), gamma(2 - s), -pi) >>> _rewrite_sin((pi, 0), s, -1, 0) (gamma(s + 1), gamma(-s), -pi) >>> _rewrite_sin((pi, pi/2), s, S(1)/2, S(3)/2) (gamma(s - 1/2), gamma(3/2 - s), -pi) >>> _rewrite_sin((pi, pi), s, 0, 1) (gamma(s), gamma(1 - s), -pi) >>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2) (gamma(2*s), gamma(1 - 2*s), pi) >>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1) (gamma(2*s - 1), gamma(2 - 2*s), -pi) """ # (This is a separate function because it is moderately complicated, # and I want to doctest it.) # We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x). # But there is one comlication: the gamma functions determine the # inegration contour in the definition of the G-function. Usually # it would not matter if this is slightly shifted, unless this way # we create an undefined function! # So we try to write this in such a way that the gammas are # eminently on the right side of the strip. from sympy import expand_mul, pi, ceiling, gamma m, n = m_n m = expand_mul(m/pi) n = expand_mul(n/pi) r = ceiling(-m*a - n.as_real_imag()[0]) # Don't use re(n), does not expand return gamma(m*s + n + r), gamma(1 - n - r - m*s), (-1)**r*pi class MellinTransformStripError(ValueError): """ Exception raised by _rewrite_gamma. Mainly for internal use. """ pass def _rewrite_gamma(f, s, a, b): """ Try to rewrite the product f(s) as a product of gamma functions, so that the inverse Mellin transform of f can be expressed as a meijer G function. Explanation =========== Return (an, ap), (bm, bq), arg, exp, fac such that G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s). Raises IntegralTransformError or MellinTransformStripError on failure. It is asserted that f has no poles in the fundamental strip designated by (a, b). One of a and b is allowed to be None. The fundamental strip is important, because it determines the inversion contour. This function can handle exponentials, linear factors, trigonometric functions. This is a helper function for inverse_mellin_transform that will not attempt any transformations on f. Examples ======== >>> from sympy.integrals.transforms import _rewrite_gamma >>> from sympy.abc import s >>> from sympy import oo >>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo) (([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1) >>> _rewrite_gamma((s-1)**2, s, -oo, oo) (([], [1, 1]), ([2, 2], []), 1, 1, 1) Importance of the fundamental strip: >>> _rewrite_gamma(1/s, s, 0, oo) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, None, oo) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, 0, None) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, -oo, 0) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(1/s, s, None, 0) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(1/s, s, -oo, None) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(2**(-s+3), s, -oo, oo) (([], []), ([], []), 1/2, 1, 8) """ from itertools import repeat from sympy import (Poly, gamma, Mul, re, CRootOf, exp as exp_, expand, roots, ilcm, pi, sin, cos, tan, cot, igcd, exp_polar) # Our strategy will be as follows: # 1) Guess a constant c such that the inversion integral should be # performed wrt s'=c*s (instead of plain s). Write s for s'. # 2) Process all factors, rewrite them independently as gamma functions in # argument s, or exponentials of s. # 3) Try to transform all gamma functions s.t. they have argument # a+s or a-s. # 4) Check that the resulting G function parameters are valid. # 5) Combine all the exponentials. a_, b_ = S([a, b]) def left(c, is_numer): """ Decide whether pole at c lies to the left of the fundamental strip. """ # heuristically, this is the best chance for us to solve the inequalities c = expand(re(c)) if a_ is None and b_ is oo: return True if a_ is None: return c < b_ if b_ is None: return c <= a_ if (c >= b_) == True: return False if (c <= a_) == True: return True if is_numer: return None if a_.free_symbols or b_.free_symbols or c.free_symbols: return None # XXX #raise IntegralTransformError('Inverse Mellin', f, # 'Could not determine position of singularity %s' # ' relative to fundamental strip' % c) raise MellinTransformStripError('Pole inside critical strip?') # 1) s_multipliers = [] for g in f.atoms(gamma): if not g.has(s): continue arg = g.args[0] if arg.is_Add: arg = arg.as_independent(s)[1] coeff, _ = arg.as_coeff_mul(s) s_multipliers += [coeff] for g in f.atoms(sin, cos, tan, cot): if not g.has(s): continue arg = g.args[0] if arg.is_Add: arg = arg.as_independent(s)[1] coeff, _ = arg.as_coeff_mul(s) s_multipliers += [coeff/pi] s_multipliers = [abs(x) if x.is_extended_real else x for x in s_multipliers] common_coefficient = S.One for x in s_multipliers: if not x.is_Rational: common_coefficient = x break s_multipliers = [x/common_coefficient for x in s_multipliers] if (any(not x.is_Rational for x in s_multipliers) or not common_coefficient.is_extended_real): raise IntegralTransformError("Gamma", None, "Nonrational multiplier") s_multiplier = common_coefficient/reduce(ilcm, [S(x.q) for x in s_multipliers], S.One) if s_multiplier == common_coefficient: if len(s_multipliers) == 0: s_multiplier = common_coefficient else: s_multiplier = common_coefficient \ *reduce(igcd, [S(x.p) for x in s_multipliers]) f = f.subs(s, s/s_multiplier) fac = S.One/s_multiplier exponent = S.One/s_multiplier if a_ is not None: a_ *= s_multiplier if b_ is not None: b_ *= s_multiplier # 2) numer, denom = f.as_numer_denom() numer = Mul.make_args(numer) denom = Mul.make_args(denom) args = list(zip(numer, repeat(True))) + list(zip(denom, repeat(False))) facs = [] dfacs = [] # *_gammas will contain pairs (a, c) representing Gamma(a*s + c) numer_gammas = [] denom_gammas = [] # exponentials will contain bases for exponentials of s exponentials = [] def exception(fact): return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact) while args: fact, is_numer = args.pop() if is_numer: ugammas, lgammas = numer_gammas, denom_gammas ufacs = facs else: ugammas, lgammas = denom_gammas, numer_gammas ufacs = dfacs def linear_arg(arg): """ Test if arg is of form a*s+b, raise exception if not. """ if not arg.is_polynomial(s): raise exception(fact) p = Poly(arg, s) if p.degree() != 1: raise exception(fact) return p.all_coeffs() # constants if not fact.has(s): ufacs += [fact] # exponentials elif fact.is_Pow or isinstance(fact, exp_): if fact.is_Pow: base = fact.base exp = fact.exp else: base = exp_polar(1) exp = fact.args[0] if exp.is_Integer: cond = is_numer if exp < 0: cond = not cond args += [(base, cond)]*abs(exp) continue elif not base.has(s): a, b = linear_arg(exp) if not is_numer: base = 1/base exponentials += [base**a] facs += [base**b] else: raise exception(fact) # linear factors elif fact.is_polynomial(s): p = Poly(fact, s) if p.degree() != 1: # We completely factor the poly. For this we need the roots. # Now roots() only works in some cases (low degree), and CRootOf # only works without parameters. So try both... coeff = p.LT()[1] rs = roots(p, s) if len(rs) != p.degree(): rs = CRootOf.all_roots(p) ufacs += [coeff] args += [(s - c, is_numer) for c in rs] continue a, c = p.all_coeffs() ufacs += [a] c /= -a # Now need to convert s - c if left(c, is_numer): ugammas += [(S.One, -c + 1)] lgammas += [(S.One, -c)] else: ufacs += [-1] ugammas += [(S.NegativeOne, c + 1)] lgammas += [(S.NegativeOne, c)] elif isinstance(fact, gamma): a, b = linear_arg(fact.args[0]) if is_numer: if (a > 0 and (left(-b/a, is_numer) == False)) or \ (a < 0 and (left(-b/a, is_numer) == True)): raise NotImplementedError( 'Gammas partially over the strip.') ugammas += [(a, b)] elif isinstance(fact, sin): # We try to re-write all trigs as gammas. This is not in # general the best strategy, since sometimes this is impossible, # but rewriting as exponentials would work. However trig functions # in inverse mellin transforms usually all come from simplifying # gamma terms, so this should work. a = fact.args[0] if is_numer: # No problem with the poles. gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi else: gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_) args += [(gamma1, not is_numer), (gamma2, not is_numer)] ufacs += [fac_] elif isinstance(fact, tan): a = fact.args[0] args += [(sin(a, evaluate=False), is_numer), (sin(pi/2 - a, evaluate=False), not is_numer)] elif isinstance(fact, cos): a = fact.args[0] args += [(sin(pi/2 - a, evaluate=False), is_numer)] elif isinstance(fact, cot): a = fact.args[0] args += [(sin(pi/2 - a, evaluate=False), is_numer), (sin(a, evaluate=False), not is_numer)] else: raise exception(fact) fac *= Mul(*facs)/Mul(*dfacs) # 3) an, ap, bm, bq = [], [], [], [] for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True), (denom_gammas, bq, ap, False)]: while gammas: a, c = gammas.pop() if a != -1 and a != +1: # We use the gamma function multiplication theorem. p = abs(S(a)) newa = a/p newc = c/p if not a.is_Integer: raise TypeError("a is not an integer") for k in range(p): gammas += [(newa, newc + k/p)] if is_numer: fac *= (2*pi)**((1 - p)/2) * p**(c - S.Half) exponentials += [p**a] else: fac /= (2*pi)**((1 - p)/2) * p**(c - S.Half) exponentials += [p**(-a)] continue if a == +1: plus.append(1 - c) else: minus.append(c) # 4) # TODO # 5) arg = Mul(*exponentials) # for testability, sort the arguments an.sort(key=default_sort_key) ap.sort(key=default_sort_key) bm.sort(key=default_sort_key) bq.sort(key=default_sort_key) return (an, ap), (bm, bq), arg, exponent, fac @_noconds_(True) def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False): """ A helper for the real inverse_mellin_transform function, this one here assumes x to be real and positive. """ from sympy import (expand, expand_mul, hyperexpand, meijerg, arg, pi, re, factor, Heaviside, gamma, Add) x = _dummy('t', 'inverse-mellin-transform', F, positive=True) # Actually, we won't try integration at all. Instead we use the definition # of the Meijer G function as a fairly general inverse mellin transform. F = F.rewrite(gamma) for g in [factor(F), expand_mul(F), expand(F)]: if g.is_Add: # do all terms separately ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg, noconds=False) for G in g.args] conds = [p[1] for p in ress] ress = [p[0] for p in ress] res = Add(*ress) if not as_meijerg: res = factor(res, gens=res.atoms(Heaviside)) return res.subs(x, x_), And(*conds) try: a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1]) except IntegralTransformError: continue try: G = meijerg(a, b, C/x**e) except ValueError: continue if as_meijerg: h = G else: try: h = hyperexpand(G) except NotImplementedError: raise IntegralTransformError( 'Inverse Mellin', F, 'Could not calculate integral') if h.is_Piecewise and len(h.args) == 3: # XXX we break modularity here! h = Heaviside(x - abs(C))*h.args[0].args[0] \ + Heaviside(abs(C) - x)*h.args[1].args[0] # We must ensure that the integral along the line we want converges, # and return that value. # See [L], 5.2 cond = [abs(arg(G.argument)) < G.delta*pi] # Note: we allow ">=" here, this corresponds to convergence if we let # limits go to oo symmetrically. ">" corresponds to absolute convergence. cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1), abs(arg(G.argument)) == G.delta*pi)] cond = Or(*cond) if cond == False: raise IntegralTransformError( 'Inverse Mellin', F, 'does not converge') return (h*fac).subs(x, x_), cond raise IntegralTransformError('Inverse Mellin', F, '') _allowed = None class InverseMellinTransform(IntegralTransform): """ Class representing unevaluated inverse Mellin transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Mellin transforms, see the :func:`inverse_mellin_transform` docstring. """ _name = 'Inverse Mellin' _none_sentinel = Dummy('None') _c = Dummy('c') def __new__(cls, F, s, x, a, b, **opts): if a is None: a = InverseMellinTransform._none_sentinel if b is None: b = InverseMellinTransform._none_sentinel return IntegralTransform.__new__(cls, F, s, x, a, b, **opts) @property def fundamental_strip(self): a, b = self.args[3], self.args[4] if a is InverseMellinTransform._none_sentinel: a = None if b is InverseMellinTransform._none_sentinel: b = None return a, b def _compute_transform(self, F, s, x, **hints): from sympy import postorder_traversal global _allowed if _allowed is None: from sympy import ( exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh, coth, factorial, rf) _allowed = { exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh, coth, factorial, rf} for f in postorder_traversal(F): if f.is_Function and f.has(s) and f.func not in _allowed: raise IntegralTransformError('Inverse Mellin', F, 'Component %s not recognised.' % f) strip = self.fundamental_strip return _inverse_mellin_transform(F, s, x, strip, **hints) def _as_integral(self, F, s, x): from sympy import I c = self.__class__._c return Integral(F*x**(-s), (s, c - I*oo, c + I*oo))/(2*S.Pi*S.ImaginaryUnit) def inverse_mellin_transform(F, s, x, strip, **hints): r""" Compute the inverse Mellin transform of `F(s)` over the fundamental strip given by ``strip=(a, b)``. Explanation =========== This can be defined as .. math:: f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s, for any `c` in the fundamental strip. Under certain regularity conditions on `F` and/or `f`, this recovers `f` from its Mellin transform `F` (and vice versa), for positive real `x`. One of `a` or `b` may be passed as ``None``; a suitable `c` will be inferred. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseMellinTransform` object. Note that this function will assume x to be positive and real, regardless of the sympy assumptions! For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy.integrals.transforms import inverse_mellin_transform >>> from sympy import oo, gamma >>> from sympy.abc import x, s >>> inverse_mellin_transform(gamma(s), s, x, (0, oo)) exp(-x) The fundamental strip matters: >>> f = 1/(s**2 - 1) >>> inverse_mellin_transform(f, s, x, (-oo, -1)) x*(1 - 1/x**2)*Heaviside(x - 1)/2 >>> inverse_mellin_transform(f, s, x, (-1, 1)) -x*Heaviside(1 - x)/2 - Heaviside(x - 1)/(2*x) >>> inverse_mellin_transform(f, s, x, (1, oo)) (1/2 - x**2/2)*Heaviside(1 - x)/x See Also ======== mellin_transform hankel_transform, inverse_hankel_transform """ return InverseMellinTransform(F, s, x, strip[0], strip[1]).doit(**hints) ########################################################################## # Laplace Transform ########################################################################## def _simplifyconds(expr, s, a): r""" Naively simplify some conditions occurring in ``expr``, given that `\operatorname{Re}(s) > a`. Examples ======== >>> from sympy.integrals.transforms import _simplifyconds as simp >>> from sympy.abc import x >>> from sympy import sympify as S >>> simp(abs(x**2) < 1, x, 1) False >>> simp(abs(x**2) < 1, x, 2) False >>> simp(abs(x**2) < 1, x, 0) Abs(x**2) < 1 >>> simp(abs(1/x**2) < 1, x, 1) True >>> simp(S(1) < abs(x), x, 1) True >>> simp(S(1) < abs(1/x), x, 1) False >>> from sympy import Ne >>> simp(Ne(1, x**3), x, 1) True >>> simp(Ne(1, x**3), x, 2) True >>> simp(Ne(1, x**3), x, 0) Ne(1, x**3) """ from sympy.core.relational import ( StrictGreaterThan, StrictLessThan, Unequality ) from sympy import Abs def power(ex): if ex == s: return 1 if ex.is_Pow and ex.base == s: return ex.exp return None def bigger(ex1, ex2): """ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. """ if ex1.has(s) and ex2.has(s): return None if isinstance(ex1, Abs): ex1 = ex1.args[0] if isinstance(ex2, Abs): ex2 = ex2.args[0] if ex1.has(s): return bigger(1/ex2, 1/ex1) n = power(ex2) if n is None: return None try: if n > 0 and (abs(ex1) <= abs(a)**n) == True: return False if n < 0 and (abs(ex1) >= abs(a)**n) == True: return True except TypeError: pass def replie(x, y): """ simplify x < y """ if not (x.is_positive or isinstance(x, Abs)) \ or not (y.is_positive or isinstance(y, Abs)): return (x < y) r = bigger(x, y) if r is not None: return not r return (x < y) def replue(x, y): b = bigger(x, y) if b == True or b == False: return True return Unequality(x, y) def repl(ex, *args): if ex == True or ex == False: return bool(ex) return ex.replace(*args) from sympy.simplify.radsimp import collect_abs expr = collect_abs(expr) expr = repl(expr, StrictLessThan, replie) expr = repl(expr, StrictGreaterThan, lambda x, y: replie(y, x)) expr = repl(expr, Unequality, replue) return S(expr) @_noconds def _laplace_transform(f, t, s_, simplify=True): """ The backend function for Laplace transforms. """ from sympy import (re, Max, exp, pi, Min, periodic_argument as arg_, arg, cos, Wild, symbols, polar_lift) s = Dummy('s') F = integrate(exp(-s*t) * f, (t, 0, oo)) if not F.has(Integral): return _simplify(F.subs(s, s_), simplify), -oo, S.true if not F.is_Piecewise: raise IntegralTransformError( 'Laplace', f, 'could not compute integral') F, cond = F.args[0] if F.has(Integral): raise IntegralTransformError( 'Laplace', f, 'integral in unexpected form') def process_conds(conds): """ Turn ``conds`` into a strip and auxiliary conditions. """ a = -oo aux = S.true conds = conjuncts(to_cnf(conds)) p, q, w1, w2, w3, w4, w5 = symbols( 'p q w1 w2 w3 w4 w5', cls=Wild, exclude=[s]) patterns = ( p*abs(arg((s + w3)*q)) < w2, p*abs(arg((s + w3)*q)) <= w2, abs(arg_((s + w3)**p*q, w1)) < w2, abs(arg_((s + w3)**p*q, w1)) <= w2, abs(arg_((polar_lift(s + w3))**p*q, w1)) < w2, abs(arg_((polar_lift(s + w3))**p*q, w1)) <= w2) for c in conds: a_ = oo aux_ = [] for d in disjuncts(c): if d.is_Relational and s in d.rhs.free_symbols: d = d.reversed if d.is_Relational and isinstance(d, (Ge, Gt)): d = d.reversedsign for pat in patterns: m = d.match(pat) if m: break if m: if m[q].is_positive and m[w2]/m[p] == pi/2: d = -re(s + m[w3]) < 0 m = d.match(p - cos(w1*abs(arg(s*w5))*w2)*abs(s**w3)**w4 < 0) if not m: m = d.match( cos(p - abs(arg_(s**w1*w5, q))*w2)*abs(s**w3)**w4 < 0) if not m: m = d.match( p - cos(abs(arg_(polar_lift(s)**w1*w5, q))*w2 )*abs(s**w3)**w4 < 0) if m and all(m[wild].is_positive for wild in [w1, w2, w3, w4, w5]): d = re(s) > m[p] d_ = d.replace( re, lambda x: x.expand().as_real_imag()[0]).subs(re(s), t) if not d.is_Relational or \ d.rel_op in ('==', '!=') \ or d_.has(s) or not d_.has(t): aux_ += [d] continue soln = _solve_inequality(d_, t) if not soln.is_Relational or \ soln.rel_op in ('==', '!='): aux_ += [d] continue if soln.lts == t: raise IntegralTransformError('Laplace', f, 'convergence not in half-plane?') else: a_ = Min(soln.lts, a_) if a_ != oo: a = Max(a_, a) else: aux = And(aux, Or(*aux_)) return a, aux conds = [process_conds(c) for c in disjuncts(cond)] conds2 = [x for x in conds if x[1] != False and x[0] != -oo] if not conds2: conds2 = [x for x in conds if x[1] != False] conds = conds2 def cnt(expr): if expr == True or expr == False: return 0 return expr.count_ops() conds.sort(key=lambda x: (-x[0], cnt(x[1]))) if not conds: raise IntegralTransformError('Laplace', f, 'no convergence found') a, aux = conds[0] def sbs(expr): return expr.subs(s, s_) if simplify: F = _simplifyconds(F, s, a) aux = _simplifyconds(aux, s, a) return _simplify(F.subs(s, s_), simplify), sbs(a), _canonical(sbs(aux)) class LaplaceTransform(IntegralTransform): """ Class representing unevaluated Laplace transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Laplace transforms, see the :func:`laplace_transform` docstring. """ _name = 'Laplace' def _compute_transform(self, f, t, s, **hints): return _laplace_transform(f, t, s, **hints) def _as_integral(self, f, t, s): from sympy import exp return Integral(f*exp(-s*t), (t, 0, oo)) def _collapse_extra(self, extra): from sympy import Max conds = [] planes = [] for plane, cond in extra: conds.append(cond) planes.append(plane) cond = And(*conds) plane = Max(*planes) if cond == False: raise IntegralTransformError( 'Laplace', None, 'No combined convergence.') return plane, cond def laplace_transform(f, t, s, **hints): r""" Compute the Laplace Transform `F(s)` of `f(t)`, .. math :: F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t. Explanation =========== For all "sensible" functions, this converges absolutely in a half plane `a < \operatorname{Re}(s)`. This function returns ``(F, a, cond)`` where ``F`` is the Laplace transform of ``f``, `\operatorname{Re}(s) > a` is the half-plane of convergence, and ``cond`` are auxiliary convergence conditions. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`LaplaceTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``, only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``). Examples ======== >>> from sympy.integrals import laplace_transform >>> from sympy.abc import t, s, a >>> laplace_transform(t**a, t, s) (s**(-a)*gamma(a + 1)/s, 0, re(a) > -1) See Also ======== inverse_laplace_transform, mellin_transform, fourier_transform hankel_transform, inverse_hankel_transform """ if isinstance(f, MatrixBase) and hasattr(f, 'applyfunc'): return f.applyfunc(lambda fij: laplace_transform(fij, t, s, **hints)) return LaplaceTransform(f, t, s).doit(**hints) @_noconds_(True) def _inverse_laplace_transform(F, s, t_, plane, simplify=True): """ The backend function for inverse Laplace transforms. """ from sympy import exp, Heaviside, log, expand_complex, Integral, Piecewise from sympy.integrals.meijerint import meijerint_inversion, _get_coeff_exp # There are two strategies we can try: # 1) Use inverse mellin transforms - related by a simple change of variables. # 2) Use the inversion integral. t = Dummy('t', real=True) def pw_simp(*args): """ Simplify a piecewise expression from hyperexpand. """ # XXX we break modularity here! if len(args) != 3: return Piecewise(*args) arg = args[2].args[0].argument coeff, exponent = _get_coeff_exp(arg, t) e1 = args[0].args[0] e2 = args[1].args[0] return Heaviside(1/abs(coeff) - t**exponent)*e1 \ + Heaviside(t**exponent - 1/abs(coeff))*e2 try: f, cond = inverse_mellin_transform(F, s, exp(-t), (None, oo), needeval=True, noconds=False) except IntegralTransformError: f = None if f is None: f = meijerint_inversion(F, s, t) if f is None: raise IntegralTransformError('Inverse Laplace', f, '') if f.is_Piecewise: f, cond = f.args[0] if f.has(Integral): raise IntegralTransformError('Inverse Laplace', f, 'inversion integral of unrecognised form.') else: cond = S.true f = f.replace(Piecewise, pw_simp) if f.is_Piecewise: # many of the functions called below can't work with piecewise # (b/c it has a bool in args) return f.subs(t, t_), cond u = Dummy('u') def simp_heaviside(arg): a = arg.subs(exp(-t), u) if a.has(t): return Heaviside(arg) rel = _solve_inequality(a > 0, u) if rel.lts == u: k = log(rel.gts) return Heaviside(t + k) else: k = log(rel.lts) return Heaviside(-(t + k)) f = f.replace(Heaviside, simp_heaviside) def simp_exp(arg): return expand_complex(exp(arg)) f = f.replace(exp, simp_exp) # TODO it would be nice to fix cosh and sinh ... simplify messes these # exponentials up return _simplify(f.subs(t, t_), simplify), cond class InverseLaplaceTransform(IntegralTransform): """ Class representing unevaluated inverse Laplace transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Laplace transforms, see the :func:`inverse_laplace_transform` docstring. """ _name = 'Inverse Laplace' _none_sentinel = Dummy('None') _c = Dummy('c') def __new__(cls, F, s, x, plane, **opts): if plane is None: plane = InverseLaplaceTransform._none_sentinel return IntegralTransform.__new__(cls, F, s, x, plane, **opts) @property def fundamental_plane(self): plane = self.args[3] if plane is InverseLaplaceTransform._none_sentinel: plane = None return plane def _compute_transform(self, F, s, t, **hints): return _inverse_laplace_transform(F, s, t, self.fundamental_plane, **hints) def _as_integral(self, F, s, t): from sympy import I, exp c = self.__class__._c return Integral(exp(s*t)*F, (s, c - I*oo, c + I*oo))/(2*S.Pi*S.ImaginaryUnit) def inverse_laplace_transform(F, s, t, plane=None, **hints): r""" Compute the inverse Laplace transform of `F(s)`, defined as .. math :: f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s, for `c` so large that `F(s)` has no singularites in the half-plane `\operatorname{Re}(s) > c-\epsilon`. Explanation =========== The plane can be specified by argument ``plane``, but will be inferred if passed as None. Under certain regularity conditions, this recovers `f(t)` from its Laplace Transform `F(s)`, for non-negative `t`, and vice versa. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseLaplaceTransform` object. Note that this function will always assume `t` to be real, regardless of the sympy assumption on `t`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy.integrals.transforms import inverse_laplace_transform >>> from sympy import exp, Symbol >>> from sympy.abc import s, t >>> a = Symbol('a', positive=True) >>> inverse_laplace_transform(exp(-a*s)/s, s, t) Heaviside(-a + t) See Also ======== laplace_transform hankel_transform, inverse_hankel_transform """ if isinstance(F, MatrixBase) and hasattr(F, 'applyfunc'): return F.applyfunc(lambda Fij: inverse_laplace_transform(Fij, s, t, plane, **hints)) return InverseLaplaceTransform(F, s, t, plane).doit(**hints) ########################################################################## # Fourier Transform ########################################################################## @_noconds_(True) def _fourier_transform(f, x, k, a, b, name, simplify=True): r""" Compute a general Fourier-type transform .. math:: F(k) = a \int_{-\infty}^{\infty} e^{bixk} f(x)\, dx. For suitable choice of *a* and *b*, this reduces to the standard Fourier and inverse Fourier transforms. """ from sympy import exp, I F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo)) if not F.has(Integral): return _simplify(F, simplify), S.true integral_f = integrate(f, (x, -oo, oo)) if integral_f in (-oo, oo, S.NaN) or integral_f.has(Integral): raise IntegralTransformError(name, f, 'function not integrable on real axis') if not F.is_Piecewise: raise IntegralTransformError(name, f, 'could not compute integral') F, cond = F.args[0] if F.has(Integral): raise IntegralTransformError(name, f, 'integral in unexpected form') return _simplify(F, simplify), cond class FourierTypeTransform(IntegralTransform): """ Base class for Fourier transforms.""" def a(self): raise NotImplementedError( "Class %s must implement a(self) but does not" % self.__class__) def b(self): raise NotImplementedError( "Class %s must implement b(self) but does not" % self.__class__) def _compute_transform(self, f, x, k, **hints): return _fourier_transform(f, x, k, self.a(), self.b(), self.__class__._name, **hints) def _as_integral(self, f, x, k): from sympy import exp, I a = self.a() b = self.b() return Integral(a*f*exp(b*I*x*k), (x, -oo, oo)) class FourierTransform(FourierTypeTransform): """ Class representing unevaluated Fourier transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Fourier transforms, see the :func:`fourier_transform` docstring. """ _name = 'Fourier' def a(self): return 1 def b(self): return -2*S.Pi def fourier_transform(f, x, k, **hints): r""" Compute the unitary, ordinary-frequency Fourier transform of ``f``, defined as .. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`FourierTransform` object. For other Fourier transform conventions, see the function :func:`sympy.integrals.transforms._fourier_transform`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import fourier_transform, exp >>> from sympy.abc import x, k >>> fourier_transform(exp(-x**2), x, k) sqrt(pi)*exp(-pi**2*k**2) >>> fourier_transform(exp(-x**2), x, k, noconds=False) (sqrt(pi)*exp(-pi**2*k**2), True) See Also ======== inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return FourierTransform(f, x, k).doit(**hints) class InverseFourierTransform(FourierTypeTransform): """ Class representing unevaluated inverse Fourier transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Fourier transforms, see the :func:`inverse_fourier_transform` docstring. """ _name = 'Inverse Fourier' def a(self): return 1 def b(self): return 2*S.Pi def inverse_fourier_transform(F, k, x, **hints): r""" Compute the unitary, ordinary-frequency inverse Fourier transform of `F`, defined as .. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseFourierTransform` object. For other Fourier transform conventions, see the function :func:`sympy.integrals.transforms._fourier_transform`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_fourier_transform, exp, sqrt, pi >>> from sympy.abc import x, k >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x) exp(-x**2) >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False) (exp(-x**2), True) See Also ======== fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return InverseFourierTransform(F, k, x).doit(**hints) ########################################################################## # Fourier Sine and Cosine Transform ########################################################################## from sympy import sin, cos, sqrt, pi @_noconds_(True) def _sine_cosine_transform(f, x, k, a, b, K, name, simplify=True): """ Compute a general sine or cosine-type transform F(k) = a int_0^oo b*sin(x*k) f(x) dx. F(k) = a int_0^oo b*cos(x*k) f(x) dx. For suitable choice of a and b, this reduces to the standard sine/cosine and inverse sine/cosine transforms. """ F = integrate(a*f*K(b*x*k), (x, 0, oo)) if not F.has(Integral): return _simplify(F, simplify), S.true if not F.is_Piecewise: raise IntegralTransformError(name, f, 'could not compute integral') F, cond = F.args[0] if F.has(Integral): raise IntegralTransformError(name, f, 'integral in unexpected form') return _simplify(F, simplify), cond class SineCosineTypeTransform(IntegralTransform): """ Base class for sine and cosine transforms. Specify cls._kern. """ def a(self): raise NotImplementedError( "Class %s must implement a(self) but does not" % self.__class__) def b(self): raise NotImplementedError( "Class %s must implement b(self) but does not" % self.__class__) def _compute_transform(self, f, x, k, **hints): return _sine_cosine_transform(f, x, k, self.a(), self.b(), self.__class__._kern, self.__class__._name, **hints) def _as_integral(self, f, x, k): a = self.a() b = self.b() K = self.__class__._kern return Integral(a*f*K(b*x*k), (x, 0, oo)) class SineTransform(SineCosineTypeTransform): """ Class representing unevaluated sine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute sine transforms, see the :func:`sine_transform` docstring. """ _name = 'Sine' _kern = sin def a(self): return sqrt(2)/sqrt(pi) def b(self): return 1 def sine_transform(f, x, k, **hints): r""" Compute the unitary, ordinary-frequency sine transform of `f`, defined as .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`SineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import sine_transform, exp >>> from sympy.abc import x, k, a >>> sine_transform(x*exp(-a*x**2), x, k) sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2)) >>> sine_transform(x**(-a), x, k) 2**(1/2 - a)*k**(a - 1)*gamma(1 - a/2)/gamma(a/2 + 1/2) See Also ======== fourier_transform, inverse_fourier_transform inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return SineTransform(f, x, k).doit(**hints) class InverseSineTransform(SineCosineTypeTransform): """ Class representing unevaluated inverse sine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse sine transforms, see the :func:`inverse_sine_transform` docstring. """ _name = 'Inverse Sine' _kern = sin def a(self): return sqrt(2)/sqrt(pi) def b(self): return 1 def inverse_sine_transform(F, k, x, **hints): r""" Compute the unitary, ordinary-frequency inverse sine transform of `F`, defined as .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseSineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_sine_transform, exp, sqrt, gamma >>> from sympy.abc import x, k, a >>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)* ... gamma(-a/2 + 1)/gamma((a+1)/2), k, x) x**(-a) >>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x) x*exp(-a*x**2) See Also ======== fourier_transform, inverse_fourier_transform sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return InverseSineTransform(F, k, x).doit(**hints) class CosineTransform(SineCosineTypeTransform): """ Class representing unevaluated cosine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute cosine transforms, see the :func:`cosine_transform` docstring. """ _name = 'Cosine' _kern = cos def a(self): return sqrt(2)/sqrt(pi) def b(self): return 1 def cosine_transform(f, x, k, **hints): r""" Compute the unitary, ordinary-frequency cosine transform of `f`, defined as .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`CosineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import cosine_transform, exp, sqrt, cos >>> from sympy.abc import x, k, a >>> cosine_transform(exp(-a*x), x, k) sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)) >>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k) a*exp(-a**2/(2*k))/(2*k**(3/2)) See Also ======== fourier_transform, inverse_fourier_transform, sine_transform, inverse_sine_transform inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return CosineTransform(f, x, k).doit(**hints) class InverseCosineTransform(SineCosineTypeTransform): """ Class representing unevaluated inverse cosine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse cosine transforms, see the :func:`inverse_cosine_transform` docstring. """ _name = 'Inverse Cosine' _kern = cos def a(self): return sqrt(2)/sqrt(pi) def b(self): return 1 def inverse_cosine_transform(F, k, x, **hints): r""" Compute the unitary, ordinary-frequency inverse cosine transform of `F`, defined as .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseCosineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_cosine_transform, sqrt, pi >>> from sympy.abc import x, k, a >>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x) exp(-a*x) >>> inverse_cosine_transform(1/sqrt(k), k, x) 1/sqrt(x) See Also ======== fourier_transform, inverse_fourier_transform, sine_transform, inverse_sine_transform cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform """ return InverseCosineTransform(F, k, x).doit(**hints) ########################################################################## # Hankel Transform ########################################################################## @_noconds_(True) def _hankel_transform(f, r, k, nu, name, simplify=True): r""" Compute a general Hankel transform .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r. """ from sympy import besselj F = integrate(f*besselj(nu, k*r)*r, (r, 0, oo)) if not F.has(Integral): return _simplify(F, simplify), S.true if not F.is_Piecewise: raise IntegralTransformError(name, f, 'could not compute integral') F, cond = F.args[0] if F.has(Integral): raise IntegralTransformError(name, f, 'integral in unexpected form') return _simplify(F, simplify), cond class HankelTypeTransform(IntegralTransform): """ Base class for Hankel transforms. """ def doit(self, **hints): return self._compute_transform(self.function, self.function_variable, self.transform_variable, self.args[3], **hints) def _compute_transform(self, f, r, k, nu, **hints): return _hankel_transform(f, r, k, nu, self._name, **hints) def _as_integral(self, f, r, k, nu): from sympy import besselj return Integral(f*besselj(nu, k*r)*r, (r, 0, oo)) @property def as_integral(self): return self._as_integral(self.function, self.function_variable, self.transform_variable, self.args[3]) class HankelTransform(HankelTypeTransform): """ Class representing unevaluated Hankel transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Hankel transforms, see the :func:`hankel_transform` docstring. """ _name = 'Hankel' def hankel_transform(f, r, k, nu, **hints): r""" Compute the Hankel transform of `f`, defined as .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`HankelTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import hankel_transform, inverse_hankel_transform >>> from sympy import exp >>> from sympy.abc import r, k, m, nu, a >>> ht = hankel_transform(1/r**m, r, k, nu) >>> ht 2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2) >>> inverse_hankel_transform(ht, k, r, nu) r**(-m) >>> ht = hankel_transform(exp(-a*r), r, k, 0) >>> ht a/(k**3*(a**2/k**2 + 1)**(3/2)) >>> inverse_hankel_transform(ht, k, r, 0) exp(-a*r) See Also ======== fourier_transform, inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform inverse_hankel_transform mellin_transform, laplace_transform """ return HankelTransform(f, r, k, nu).doit(**hints) class InverseHankelTransform(HankelTypeTransform): """ Class representing unevaluated inverse Hankel transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Hankel transforms, see the :func:`inverse_hankel_transform` docstring. """ _name = 'Inverse Hankel' def inverse_hankel_transform(F, k, r, nu, **hints): r""" Compute the inverse Hankel transform of `F` defined as .. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseHankelTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import hankel_transform, inverse_hankel_transform >>> from sympy import exp >>> from sympy.abc import r, k, m, nu, a >>> ht = hankel_transform(1/r**m, r, k, nu) >>> ht 2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2) >>> inverse_hankel_transform(ht, k, r, nu) r**(-m) >>> ht = hankel_transform(exp(-a*r), r, k, 0) >>> ht a/(k**3*(a**2/k**2 + 1)**(3/2)) >>> inverse_hankel_transform(ht, k, r, 0) exp(-a*r) See Also ======== fourier_transform, inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform mellin_transform, laplace_transform """ return InverseHankelTransform(F, k, r, nu).doit(**hints)
35745b8be8949859fc93d18480bc44bd9898f9d9535d7d8abad8e5978de7f5d7
from sympy.core import S, Dummy, pi from sympy.functions.combinatorial.factorials import factorial from sympy.functions.elementary.trigonometric import sin, cos from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.special.gamma_functions import gamma from sympy.polys.orthopolys import (legendre_poly, laguerre_poly, hermite_poly, jacobi_poly) from sympy.polys.rootoftools import RootOf def gauss_legendre(n, n_digits): r""" Computes the Gauss-Legendre quadrature [1]_ points and weights. Explanation =========== The Gauss-Legendre quadrature approximates the integral: .. math:: \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `P_n` and the weights `w_i` are given by: .. math:: w_i = \frac{2}{\left(1-x_i^2\right) \left(P'_n(x_i)\right)^2} Parameters ========== n : The order of quadrature. n_digits : Number of significant digits of the points and weights to return. Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_legendre >>> x, w = gauss_legendre(3, 5) >>> x [-0.7746, 0, 0.7746] >>> w [0.55556, 0.88889, 0.55556] >>> x, w = gauss_legendre(4, 5) >>> x [-0.86114, -0.33998, 0.33998, 0.86114] >>> w [0.34785, 0.65215, 0.65215, 0.34785] See Also ======== gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule/legendre_rule.html """ x = Dummy("x") p = legendre_poly(n, x, polys=True) pd = p.diff(x) xi = [] w = [] for r in p.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append((2/((1-r**2) * pd.subs(x, r)**2)).n(n_digits)) return xi, w def gauss_laguerre(n, n_digits): r""" Computes the Gauss-Laguerre quadrature [1]_ points and weights. Explanation =========== The Gauss-Laguerre quadrature approximates the integral: .. math:: \int_0^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `L_n` and the weights `w_i` are given by: .. math:: w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2} Parameters ========== n : The order of quadrature. n_digits : Number of significant digits of the points and weights to return. Returns ======= (x, w) : The ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_laguerre >>> x, w = gauss_laguerre(3, 5) >>> x [0.41577, 2.2943, 6.2899] >>> w [0.71109, 0.27852, 0.010389] >>> x, w = gauss_laguerre(6, 5) >>> x [0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983] >>> w [0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7] See Also ======== gauss_legendre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/laguerre_rule/laguerre_rule.html """ x = Dummy("x") p = laguerre_poly(n, x, polys=True) p1 = laguerre_poly(n+1, x, polys=True) xi = [] w = [] for r in p.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append((r/((n+1)**2 * p1.subs(x, r)**2)).n(n_digits)) return xi, w def gauss_hermite(n, n_digits): r""" Computes the Gauss-Hermite quadrature [1]_ points and weights. Explanation =========== The Gauss-Hermite quadrature approximates the integral: .. math:: \int_{-\infty}^{\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `H_n` and the weights `w_i` are given by: .. math:: w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 \left(H_{n-1}(x_i)\right)^2} Parameters ========== n : The order of quadrature. n_digits : Number of significant digits of the points and weights to return. Returns ======= (x, w) : The ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_hermite >>> x, w = gauss_hermite(3, 5) >>> x [-1.2247, 0, 1.2247] >>> w [0.29541, 1.1816, 0.29541] >>> x, w = gauss_hermite(6, 5) >>> x [-2.3506, -1.3358, -0.43608, 0.43608, 1.3358, 2.3506] >>> w [0.00453, 0.15707, 0.72463, 0.72463, 0.15707, 0.00453] See Also ======== gauss_legendre, gauss_laguerre, gauss_gen_laguerre, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Gauss-Hermite_Quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/hermite_rule/hermite_rule.html .. [3] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html """ x = Dummy("x") p = hermite_poly(n, x, polys=True) p1 = hermite_poly(n-1, x, polys=True) xi = [] w = [] for r in p.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append(((2**(n-1) * factorial(n) * sqrt(pi)) / (n**2 * p1.subs(x, r)**2)).n(n_digits)) return xi, w def gauss_gen_laguerre(n, alpha, n_digits): r""" Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights. Explanation =========== The generalized Gauss-Laguerre quadrature approximates the integral: .. math:: \int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `L^{\alpha}_n` and the weights `w_i` are given by: .. math:: w_i = \frac{\Gamma(\alpha+n)} {n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)} Parameters ========== n : The order of quadrature. alpha : The exponent of the singularity, `\alpha > -1`. n_digits : Number of significant digits of the points and weights to return. Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy import S >>> from sympy.integrals.quadrature import gauss_gen_laguerre >>> x, w = gauss_gen_laguerre(3, -S.Half, 5) >>> x [0.19016, 1.7845, 5.5253] >>> w [1.4493, 0.31413, 0.00906] >>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5) >>> x [0.97851, 2.9904, 6.3193, 11.712] >>> w [0.53087, 0.67721, 0.11895, 0.0023152] See Also ======== gauss_legendre, gauss_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html """ x = Dummy("x") p = laguerre_poly(n, x, alpha=alpha, polys=True) p1 = laguerre_poly(n-1, x, alpha=alpha, polys=True) p2 = laguerre_poly(n-1, x, alpha=alpha+1, polys=True) xi = [] w = [] for r in p.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append((gamma(alpha+n) / (n*gamma(n)*p1.subs(x, r)*p2.subs(x, r))).n(n_digits)) return xi, w def gauss_chebyshev_t(n, n_digits): r""" Computes the Gauss-Chebyshev quadrature [1]_ points and weights of the first kind. Explanation =========== The Gauss-Chebyshev quadrature of the first kind approximates the integral: .. math:: \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `T_n` and the weights `w_i` are given by: .. math:: w_i = \frac{\pi}{n} Parameters ========== n : The order of quadrature. n_digits : Number of significant digits of the points and weights to return. Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_chebyshev_t >>> x, w = gauss_chebyshev_t(3, 5) >>> x [0.86602, 0, -0.86602] >>> w [1.0472, 1.0472, 1.0472] >>> x, w = gauss_chebyshev_t(6, 5) >>> x [0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593] >>> w [0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236] See Also ======== gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html """ xi = [] w = [] for i in range(1, n+1): xi.append((cos((2*i-S.One)/(2*n)*S.Pi)).n(n_digits)) w.append((S.Pi/n).n(n_digits)) return xi, w def gauss_chebyshev_u(n, n_digits): r""" Computes the Gauss-Chebyshev quadrature [1]_ points and weights of the second kind. Explanation =========== The Gauss-Chebyshev quadrature of the second kind approximates the integral: .. math:: \int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `U_n` and the weights `w_i` are given by: .. math:: w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right) Parameters ========== n : the order of quadrature n_digits : number of significant digits of the points and weights to return Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_chebyshev_u >>> x, w = gauss_chebyshev_u(3, 5) >>> x [0.70711, 0, -0.70711] >>> w [0.3927, 0.7854, 0.3927] >>> x, w = gauss_chebyshev_u(6, 5) >>> x [0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097] >>> w [0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489] See Also ======== gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_jacobi, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html """ xi = [] w = [] for i in range(1, n+1): xi.append((cos(i/(n+S.One)*S.Pi)).n(n_digits)) w.append((S.Pi/(n+S.One)*sin(i*S.Pi/(n+S.One))**2).n(n_digits)) return xi, w def gauss_jacobi(n, alpha, beta, n_digits): r""" Computes the Gauss-Jacobi quadrature [1]_ points and weights. Explanation =========== The Gauss-Jacobi quadrature of the first kind approximates the integral: .. math:: \int_{-1}^1 (1-x)^\alpha (1+x)^\beta f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `P^{(\alpha,\beta)}_n` and the weights `w_i` are given by: .. math:: w_i = -\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)} {\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i) P^{(\alpha,\beta)}_{n+1}(x_i)} Parameters ========== n : the order of quadrature alpha : the first parameter of the Jacobi Polynomial, `\alpha > -1` beta : the second parameter of the Jacobi Polynomial, `\beta > -1` n_digits : number of significant digits of the points and weights to return Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy import S >>> from sympy.integrals.quadrature import gauss_jacobi >>> x, w = gauss_jacobi(3, S.Half, -S.Half, 5) >>> x [-0.90097, -0.22252, 0.62349] >>> w [1.7063, 1.0973, 0.33795] >>> x, w = gauss_jacobi(6, 1, 1, 5) >>> x [-0.87174, -0.5917, -0.2093, 0.2093, 0.5917, 0.87174] >>> w [0.050584, 0.22169, 0.39439, 0.39439, 0.22169, 0.050584] See Also ======== gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_chebyshev_u, gauss_lobatto References ========== .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/jacobi_rule/jacobi_rule.html .. [3] http://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html """ x = Dummy("x") p = jacobi_poly(n, alpha, beta, x, polys=True) pd = p.diff(x) pn = jacobi_poly(n+1, alpha, beta, x, polys=True) xi = [] w = [] for r in p.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append(( - (2*n+alpha+beta+2) / (n+alpha+beta+S.One) * (gamma(n+alpha+1)*gamma(n+beta+1)) / (gamma(n+alpha+beta+S.One)*gamma(n+2)) * 2**(alpha+beta) / (pd.subs(x, r) * pn.subs(x, r))).n(n_digits)) return xi, w def gauss_lobatto(n, n_digits): r""" Computes the Gauss-Lobatto quadrature [1]_ points and weights. Explanation =========== The Gauss-Lobatto quadrature approximates the integral: .. math:: \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) The nodes `x_i` of an order `n` quadrature rule are the roots of `P'_(n-1)` and the weights `w_i` are given by: .. math:: &w_i = \frac{2}{n(n-1) \left[P_{n-1}(x_i)\right]^2},\quad x\neq\pm 1\\ &w_i = \frac{2}{n(n-1)},\quad x=\pm 1 Parameters ========== n : the order of quadrature n_digits : number of significant digits of the points and weights to return Returns ======= (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats. The points `x_i` and weights `w_i` are returned as ``(x, w)`` tuple of lists. Examples ======== >>> from sympy.integrals.quadrature import gauss_lobatto >>> x, w = gauss_lobatto(3, 5) >>> x [-1, 0, 1] >>> w [0.33333, 1.3333, 0.33333] >>> x, w = gauss_lobatto(4, 5) >>> x [-1, -0.44721, 0.44721, 1] >>> w [0.16667, 0.83333, 0.83333, 0.16667] See Also ======== gauss_legendre,gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi References ========== .. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules .. [2] http://people.math.sfu.ca/~cbm/aands/page_888.htm """ x = Dummy("x") p = legendre_poly(n-1, x, polys=True) pd = p.diff(x) xi = [] w = [] for r in pd.real_roots(): if isinstance(r, RootOf): r = r.eval_rational(S.One/10**(n_digits+2)) xi.append(r.n(n_digits)) w.append((2/(n*(n-1) * p.subs(x, r)**2)).n(n_digits)) xi.insert(0, -1) xi.append(1) w.insert(0, (S(2)/(n*(n-1))).n(n_digits)) w.append((S(2)/(n*(n-1))).n(n_digits)) return xi, w
696db6b6b017255fbc1bc08c43ee546d4baedf56667a4159621f6227835bfed0
""" Module to implement integration of uni/bivariate polynomials over 2D Polytopes and uni/bi/trivariate polynomials over 3D Polytopes. Uses evaluation techniques as described in Chin et al. (2015) [1]. References =========== .. [1] Chin, Eric B., Jean B. Lasserre, and N. Sukumar. "Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra." Computational Mechanics 56.6 (2015): 967-981 PDF link : http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf """ from functools import cmp_to_key from sympy.abc import x, y, z from sympy.core import S, diff, Expr, Symbol from sympy.core.sympify import _sympify from sympy.geometry import Segment2D, Polygon, Point, Point2D from sympy.polys.polytools import LC, gcd_list, degree_list from sympy.simplify.simplify import nsimplify def polytope_integrate(poly, expr=None, *, clockwise=False, max_degree=None): """Integrates polynomials over 2/3-Polytopes. Explanation =========== This function accepts the polytope in ``poly`` and the function in ``expr`` (uni/bi/trivariate polynomials are implemented) and returns the exact integral of ``expr`` over ``poly``. Parameters ========== poly : The input Polygon. expr : The input polynomial. clockwise : Binary value to sort input points of 2-Polytope clockwise.(Optional) max_degree : The maximum degree of any monomial of the input polynomial.(Optional) Examples ======== >>> from sympy.abc import x, y >>> from sympy.geometry.polygon import Polygon >>> from sympy.geometry.point import Point >>> from sympy.integrals.intpoly import polytope_integrate >>> polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0)) >>> polys = [1, x, y, x*y, x**2*y, x*y**2] >>> expr = x*y >>> polytope_integrate(polygon, expr) 1/4 >>> polytope_integrate(polygon, polys, max_degree=3) {1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6} """ if clockwise: if isinstance(poly, Polygon): poly = Polygon(*point_sort(poly.vertices), evaluate=False) else: raise TypeError("clockwise=True works for only 2-Polytope" "V-representation input") if isinstance(poly, Polygon): # For Vertex Representation(2D case) hp_params = hyperplane_parameters(poly) facets = poly.sides elif len(poly[0]) == 2: # For Hyperplane Representation(2D case) plen = len(poly) if len(poly[0][0]) == 2: intersections = [intersection(poly[(i - 1) % plen], poly[i], "plane2D") for i in range(0, plen)] hp_params = poly lints = len(intersections) facets = [Segment2D(intersections[i], intersections[(i + 1) % lints]) for i in range(0, lints)] else: raise NotImplementedError("Integration for H-representation 3D" "case not implemented yet.") else: # For Vertex Representation(3D case) vertices = poly[0] facets = poly[1:] hp_params = hyperplane_parameters(facets, vertices) if max_degree is None: if expr is None: raise TypeError('Input expression be must' 'be a valid SymPy expression') return main_integrate3d(expr, facets, vertices, hp_params) if max_degree is not None: result = {} if not isinstance(expr, list) and expr is not None: raise TypeError('Input polynomials must be list of expressions') if len(hp_params[0][0]) == 3: result_dict = main_integrate3d(0, facets, vertices, hp_params, max_degree) else: result_dict = main_integrate(0, facets, hp_params, max_degree) if expr is None: return result_dict for poly in expr: poly = _sympify(poly) if poly not in result: if poly.is_zero: result[S.Zero] = S.Zero continue integral_value = S.Zero monoms = decompose(poly, separate=True) for monom in monoms: monom = nsimplify(monom) coeff, m = strip(monom) integral_value += result_dict[m] * coeff result[poly] = integral_value return result if expr is None: raise TypeError('Input expression be must' 'be a valid SymPy expression') return main_integrate(expr, facets, hp_params) def strip(monom): if monom.is_zero: return 0, 0 elif monom.is_number: return monom, 1 else: coeff = LC(monom) return coeff, S(monom) / coeff def main_integrate3d(expr, facets, vertices, hp_params, max_degree=None): """Function to translate the problem of integrating uni/bi/tri-variate polynomials over a 3-Polytope to integrating over its faces. This is done using Generalized Stokes' Theorem and Euler's Theorem. Parameters ========== expr : The input polynomial. facets : Faces of the 3-Polytope(expressed as indices of `vertices`). vertices : Vertices that constitute the Polytope. hp_params : Hyperplane Parameters of the facets. max_degree : optional Max degree of constituent monomial in given list of polynomial. Examples ======== >>> from sympy.integrals.intpoly import main_integrate3d, \ hyperplane_parameters >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\ (5, 0, 5), (5, 5, 0), (5, 5, 5)],\ [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\ [3, 1, 0, 2], [0, 4, 6, 2]] >>> vertices = cube[0] >>> faces = cube[1:] >>> hp_params = hyperplane_parameters(faces, vertices) >>> main_integrate3d(1, faces, vertices, hp_params) -125 """ result = {} dims = (x, y, z) dim_length = len(dims) if max_degree: grad_terms = gradient_terms(max_degree, 3) flat_list = [term for z_terms in grad_terms for x_term in z_terms for term in x_term] for term in flat_list: result[term[0]] = 0 for facet_count, hp in enumerate(hp_params): a, b = hp[0], hp[1] x0 = vertices[facets[facet_count][0]] for i, monom in enumerate(flat_list): # Every monomial is a tuple : # (term, x_degree, y_degree, z_degree, value over boundary) expr, x_d, y_d, z_d, z_index, y_index, x_index, _ = monom degree = x_d + y_d + z_d if b.is_zero: value_over_face = S.Zero else: value_over_face = \ integration_reduction_dynamic(facets, facet_count, a, b, expr, degree, dims, x_index, y_index, z_index, x0, grad_terms, i, vertices, hp) monom[7] = value_over_face result[expr] += value_over_face * \ (b / norm(a)) / (dim_length + x_d + y_d + z_d) return result else: integral_value = S.Zero polynomials = decompose(expr) for deg in polynomials: poly_contribute = S.Zero facet_count = 0 for i, facet in enumerate(facets): hp = hp_params[i] if hp[1].is_zero: continue pi = polygon_integrate(facet, hp, i, facets, vertices, expr, deg) poly_contribute += pi *\ (hp[1] / norm(tuple(hp[0]))) facet_count += 1 poly_contribute /= (dim_length + deg) integral_value += poly_contribute return integral_value def main_integrate(expr, facets, hp_params, max_degree=None): """Function to translate the problem of integrating univariate/bivariate polynomials over a 2-Polytope to integrating over its boundary facets. This is done using Generalized Stokes's Theorem and Euler's Theorem. Parameters ========== expr : The input polynomial. facets : Facets(Line Segments) of the 2-Polytope. hp_params : Hyperplane Parameters of the facets. max_degree : optional The maximum degree of any monomial of the input polynomial. >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import main_integrate,\ hyperplane_parameters >>> from sympy.geometry.polygon import Polygon >>> from sympy.geometry.point import Point >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> hp_params = hyperplane_parameters(triangle) >>> main_integrate(x**2 + y**2, facets, hp_params) 325/6 """ dims = (x, y) dim_length = len(dims) result = {} integral_value = S.Zero if max_degree: grad_terms = [[0, 0, 0, 0]] + gradient_terms(max_degree) for facet_count, hp in enumerate(hp_params): a, b = hp[0], hp[1] x0 = facets[facet_count].points[0] for i, monom in enumerate(grad_terms): # Every monomial is a tuple : # (term, x_degree, y_degree, value over boundary) m, x_d, y_d, _ = monom value = result.get(m, None) degree = S.Zero if b.is_zero: value_over_boundary = S.Zero else: degree = x_d + y_d value_over_boundary = \ integration_reduction_dynamic(facets, facet_count, a, b, m, degree, dims, x_d, y_d, max_degree, x0, grad_terms, i) monom[3] = value_over_boundary if value is not None: result[m] += value_over_boundary * \ (b / norm(a)) / (dim_length + degree) else: result[m] = value_over_boundary * \ (b / norm(a)) / (dim_length + degree) return result else: polynomials = decompose(expr) for deg in polynomials: poly_contribute = S.Zero facet_count = 0 for hp in hp_params: value_over_boundary = integration_reduction(facets, facet_count, hp[0], hp[1], polynomials[deg], dims, deg) poly_contribute += value_over_boundary * (hp[1] / norm(hp[0])) facet_count += 1 poly_contribute /= (dim_length + deg) integral_value += poly_contribute return integral_value def polygon_integrate(facet, hp_param, index, facets, vertices, expr, degree): """Helper function to integrate the input uni/bi/trivariate polynomial over a certain face of the 3-Polytope. Parameters ========== facet : Particular face of the 3-Polytope over which ``expr`` is integrated. index : The index of ``facet`` in ``facets``. facets : Faces of the 3-Polytope(expressed as indices of `vertices`). vertices : Vertices that constitute the facet. expr : The input polynomial. degree : Degree of ``expr``. Examples ======== >>> from sympy.integrals.intpoly import polygon_integrate >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\ (5, 0, 5), (5, 5, 0), (5, 5, 5)],\ [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\ [3, 1, 0, 2], [0, 4, 6, 2]] >>> facet = cube[1] >>> facets = cube[1:] >>> vertices = cube[0] >>> polygon_integrate(facet, [(0, 1, 0), 5], 0, facets, vertices, 1, 0) -25 """ expr = S(expr) if expr.is_zero: return S.Zero result = S.Zero x0 = vertices[facet[0]] for i in range(len(facet)): side = (vertices[facet[i]], vertices[facet[(i + 1) % len(facet)]]) result += distance_to_side(x0, side, hp_param[0]) *\ lineseg_integrate(facet, i, side, expr, degree) if not expr.is_number: expr = diff(expr, x) * x0[0] + diff(expr, y) * x0[1] +\ diff(expr, z) * x0[2] result += polygon_integrate(facet, hp_param, index, facets, vertices, expr, degree - 1) result /= (degree + 2) return result def distance_to_side(point, line_seg, A): """Helper function to compute the signed distance between given 3D point and a line segment. Parameters ========== point : 3D Point line_seg : Line Segment Examples ======== >>> from sympy.integrals.intpoly import distance_to_side >>> point = (0, 0, 0) >>> distance_to_side(point, [(0, 0, 1), (0, 1, 0)], (1, 0, 0)) -sqrt(2)/2 """ x1, x2 = line_seg rev_normal = [-1 * S(i)/norm(A) for i in A] vector = [x2[i] - x1[i] for i in range(0, 3)] vector = [vector[i]/norm(vector) for i in range(0, 3)] n_side = cross_product((0, 0, 0), rev_normal, vector) vectorx0 = [line_seg[0][i] - point[i] for i in range(0, 3)] dot_product = sum([vectorx0[i] * n_side[i] for i in range(0, 3)]) return dot_product def lineseg_integrate(polygon, index, line_seg, expr, degree): """Helper function to compute the line integral of ``expr`` over ``line_seg``. Parameters =========== polygon : Face of a 3-Polytope. index : Index of line_seg in polygon. line_seg : Line Segment. Examples ======== >>> from sympy.integrals.intpoly import lineseg_integrate >>> polygon = [(0, 5, 0), (5, 5, 0), (5, 5, 5), (0, 5, 5)] >>> line_seg = [(0, 5, 0), (5, 5, 0)] >>> lineseg_integrate(polygon, 0, line_seg, 1, 0) 5 """ expr = _sympify(expr) if expr.is_zero: return S.Zero result = S.Zero x0 = line_seg[0] distance = norm(tuple([line_seg[1][i] - line_seg[0][i] for i in range(3)])) if isinstance(expr, Expr): expr_dict = {x: line_seg[1][0], y: line_seg[1][1], z: line_seg[1][2]} result += distance * expr.subs(expr_dict) else: result += distance * expr expr = diff(expr, x) * x0[0] + diff(expr, y) * x0[1] +\ diff(expr, z) * x0[2] result += lineseg_integrate(polygon, index, line_seg, expr, degree - 1) result /= (degree + 1) return result def integration_reduction(facets, index, a, b, expr, dims, degree): """Helper method for main_integrate. Returns the value of the input expression evaluated over the polytope facet referenced by a given index. Parameters =========== facets : List of facets of the polytope. index : Index referencing the facet to integrate the expression over. a : Hyperplane parameter denoting direction. b : Hyperplane parameter denoting distance. expr : The expression to integrate over the facet. dims : List of symbols denoting axes. degree : Degree of the homogeneous polynomial. Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import integration_reduction,\ hyperplane_parameters >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> a, b = hyperplane_parameters(triangle)[0] >>> integration_reduction(facets, 0, a, b, 1, (x, y), 0) 5 """ expr = _sympify(expr) if expr.is_zero: return expr value = S.Zero x0 = facets[index].points[0] m = len(facets) gens = (x, y) inner_product = diff(expr, gens[0]) * x0[0] + diff(expr, gens[1]) * x0[1] if inner_product != 0: value += integration_reduction(facets, index, a, b, inner_product, dims, degree - 1) value += left_integral2D(m, index, facets, x0, expr, gens) return value/(len(dims) + degree - 1) def left_integral2D(m, index, facets, x0, expr, gens): """Computes the left integral of Eq 10 in Chin et al. For the 2D case, the integral is just an evaluation of the polynomial at the intersection of two facets which is multiplied by the distance between the first point of facet and that intersection. Parameters ========== m : No. of hyperplanes. index : Index of facet to find intersections with. facets : List of facets(Line Segments in 2D case). x0 : First point on facet referenced by index. expr : Input polynomial gens : Generators which generate the polynomial Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import left_integral2D >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> left_integral2D(3, 0, facets, facets[0].points[0], 1, (x, y)) 5 """ value = S.Zero for j in range(0, m): intersect = () if j == (index - 1) % m or j == (index + 1) % m: intersect = intersection(facets[index], facets[j], "segment2D") if intersect: distance_origin = norm(tuple(map(lambda x, y: x - y, intersect, x0))) if is_vertex(intersect): if isinstance(expr, Expr): if len(gens) == 3: expr_dict = {gens[0]: intersect[0], gens[1]: intersect[1], gens[2]: intersect[2]} else: expr_dict = {gens[0]: intersect[0], gens[1]: intersect[1]} value += distance_origin * expr.subs(expr_dict) else: value += distance_origin * expr return value def integration_reduction_dynamic(facets, index, a, b, expr, degree, dims, x_index, y_index, max_index, x0, monomial_values, monom_index, vertices=None, hp_param=None): """The same integration_reduction function which uses a dynamic programming approach to compute terms by using the values of the integral of previously computed terms. Parameters ========== facets : Facets of the Polytope. index : Index of facet to find intersections with.(Used in left_integral()). a, b : Hyperplane parameters. expr : Input monomial. degree : Total degree of ``expr``. dims : Tuple denoting axes variables. x_index : Exponent of 'x' in ``expr``. y_index : Exponent of 'y' in ``expr``. max_index : Maximum exponent of any monomial in ``monomial_values``. x0 : First point on ``facets[index]``. monomial_values : List of monomial values constituting the polynomial. monom_index : Index of monomial whose integration is being found. vertices : optional Coordinates of vertices constituting the 3-Polytope. hp_param : optional Hyperplane Parameter of the face of the facets[index]. Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import (integration_reduction_dynamic, \ hyperplane_parameters) >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1)) >>> facets = triangle.sides >>> a, b = hyperplane_parameters(triangle)[0] >>> x0 = facets[0].points[0] >>> monomial_values = [[0, 0, 0, 0], [1, 0, 0, 5],\ [y, 0, 1, 15], [x, 1, 0, None]] >>> integration_reduction_dynamic(facets, 0, a, b, x, 1, (x, y), 1, 0, 1,\ x0, monomial_values, 3) 25/2 """ value = S.Zero m = len(facets) if expr == S.Zero: return expr if len(dims) == 2: if not expr.is_number: _, x_degree, y_degree, _ = monomial_values[monom_index] x_index = monom_index - max_index + \ x_index - 2 if x_degree > 0 else 0 y_index = monom_index - 1 if y_degree > 0 else 0 x_value, y_value =\ monomial_values[x_index][3], monomial_values[y_index][3] value += x_degree * x_value * x0[0] + y_degree * y_value * x0[1] value += left_integral2D(m, index, facets, x0, expr, dims) else: # For 3D use case the max_index contains the z_degree of the term z_index = max_index if not expr.is_number: x_degree, y_degree, z_degree = y_index,\ z_index - x_index - y_index, x_index x_value = monomial_values[z_index - 1][y_index - 1][x_index][7]\ if x_degree > 0 else 0 y_value = monomial_values[z_index - 1][y_index][x_index][7]\ if y_degree > 0 else 0 z_value = monomial_values[z_index - 1][y_index][x_index - 1][7]\ if z_degree > 0 else 0 value += x_degree * x_value * x0[0] + y_degree * y_value * x0[1] \ + z_degree * z_value * x0[2] value += left_integral3D(facets, index, expr, vertices, hp_param, degree) return value / (len(dims) + degree - 1) def left_integral3D(facets, index, expr, vertices, hp_param, degree): """Computes the left integral of Eq 10 in Chin et al. Explanation =========== For the 3D case, this is the sum of the integral values over constituting line segments of the face (which is accessed by facets[index]) multiplied by the distance between the first point of facet and that line segment. Parameters ========== facets : List of faces of the 3-Polytope. index : Index of face over which integral is to be calculated. expr : Input polynomial. vertices : List of vertices that constitute the 3-Polytope. hp_param : The hyperplane parameters of the face. degree : Degree of the ``expr``. Examples ======== >>> from sympy.integrals.intpoly import left_integral3D >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\ (5, 0, 5), (5, 5, 0), (5, 5, 5)],\ [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\ [3, 1, 0, 2], [0, 4, 6, 2]] >>> facets = cube[1:] >>> vertices = cube[0] >>> left_integral3D(facets, 3, 1, vertices, ([0, -1, 0], -5), 0) -50 """ value = S.Zero facet = facets[index] x0 = vertices[facet[0]] for i in range(len(facet)): side = (vertices[facet[i]], vertices[facet[(i + 1) % len(facet)]]) value += distance_to_side(x0, side, hp_param[0]) * \ lineseg_integrate(facet, i, side, expr, degree) return value def gradient_terms(binomial_power=0, no_of_gens=2): """Returns a list of all the possible monomials between 0 and y**binomial_power for 2D case and z**binomial_power for 3D case. Parameters ========== binomial_power : Power upto which terms are generated. no_of_gens : Denotes whether terms are being generated for 2D or 3D case. Examples ======== >>> from sympy.integrals.intpoly import gradient_terms >>> gradient_terms(2) [[1, 0, 0, 0], [y, 0, 1, 0], [y**2, 0, 2, 0], [x, 1, 0, 0], [x*y, 1, 1, 0], [x**2, 2, 0, 0]] >>> gradient_terms(2, 3) [[[[1, 0, 0, 0, 0, 0, 0, 0]]], [[[y, 0, 1, 0, 1, 0, 0, 0], [z, 0, 0, 1, 1, 0, 1, 0]], [[x, 1, 0, 0, 1, 1, 0, 0]]], [[[y**2, 0, 2, 0, 2, 0, 0, 0], [y*z, 0, 1, 1, 2, 0, 1, 0], [z**2, 0, 0, 2, 2, 0, 2, 0]], [[x*y, 1, 1, 0, 2, 1, 0, 0], [x*z, 1, 0, 1, 2, 1, 1, 0]], [[x**2, 2, 0, 0, 2, 2, 0, 0]]]] """ if no_of_gens == 2: count = 0 terms = [None] * int((binomial_power ** 2 + 3 * binomial_power + 2) / 2) for x_count in range(0, binomial_power + 1): for y_count in range(0, binomial_power - x_count + 1): terms[count] = [x**x_count*y**y_count, x_count, y_count, 0] count += 1 else: terms = [[[[x ** x_count * y ** y_count * z ** (z_count - y_count - x_count), x_count, y_count, z_count - y_count - x_count, z_count, x_count, z_count - y_count - x_count, 0] for y_count in range(z_count - x_count, -1, -1)] for x_count in range(0, z_count + 1)] for z_count in range(0, binomial_power + 1)] return terms def hyperplane_parameters(poly, vertices=None): """A helper function to return the hyperplane parameters of which the facets of the polytope are a part of. Parameters ========== poly : The input 2/3-Polytope. vertices : Vertex indices of 3-Polytope. Examples ======== >>> from sympy.geometry.point import Point >>> from sympy.geometry.polygon import Polygon >>> from sympy.integrals.intpoly import hyperplane_parameters >>> hyperplane_parameters(Polygon(Point(0, 3), Point(5, 3), Point(1, 1))) [((0, 1), 3), ((1, -2), -1), ((-2, -1), -3)] >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\ (5, 0, 5), (5, 5, 0), (5, 5, 5)],\ [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\ [3, 1, 0, 2], [0, 4, 6, 2]] >>> hyperplane_parameters(cube[1:], cube[0]) [([0, -1, 0], -5), ([0, 0, -1], -5), ([-1, 0, 0], -5), ([0, 1, 0], 0), ([1, 0, 0], 0), ([0, 0, 1], 0)] """ if isinstance(poly, Polygon): vertices = list(poly.vertices) + [poly.vertices[0]] # Close the polygon params = [None] * (len(vertices) - 1) for i in range(len(vertices) - 1): v1 = vertices[i] v2 = vertices[i + 1] a1 = v1[1] - v2[1] a2 = v2[0] - v1[0] b = v2[0] * v1[1] - v2[1] * v1[0] factor = gcd_list([a1, a2, b]) b = S(b) / factor a = (S(a1) / factor, S(a2) / factor) params[i] = (a, b) else: params = [None] * len(poly) for i, polygon in enumerate(poly): v1, v2, v3 = [vertices[vertex] for vertex in polygon[:3]] normal = cross_product(v1, v2, v3) b = sum([normal[j] * v1[j] for j in range(0, 3)]) fac = gcd_list(normal) if fac.is_zero: fac = 1 normal = [j / fac for j in normal] b = b / fac params[i] = (normal, b) return params def cross_product(v1, v2, v3): """Returns the cross-product of vectors (v2 - v1) and (v3 - v1) That is : (v2 - v1) X (v3 - v1) """ v2 = [v2[j] - v1[j] for j in range(0, 3)] v3 = [v3[j] - v1[j] for j in range(0, 3)] return [v3[2] * v2[1] - v3[1] * v2[2], v3[0] * v2[2] - v3[2] * v2[0], v3[1] * v2[0] - v3[0] * v2[1]] def best_origin(a, b, lineseg, expr): """Helper method for polytope_integrate. Currently not used in the main algorithm. Explanation =========== Returns a point on the lineseg whose vector inner product with the divergence of `expr` yields an expression with the least maximum total power. Parameters ========== a : Hyperplane parameter denoting direction. b : Hyperplane parameter denoting distance. lineseg : Line segment on which to find the origin. expr : The expression which determines the best point. Algorithm(currently works only for 2D use case) =============================================== 1 > Firstly, check for edge cases. Here that would refer to vertical or horizontal lines. 2 > If input expression is a polynomial containing more than one generator then find out the total power of each of the generators. x**2 + 3 + x*y + x**4*y**5 ---> {x: 7, y: 6} If expression is a constant value then pick the first boundary point of the line segment. 3 > First check if a point exists on the line segment where the value of the highest power generator becomes 0. If not check if the value of the next highest becomes 0. If none becomes 0 within line segment constraints then pick the first boundary point of the line segment. Actually, any point lying on the segment can be picked as best origin in the last case. Examples ======== >>> from sympy.integrals.intpoly import best_origin >>> from sympy.abc import x, y >>> from sympy.geometry.line import Segment2D >>> from sympy.geometry.point import Point >>> l = Segment2D(Point(0, 3), Point(1, 1)) >>> expr = x**3*y**7 >>> best_origin((2, 1), 3, l, expr) (0, 3.0) """ a1, b1 = lineseg.points[0] def x_axis_cut(ls): """Returns the point where the input line segment intersects the x-axis. Parameters ========== ls : Line segment """ p, q = ls.points if p.y.is_zero: return tuple(p) elif q.y.is_zero: return tuple(q) elif p.y/q.y < S.Zero: return p.y * (p.x - q.x)/(q.y - p.y) + p.x, S.Zero else: return () def y_axis_cut(ls): """Returns the point where the input line segment intersects the y-axis. Parameters ========== ls : Line segment """ p, q = ls.points if p.x.is_zero: return tuple(p) elif q.x.is_zero: return tuple(q) elif p.x/q.x < S.Zero: return S.Zero, p.x * (p.y - q.y)/(q.x - p.x) + p.y else: return () gens = (x, y) power_gens = {} for i in gens: power_gens[i] = S.Zero if len(gens) > 1: # Special case for vertical and horizontal lines if len(gens) == 2: if a[0] == 0: if y_axis_cut(lineseg): return S.Zero, b/a[1] else: return a1, b1 elif a[1] == 0: if x_axis_cut(lineseg): return b/a[0], S.Zero else: return a1, b1 if isinstance(expr, Expr): # Find the sum total of power of each if expr.is_Add: # generator and store in a dictionary. for monomial in expr.args: if monomial.is_Pow: if monomial.args[0] in gens: power_gens[monomial.args[0]] += monomial.args[1] else: for univariate in monomial.args: term_type = len(univariate.args) if term_type == 0 and univariate in gens: power_gens[univariate] += 1 elif term_type == 2 and univariate.args[0] in gens: power_gens[univariate.args[0]] +=\ univariate.args[1] elif expr.is_Mul: for term in expr.args: term_type = len(term.args) if term_type == 0 and term in gens: power_gens[term] += 1 elif term_type == 2 and term.args[0] in gens: power_gens[term.args[0]] += term.args[1] elif expr.is_Pow: power_gens[expr.args[0]] = expr.args[1] elif expr.is_Symbol: power_gens[expr] += 1 else: # If `expr` is a constant take first vertex of the line segment. return a1, b1 # TODO : This part is quite hacky. Should be made more robust with # TODO : respect to symbol names and scalable w.r.t higher dimensions. power_gens = sorted(power_gens.items(), key=lambda k: str(k[0])) if power_gens[0][1] >= power_gens[1][1]: if y_axis_cut(lineseg): x0 = (S.Zero, b / a[1]) elif x_axis_cut(lineseg): x0 = (b / a[0], S.Zero) else: x0 = (a1, b1) else: if x_axis_cut(lineseg): x0 = (b/a[0], S.Zero) elif y_axis_cut(lineseg): x0 = (S.Zero, b/a[1]) else: x0 = (a1, b1) else: x0 = (b/a[0]) return x0 def decompose(expr, separate=False): """Decomposes an input polynomial into homogeneous ones of smaller or equal degree. Explanation =========== Returns a dictionary with keys as the degree of the smaller constituting polynomials. Values are the constituting polynomials. Parameters ========== expr : Expr Polynomial(SymPy expression). separate : bool If True then simply return a list of the constituent monomials If not then break up the polynomial into constituent homogeneous polynomials. Examples ======== >>> from sympy.abc import x, y >>> from sympy.integrals.intpoly import decompose >>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5) {1: x + y, 2: x**2 + x*y, 5: x**3*y**2 + y**5} >>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5, True) {x, x**2, y, y**5, x*y, x**3*y**2} """ poly_dict = {} if isinstance(expr, Expr) and not expr.is_number: if expr.is_Symbol: poly_dict[1] = expr elif expr.is_Add: symbols = expr.atoms(Symbol) degrees = [(sum(degree_list(monom, *symbols)), monom) for monom in expr.args] if separate: return {monom[1] for monom in degrees} else: for monom in degrees: degree, term = monom if poly_dict.get(degree): poly_dict[degree] += term else: poly_dict[degree] = term elif expr.is_Pow: _, degree = expr.args poly_dict[degree] = expr else: # Now expr can only be of `Mul` type degree = 0 for term in expr.args: term_type = len(term.args) if term_type == 0 and term.is_Symbol: degree += 1 elif term_type == 2: degree += term.args[1] poly_dict[degree] = expr else: poly_dict[0] = expr if separate: return set(poly_dict.values()) return poly_dict def point_sort(poly, normal=None, clockwise=True): """Returns the same polygon with points sorted in clockwise or anti-clockwise order. Note that it's necessary for input points to be sorted in some order (clockwise or anti-clockwise) for the integration algorithm to work. As a convention algorithm has been implemented keeping clockwise orientation in mind. Parameters ========== poly: 2D or 3D Polygon. normal : optional The normal of the plane which the 3-Polytope is a part of. clockwise : bool, optional Returns points sorted in clockwise order if True and anti-clockwise if False. Examples ======== >>> from sympy.integrals.intpoly import point_sort >>> from sympy.geometry.point import Point >>> point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) [Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)] """ pts = poly.vertices if isinstance(poly, Polygon) else poly n = len(pts) if n < 2: return list(pts) order = S.One if clockwise else S.NegativeOne dim = len(pts[0]) if dim == 2: center = Point(sum(map(lambda vertex: vertex.x, pts)) / n, sum(map(lambda vertex: vertex.y, pts)) / n) else: center = Point(sum(map(lambda vertex: vertex.x, pts)) / n, sum(map(lambda vertex: vertex.y, pts)) / n, sum(map(lambda vertex: vertex.z, pts)) / n) def compare(a, b): if a.x - center.x >= S.Zero and b.x - center.x < S.Zero: return -order elif a.x - center.x < 0 and b.x - center.x >= 0: return order elif a.x - center.x == 0 and b.x - center.x == 0: if a.y - center.y >= 0 or b.y - center.y >= 0: return -order if a.y > b.y else order return -order if b.y > a.y else order det = (a.x - center.x) * (b.y - center.y) -\ (b.x - center.x) * (a.y - center.y) if det < 0: return -order elif det > 0: return order first = (a.x - center.x) * (a.x - center.x) +\ (a.y - center.y) * (a.y - center.y) second = (b.x - center.x) * (b.x - center.x) +\ (b.y - center.y) * (b.y - center.y) return -order if first > second else order def compare3d(a, b): det = cross_product(center, a, b) dot_product = sum([det[i] * normal[i] for i in range(0, 3)]) if dot_product < 0: return -order elif dot_product > 0: return order return sorted(pts, key=cmp_to_key(compare if dim==2 else compare3d)) def norm(point): """Returns the Euclidean norm of a point from origin. Parameters ========== point: This denotes a point in the dimension_al spac_e. Examples ======== >>> from sympy.integrals.intpoly import norm >>> from sympy.geometry.point import Point >>> norm(Point(2, 7)) sqrt(53) """ half = S.Half if isinstance(point, (list, tuple)): return sum([coord ** 2 for coord in point]) ** half elif isinstance(point, Point): if isinstance(point, Point2D): return (point.x ** 2 + point.y ** 2) ** half else: return (point.x ** 2 + point.y ** 2 + point.z) ** half elif isinstance(point, dict): return sum(i**2 for i in point.values()) ** half def intersection(geom_1, geom_2, intersection_type): """Returns intersection between geometric objects. Explanation =========== Note that this function is meant for use in integration_reduction and at that point in the calling function the lines denoted by the segments surely intersect within segment boundaries. Coincident lines are taken to be non-intersecting. Also, the hyperplane intersection for 2D case is also implemented. Parameters ========== geom_1, geom_2: The input line segments. Examples ======== >>> from sympy.integrals.intpoly import intersection >>> from sympy.geometry.point import Point >>> from sympy.geometry.line import Segment2D >>> l1 = Segment2D(Point(1, 1), Point(3, 5)) >>> l2 = Segment2D(Point(2, 0), Point(2, 5)) >>> intersection(l1, l2, "segment2D") (2, 3) >>> p1 = ((-1, 0), 0) >>> p2 = ((0, 1), 1) >>> intersection(p1, p2, "plane2D") (0, 1) """ if intersection_type[:-2] == "segment": if intersection_type == "segment2D": x1, y1 = geom_1.points[0] x2, y2 = geom_1.points[1] x3, y3 = geom_2.points[0] x4, y4 = geom_2.points[1] elif intersection_type == "segment3D": x1, y1, z1 = geom_1.points[0] x2, y2, z2 = geom_1.points[1] x3, y3, z3 = geom_2.points[0] x4, y4, z4 = geom_2.points[1] denom = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4) if denom: t1 = x1 * y2 - y1 * x2 t2 = x3 * y4 - x4 * y3 return (S(t1 * (x3 - x4) - t2 * (x1 - x2)) / denom, S(t1 * (y3 - y4) - t2 * (y1 - y2)) / denom) if intersection_type[:-2] == "plane": if intersection_type == "plane2D": # Intersection of hyperplanes a1x, a1y = geom_1[0] a2x, a2y = geom_2[0] b1, b2 = geom_1[1], geom_2[1] denom = a1x * a2y - a2x * a1y if denom: return (S(b1 * a2y - b2 * a1y) / denom, S(b2 * a1x - b1 * a2x) / denom) def is_vertex(ent): """If the input entity is a vertex return True. Parameter ========= ent : Denotes a geometric entity representing a point. Examples ======== >>> from sympy.geometry.point import Point >>> from sympy.integrals.intpoly import is_vertex >>> is_vertex((2, 3)) True >>> is_vertex((2, 3, 6)) True >>> is_vertex(Point(2, 3)) True """ if isinstance(ent, tuple): if len(ent) in [2, 3]: return True elif isinstance(ent, Point): return True return False def plot_polytope(poly): """Plots the 2D polytope using the functions written in plotting module which in turn uses matplotlib backend. Parameter ========= poly: Denotes a 2-Polytope. """ from sympy.plotting.plot import Plot, List2DSeries xl = list(map(lambda vertex: vertex.x, poly.vertices)) yl = list(map(lambda vertex: vertex.y, poly.vertices)) xl.append(poly.vertices[0].x) # Closing the polygon yl.append(poly.vertices[0].y) l2ds = List2DSeries(xl, yl) p = Plot(l2ds, axes='label_axes=True') p.show() def plot_polynomial(expr): """Plots the polynomial using the functions written in plotting module which in turn uses matplotlib backend. Parameter ========= expr: Denotes a polynomial(SymPy expression). """ from sympy.plotting.plot import plot3d, plot gens = expr.free_symbols if len(gens) == 2: plot3d(expr) else: plot(expr)
4fb3ffe781598da07cf636e830d8d419458ef7bc7d05a876117359f5f73e5a6e
from sympy.concrete.expr_with_limits import AddWithLimits from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.compatibility import is_sequence from sympy.core.containers import Tuple from sympy.core.expr import Expr from sympy.core.function import diff from sympy.core.logic import fuzzy_bool from sympy.core.mul import Mul from sympy.core.numbers import oo, pi from sympy.core.relational import Ne from sympy.core.singleton import S from sympy.core.symbol import (Dummy, Symbol, Wild) from sympy.core.sympify import sympify from sympy.functions import Piecewise, sqrt, piecewise_fold, tan, cot, atan from sympy.functions.elementary.exponential import log from sympy.functions.elementary.integers import floor from sympy.functions.elementary.complexes import Abs, sign from sympy.functions.elementary.miscellaneous import Min, Max from sympy.integrals.manualintegrate import manualintegrate from sympy.integrals.trigonometry import trigintegrate from sympy.integrals.meijerint import meijerint_definite, meijerint_indefinite from sympy.matrices import MatrixBase from sympy.polys import Poly, PolynomialError from sympy.series import limit from sympy.series.order import Order from sympy.series.formal import FormalPowerSeries from sympy.simplify.fu import sincos_to_sum from sympy.utilities.misc import filldedent from sympy.utilities.exceptions import SymPyDeprecationWarning class Integral(AddWithLimits): """Represents unevaluated integral.""" __slots__ = ('is_commutative',) def __new__(cls, function, *symbols, **assumptions): """Create an unevaluated integral. Explanation =========== Arguments are an integrand followed by one or more limits. If no limits are given and there is only one free symbol in the expression, that symbol will be used, otherwise an error will be raised. >>> from sympy import Integral >>> from sympy.abc import x, y >>> Integral(x) Integral(x, x) >>> Integral(y) Integral(y, y) When limits are provided, they are interpreted as follows (using ``x`` as though it were the variable of integration): (x,) or x - indefinite integral (x, a) - "evaluate at" integral is an abstract antiderivative (x, a, b) - definite integral The ``as_dummy`` method can be used to see which symbols cannot be targeted by subs: those with a prepended underscore cannot be changed with ``subs``. (Also, the integration variables themselves -- the first element of a limit -- can never be changed by subs.) >>> i = Integral(x, x) >>> at = Integral(x, (x, x)) >>> i.as_dummy() Integral(x, x) >>> at.as_dummy() Integral(_0, (_0, x)) """ #This will help other classes define their own definitions #of behaviour with Integral. if hasattr(function, '_eval_Integral'): return function._eval_Integral(*symbols, **assumptions) if isinstance(function, Poly): SymPyDeprecationWarning( feature="Using integrate/Integral with Poly", issue=18613, deprecated_since_version="1.6", useinstead="the as_expr or integrate methods of Poly").warn() obj = AddWithLimits.__new__(cls, function, *symbols, **assumptions) return obj def __getnewargs__(self): return (self.function,) + tuple([tuple(xab) for xab in self.limits]) @property def free_symbols(self): """ This method returns the symbols that will exist when the integral is evaluated. This is useful if one is trying to determine whether an integral depends on a certain symbol or not. Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, y >>> Integral(x, (x, y, 1)).free_symbols {y} See Also ======== sympy.concrete.expr_with_limits.ExprWithLimits.function sympy.concrete.expr_with_limits.ExprWithLimits.limits sympy.concrete.expr_with_limits.ExprWithLimits.variables """ return AddWithLimits.free_symbols.fget(self) def _eval_is_zero(self): # This is a very naive and quick test, not intended to do the integral to # answer whether it is zero or not, e.g. Integral(sin(x), (x, 0, 2*pi)) # is zero but this routine should return None for that case. But, like # Mul, there are trivial situations for which the integral will be # zero so we check for those. if self.function.is_zero: return True got_none = False for l in self.limits: if len(l) == 3: z = (l[1] == l[2]) or (l[1] - l[2]).is_zero if z: return True elif z is None: got_none = True free = self.function.free_symbols for xab in self.limits: if len(xab) == 1: free.add(xab[0]) continue if len(xab) == 2 and xab[0] not in free: if xab[1].is_zero: return True elif xab[1].is_zero is None: got_none = True # take integration symbol out of free since it will be replaced # with the free symbols in the limits free.discard(xab[0]) # add in the new symbols for i in xab[1:]: free.update(i.free_symbols) if self.function.is_zero is False and got_none is False: return False def transform(self, x, u): r""" Performs a change of variables from `x` to `u` using the relationship given by `x` and `u` which will define the transformations `f` and `F` (which are inverses of each other) as follows: 1) If `x` is a Symbol (which is a variable of integration) then `u` will be interpreted as some function, f(u), with inverse F(u). This, in effect, just makes the substitution of x with f(x). 2) If `u` is a Symbol then `x` will be interpreted as some function, F(x), with inverse f(u). This is commonly referred to as u-substitution. Once f and F have been identified, the transformation is made as follows: .. math:: \int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x) \frac{\mathrm{d}}{\mathrm{d}x} where `F(x)` is the inverse of `f(x)` and the limits and integrand have been corrected so as to retain the same value after integration. Notes ===== The mappings, F(x) or f(u), must lead to a unique integral. Linear or rational linear expression, ``2*x``, ``1/x`` and ``sqrt(x)``, will always work; quadratic expressions like ``x**2 - 1`` are acceptable as long as the resulting integrand does not depend on the sign of the solutions (see examples). The integral will be returned unchanged if ``x`` is not a variable of integration. ``x`` must be (or contain) only one of of the integration variables. If ``u`` has more than one free symbol then it should be sent as a tuple (``u``, ``uvar``) where ``uvar`` identifies which variable is replacing the integration variable. XXX can it contain another integration variable? Examples ======== >>> from sympy.abc import a, x, u >>> from sympy import Integral, cos, sqrt >>> i = Integral(x*cos(x**2 - 1), (x, 0, 1)) transform can change the variable of integration >>> i.transform(x, u) Integral(u*cos(u**2 - 1), (u, 0, 1)) transform can perform u-substitution as long as a unique integrand is obtained: >>> i.transform(x**2 - 1, u) Integral(cos(u)/2, (u, -1, 0)) This attempt fails because x = +/-sqrt(u + 1) and the sign does not cancel out of the integrand: >>> Integral(cos(x**2 - 1), (x, 0, 1)).transform(x**2 - 1, u) Traceback (most recent call last): ... ValueError: The mapping between F(x) and f(u) did not give a unique integrand. transform can do a substitution. Here, the previous result is transformed back into the original expression using "u-substitution": >>> ui = _ >>> _.transform(sqrt(u + 1), x) == i True We can accomplish the same with a regular substitution: >>> ui.transform(u, x**2 - 1) == i True If the `x` does not contain a symbol of integration then the integral will be returned unchanged. Integral `i` does not have an integration variable `a` so no change is made: >>> i.transform(a, x) == i True When `u` has more than one free symbol the symbol that is replacing `x` must be identified by passing `u` as a tuple: >>> Integral(x, (x, 0, 1)).transform(x, (u + a, u)) Integral(a + u, (u, -a, 1 - a)) >>> Integral(x, (x, 0, 1)).transform(x, (u + a, a)) Integral(a + u, (a, -u, 1 - u)) See Also ======== sympy.concrete.expr_with_limits.ExprWithLimits.variables : Lists the integration variables as_dummy : Replace integration variables with dummy ones """ from sympy.solvers.solvers import solve, posify d = Dummy('d') xfree = x.free_symbols.intersection(self.variables) if len(xfree) > 1: raise ValueError( 'F(x) can only contain one of: %s' % self.variables) xvar = xfree.pop() if xfree else d if xvar not in self.variables: return self u = sympify(u) if isinstance(u, Expr): ufree = u.free_symbols if len(ufree) == 0: raise ValueError(filldedent(''' f(u) cannot be a constant''')) if len(ufree) > 1: raise ValueError(filldedent(''' When f(u) has more than one free symbol, the one replacing x must be identified: pass f(u) as (f(u), u)''')) uvar = ufree.pop() else: u, uvar = u if uvar not in u.free_symbols: raise ValueError(filldedent(''' Expecting a tuple (expr, symbol) where symbol identified a free symbol in expr, but symbol is not in expr's free symbols.''')) if not isinstance(uvar, Symbol): # This probably never evaluates to True raise ValueError(filldedent(''' Expecting a tuple (expr, symbol) but didn't get a symbol; got %s''' % uvar)) if x.is_Symbol and u.is_Symbol: return self.xreplace({x: u}) if not x.is_Symbol and not u.is_Symbol: raise ValueError('either x or u must be a symbol') if uvar == xvar: return self.transform(x, (u.subs(uvar, d), d)).xreplace({d: uvar}) if uvar in self.limits: raise ValueError(filldedent(''' u must contain the same variable as in x or a variable that is not already an integration variable''')) if not x.is_Symbol: F = [x.subs(xvar, d)] soln = solve(u - x, xvar, check=False) if not soln: raise ValueError('no solution for solve(F(x) - f(u), x)') f = [fi.subs(uvar, d) for fi in soln] else: f = [u.subs(uvar, d)] pdiff, reps = posify(u - x) puvar = uvar.subs([(v, k) for k, v in reps.items()]) soln = [s.subs(reps) for s in solve(pdiff, puvar)] if not soln: raise ValueError('no solution for solve(F(x) - f(u), u)') F = [fi.subs(xvar, d) for fi in soln] newfuncs = {(self.function.subs(xvar, fi)*fi.diff(d) ).subs(d, uvar) for fi in f} if len(newfuncs) > 1: raise ValueError(filldedent(''' The mapping between F(x) and f(u) did not give a unique integrand.''')) newfunc = newfuncs.pop() def _calc_limit_1(F, a, b): """ replace d with a, using subs if possible, otherwise limit where sign of b is considered """ wok = F.subs(d, a) if wok is S.NaN or wok.is_finite is False and a.is_finite: return limit(sign(b)*F, d, a) return wok def _calc_limit(a, b): """ replace d with a, using subs if possible, otherwise limit where sign of b is considered """ avals = list({_calc_limit_1(Fi, a, b) for Fi in F}) if len(avals) > 1: raise ValueError(filldedent(''' The mapping between F(x) and f(u) did not give a unique limit.''')) return avals[0] newlimits = [] for xab in self.limits: sym = xab[0] if sym == xvar: if len(xab) == 3: a, b = xab[1:] a, b = _calc_limit(a, b), _calc_limit(b, a) if fuzzy_bool(a - b > 0): a, b = b, a newfunc = -newfunc newlimits.append((uvar, a, b)) elif len(xab) == 2: a = _calc_limit(xab[1], 1) newlimits.append((uvar, a)) else: newlimits.append(uvar) else: newlimits.append(xab) return self.func(newfunc, *newlimits) def doit(self, **hints): """ Perform the integration using any hints given. Examples ======== >>> from sympy import Piecewise, S >>> from sympy.abc import x, t >>> p = x**2 + Piecewise((0, x/t < 0), (1, True)) >>> p.integrate((t, S(4)/5, 1), (x, -1, 1)) 1/3 See Also ======== sympy.integrals.trigonometry.trigintegrate sympy.integrals.heurisch.heurisch sympy.integrals.rationaltools.ratint as_sum : Approximate the integral using a sum """ from sympy.concrete.summations import Sum if not hints.get('integrals', True): return self deep = hints.get('deep', True) meijerg = hints.get('meijerg', None) conds = hints.get('conds', 'piecewise') risch = hints.get('risch', None) heurisch = hints.get('heurisch', None) manual = hints.get('manual', None) if len(list(filter(None, (manual, meijerg, risch, heurisch)))) > 1: raise ValueError("At most one of manual, meijerg, risch, heurisch can be True") elif manual: meijerg = risch = heurisch = False elif meijerg: manual = risch = heurisch = False elif risch: manual = meijerg = heurisch = False elif heurisch: manual = meijerg = risch = False eval_kwargs = dict(meijerg=meijerg, risch=risch, manual=manual, heurisch=heurisch, conds=conds) if conds not in ['separate', 'piecewise', 'none']: raise ValueError('conds must be one of "separate", "piecewise", ' '"none", got: %s' % conds) if risch and any(len(xab) > 1 for xab in self.limits): raise ValueError('risch=True is only allowed for indefinite integrals.') # check for the trivial zero if self.is_zero: return S.Zero # hacks to handle integrals of # nested summations if isinstance(self.function, Sum): if any(v in self.function.limits[0] for v in self.variables): raise ValueError('Limit of the sum cannot be an integration variable.') if any(l.is_infinite for l in self.function.limits[0][1:]): return self _i = self _sum = self.function return _sum.func(_i.func(_sum.function, *_i.limits).doit(), *_sum.limits).doit() # now compute and check the function function = self.function if deep: function = function.doit(**hints) if function.is_zero: return S.Zero # hacks to handle special cases if isinstance(function, MatrixBase): return function.applyfunc( lambda f: self.func(f, self.limits).doit(**hints)) if isinstance(function, FormalPowerSeries): if len(self.limits) > 1: raise NotImplementedError xab = self.limits[0] if len(xab) > 1: return function.integrate(xab, **eval_kwargs) else: return function.integrate(xab[0], **eval_kwargs) # There is no trivial answer and special handling # is done so continue # first make sure any definite limits have integration # variables with matching assumptions reps = {} for xab in self.limits: if len(xab) != 3: continue x, a, b = xab l = (a, b) if all(i.is_nonnegative for i in l) and not x.is_nonnegative: d = Dummy(positive=True) elif all(i.is_nonpositive for i in l) and not x.is_nonpositive: d = Dummy(negative=True) elif all(i.is_real for i in l) and not x.is_real: d = Dummy(real=True) else: d = None if d: reps[x] = d if reps: undo = {v: k for k, v in reps.items()} did = self.xreplace(reps).doit(**hints) if type(did) is tuple: # when separate=True did = tuple([i.xreplace(undo) for i in did]) else: did = did.xreplace(undo) return did # continue with existing assumptions undone_limits = [] # ulj = free symbols of any undone limits' upper and lower limits ulj = set() for xab in self.limits: # compute uli, the free symbols in the # Upper and Lower limits of limit I if len(xab) == 1: uli = set(xab[:1]) elif len(xab) == 2: uli = xab[1].free_symbols elif len(xab) == 3: uli = xab[1].free_symbols.union(xab[2].free_symbols) # this integral can be done as long as there is no blocking # limit that has been undone. An undone limit is blocking if # it contains an integration variable that is in this limit's # upper or lower free symbols or vice versa if xab[0] in ulj or any(v[0] in uli for v in undone_limits): undone_limits.append(xab) ulj.update(uli) function = self.func(*([function] + [xab])) factored_function = function.factor() if not isinstance(factored_function, Integral): function = factored_function continue if function.has(Abs, sign) and ( (len(xab) < 3 and all(x.is_extended_real for x in xab)) or (len(xab) == 3 and all(x.is_extended_real and not x.is_infinite for x in xab[1:]))): # some improper integrals are better off with Abs xr = Dummy("xr", real=True) function = (function.xreplace({xab[0]: xr}) .rewrite(Piecewise).xreplace({xr: xab[0]})) elif function.has(Min, Max): function = function.rewrite(Piecewise) if (function.has(Piecewise) and not isinstance(function, Piecewise)): function = piecewise_fold(function) if isinstance(function, Piecewise): if len(xab) == 1: antideriv = function._eval_integral(xab[0], **eval_kwargs) else: antideriv = self._eval_integral( function, xab[0], **eval_kwargs) else: # There are a number of tradeoffs in using the # Meijer G method. It can sometimes be a lot faster # than other methods, and sometimes slower. And # there are certain types of integrals for which it # is more likely to work than others. These # heuristics are incorporated in deciding what # integration methods to try, in what order. See the # integrate() docstring for details. def try_meijerg(function, xab): ret = None if len(xab) == 3 and meijerg is not False: x, a, b = xab try: res = meijerint_definite(function, x, a, b) except NotImplementedError: from sympy.integrals.meijerint import _debug _debug('NotImplementedError ' 'from meijerint_definite') res = None if res is not None: f, cond = res if conds == 'piecewise': ret = Piecewise( (f, cond), (self.func( function, (x, a, b)), True)) elif conds == 'separate': if len(self.limits) != 1: raise ValueError(filldedent(''' conds=separate not supported in multiple integrals''')) ret = f, cond else: ret = f return ret meijerg1 = meijerg if (meijerg is not False and len(xab) == 3 and xab[1].is_extended_real and xab[2].is_extended_real and not function.is_Poly and (xab[1].has(oo, -oo) or xab[2].has(oo, -oo))): ret = try_meijerg(function, xab) if ret is not None: function = ret continue meijerg1 = False # If the special meijerg code did not succeed in # finding a definite integral, then the code using # meijerint_indefinite will not either (it might # find an antiderivative, but the answer is likely # to be nonsensical). Thus if we are requested to # only use Meijer G-function methods, we give up at # this stage. Otherwise we just disable G-function # methods. if meijerg1 is False and meijerg is True: antideriv = None else: antideriv = self._eval_integral( function, xab[0], **eval_kwargs) if antideriv is None and meijerg is True: ret = try_meijerg(function, xab) if ret is not None: function = ret continue if not isinstance(antideriv, Integral) and antideriv is not None: for atan_term in antideriv.atoms(atan): atan_arg = atan_term.args[0] # Checking `atan_arg` to be linear combination of `tan` or `cot` for tan_part in atan_arg.atoms(tan): x1 = Dummy('x1') tan_exp1 = atan_arg.subs(tan_part, x1) # The coefficient of `tan` should be constant coeff = tan_exp1.diff(x1) if x1 not in coeff.free_symbols: a = tan_part.args[0] antideriv = antideriv.subs(atan_term, Add(atan_term, sign(coeff)*pi*floor((a-pi/2)/pi))) for cot_part in atan_arg.atoms(cot): x1 = Dummy('x1') cot_exp1 = atan_arg.subs(cot_part, x1) # The coefficient of `cot` should be constant coeff = cot_exp1.diff(x1) if x1 not in coeff.free_symbols: a = cot_part.args[0] antideriv = antideriv.subs(atan_term, Add(atan_term, sign(coeff)*pi*floor((a)/pi))) if antideriv is None: undone_limits.append(xab) function = self.func(*([function] + [xab])).factor() factored_function = function.factor() if not isinstance(factored_function, Integral): function = factored_function continue else: if len(xab) == 1: function = antideriv else: if len(xab) == 3: x, a, b = xab elif len(xab) == 2: x, b = xab a = None else: raise NotImplementedError if deep: if isinstance(a, Basic): a = a.doit(**hints) if isinstance(b, Basic): b = b.doit(**hints) if antideriv.is_Poly: gens = list(antideriv.gens) gens.remove(x) antideriv = antideriv.as_expr() function = antideriv._eval_interval(x, a, b) function = Poly(function, *gens) else: def is_indef_int(g, x): return (isinstance(g, Integral) and any(i == (x,) for i in g.limits)) def eval_factored(f, x, a, b): # _eval_interval for integrals with # (constant) factors # a single indefinite integral is assumed args = [] for g in Mul.make_args(f): if is_indef_int(g, x): args.append(g._eval_interval(x, a, b)) else: args.append(g) return Mul(*args) integrals, others, piecewises = [], [], [] for f in Add.make_args(antideriv): if any(is_indef_int(g, x) for g in Mul.make_args(f)): integrals.append(f) elif any(isinstance(g, Piecewise) for g in Mul.make_args(f)): piecewises.append(piecewise_fold(f)) else: others.append(f) uneval = Add(*[eval_factored(f, x, a, b) for f in integrals]) try: evalued = Add(*others)._eval_interval(x, a, b) evalued_pw = piecewise_fold(Add(*piecewises))._eval_interval(x, a, b) function = uneval + evalued + evalued_pw except NotImplementedError: # This can happen if _eval_interval depends in a # complicated way on limits that cannot be computed undone_limits.append(xab) function = self.func(*([function] + [xab])) factored_function = function.factor() if not isinstance(factored_function, Integral): function = factored_function return function def _eval_derivative(self, sym): """Evaluate the derivative of the current Integral object by differentiating under the integral sign [1], using the Fundamental Theorem of Calculus [2] when possible. Explanation =========== Whenever an Integral is encountered that is equivalent to zero or has an integrand that is independent of the variable of integration those integrals are performed. All others are returned as Integral instances which can be resolved with doit() (provided they are integrable). References ========== .. [1] https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign .. [2] https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus Examples ======== >>> from sympy import Integral >>> from sympy.abc import x, y >>> i = Integral(x + y, y, (y, 1, x)) >>> i.diff(x) Integral(x + y, (y, x)) + Integral(1, y, (y, 1, x)) >>> i.doit().diff(x) == i.diff(x).doit() True >>> i.diff(y) 0 The previous must be true since there is no y in the evaluated integral: >>> i.free_symbols {x} >>> i.doit() 2*x**3/3 - x/2 - 1/6 """ # differentiate under the integral sign; we do not # check for regularity conditions (TODO), see issue 4215 # get limits and the function f, limits = self.function, list(self.limits) # the order matters if variables of integration appear in the limits # so work our way in from the outside to the inside. limit = limits.pop(-1) if len(limit) == 3: x, a, b = limit elif len(limit) == 2: x, b = limit a = None else: a = b = None x = limit[0] if limits: # f is the argument to an integral f = self.func(f, *tuple(limits)) # assemble the pieces def _do(f, ab): dab_dsym = diff(ab, sym) if not dab_dsym: return S.Zero if isinstance(f, Integral): limits = [(x, x) if (len(l) == 1 and l[0] == x) else l for l in f.limits] f = self.func(f.function, *limits) return f.subs(x, ab)*dab_dsym rv = S.Zero if b is not None: rv += _do(f, b) if a is not None: rv -= _do(f, a) if len(limit) == 1 and sym == x: # the dummy variable *is* also the real-world variable arg = f rv += arg else: # the dummy variable might match sym but it's # only a dummy and the actual variable is determined # by the limits, so mask off the variable of integration # while differentiating u = Dummy('u') arg = f.subs(x, u).diff(sym).subs(u, x) if arg: rv += self.func(arg, Tuple(x, a, b)) return rv def _eval_integral(self, f, x, meijerg=None, risch=None, manual=None, heurisch=None, conds='piecewise'): """ Calculate the anti-derivative to the function f(x). Explanation =========== The following algorithms are applied (roughly in this order): 1. Simple heuristics (based on pattern matching and integral table): - most frequently used functions (e.g. polynomials, products of trig functions) 2. Integration of rational functions: - A complete algorithm for integrating rational functions is implemented (the Lazard-Rioboo-Trager algorithm). The algorithm also uses the partial fraction decomposition algorithm implemented in apart() as a preprocessor to make this process faster. Note that the integral of a rational function is always elementary, but in general, it may include a RootSum. 3. Full Risch algorithm: - The Risch algorithm is a complete decision procedure for integrating elementary functions, which means that given any elementary function, it will either compute an elementary antiderivative, or else prove that none exists. Currently, part of transcendental case is implemented, meaning elementary integrals containing exponentials, logarithms, and (soon!) trigonometric functions can be computed. The algebraic case, e.g., functions containing roots, is much more difficult and is not implemented yet. - If the routine fails (because the integrand is not elementary, or because a case is not implemented yet), it continues on to the next algorithms below. If the routine proves that the integrals is nonelementary, it still moves on to the algorithms below, because we might be able to find a closed-form solution in terms of special functions. If risch=True, however, it will stop here. 4. The Meijer G-Function algorithm: - This algorithm works by first rewriting the integrand in terms of very general Meijer G-Function (meijerg in SymPy), integrating it, and then rewriting the result back, if possible. This algorithm is particularly powerful for definite integrals (which is actually part of a different method of Integral), since it can compute closed-form solutions of definite integrals even when no closed-form indefinite integral exists. But it also is capable of computing many indefinite integrals as well. - Another advantage of this method is that it can use some results about the Meijer G-Function to give a result in terms of a Piecewise expression, which allows to express conditionally convergent integrals. - Setting meijerg=True will cause integrate() to use only this method. 5. The "manual integration" algorithm: - This algorithm tries to mimic how a person would find an antiderivative by hand, for example by looking for a substitution or applying integration by parts. This algorithm does not handle as many integrands but can return results in a more familiar form. - Sometimes this algorithm can evaluate parts of an integral; in this case integrate() will try to evaluate the rest of the integrand using the other methods here. - Setting manual=True will cause integrate() to use only this method. 6. The Heuristic Risch algorithm: - This is a heuristic version of the Risch algorithm, meaning that it is not deterministic. This is tried as a last resort because it can be very slow. It is still used because not enough of the full Risch algorithm is implemented, so that there are still some integrals that can only be computed using this method. The goal is to implement enough of the Risch and Meijer G-function methods so that this can be deleted. Setting heurisch=True will cause integrate() to use only this method. Set heurisch=False to not use it. """ from sympy.integrals.deltafunctions import deltaintegrate from sympy.integrals.singularityfunctions import singularityintegrate from sympy.integrals.heurisch import heurisch as heurisch_, heurisch_wrapper from sympy.integrals.rationaltools import ratint from sympy.integrals.risch import risch_integrate if risch: try: return risch_integrate(f, x, conds=conds) except NotImplementedError: return None if manual: try: result = manualintegrate(f, x) if result is not None and result.func != Integral: return result except (ValueError, PolynomialError): pass eval_kwargs = dict(meijerg=meijerg, risch=risch, manual=manual, heurisch=heurisch, conds=conds) # if it is a poly(x) then let the polynomial integrate itself (fast) # # It is important to make this check first, otherwise the other code # will return a sympy expression instead of a Polynomial. # # see Polynomial for details. if isinstance(f, Poly) and not (manual or meijerg or risch): SymPyDeprecationWarning( feature="Using integrate/Integral with Poly", issue=18613, deprecated_since_version="1.6", useinstead="the as_expr or integrate methods of Poly").warn() return f.integrate(x) # Piecewise antiderivatives need to call special integrate. if isinstance(f, Piecewise): return f.piecewise_integrate(x, **eval_kwargs) # let's cut it short if `f` does not depend on `x`; if # x is only a dummy, that will be handled below if not f.has(x): return f*x # try to convert to poly(x) and then integrate if successful (fast) poly = f.as_poly(x) if poly is not None and not (manual or meijerg or risch): return poly.integrate().as_expr() if risch is not False: try: result, i = risch_integrate(f, x, separate_integral=True, conds=conds) except NotImplementedError: pass else: if i: # There was a nonelementary integral. Try integrating it. # if no part of the NonElementaryIntegral is integrated by # the Risch algorithm, then use the original function to # integrate, instead of re-written one if result == 0: from sympy.integrals.risch import NonElementaryIntegral return NonElementaryIntegral(f, x).doit(risch=False) else: return result + i.doit(risch=False) else: return result # since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ... # we are going to handle Add terms separately, # if `f` is not Add -- we only have one term # Note that in general, this is a bad idea, because Integral(g1) + # Integral(g2) might not be computable, even if Integral(g1 + g2) is. # For example, Integral(x**x + x**x*log(x)). But many heuristics only # work term-wise. So we compute this step last, after trying # risch_integrate. We also try risch_integrate again in this loop, # because maybe the integral is a sum of an elementary part and a # nonelementary part (like erf(x) + exp(x)). risch_integrate() is # quite fast, so this is acceptable. parts = [] args = Add.make_args(f) for g in args: coeff, g = g.as_independent(x) # g(x) = const if g is S.One and not meijerg: parts.append(coeff*x) continue # g(x) = expr + O(x**n) order_term = g.getO() if order_term is not None: h = self._eval_integral(g.removeO(), x, **eval_kwargs) if h is not None: h_order_expr = self._eval_integral(order_term.expr, x, **eval_kwargs) if h_order_expr is not None: h_order_term = order_term.func( h_order_expr, *order_term.variables) parts.append(coeff*(h + h_order_term)) continue # NOTE: if there is O(x**n) and we fail to integrate then # there is no point in trying other methods because they # will fail, too. return None # c # g(x) = (a*x+b) if g.is_Pow and not g.exp.has(x) and not meijerg: a = Wild('a', exclude=[x]) b = Wild('b', exclude=[x]) M = g.base.match(a*x + b) if M is not None: if g.exp == -1: h = log(g.base) elif conds != 'piecewise': h = g.base**(g.exp + 1) / (g.exp + 1) else: h1 = log(g.base) h2 = g.base**(g.exp + 1) / (g.exp + 1) h = Piecewise((h2, Ne(g.exp, -1)), (h1, True)) parts.append(coeff * h / M[a]) continue # poly(x) # g(x) = ------- # poly(x) if g.is_rational_function(x) and not (manual or meijerg or risch): parts.append(coeff * ratint(g, x)) continue if not (manual or meijerg or risch): # g(x) = Mul(trig) h = trigintegrate(g, x, conds=conds) if h is not None: parts.append(coeff * h) continue # g(x) has at least a DiracDelta term h = deltaintegrate(g, x) if h is not None: parts.append(coeff * h) continue # g(x) has at least a Singularity Function term h = singularityintegrate(g, x) if h is not None: parts.append(coeff * h) continue # Try risch again. if risch is not False: try: h, i = risch_integrate(g, x, separate_integral=True, conds=conds) except NotImplementedError: h = None else: if i: h = h + i.doit(risch=False) parts.append(coeff*h) continue # fall back to heurisch if heurisch is not False: try: if conds == 'piecewise': h = heurisch_wrapper(g, x, hints=[]) else: h = heurisch_(g, x, hints=[]) except PolynomialError: # XXX: this exception means there is a bug in the # implementation of heuristic Risch integration # algorithm. h = None else: h = None if meijerg is not False and h is None: # rewrite using G functions try: h = meijerint_indefinite(g, x) except NotImplementedError: from sympy.integrals.meijerint import _debug _debug('NotImplementedError from meijerint_definite') if h is not None: parts.append(coeff * h) continue if h is None and manual is not False: try: result = manualintegrate(g, x) if result is not None and not isinstance(result, Integral): if result.has(Integral) and not manual: # Try to have other algorithms do the integrals # manualintegrate can't handle, # unless we were asked to use manual only. # Keep the rest of eval_kwargs in case another # method was set to False already new_eval_kwargs = eval_kwargs new_eval_kwargs["manual"] = False result = result.func(*[ arg.doit(**new_eval_kwargs) if arg.has(Integral) else arg for arg in result.args ]).expand(multinomial=False, log=False, power_exp=False, power_base=False) if not result.has(Integral): parts.append(coeff * result) continue except (ValueError, PolynomialError): # can't handle some SymPy expressions pass # if we failed maybe it was because we had # a product that could have been expanded, # so let's try an expansion of the whole # thing before giving up; we don't try this # at the outset because there are things # that cannot be solved unless they are # NOT expanded e.g., x**x*(1+log(x)). There # should probably be a checker somewhere in this # routine to look for such cases and try to do # collection on the expressions if they are already # in an expanded form if not h and len(args) == 1: f = sincos_to_sum(f).expand(mul=True, deep=False) if f.is_Add: # Note: risch will be identical on the expanded # expression, but maybe it will be able to pick out parts, # like x*(exp(x) + erf(x)). return self._eval_integral(f, x, **eval_kwargs) if h is not None: parts.append(coeff * h) else: return None return Add(*parts) def _eval_lseries(self, x, logx, cdir=0): expr = self.as_dummy() symb = x for l in expr.limits: if x in l[1:]: symb = l[0] break for term in expr.function.lseries(symb, logx): yield integrate(term, *expr.limits) def _eval_nseries(self, x, n, logx, cdir=0): expr = self.as_dummy() symb = x for l in expr.limits: if x in l[1:]: symb = l[0] break terms, order = expr.function.nseries( x=symb, n=n, logx=logx).as_coeff_add(Order) order = [o.subs(symb, x) for o in order] return integrate(terms, *expr.limits) + Add(*order)*x def _eval_as_leading_term(self, x, cdir=0): series_gen = self.args[0].lseries(x) for leading_term in series_gen: if leading_term != 0: break return integrate(leading_term, *self.args[1:]) def _eval_simplify(self, **kwargs): from sympy.core.exprtools import factor_terms from sympy.simplify.simplify import simplify expr = factor_terms(self) if isinstance(expr, Integral): return expr.func(*[simplify(i, **kwargs) for i in expr.args]) return expr.simplify(**kwargs) def as_sum(self, n=None, method="midpoint", evaluate=True): """ Approximates a definite integral by a sum. Parameters ========== n : The number of subintervals to use, optional. method : One of: 'left', 'right', 'midpoint', 'trapezoid'. evaluate : bool If False, returns an unevaluated Sum expression. The default is True, evaluate the sum. Notes ===== These methods of approximate integration are described in [1]. Examples ======== >>> from sympy import sin, sqrt >>> from sympy.abc import x, n >>> from sympy.integrals import Integral >>> e = Integral(sin(x), (x, 3, 7)) >>> e Integral(sin(x), (x, 3, 7)) For demonstration purposes, this interval will only be split into 2 regions, bounded by [3, 5] and [5, 7]. The left-hand rule uses function evaluations at the left of each interval: >>> e.as_sum(2, 'left') 2*sin(5) + 2*sin(3) The midpoint rule uses evaluations at the center of each interval: >>> e.as_sum(2, 'midpoint') 2*sin(4) + 2*sin(6) The right-hand rule uses function evaluations at the right of each interval: >>> e.as_sum(2, 'right') 2*sin(5) + 2*sin(7) The trapezoid rule uses function evaluations on both sides of the intervals. This is equivalent to taking the average of the left and right hand rule results: >>> e.as_sum(2, 'trapezoid') 2*sin(5) + sin(3) + sin(7) >>> (e.as_sum(2, 'left') + e.as_sum(2, 'right'))/2 == _ True Here, the discontinuity at x = 0 can be avoided by using the midpoint or right-hand method: >>> e = Integral(1/sqrt(x), (x, 0, 1)) >>> e.as_sum(5).n(4) 1.730 >>> e.as_sum(10).n(4) 1.809 >>> e.doit().n(4) # the actual value is 2 2.000 The left- or trapezoid method will encounter the discontinuity and return infinity: >>> e.as_sum(5, 'left') zoo The number of intervals can be symbolic. If omitted, a dummy symbol will be used for it. >>> e = Integral(x**2, (x, 0, 2)) >>> e.as_sum(n, 'right').expand() 8/3 + 4/n + 4/(3*n**2) This shows that the midpoint rule is more accurate, as its error term decays as the square of n: >>> e.as_sum(method='midpoint').expand() 8/3 - 2/(3*_n**2) A symbolic sum is returned with evaluate=False: >>> e.as_sum(n, 'midpoint', evaluate=False) 2*Sum((2*_k/n - 1/n)**2, (_k, 1, n))/n See Also ======== Integral.doit : Perform the integration using any hints References ========== .. [1] https://en.wikipedia.org/wiki/Riemann_sum#Methods """ from sympy.concrete.summations import Sum limits = self.limits if len(limits) > 1: raise NotImplementedError( "Multidimensional midpoint rule not implemented yet") else: limit = limits[0] if (len(limit) != 3 or limit[1].is_finite is False or limit[2].is_finite is False): raise ValueError("Expecting a definite integral over " "a finite interval.") if n is None: n = Dummy('n', integer=True, positive=True) else: n = sympify(n) if (n.is_positive is False or n.is_integer is False or n.is_finite is False): raise ValueError("n must be a positive integer, got %s" % n) x, a, b = limit dx = (b - a)/n k = Dummy('k', integer=True, positive=True) f = self.function if method == "left": result = dx*Sum(f.subs(x, a + (k-1)*dx), (k, 1, n)) elif method == "right": result = dx*Sum(f.subs(x, a + k*dx), (k, 1, n)) elif method == "midpoint": result = dx*Sum(f.subs(x, a + k*dx - dx/2), (k, 1, n)) elif method == "trapezoid": result = dx*((f.subs(x, a) + f.subs(x, b))/2 + Sum(f.subs(x, a + k*dx), (k, 1, n - 1))) else: raise ValueError("Unknown method %s" % method) return result.doit() if evaluate else result def _sage_(self): import sage.all as sage f, limits = self.function._sage_(), list(self.limits) for limit_ in limits: if len(limit_) == 1: x = limit_[0] f = sage.integral(f, x._sage_(), hold=True) elif len(limit_) == 2: x, b = limit_ f = sage.integral(f, x._sage_(), b._sage_(), hold=True) else: x, a, b = limit_ f = sage.integral(f, (x._sage_(), a._sage_(), b._sage_()), hold=True) return f def principal_value(self, **kwargs): """ Compute the Cauchy Principal Value of the definite integral of a real function in the given interval on the real axis. Explanation =========== In mathematics, the Cauchy principal value, is a method for assigning values to certain improper integrals which would otherwise be undefined. Examples ======== >>> from sympy import oo >>> from sympy.integrals.integrals import Integral >>> from sympy.abc import x >>> Integral(x+1, (x, -oo, oo)).principal_value() oo >>> f = 1 / (x**3) >>> Integral(f, (x, -oo, oo)).principal_value() 0 >>> Integral(f, (x, -10, 10)).principal_value() 0 >>> Integral(f, (x, -10, oo)).principal_value() + Integral(f, (x, -oo, 10)).principal_value() 0 References ========== .. [1] https://en.wikipedia.org/wiki/Cauchy_principal_value .. [2] http://mathworld.wolfram.com/CauchyPrincipalValue.html """ from sympy.calculus import singularities if len(self.limits) != 1 or len(list(self.limits[0])) != 3: raise ValueError("You need to insert a variable, lower_limit, and upper_limit correctly to calculate " "cauchy's principal value") x, a, b = self.limits[0] if not (a.is_comparable and b.is_comparable and a <= b): raise ValueError("The lower_limit must be smaller than or equal to the upper_limit to calculate " "cauchy's principal value. Also, a and b need to be comparable.") if a == b: return 0 r = Dummy('r') f = self.function singularities_list = [s for s in singularities(f, x) if s.is_comparable and a <= s <= b] for i in singularities_list: if (i == b) or (i == a): raise ValueError( 'The principal value is not defined in the given interval due to singularity at %d.' % (i)) F = integrate(f, x, **kwargs) if F.has(Integral): return self if a is -oo and b is oo: I = limit(F - F.subs(x, -x), x, oo) else: I = limit(F, x, b, '-') - limit(F, x, a, '+') for s in singularities_list: I += limit(((F.subs(x, s - r)) - F.subs(x, s + r)), r, 0, '+') return I def integrate(*args, meijerg=None, conds='piecewise', risch=None, heurisch=None, manual=None, **kwargs): """integrate(f, var, ...) Explanation =========== Compute definite or indefinite integral of one or more variables using Risch-Norman algorithm and table lookup. This procedure is able to handle elementary algebraic and transcendental functions and also a huge class of special functions, including Airy, Bessel, Whittaker and Lambert. var can be: - a symbol -- indefinite integration - a tuple (symbol, a) -- indefinite integration with result given with `a` replacing `symbol` - a tuple (symbol, a, b) -- definite integration Several variables can be specified, in which case the result is multiple integration. (If var is omitted and the integrand is univariate, the indefinite integral in that variable will be performed.) Indefinite integrals are returned without terms that are independent of the integration variables. (see examples) Definite improper integrals often entail delicate convergence conditions. Pass conds='piecewise', 'separate' or 'none' to have these returned, respectively, as a Piecewise function, as a separate result (i.e. result will be a tuple), or not at all (default is 'piecewise'). **Strategy** SymPy uses various approaches to definite integration. One method is to find an antiderivative for the integrand, and then use the fundamental theorem of calculus. Various functions are implemented to integrate polynomial, rational and trigonometric functions, and integrands containing DiracDelta terms. SymPy also implements the part of the Risch algorithm, which is a decision procedure for integrating elementary functions, i.e., the algorithm can either find an elementary antiderivative, or prove that one does not exist. There is also a (very successful, albeit somewhat slow) general implementation of the heuristic Risch algorithm. This algorithm will eventually be phased out as more of the full Risch algorithm is implemented. See the docstring of Integral._eval_integral() for more details on computing the antiderivative using algebraic methods. The option risch=True can be used to use only the (full) Risch algorithm. This is useful if you want to know if an elementary function has an elementary antiderivative. If the indefinite Integral returned by this function is an instance of NonElementaryIntegral, that means that the Risch algorithm has proven that integral to be non-elementary. Note that by default, additional methods (such as the Meijer G method outlined below) are tried on these integrals, as they may be expressible in terms of special functions, so if you only care about elementary answers, use risch=True. Also note that an unevaluated Integral returned by this function is not necessarily a NonElementaryIntegral, even with risch=True, as it may just be an indication that the particular part of the Risch algorithm needed to integrate that function is not yet implemented. Another family of strategies comes from re-writing the integrand in terms of so-called Meijer G-functions. Indefinite integrals of a single G-function can always be computed, and the definite integral of a product of two G-functions can be computed from zero to infinity. Various strategies are implemented to rewrite integrands as G-functions, and use this information to compute integrals (see the ``meijerint`` module). The option manual=True can be used to use only an algorithm that tries to mimic integration by hand. This algorithm does not handle as many integrands as the other algorithms implemented but may return results in a more familiar form. The ``manualintegrate`` module has functions that return the steps used (see the module docstring for more information). In general, the algebraic methods work best for computing antiderivatives of (possibly complicated) combinations of elementary functions. The G-function methods work best for computing definite integrals from zero to infinity of moderately complicated combinations of special functions, or indefinite integrals of very simple combinations of special functions. The strategy employed by the integration code is as follows: - If computing a definite integral, and both limits are real, and at least one limit is +- oo, try the G-function method of definite integration first. - Try to find an antiderivative, using all available methods, ordered by performance (that is try fastest method first, slowest last; in particular polynomial integration is tried first, Meijer G-functions second to last, and heuristic Risch last). - If still not successful, try G-functions irrespective of the limits. The option meijerg=True, False, None can be used to, respectively: always use G-function methods and no others, never use G-function methods, or use all available methods (in order as described above). It defaults to None. Examples ======== >>> from sympy import integrate, log, exp, oo >>> from sympy.abc import a, x, y >>> integrate(x*y, x) x**2*y/2 >>> integrate(log(x), x) x*log(x) - x >>> integrate(log(x), (x, 1, a)) a*log(a) - a + 1 >>> integrate(x) x**2/2 Terms that are independent of x are dropped by indefinite integration: >>> from sympy import sqrt >>> integrate(sqrt(1 + x), (x, 0, x)) 2*(x + 1)**(3/2)/3 - 2/3 >>> integrate(sqrt(1 + x), x) 2*(x + 1)**(3/2)/3 >>> integrate(x*y) Traceback (most recent call last): ... ValueError: specify integration variables to integrate x*y Note that ``integrate(x)`` syntax is meant only for convenience in interactive sessions and should be avoided in library code. >>> integrate(x**a*exp(-x), (x, 0, oo)) # same as conds='piecewise' Piecewise((gamma(a + 1), re(a) > -1), (Integral(x**a*exp(-x), (x, 0, oo)), True)) >>> integrate(x**a*exp(-x), (x, 0, oo), conds='none') gamma(a + 1) >>> integrate(x**a*exp(-x), (x, 0, oo), conds='separate') (gamma(a + 1), -re(a) < 1) See Also ======== Integral, Integral.doit """ doit_flags = { 'deep': False, 'meijerg': meijerg, 'conds': conds, 'risch': risch, 'heurisch': heurisch, 'manual': manual } integral = Integral(*args, **kwargs) if isinstance(integral, Integral): return integral.doit(**doit_flags) else: new_args = [a.doit(**doit_flags) if isinstance(a, Integral) else a for a in integral.args] return integral.func(*new_args) def line_integrate(field, curve, vars): """line_integrate(field, Curve, variables) Compute the line integral. Examples ======== >>> from sympy import Curve, line_integrate, E, ln >>> from sympy.abc import x, y, t >>> C = Curve([E**t + 1, E**t - 1], (t, 0, ln(2))) >>> line_integrate(x + y, C, [x, y]) 3*sqrt(2) See Also ======== sympy.integrals.integrals.integrate, Integral """ from sympy.geometry import Curve F = sympify(field) if not F: raise ValueError( "Expecting function specifying field as first argument.") if not isinstance(curve, Curve): raise ValueError("Expecting Curve entity as second argument.") if not is_sequence(vars): raise ValueError("Expecting ordered iterable for variables.") if len(curve.functions) != len(vars): raise ValueError("Field variable size does not match curve dimension.") if curve.parameter in vars: raise ValueError("Curve parameter clashes with field parameters.") # Calculate derivatives for line parameter functions # F(r) -> F(r(t)) and finally F(r(t)*r'(t)) Ft = F dldt = 0 for i, var in enumerate(vars): _f = curve.functions[i] _dn = diff(_f, curve.parameter) # ...arc length dldt = dldt + (_dn * _dn) Ft = Ft.subs(var, _f) Ft = Ft * sqrt(dldt) integral = Integral(Ft, curve.limits).doit(deep=False) return integral
8b3c8be3a3c695794b42ec6cfb593b8eede174e4f4ddd2e7b6f25ba3ebeb26bb
from typing import Dict, List from itertools import permutations from sympy.core.add import Add from sympy.core.basic import Basic from sympy.core.mul import Mul from sympy.core.symbol import Wild, Dummy from sympy.core.basic import sympify from sympy.core.numbers import Rational, pi, I from sympy.core.relational import Eq, Ne from sympy.core.singleton import S from sympy.functions import exp, sin, cos, tan, cot, asin, atan from sympy.functions import log, sinh, cosh, tanh, coth, asinh, acosh from sympy.functions import sqrt, erf, erfi, li, Ei from sympy.functions import besselj, bessely, besseli, besselk from sympy.functions import hankel1, hankel2, jn, yn from sympy.functions.elementary.complexes import Abs, re, im, sign, arg from sympy.functions.elementary.exponential import LambertW from sympy.functions.elementary.integers import floor, ceiling from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.special.delta_functions import Heaviside, DiracDelta from sympy.simplify.radsimp import collect from sympy.logic.boolalg import And, Or from sympy.utilities.iterables import uniq from sympy.polys import quo, gcd, lcm, factor, cancel, PolynomialError from sympy.polys.monomials import itermonomials from sympy.polys.polyroots import root_factors from sympy.polys.rings import PolyRing from sympy.polys.solvers import solve_lin_sys from sympy.polys.constructor import construct_domain from sympy.core.compatibility import reduce, ordered from sympy.integrals.integrals import integrate def components(f, x): """ Returns a set of all functional components of the given expression which includes symbols, function applications and compositions and non-integer powers. Fractional powers are collected with minimal, positive exponents. Examples ======== >>> from sympy import cos, sin >>> from sympy.abc import x >>> from sympy.integrals.heurisch import components >>> components(sin(x)*cos(x)**2, x) {x, sin(x), cos(x)} See Also ======== heurisch """ result = set() if x in f.free_symbols: if f.is_symbol and f.is_commutative: result.add(f) elif f.is_Function or f.is_Derivative: for g in f.args: result |= components(g, x) result.add(f) elif f.is_Pow: result |= components(f.base, x) if not f.exp.is_Integer: if f.exp.is_Rational: result.add(f.base**Rational(1, f.exp.q)) else: result |= components(f.exp, x) | {f} else: for g in f.args: result |= components(g, x) return result # name -> [] of symbols _symbols_cache = {} # type: Dict[str, List[Dummy]] # NB @cacheit is not convenient here def _symbols(name, n): """get vector of symbols local to this module""" try: lsyms = _symbols_cache[name] except KeyError: lsyms = [] _symbols_cache[name] = lsyms while len(lsyms) < n: lsyms.append( Dummy('%s%i' % (name, len(lsyms))) ) return lsyms[:n] def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None, _try_heurisch=None): """ A wrapper around the heurisch integration algorithm. Explanation =========== This method takes the result from heurisch and checks for poles in the denominator. For each of these poles, the integral is reevaluated, and the final integration result is given in terms of a Piecewise. Examples ======== >>> from sympy.core import symbols >>> from sympy.functions import cos >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper >>> n, x = symbols('n x') >>> heurisch(cos(n*x), x) sin(n*x)/n >>> heurisch_wrapper(cos(n*x), x) Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True)) See Also ======== heurisch """ from sympy.solvers.solvers import solve, denoms f = sympify(f) if x not in f.free_symbols: return f*x res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations, _try_heurisch) if not isinstance(res, Basic): return res # We consider each denominator in the expression, and try to find # cases where one or more symbolic denominator might be zero. The # conditions for these cases are stored in the list slns. slns = [] for d in denoms(res): try: slns += solve(d, dict=True, exclude=(x,)) except NotImplementedError: pass if not slns: return res slns = list(uniq(slns)) # Remove the solutions corresponding to poles in the original expression. slns0 = [] for d in denoms(f): try: slns0 += solve(d, dict=True, exclude=(x,)) except NotImplementedError: pass slns = [s for s in slns if s not in slns0] if not slns: return res if len(slns) > 1: eqs = [] for sub_dict in slns: eqs.extend([Eq(key, value) for key, value in sub_dict.items()]) slns = solve(eqs, dict=True, exclude=(x,)) + slns # For each case listed in the list slns, we reevaluate the integral. pairs = [] for sub_dict in slns: expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations, _try_heurisch) cond = And(*[Eq(key, value) for key, value in sub_dict.items()]) generic = Or(*[Ne(key, value) for key, value in sub_dict.items()]) if expr is None: expr = integrate(f.subs(sub_dict),x) pairs.append((expr, cond)) # If there is one condition, put the generic case first. Otherwise, # doing so may lead to longer Piecewise formulas if len(pairs) == 1: pairs = [(heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations, _try_heurisch), generic), (pairs[0][0], True)] else: pairs.append((heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations, _try_heurisch), True)) return Piecewise(*pairs) class BesselTable: """ Derivatives of Bessel functions of orders n and n-1 in terms of each other. See the docstring of DiffCache. """ def __init__(self): self.table = {} self.n = Dummy('n') self.z = Dummy('z') self._create_table() def _create_table(t): table, n, z = t.table, t.n, t.z for f in (besselj, bessely, hankel1, hankel2): table[f] = (f(n-1, z) - n*f(n, z)/z, (n-1)*f(n-1, z)/z - f(n, z)) f = besseli table[f] = (f(n-1, z) - n*f(n, z)/z, (n-1)*f(n-1, z)/z + f(n, z)) f = besselk table[f] = (-f(n-1, z) - n*f(n, z)/z, (n-1)*f(n-1, z)/z - f(n, z)) for f in (jn, yn): table[f] = (f(n-1, z) - (n+1)*f(n, z)/z, (n-1)*f(n-1, z)/z - f(n, z)) def diffs(t, f, n, z): if f in t.table: diff0, diff1 = t.table[f] repl = [(t.n, n), (t.z, z)] return (diff0.subs(repl), diff1.subs(repl)) def has(t, f): return f in t.table _bessel_table = None class DiffCache: """ Store for derivatives of expressions. Explanation =========== The standard form of the derivative of a Bessel function of order n contains two Bessel functions of orders n-1 and n+1, respectively. Such forms cannot be used in parallel Risch algorithm, because there is a linear recurrence relation between the three functions while the algorithm expects that functions and derivatives are represented in terms of algebraically independent transcendentals. The solution is to take two of the functions, e.g., those of orders n and n-1, and to express the derivatives in terms of the pair. To guarantee that the proper form is used the two derivatives are cached as soon as one is encountered. Derivatives of other functions are also cached at no extra cost. All derivatives are with respect to the same variable `x`. """ def __init__(self, x): self.cache = {} self.x = x global _bessel_table if not _bessel_table: _bessel_table = BesselTable() def get_diff(self, f): cache = self.cache if f in cache: pass elif (not hasattr(f, 'func') or not _bessel_table.has(f.func)): cache[f] = cancel(f.diff(self.x)) else: n, z = f.args d0, d1 = _bessel_table.diffs(f.func, n, z) dz = self.get_diff(z) cache[f] = d0*dz cache[f.func(n-1, z)] = d1*dz return cache[f] def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None, _try_heurisch=None): """ Compute indefinite integral using heuristic Risch algorithm. Explanation =========== This is a heuristic approach to indefinite integration in finite terms using the extended heuristic (parallel) Risch algorithm, based on Manuel Bronstein's "Poor Man's Integrator". The algorithm supports various classes of functions including transcendental elementary or special functions like Airy, Bessel, Whittaker and Lambert. Note that this algorithm is not a decision procedure. If it isn't able to compute the antiderivative for a given function, then this is not a proof that such a functions does not exist. One should use recursive Risch algorithm in such case. It's an open question if this algorithm can be made a full decision procedure. This is an internal integrator procedure. You should use toplevel 'integrate' function in most cases, as this procedure needs some preprocessing steps and otherwise may fail. Specification ============= heurisch(f, x, rewrite=False, hints=None) where f : expression x : symbol rewrite -> force rewrite 'f' in terms of 'tan' and 'tanh' hints -> a list of functions that may appear in anti-derivate - hints = None --> no suggestions at all - hints = [ ] --> try to figure out - hints = [f1, ..., fn] --> we know better Examples ======== >>> from sympy import tan >>> from sympy.integrals.heurisch import heurisch >>> from sympy.abc import x, y >>> heurisch(y*tan(x), x) y*log(tan(x)**2 + 1)/2 See Manuel Bronstein's "Poor Man's Integrator": References ========== .. [1] http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html For more information on the implemented algorithm refer to: .. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration Method and its Implementation in Maple, Proceedings of ISSAC'89, ACM Press, 212-217. .. [3] J. H. Davenport, On the Parallel Risch Algorithm (I), Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157. .. [4] J. H. Davenport, On the Parallel Risch Algorithm (III): Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6. .. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch Algorithm (II), ACM Transactions on Mathematical Software 11 (1985), 356-362. See Also ======== sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral sympy.integrals.heurisch.components """ f = sympify(f) # There are some functions that Heurisch cannot currently handle, # so do not even try. # Set _try_heurisch=True to skip this check if _try_heurisch is not True: if f.has(Abs, re, im, sign, Heaviside, DiracDelta, floor, ceiling, arg): return if x not in f.free_symbols: return f*x if not f.is_Add: indep, f = f.as_independent(x) else: indep = S.One rewritables = { (sin, cos, cot): tan, (sinh, cosh, coth): tanh, } if rewrite: for candidates, rule in rewritables.items(): f = f.rewrite(candidates, rule) else: for candidates in rewritables.keys(): if f.has(*candidates): break else: rewrite = True terms = components(f, x) if hints is not None: if not hints: a = Wild('a', exclude=[x]) b = Wild('b', exclude=[x]) c = Wild('c', exclude=[x]) for g in set(terms): # using copy of terms if g.is_Function: if isinstance(g, li): M = g.args[0].match(a*x**b) if M is not None: terms.add( x*(li(M[a]*x**M[b]) - (M[a]*x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) ) #terms.add( x*(li(M[a]*x**M[b]) - (x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) ) #terms.add( x*(li(M[a]*x**M[b]) - x*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) ) #terms.add( li(M[a]*x**M[b]) - Ei((M[b]+1)*log(M[a]*x**M[b])/M[b]) ) elif isinstance(g, exp): M = g.args[0].match(a*x**2) if M is not None: if M[a].is_positive: terms.add(erfi(sqrt(M[a])*x)) else: # M[a].is_negative or unknown terms.add(erf(sqrt(-M[a])*x)) M = g.args[0].match(a*x**2 + b*x + c) if M is not None: if M[a].is_positive: terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))* erfi(sqrt(M[a])*x + M[b]/(2*sqrt(M[a])))) elif M[a].is_negative: terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))* erf(sqrt(-M[a])*x - M[b]/(2*sqrt(-M[a])))) M = g.args[0].match(a*log(x)**2) if M is not None: if M[a].is_positive: terms.add(erfi(sqrt(M[a])*log(x) + 1/(2*sqrt(M[a])))) if M[a].is_negative: terms.add(erf(sqrt(-M[a])*log(x) - 1/(2*sqrt(-M[a])))) elif g.is_Pow: if g.exp.is_Rational and g.exp.q == 2: M = g.base.match(a*x**2 + b) if M is not None and M[b].is_positive: if M[a].is_positive: terms.add(asinh(sqrt(M[a]/M[b])*x)) elif M[a].is_negative: terms.add(asin(sqrt(-M[a]/M[b])*x)) M = g.base.match(a*x**2 - b) if M is not None and M[b].is_positive: if M[a].is_positive: terms.add(acosh(sqrt(M[a]/M[b])*x)) elif M[a].is_negative: terms.add(-M[b]/2*sqrt(-M[a])* atan(sqrt(-M[a])*x/sqrt(M[a]*x**2 - M[b]))) else: terms |= set(hints) dcache = DiffCache(x) for g in set(terms): # using copy of terms terms |= components(dcache.get_diff(g), x) # TODO: caching is significant factor for why permutations work at all. Change this. V = _symbols('x', len(terms)) # sort mapping expressions from largest to smallest (last is always x). mapping = list(reversed(list(zip(*ordered( # [(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1])) # rev_mapping = {v: k for k, v in mapping} # if mappings is None: # # optimizing the number of permutations of mapping # assert mapping[-1][0] == x # if not, find it and correct this comment unnecessary_permutations = [mapping.pop(-1)] mappings = permutations(mapping) else: unnecessary_permutations = unnecessary_permutations or [] def _substitute(expr): return expr.subs(mapping) for mapping in mappings: mapping = list(mapping) mapping = mapping + unnecessary_permutations diffs = [ _substitute(dcache.get_diff(g)) for g in terms ] denoms = [ g.as_numer_denom()[1] for g in diffs ] if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V): denom = reduce(lambda p, q: lcm(p, q, *V), denoms) break else: if not rewrite: result = heurisch(f, x, rewrite=True, hints=hints, unnecessary_permutations=unnecessary_permutations) if result is not None: return indep*result return None numers = [ cancel(denom*g) for g in diffs ] def _derivation(h): return Add(*[ d * h.diff(v) for d, v in zip(numers, V) ]) def _deflation(p): for y in V: if not p.has(y): continue if _derivation(p) is not S.Zero: c, q = p.as_poly(y).primitive() return _deflation(c)*gcd(q, q.diff(y)).as_expr() return p def _splitter(p): for y in V: if not p.has(y): continue if _derivation(y) is not S.Zero: c, q = p.as_poly(y).primitive() q = q.as_expr() h = gcd(q, _derivation(q), y) s = quo(h, gcd(q, q.diff(y), y), y) c_split = _splitter(c) if s.as_poly(y).degree() == 0: return (c_split[0], q * c_split[1]) q_split = _splitter(cancel(q / s)) return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1]) return (S.One, p) special = {} for term in terms: if term.is_Function: if isinstance(term, tan): special[1 + _substitute(term)**2] = False elif isinstance(term, tanh): special[1 + _substitute(term)] = False special[1 - _substitute(term)] = False elif isinstance(term, LambertW): special[_substitute(term)] = True F = _substitute(f) P, Q = F.as_numer_denom() u_split = _splitter(denom) v_split = _splitter(Q) polys = set(list(v_split) + [ u_split[0] ] + list(special.keys())) s = u_split[0] * Mul(*[ k for k, v in special.items() if v ]) polified = [ p.as_poly(*V) for p in [s, P, Q] ] if None in polified: return None #--- definitions for _integrate a, b, c = [ p.total_degree() for p in polified ] poly_denom = (s * v_split[0] * _deflation(v_split[1])).as_expr() def _exponent(g): if g.is_Pow: if g.exp.is_Rational and g.exp.q != 1: if g.exp.p > 0: return g.exp.p + g.exp.q - 1 else: return abs(g.exp.p + g.exp.q) else: return 1 elif not g.is_Atom and g.args: return max([ _exponent(h) for h in g.args ]) else: return 1 A, B = _exponent(f), a + max(b, c) if A > 1 and B > 1: monoms = tuple(ordered(itermonomials(V, A + B - 1 + degree_offset))) else: monoms = tuple(ordered(itermonomials(V, A + B + degree_offset))) poly_coeffs = _symbols('A', len(monoms)) poly_part = Add(*[ poly_coeffs[i]*monomial for i, monomial in enumerate(monoms) ]) reducibles = set() for poly in polys: if poly.has(*V): try: factorization = factor(poly, greedy=True) except PolynomialError: factorization = poly if factorization.is_Mul: factors = factorization.args else: factors = (factorization, ) for fact in factors: if fact.is_Pow: reducibles.add(fact.base) else: reducibles.add(fact) def _integrate(field=None): irreducibles = set() atans = set() pairs = set() for poly in reducibles: for z in poly.free_symbols: if z in V: break # should this be: `irreducibles |= \ else: # set(root_factors(poly, z, filter=field))` continue # and the line below deleted? # | # V irreducibles |= set(root_factors(poly, z, filter=field)) log_part, atan_part = [], [] for poly in list(irreducibles): m = collect(poly, I, evaluate=False) y = m.get(I, S.Zero) if y: x = m.get(S.One, S.Zero) if x.has(I) or y.has(I): continue # nontrivial x + I*y pairs.add((x, y)) irreducibles.remove(poly) while pairs: x, y = pairs.pop() if (x, -y) in pairs: pairs.remove((x, -y)) # Choosing b with no minus sign if y.could_extract_minus_sign(): y = -y irreducibles.add(x*x + y*y) atans.add(atan(x/y)) else: irreducibles.add(x + I*y) B = _symbols('B', len(irreducibles)) C = _symbols('C', len(atans)) # Note: the ordering matters here for poly, b in reversed(list(zip(ordered(irreducibles), B))): if poly.has(*V): poly_coeffs.append(b) log_part.append(b * log(poly)) for poly, c in reversed(list(zip(ordered(atans), C))): if poly.has(*V): poly_coeffs.append(c) atan_part.append(c * poly) # TODO: Currently it's better to use symbolic expressions here instead # of rational functions, because it's simpler and FracElement doesn't # give big speed improvement yet. This is because cancellation is slow # due to slow polynomial GCD algorithms. If this gets improved then # revise this code. candidate = poly_part/poly_denom + Add(*log_part) + Add(*atan_part) h = F - _derivation(candidate) / denom raw_numer = h.as_numer_denom()[0] # Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field # that we have to determine. We can't use simply atoms() because log(3), # sqrt(y) and similar expressions can appear, leading to non-trivial # domains. syms = set(poly_coeffs) | set(V) non_syms = set() def find_non_syms(expr): if expr.is_Integer or expr.is_Rational: pass # ignore trivial numbers elif expr in syms: pass # ignore variables elif not expr.has(*syms): non_syms.add(expr) elif expr.is_Add or expr.is_Mul or expr.is_Pow: list(map(find_non_syms, expr.args)) else: # TODO: Non-polynomial expression. This should have been # filtered out at an earlier stage. raise PolynomialError try: find_non_syms(raw_numer) except PolynomialError: return None else: ground, _ = construct_domain(non_syms, field=True) coeff_ring = PolyRing(poly_coeffs, ground) ring = PolyRing(V, coeff_ring) try: numer = ring.from_expr(raw_numer) except ValueError: raise PolynomialError solution = solve_lin_sys(numer.coeffs(), coeff_ring, _raw=False) if solution is None: return None else: return candidate.subs(solution).subs( list(zip(poly_coeffs, [S.Zero]*len(poly_coeffs)))) if not (F.free_symbols - set(V)): solution = _integrate('Q') if solution is None: solution = _integrate() else: solution = _integrate() if solution is not None: antideriv = solution.subs(rev_mapping) antideriv = cancel(antideriv).expand(force=True) if antideriv.is_Add: antideriv = antideriv.as_independent(x)[1] return indep*antideriv else: if retries >= 0: result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) if result is not None: return indep*result return None
c6f185468b176acece711db7e0d4a7c7d0eed99dbc767743fbb0d1c9e5a10841
""" Algorithms for solving the Risch differential equation. Given a differential field K of characteristic 0 that is a simple monomial extension of a base field k and f, g in K, the Risch Differential Equation problem is to decide if there exist y in K such that Dy + f*y == g and to find one if there are some. If t is a monomial over k and the coefficients of f and g are in k(t), then y is in k(t), and the outline of the algorithm here is given as: 1. Compute the normal part n of the denominator of y. The problem is then reduced to finding y' in k<t>, where y == y'/n. 2. Compute the special part s of the denominator of y. The problem is then reduced to finding y'' in k[t], where y == y''/(n*s) 3. Bound the degree of y''. 4. Reduce the equation Dy + f*y == g to a similar equation with f, g in k[t]. 5. Find the solutions in k[t] of bounded degree of the reduced equation. See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by Manuel Bronstein. See also the docstring of risch.py. """ from operator import mul from sympy.core import oo from sympy.core.compatibility import reduce from sympy.core.symbol import Dummy from sympy.polys import Poly, gcd, ZZ, cancel from sympy import sqrt, re, im from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation, splitfactor, NonElementaryIntegralException, DecrementLevel, recognize_log_derivative) # TODO: Add messages to NonElementaryIntegralException errors def order_at(a, p, t): """ Computes the order of a at p, with respect to t. Explanation =========== For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n in Z+ such that p**n|a}), where a != 0. If a == 0, nu_p(a) = +oo. To compute the order at a rational function, a/b, use the fact that nu_p(a/b) == nu_p(a) - nu_p(b). """ if a.is_zero: return oo if p == Poly(t, t): return a.as_poly(t).ET()[0][0] # Uses binary search for calculating the power. power_list collects the tuples # (p^k,k) where each k is some power of 2. After deciding the largest k # such that k is power of 2 and p^k|a the loop iteratively calculates # the actual power. power_list = [] p1 = p r = a.rem(p1) tracks_power = 1 while r.is_zero: power_list.append((p1,tracks_power)) p1 = p1*p1 tracks_power *= 2 r = a.rem(p1) n = 0 product = Poly(1, t) while len(power_list) != 0: final = power_list.pop() productf = product*final[0] r = a.rem(productf) if r.is_zero: n += final[1] product = productf return n def order_at_oo(a, d, t): """ Computes the order of a/d at oo (infinity), with respect to t. For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where f == a/d. """ if a.is_zero: return oo return d.degree(t) - a.degree(t) def weak_normalizer(a, d, DE, z=None): """ Weak normalization. Explanation =========== Given a derivation D on k[t] and f == a/d in k(t), return q in k[t] such that f - Dq/q is weakly normalized with respect to t. f in k(t) is said to be "weakly normalized" with respect to t if residue_p(f) is not a positive integer for any normal irreducible p in k[t] such that f is in R_p (Definition 6.1.1). If f has an elementary integral, this is equivalent to no logarithm of integral(f) whose argument depends on t has a positive integer coefficient, where the arguments of the logarithms not in k(t) are in k[t]. Returns (q, f - Dq/q) """ z = z or Dummy('z') dn, ds = splitfactor(d, DE) # Compute d1, where dn == d1*d2**2*...*dn**n is a square-free # factorization of d. g = gcd(dn, dn.diff(DE.t)) d_sqf_part = dn.quo(g) d1 = d_sqf_part.quo(gcd(d_sqf_part, g)) a1, b = gcdex_diophantine(d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t), a.as_poly(DE.t)) r = (a - Poly(z, DE.t)*derivation(d1, DE)).as_poly(DE.t).resultant( d1.as_poly(DE.t)) r = Poly(r, z) if not r.expr.has(z): return (Poly(1, DE.t), (a, d)) N = [i for i in r.real_roots() if i in ZZ and i > 0] q = reduce(mul, [gcd(a - Poly(n, DE.t)*derivation(d1, DE), d1) for n in N], Poly(1, DE.t)) dq = derivation(q, DE) sn = q*a - d*dq sd = q*d sn, sd = sn.cancel(sd, include=True) return (q, (sn, sd)) def normal_denom(fa, fd, ga, gd, DE): """ Normal part of the denominator. Explanation =========== Given a derivation D on k[t] and f, g in k(t) with f weakly normalized with respect to t, either raise NonElementaryIntegralException, in which case the equation Dy + f*y == g has no solution in k(t), or the quadruplet (a, b, c, h) such that a, h in k[t], b, c in k<t>, and for any solution y in k(t) of Dy + f*y == g, q = y*h in k<t> satisfies a*Dq + b*q == c. This constitutes step 1 in the outline given in the rde.py docstring. """ dn, ds = splitfactor(fd, DE) en, es = splitfactor(gd, DE) p = dn.gcd(en) h = en.gcd(en.diff(DE.t)).quo(p.gcd(p.diff(DE.t))) a = dn*h c = a*h if c.div(en)[1]: # en does not divide dn*h**2 raise NonElementaryIntegralException ca = c*ga ca, cd = ca.cancel(gd, include=True) ba = a*fa - dn*derivation(h, DE)*fd ba, bd = ba.cancel(fd, include=True) # (dn*h, dn*h*f - dn*Dh, dn*h**2*g, h) return (a, (ba, bd), (ca, cd), h) def special_denom(a, ba, bd, ca, cd, DE, case='auto'): """ Special part of the denominator. Explanation =========== case is one of {'exp', 'tan', 'primitive'} for the hyperexponential, hypertangent, and primitive cases, respectively. For the hyperexponential (resp. hypertangent) case, given a derivation D on k[t] and a in k[t], b, c, in k<t> with Dt/t in k (resp. Dt/(t**2 + 1) in k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp. gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that A, B, C, h in k[t] and for any solution q in k<t> of a*Dq + b*q == c, r = qh in k[t] satisfies A*Dr + B*r == C. For ``case == 'primitive'``, k<t> == k[t], so it returns (a, b, c, 1) in this case. This constitutes step 2 of the outline given in the rde.py docstring. """ from sympy.integrals.prde import parametric_log_deriv # TODO: finish writing this and write tests if case == 'auto': case = DE.case if case == 'exp': p = Poly(DE.t, DE.t) elif case == 'tan': p = Poly(DE.t**2 + 1, DE.t) elif case in ['primitive', 'base']: B = ba.to_field().quo(bd) C = ca.to_field().quo(cd) return (a, B, C, Poly(1, DE.t)) else: raise ValueError("case must be one of {'exp', 'tan', 'primitive', " "'base'}, not %s." % case) nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t) nc = order_at(ca, p, DE.t) - order_at(cd, p, DE.t) n = min(0, nc - min(0, nb)) if not nb: # Possible cancellation. if case == 'exp': dcoeff = DE.d.quo(Poly(DE.t, DE.t)) with DecrementLevel(DE): # We are guaranteed to not have problems, # because case != 'base'. alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t) etaa, etad = frac_in(dcoeff, DE.t) A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) if A is not None: Q, m, z = A if Q == 1: n = min(n, m) elif case == 'tan': dcoeff = DE.d.quo(Poly(DE.t**2+1, DE.t)) with DecrementLevel(DE): # We are guaranteed to not have problems, # because case != 'base'. alphaa, alphad = frac_in(im(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t) betaa, betad = frac_in(re(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t) etaa, etad = frac_in(dcoeff, DE.t) if recognize_log_derivative(Poly(2, DE.t)*betaa, betad, DE): A = parametric_log_deriv(alphaa*Poly(sqrt(-1), DE.t)*betad+alphad*betaa, alphad*betad, etaa, etad, DE) if A is not None: Q, m, z = A if Q == 1: n = min(n, m) N = max(0, -nb, n - nc) pN = p**N pn = p**-n A = a*pN B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN C = (ca*pN*pn).quo(cd) h = pn # (a*p**N, (b + n*a*Dp/p)*p**N, c*p**(N - n), p**-n) return (A, B, C, h) def bound_degree(a, b, cQ, DE, case='auto', parametric=False): """ Bound on polynomial solutions. Explanation =========== Given a derivation D on k[t] and ``a``, ``b``, ``c`` in k[t] with ``a != 0``, return n in ZZ such that deg(q) <= n for any solution q in k[t] of a*Dq + b*q == c, when parametric=False, or deg(q) <= n for any solution c1, ..., cm in Const(k) and q in k[t] of a*Dq + b*q == Sum(ci*gi, (i, 1, m)) when parametric=True. For ``parametric=False``, ``cQ`` is ``c``, a ``Poly``; for ``parametric=True``, ``cQ`` is Q == [q1, ..., qm], a list of Polys. This constitutes step 3 of the outline given in the rde.py docstring. """ from sympy.integrals.prde import (parametric_log_deriv, limited_integrate, is_log_deriv_k_t_radical_in_field) # TODO: finish writing this and write tests if case == 'auto': case = DE.case da = a.degree(DE.t) db = b.degree(DE.t) # The parametric and regular cases are identical, except for this part if parametric: dc = max([i.degree(DE.t) for i in cQ]) else: dc = cQ.degree(DE.t) alpha = cancel(-b.as_poly(DE.t).LC().as_expr()/ a.as_poly(DE.t).LC().as_expr()) if case == 'base': n = max(0, dc - max(db, da - 1)) if db == da - 1 and alpha.is_Integer: n = max(0, alpha, dc - db) elif case == 'primitive': if db > da: n = max(0, dc - db) else: n = max(0, dc - da + 1) etaa, etad = frac_in(DE.d, DE.T[DE.level - 1]) t1 = DE.t with DecrementLevel(DE): alphaa, alphad = frac_in(alpha, DE.t) if db == da - 1: # if alpha == m*Dt + Dz for z in k and m in ZZ: try: (za, zd), m = limited_integrate(alphaa, alphad, [(etaa, etad)], DE) except NonElementaryIntegralException: pass else: if len(m) != 1: raise ValueError("Length of m should be 1") n = max(n, m[0]) elif db == da: # if alpha == Dz/z for z in k*: # beta = -lc(a*Dz + b*z)/(z*lc(a)) # if beta == m*Dt + Dw for w in k and m in ZZ: # n = max(n, m) A = is_log_deriv_k_t_radical_in_field(alphaa, alphad, DE) if A is not None: aa, z = A if aa == 1: beta = -(a*derivation(z, DE).as_poly(t1) + b*z.as_poly(t1)).LC()/(z.as_expr()*a.LC()) betaa, betad = frac_in(beta, DE.t) try: (za, zd), m = limited_integrate(betaa, betad, [(etaa, etad)], DE) except NonElementaryIntegralException: pass else: if len(m) != 1: raise ValueError("Length of m should be 1") n = max(n, m[0]) elif case == 'exp': n = max(0, dc - max(db, da)) if da == db: etaa, etad = frac_in(DE.d.quo(Poly(DE.t, DE.t)), DE.T[DE.level - 1]) with DecrementLevel(DE): alphaa, alphad = frac_in(alpha, DE.t) A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) if A is not None: # if alpha == m*Dt/t + Dz/z for z in k* and m in ZZ: # n = max(n, m) a, m, z = A if a == 1: n = max(n, m) elif case in ['tan', 'other_nonlinear']: delta = DE.d.degree(DE.t) lam = DE.d.LC() alpha = cancel(alpha/lam) n = max(0, dc - max(da + delta - 1, db)) if db == da + delta - 1 and alpha.is_Integer: n = max(0, alpha, dc - db) else: raise ValueError("case must be one of {'exp', 'tan', 'primitive', " "'other_nonlinear', 'base'}, not %s." % case) return n def spde(a, b, c, n, DE): """ Rothstein's Special Polynomial Differential Equation algorithm. Explanation =========== Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with ``a != 0``, either raise NonElementaryIntegralException, in which case the equation a*Dq + b*q == c has no solution of degree at most ``n`` in k[t], or return the tuple (B, C, m, alpha, beta) such that B, C, alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree at most n of a*Dq + b*q == c must be of the form q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C. This constitutes step 4 of the outline given in the rde.py docstring. """ zero = Poly(0, DE.t) alpha = Poly(1, DE.t) beta = Poly(0, DE.t) while True: if c.is_zero: return (zero, zero, 0, zero, beta) # -1 is more to the point if (n < 0) is True: raise NonElementaryIntegralException g = a.gcd(b) if not c.rem(g).is_zero: # g does not divide c raise NonElementaryIntegralException a, b, c = a.quo(g), b.quo(g), c.quo(g) if a.degree(DE.t) == 0: b = b.to_field().quo(a) c = c.to_field().quo(a) return (b, c, n, alpha, beta) r, z = gcdex_diophantine(b, a, c) b += derivation(a, DE) c = z - derivation(r, DE) n -= a.degree(DE.t) beta += alpha * r alpha *= a def no_cancel_b_large(b, c, n, DE): """ Poly Risch Differential Equation - No cancellation: deg(b) large enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with ``b != 0`` and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation ``Dq + b*q == c`` has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with ``deg(q) < n``. """ q = Poly(0, DE.t) while not c.is_zero: m = c.degree(DE.t) - b.degree(DE.t) if not 0 <= m <= n: # n < 0 or m < 0 or m > n raise NonElementaryIntegralException p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC()*DE.t**m, DE.t, expand=False) q = q + p n = m - 1 c = c - derivation(p, DE) - b*p return q def no_cancel_b_small(b, c, n, DE): """ Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any solution q in k[t] of degree at most n of Dq + bq == c, y == q - h is a solution in k of Dy + b0*y == c0. """ q = Poly(0, DE.t) while not c.is_zero: if n == 0: m = 0 else: m = c.degree(DE.t) - DE.d.degree(DE.t) + 1 if not 0 <= m <= n: # n < 0 or m < 0 or m > n raise NonElementaryIntegralException if m > 0: p = Poly(c.as_poly(DE.t).LC()/(m*DE.d.as_poly(DE.t).LC())*DE.t**m, DE.t, expand=False) else: if b.degree(DE.t) != c.degree(DE.t): raise NonElementaryIntegralException if b.degree(DE.t) == 0: return (q, b.as_poly(DE.T[DE.level - 1]), c.as_poly(DE.T[DE.level - 1])) p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC(), DE.t, expand=False) q = q + p n = m - 1 c = c - derivation(p, DE) - b*p return q # TODO: better name for this function def no_cancel_equal(b, c, n, DE): """ Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1 Explanation =========== Given a derivation D on k[t] with deg(D) >= 2, n either an integer or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, m, C) such that h in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of degree at most n of Dq + b*q == c, y == q - h is a solution in k[t] of degree at most m of Dy + b*y == C. """ q = Poly(0, DE.t) lc = cancel(-b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC()) if lc.is_Integer and lc.is_positive: M = lc else: M = -1 while not c.is_zero: m = max(M, c.degree(DE.t) - DE.d.degree(DE.t) + 1) if not 0 <= m <= n: # n < 0 or m < 0 or m > n raise NonElementaryIntegralException u = cancel(m*DE.d.as_poly(DE.t).LC() + b.as_poly(DE.t).LC()) if u.is_zero: return (q, m, c) if m > 0: p = Poly(c.as_poly(DE.t).LC()/u*DE.t**m, DE.t, expand=False) else: if c.degree(DE.t) != DE.d.degree(DE.t) - 1: raise NonElementaryIntegralException else: p = c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC() q = q + p n = m - 1 c = c - derivation(p, DE) - b*p return q def cancel_primitive(b, c, n, DE): """ Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. """ from sympy.integrals.prde import is_log_deriv_k_t_radical_in_field with DecrementLevel(DE): ba, bd = frac_in(b, DE.t) A = is_log_deriv_k_t_radical_in_field(ba, bd, DE) if A is not None: n, z = A if n == 1: # b == Dz/z raise NotImplementedError("is_deriv_in_field() is required to " " solve this problem.") # if z*c == Dp for p in k[t] and deg(p) <= n: # return p/z # else: # raise NonElementaryIntegralException if c.is_zero: return c # return 0 if n < c.degree(DE.t): raise NonElementaryIntegralException q = Poly(0, DE.t) while not c.is_zero: m = c.degree(DE.t) if n < m: raise NonElementaryIntegralException with DecrementLevel(DE): a2a, a2d = frac_in(c.LC(), DE.t) sa, sd = rischDE(ba, bd, a2a, a2d, DE) stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False) q += stm n = m - 1 c -= b*stm + derivation(stm, DE) return q def cancel_exp(b, c, n, DE): """ Poly Risch Differential Equation - Cancellation: Hyperexponential case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt/t in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. """ from sympy.integrals.prde import parametric_log_deriv eta = DE.d.quo(Poly(DE.t, DE.t)).as_expr() with DecrementLevel(DE): etaa, etad = frac_in(eta, DE.t) ba, bd = frac_in(b, DE.t) A = parametric_log_deriv(ba, bd, etaa, etad, DE) if A is not None: a, m, z = A if a == 1: raise NotImplementedError("is_deriv_in_field() is required to " "solve this problem.") # if c*z*t**m == Dp for p in k<t> and q = p/(z*t**m) in k[t] and # deg(q) <= n: # return q # else: # raise NonElementaryIntegralException if c.is_zero: return c # return 0 if n < c.degree(DE.t): raise NonElementaryIntegralException q = Poly(0, DE.t) while not c.is_zero: m = c.degree(DE.t) if n < m: raise NonElementaryIntegralException # a1 = b + m*Dt/t a1 = b.as_expr() with DecrementLevel(DE): # TODO: Write a dummy function that does this idiom a1a, a1d = frac_in(a1, DE.t) a1a = a1a*etad + etaa*a1d*Poly(m, DE.t) a1d = a1d*etad a2a, a2d = frac_in(c.LC(), DE.t) sa, sd = rischDE(a1a, a1d, a2a, a2d, DE) stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False) q += stm n = m - 1 c -= b*stm + derivation(stm, DE) # deg(c) becomes smaller return q def solve_poly_rde(b, cQ, n, DE, parametric=False): """ Solve a Polynomial Risch Differential Equation with degree bound ``n``. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. """ from sympy.integrals.prde import (prde_no_cancel_b_large, prde_no_cancel_b_small) # No cancellation if not b.is_zero and (DE.case == 'base' or b.degree(DE.t) > max(0, DE.d.degree(DE.t) - 1)): if parametric: return prde_no_cancel_b_large(b, cQ, n, DE) return no_cancel_b_large(b, cQ, n, DE) elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \ (DE.case == 'base' or DE.d.degree(DE.t) >= 2): if parametric: return prde_no_cancel_b_small(b, cQ, n, DE) R = no_cancel_b_small(b, cQ, n, DE) if isinstance(R, Poly): return R else: # XXX: Might k be a field? (pg. 209) h, b0, c0 = R with DecrementLevel(DE): b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t) if b0 is None: # See above comment raise ValueError("b0 should be a non-Null value") if c0 is None: raise ValueError("c0 should be a non-Null value") y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t) return h + y elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \ n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC(): # TODO: Is this check necessary, and if so, what should it do if it fails? # b comes from the first element returned from spde() if not b.as_poly(DE.t).LC().is_number: raise TypeError("Result should be a number") if parametric: raise NotImplementedError("prde_no_cancel_b_equal() is not yet " "implemented.") R = no_cancel_equal(b, cQ, n, DE) if isinstance(R, Poly): return R else: h, m, C = R # XXX: Or should it be rischDE()? y = solve_poly_rde(b, C, m, DE) return h + y else: # Cancellation if b.is_zero: raise NotImplementedError("Remaining cases for Poly (P)RDE are " "not yet implemented (is_deriv_in_field() required).") else: if DE.case == 'exp': if parametric: raise NotImplementedError("Parametric RDE cancellation " "hyperexponential case is not yet implemented.") return cancel_exp(b, cQ, n, DE) elif DE.case == 'primitive': if parametric: raise NotImplementedError("Parametric RDE cancellation " "primitive case is not yet implemented.") return cancel_primitive(b, cQ, n, DE) else: raise NotImplementedError("Other Poly (P)RDE cancellation " "cases are not yet implemented (%s)." % DE.case) if parametric: raise NotImplementedError("Remaining cases for Poly PRDE not yet " "implemented.") raise NotImplementedError("Remaining cases for Poly RDE not yet " "implemented.") def rischDE(fa, fd, ga, gd, DE): """ Solve a Risch Differential Equation: Dy + f*y == g. Explanation =========== See the outline in the docstring of rde.py for more information about the procedure used. Either raise NonElementaryIntegralException, in which case there is no solution y in the given differential field, or return y in k(t) satisfying Dy + f*y == g, or raise NotImplementedError, in which case, the algorithms necessary to solve the given Risch Differential Equation have not yet been implemented. """ _, (fa, fd) = weak_normalizer(fa, fd, DE) a, (ba, bd), (ca, cd), hn = normal_denom(fa, fd, ga, gd, DE) A, B, C, hs = special_denom(a, ba, bd, ca, cd, DE) try: # Until this is fully implemented, use oo. Note that this will almost # certainly cause non-termination in spde() (unless A == 1), and # *might* lead to non-termination in the next step for a nonelementary # integral (I don't know for certain yet). Fortunately, spde() is # currently written recursively, so this will just give # RuntimeError: maximum recursion depth exceeded. n = bound_degree(A, B, C, DE) except NotImplementedError: # Useful for debugging: # import warnings # warnings.warn("rischDE: Proceeding with n = oo; may cause " # "non-termination.") n = oo B, C, m, alpha, beta = spde(A, B, C, n, DE) if C.is_zero: y = C else: y = solve_poly_rde(B, C, m, DE) return (alpha*y + beta, hn*hs)
c07653a8ffab2efbb7bf1c3f2988ab6cb872d6c9fca907b4d9455fb50c51df57
""" The Risch Algorithm for transcendental function integration. The core algorithms for the Risch algorithm are here. The subproblem algorithms are in the rde.py and prde.py files for the Risch Differential Equation solver and the parametric problems solvers, respectively. All important information concerning the differential extension for an integrand is stored in a DifferentialExtension object, which in the code is usually called DE. Throughout the code and Inside the DifferentialExtension object, the conventions/attribute names are that the base domain is QQ and each differential extension is x, t0, t1, ..., tn-1 = DE.t. DE.x is the variable of integration (Dx == 1), DE.D is a list of the derivatives of x, t1, t2, ..., tn-1 = t, DE.T is the list [x, t1, t2, ..., tn-1], DE.t is the outer-most variable of the differential extension at the given level (the level can be adjusted using DE.increment_level() and DE.decrement_level()), k is the field C(x, t0, ..., tn-2), where C is the constant field. The numerator of a fraction is denoted by a and the denominator by d. If the fraction is named f, fa == numer(f) and fd == denom(f). Fractions are returned as tuples (fa, fd). DE.d and DE.t are used to represent the topmost derivation and extension variable, respectively. The docstring of a function signifies whether an argument is in k[t], in which case it will just return a Poly in t, or in k(t), in which case it will return the fraction (fa, fd). Other variable names probably come from the names used in Bronstein's book. """ from sympy import real_roots, default_sort_key from sympy.abc import z from sympy.core.function import Lambda from sympy.core.numbers import ilcm, oo, I from sympy.core.mul import Mul from sympy.core.power import Pow from sympy.core.relational import Ne from sympy.core.singleton import S from sympy.core.symbol import Symbol, Dummy from sympy.core.compatibility import reduce, ordered from sympy.integrals.heurisch import _symbols from sympy.functions import (acos, acot, asin, atan, cos, cot, exp, log, Piecewise, sin, tan) from sympy.functions import sinh, cosh, tanh, coth from sympy.integrals import Integral, integrate from sympy.polys import gcd, cancel, PolynomialError, Poly, reduced, RootSum, DomainError from sympy.utilities.iterables import numbered_symbols from types import GeneratorType def integer_powers(exprs): """ Rewrites a list of expressions as integer multiples of each other. Explanation =========== For example, if you have [x, x/2, x**2 + 1, 2*x/3], then you can rewrite this as [(x/6) * 6, (x/6) * 3, (x**2 + 1) * 1, (x/6) * 4]. This is useful in the Risch integration algorithm, where we must write exp(x) + exp(x/2) as (exp(x/2))**2 + exp(x/2), but not as exp(x) + sqrt(exp(x)) (this is because only the transcendental case is implemented and we therefore cannot integrate algebraic extensions). The integer multiples returned by this function for each term are the smallest possible (their content equals 1). Returns a list of tuples where the first element is the base term and the second element is a list of `(item, factor)` terms, where `factor` is the integer multiplicative factor that must multiply the base term to obtain the original item. The easiest way to understand this is to look at an example: >>> from sympy.abc import x >>> from sympy.integrals.risch import integer_powers >>> integer_powers([x, x/2, x**2 + 1, 2*x/3]) [(x/6, [(x, 6), (x/2, 3), (2*x/3, 4)]), (x**2 + 1, [(x**2 + 1, 1)])] We can see how this relates to the example at the beginning of the docstring. It chose x/6 as the first base term. Then, x can be written as (x/2) * 2, so we get (0, 2), and so on. Now only element (x**2 + 1) remains, and there are no other terms that can be written as a rational multiple of that, so we get that it can be written as (x**2 + 1) * 1. """ # Here is the strategy: # First, go through each term and determine if it can be rewritten as a # rational multiple of any of the terms gathered so far. # cancel(a/b).is_Rational is sufficient for this. If it is a multiple, we # add its multiple to the dictionary. terms = {} for term in exprs: for j in terms: a = cancel(term/j) if a.is_Rational: terms[j].append((term, a)) break else: terms[term] = [(term, S.One)] # After we have done this, we have all the like terms together, so we just # need to find a common denominator so that we can get the base term and # integer multiples such that each term can be written as an integer # multiple of the base term, and the content of the integers is 1. newterms = {} for term in terms: common_denom = reduce(ilcm, [i.as_numer_denom()[1] for _, i in terms[term]]) newterm = term/common_denom newmults = [(i, j*common_denom) for i, j in terms[term]] newterms[newterm] = newmults return sorted(iter(newterms.items()), key=lambda item: item[0].sort_key()) class DifferentialExtension: """ A container for all the information relating to a differential extension. Explanation =========== The attributes of this object are (see also the docstring of __init__): - f: The original (Expr) integrand. - x: The variable of integration. - T: List of variables in the extension. - D: List of derivations in the extension; corresponds to the elements of T. - fa: Poly of the numerator of the integrand. - fd: Poly of the denominator of the integrand. - Tfuncs: Lambda() representations of each element of T (except for x). For back-substitution after integration. - backsubs: A (possibly empty) list of further substitutions to be made on the final integral to make it look more like the integrand. - exts: - extargs: - cases: List of string representations of the cases of T. - t: The top level extension variable, as defined by the current level (see level below). - d: The top level extension derivation, as defined by the current derivation (see level below). - case: The string representation of the case of self.d. (Note that self.T and self.D will always contain the complete extension, regardless of the level. Therefore, you should ALWAYS use DE.t and DE.d instead of DE.T[-1] and DE.D[-1]. If you want to have a list of the derivations or variables only up to the current level, use DE.D[:len(DE.D) + DE.level + 1] and DE.T[:len(DE.T) + DE.level + 1]. Note that, in particular, the derivation() function does this.) The following are also attributes, but will probably not be useful other than in internal use: - newf: Expr form of fa/fd. - level: The number (between -1 and -len(self.T)) such that self.T[self.level] == self.t and self.D[self.level] == self.d. Use the methods self.increment_level() and self.decrement_level() to change the current level. """ # __slots__ is defined mainly so we can iterate over all the attributes # of the class easily (the memory use doesn't matter too much, since we # only create one DifferentialExtension per integration). Also, it's nice # to have a safeguard when debugging. __slots__ = ('f', 'x', 'T', 'D', 'fa', 'fd', 'Tfuncs', 'backsubs', 'exts', 'extargs', 'cases', 'case', 't', 'd', 'newf', 'level', 'ts', 'dummy') def __init__(self, f=None, x=None, handle_first='log', dummy=False, extension=None, rewrite_complex=None): """ Tries to build a transcendental extension tower from ``f`` with respect to ``x``. Explanation =========== If it is successful, creates a DifferentialExtension object with, among others, the attributes fa, fd, D, T, Tfuncs, and backsubs such that fa and fd are Polys in T[-1] with rational coefficients in T[:-1], fa/fd == f, and D[i] is a Poly in T[i] with rational coefficients in T[:i] representing the derivative of T[i] for each i from 1 to len(T). Tfuncs is a list of Lambda objects for back replacing the functions after integrating. Lambda() is only used (instead of lambda) to make them easier to test and debug. Note that Tfuncs corresponds to the elements of T, except for T[0] == x, but they should be back-substituted in reverse order. backsubs is a (possibly empty) back-substitution list that should be applied on the completed integral to make it look more like the original integrand. If it is unsuccessful, it raises NotImplementedError. You can also create an object by manually setting the attributes as a dictionary to the extension keyword argument. You must include at least D. Warning, any attribute that is not given will be set to None. The attributes T, t, d, cases, case, x, and level are set automatically and do not need to be given. The functions in the Risch Algorithm will NOT check to see if an attribute is None before using it. This also does not check to see if the extension is valid (non-algebraic) or even if it is self-consistent. Therefore, this should only be used for testing/debugging purposes. """ # XXX: If you need to debug this function, set the break point here if extension: if 'D' not in extension: raise ValueError("At least the key D must be included with " "the extension flag to DifferentialExtension.") for attr in extension: setattr(self, attr, extension[attr]) self._auto_attrs() return elif f is None or x is None: raise ValueError("Either both f and x or a manual extension must " "be given.") if handle_first not in ['log', 'exp']: raise ValueError("handle_first must be 'log' or 'exp', not %s." % str(handle_first)) # f will be the original function, self.f might change if we reset # (e.g., we pull out a constant from an exponential) self.f = f self.x = x # setting the default value 'dummy' self.dummy = dummy self.reset() exp_new_extension, log_new_extension = True, True # case of 'automatic' choosing if rewrite_complex is None: rewrite_complex = I in self.f.atoms() if rewrite_complex: rewritables = { (sin, cos, cot, tan, sinh, cosh, coth, tanh): exp, (asin, acos, acot, atan): log, } # rewrite the trigonometric components for candidates, rule in rewritables.items(): self.newf = self.newf.rewrite(candidates, rule) self.newf = cancel(self.newf) else: if any(i.has(x) for i in self.f.atoms(sin, cos, tan, atan, asin, acos)): raise NotImplementedError("Trigonometric extensions are not " "supported (yet!)") exps = set() pows = set() numpows = set() sympows = set() logs = set() symlogs = set() while True: if self.newf.is_rational_function(*self.T): break if not exp_new_extension and not log_new_extension: # We couldn't find a new extension on the last pass, so I guess # we can't do it. raise NotImplementedError("Couldn't find an elementary " "transcendental extension for %s. Try using a " % str(f) + "manual extension with the extension flag.") exps, pows, numpows, sympows, log_new_extension = \ self._rewrite_exps_pows(exps, pows, numpows, sympows, log_new_extension) logs, symlogs = self._rewrite_logs(logs, symlogs) if handle_first == 'exp' or not log_new_extension: exp_new_extension = self._exp_part(exps) if exp_new_extension is None: # reset and restart self.f = self.newf self.reset() exp_new_extension = True continue if handle_first == 'log' or not exp_new_extension: log_new_extension = self._log_part(logs) self.fa, self.fd = frac_in(self.newf, self.t) self._auto_attrs() return def __getattr__(self, attr): # Avoid AttributeErrors when debugging if attr not in self.__slots__: raise AttributeError("%s has no attribute %s" % (repr(self), repr(attr))) return None def _rewrite_exps_pows(self, exps, pows, numpows, sympows, log_new_extension): """ Rewrite exps/pows for better processing. """ # Pre-preparsing. ################# # Get all exp arguments, so we can avoid ahead of time doing # something like t1 = exp(x), t2 = exp(x/2) == sqrt(t1). # Things like sqrt(exp(x)) do not automatically simplify to # exp(x/2), so they will be viewed as algebraic. The easiest way # to handle this is to convert all instances of (a**b)**Rational # to a**(Rational*b) before doing anything else. Note that the # _exp_part code can generate terms of this form, so we do need to # do this at each pass (or else modify it to not do that). from sympy.integrals.prde import is_deriv_k ratpows = [i for i in self.newf.atoms(Pow).union(self.newf.atoms(exp)) if (i.base.is_Pow or isinstance(i.base, exp) and i.exp.is_Rational)] ratpows_repl = [ (i, i.base.base**(i.exp*i.base.exp)) for i in ratpows] self.backsubs += [(j, i) for i, j in ratpows_repl] self.newf = self.newf.xreplace(dict(ratpows_repl)) # To make the process deterministic, the args are sorted # so that functions with smaller op-counts are processed first. # Ties are broken with the default_sort_key. # XXX Although the method is deterministic no additional work # has been done to guarantee that the simplest solution is # returned and that it would be affected be using different # variables. Though it is possible that this is the case # one should know that it has not been done intentionally, so # further improvements may be possible. # TODO: This probably doesn't need to be completely recomputed at # each pass. exps = update_sets(exps, self.newf.atoms(exp), lambda i: i.exp.is_rational_function(*self.T) and i.exp.has(*self.T)) pows = update_sets(pows, self.newf.atoms(Pow), lambda i: i.exp.is_rational_function(*self.T) and i.exp.has(*self.T)) numpows = update_sets(numpows, set(pows), lambda i: not i.base.has(*self.T)) sympows = update_sets(sympows, set(pows) - set(numpows), lambda i: i.base.is_rational_function(*self.T) and not i.exp.is_Integer) # The easiest way to deal with non-base E powers is to convert them # into base E, integrate, and then convert back. for i in ordered(pows): old = i new = exp(i.exp*log(i.base)) # If exp is ever changed to automatically reduce exp(x*log(2)) # to 2**x, then this will break. The solution is to not change # exp to do that :) if i in sympows: if i.exp.is_Rational: raise NotImplementedError("Algebraic extensions are " "not supported (%s)." % str(i)) # We can add a**b only if log(a) in the extension, because # a**b == exp(b*log(a)). basea, based = frac_in(i.base, self.t) A = is_deriv_k(basea, based, self) if A is None: # Nonelementary monomial (so far) # TODO: Would there ever be any benefit from just # adding log(base) as a new monomial? # ANSWER: Yes, otherwise we can't integrate x**x (or # rather prove that it has no elementary integral) # without first manually rewriting it as exp(x*log(x)) self.newf = self.newf.xreplace({old: new}) self.backsubs += [(new, old)] log_new_extension = self._log_part([log(i.base)]) exps = update_sets(exps, self.newf.atoms(exp), lambda i: i.exp.is_rational_function(*self.T) and i.exp.has(*self.T)) continue ans, u, const = A newterm = exp(i.exp*(log(const) + u)) # Under the current implementation, exp kills terms # only if they are of the form a*log(x), where a is a # Number. This case should have already been killed by the # above tests. Again, if this changes to kill more than # that, this will break, which maybe is a sign that you # shouldn't be changing that. Actually, if anything, this # auto-simplification should be removed. See # http://groups.google.com/group/sympy/browse_thread/thread/a61d48235f16867f self.newf = self.newf.xreplace({i: newterm}) elif i not in numpows: continue else: # i in numpows newterm = new # TODO: Just put it in self.Tfuncs self.backsubs.append((new, old)) self.newf = self.newf.xreplace({old: newterm}) exps.append(newterm) return exps, pows, numpows, sympows, log_new_extension def _rewrite_logs(self, logs, symlogs): """ Rewrite logs for better processing. """ atoms = self.newf.atoms(log) logs = update_sets(logs, atoms, lambda i: i.args[0].is_rational_function(*self.T) and i.args[0].has(*self.T)) symlogs = update_sets(symlogs, atoms, lambda i: i.has(*self.T) and i.args[0].is_Pow and i.args[0].base.is_rational_function(*self.T) and not i.args[0].exp.is_Integer) # We can handle things like log(x**y) by converting it to y*log(x) # This will fix not only symbolic exponents of the argument, but any # non-Integer exponent, like log(sqrt(x)). The exponent can also # depend on x, like log(x**x). for i in ordered(symlogs): # Unlike in the exponential case above, we do not ever # potentially add new monomials (above we had to add log(a)). # Therefore, there is no need to run any is_deriv functions # here. Just convert log(a**b) to b*log(a) and let # log_new_extension() handle it from there. lbase = log(i.args[0].base) logs.append(lbase) new = i.args[0].exp*lbase self.newf = self.newf.xreplace({i: new}) self.backsubs.append((new, i)) # remove any duplicates logs = sorted(set(logs), key=default_sort_key) return logs, symlogs def _auto_attrs(self): """ Set attributes that are generated automatically. """ if not self.T: # i.e., when using the extension flag and T isn't given self.T = [i.gen for i in self.D] if not self.x: self.x = self.T[0] self.cases = [get_case(d, t) for d, t in zip(self.D, self.T)] self.level = -1 self.t = self.T[self.level] self.d = self.D[self.level] self.case = self.cases[self.level] def _exp_part(self, exps): """ Try to build an exponential extension. Returns ======= Returns True if there was a new extension, False if there was no new extension but it was able to rewrite the given exponentials in terms of the existing extension, and None if the entire extension building process should be restarted. If the process fails because there is no way around an algebraic extension (e.g., exp(log(x)/2)), it will raise NotImplementedError. """ from sympy.integrals.prde import is_log_deriv_k_t_radical new_extension = False restart = False expargs = [i.exp for i in exps] ip = integer_powers(expargs) for arg, others in ip: # Minimize potential problems with algebraic substitution others.sort(key=lambda i: i[1]) arga, argd = frac_in(arg, self.t) A = is_log_deriv_k_t_radical(arga, argd, self) if A is not None: ans, u, n, const = A # if n is 1 or -1, it's algebraic, but we can handle it if n == -1: # This probably will never happen, because # Rational.as_numer_denom() returns the negative term in # the numerator. But in case that changes, reduce it to # n == 1. n = 1 u **= -1 const *= -1 ans = [(i, -j) for i, j in ans] if n == 1: # Example: exp(x + x**2) over QQ(x, exp(x), exp(x**2)) self.newf = self.newf.xreplace({exp(arg): exp(const)*Mul(*[ u**power for u, power in ans])}) self.newf = self.newf.xreplace({exp(p*exparg): exp(const*p) * Mul(*[u**power for u, power in ans]) for exparg, p in others}) # TODO: Add something to backsubs to put exp(const*p) # back together. continue else: # Bad news: we have an algebraic radical. But maybe we # could still avoid it by choosing a different extension. # For example, integer_powers() won't handle exp(x/2 + 1) # over QQ(x, exp(x)), but if we pull out the exp(1), it # will. Or maybe we have exp(x + x**2/2), over # QQ(x, exp(x), exp(x**2)), which is exp(x)*sqrt(exp(x**2)), # but if we use QQ(x, exp(x), exp(x**2/2)), then they will # all work. # # So here is what we do: If there is a non-zero const, pull # it out and retry. Also, if len(ans) > 1, then rewrite # exp(arg) as the product of exponentials from ans, and # retry that. If const == 0 and len(ans) == 1, then we # assume that it would have been handled by either # integer_powers() or n == 1 above if it could be handled, # so we give up at that point. For example, you can never # handle exp(log(x)/2) because it equals sqrt(x). if const or len(ans) > 1: rad = Mul(*[term**(power/n) for term, power in ans]) self.newf = self.newf.xreplace({exp(p*exparg): exp(const*p)*rad for exparg, p in others}) self.newf = self.newf.xreplace(dict(list(zip(reversed(self.T), reversed([f(self.x) for f in self.Tfuncs]))))) restart = True break else: # TODO: give algebraic dependence in error string raise NotImplementedError("Cannot integrate over " "algebraic extensions.") else: arga, argd = frac_in(arg, self.t) darga = (argd*derivation(Poly(arga, self.t), self) - arga*derivation(Poly(argd, self.t), self)) dargd = argd**2 darga, dargd = darga.cancel(dargd, include=True) darg = darga.as_expr()/dargd.as_expr() self.t = next(self.ts) self.T.append(self.t) self.extargs.append(arg) self.exts.append('exp') self.D.append(darg.as_poly(self.t, expand=False)*Poly(self.t, self.t, expand=False)) if self.dummy: i = Dummy("i") else: i = Symbol('i') self.Tfuncs += [Lambda(i, exp(arg.subs(self.x, i)))] self.newf = self.newf.xreplace( {exp(exparg): self.t**p for exparg, p in others}) new_extension = True if restart: return None return new_extension def _log_part(self, logs): """ Try to build a logarithmic extension. Returns ======= Returns True if there was a new extension and False if there was no new extension but it was able to rewrite the given logarithms in terms of the existing extension. Unlike with exponential extensions, there is no way that a logarithm is not transcendental over and cannot be rewritten in terms of an already existing extension in a non-algebraic way, so this function does not ever return None or raise NotImplementedError. """ from sympy.integrals.prde import is_deriv_k new_extension = False logargs = [i.args[0] for i in logs] for arg in ordered(logargs): # The log case is easier, because whenever a logarithm is algebraic # over the base field, it is of the form a1*t1 + ... an*tn + c, # which is a polynomial, so we can just replace it with that. # In other words, we don't have to worry about radicals. arga, argd = frac_in(arg, self.t) A = is_deriv_k(arga, argd, self) if A is not None: ans, u, const = A newterm = log(const) + u self.newf = self.newf.xreplace({log(arg): newterm}) continue else: arga, argd = frac_in(arg, self.t) darga = (argd*derivation(Poly(arga, self.t), self) - arga*derivation(Poly(argd, self.t), self)) dargd = argd**2 darg = darga.as_expr()/dargd.as_expr() self.t = next(self.ts) self.T.append(self.t) self.extargs.append(arg) self.exts.append('log') self.D.append(cancel(darg.as_expr()/arg).as_poly(self.t, expand=False)) if self.dummy: i = Dummy("i") else: i = Symbol('i') self.Tfuncs += [Lambda(i, log(arg.subs(self.x, i)))] self.newf = self.newf.xreplace({log(arg): self.t}) new_extension = True return new_extension @property def _important_attrs(self): """ Returns some of the more important attributes of self. Explanation =========== Used for testing and debugging purposes. The attributes are (fa, fd, D, T, Tfuncs, backsubs, exts, extargs). """ return (self.fa, self.fd, self.D, self.T, self.Tfuncs, self.backsubs, self.exts, self.extargs) # NOTE: this printing doesn't follow the Python's standard # eval(repr(DE)) == DE, where DE is the DifferentialExtension object # , also this printing is supposed to contain all the important # attributes of a DifferentialExtension object def __repr__(self): # no need to have GeneratorType object printed in it r = [(attr, getattr(self, attr)) for attr in self.__slots__ if not isinstance(getattr(self, attr), GeneratorType)] return self.__class__.__name__ + '(dict(%r))' % (r) # fancy printing of DifferentialExtension object def __str__(self): return (self.__class__.__name__ + '({fa=%s, fd=%s, D=%s})' % (self.fa, self.fd, self.D)) # should only be used for debugging purposes, internally # f1 = f2 = log(x) at different places in code execution # may return D1 != D2 as True, since 'level' or other attribute # may differ def __eq__(self, other): for attr in self.__class__.__slots__: d1, d2 = getattr(self, attr), getattr(other, attr) if not (isinstance(d1, GeneratorType) or d1 == d2): return False return True def reset(self): """ Reset self to an initial state. Used by __init__. """ self.t = self.x self.T = [self.x] self.D = [Poly(1, self.x)] self.level = -1 self.exts = [None] self.extargs = [None] if self.dummy: self.ts = numbered_symbols('t', cls=Dummy) else: # For testing self.ts = numbered_symbols('t') # For various things that we change to make things work that we need to # change back when we are done. self.backsubs = [] self.Tfuncs = [] self.newf = self.f def indices(self, extension): """ Parameters ========== extension : str Represents a valid extension type. Returns ======= list: A list of indices of 'exts' where extension of type 'extension' is present. Examples ======== >>> from sympy.integrals.risch import DifferentialExtension >>> from sympy import log, exp >>> from sympy.abc import x >>> DE = DifferentialExtension(log(x) + exp(x), x, handle_first='exp') >>> DE.indices('log') [2] >>> DE.indices('exp') [1] """ return [i for i, ext in enumerate(self.exts) if ext == extension] def increment_level(self): """ Increment the level of self. Explanation =========== This makes the working differential extension larger. self.level is given relative to the end of the list (-1, -2, etc.), so we don't need do worry about it when building the extension. """ if self.level >= -1: raise ValueError("The level of the differential extension cannot " "be incremented any further.") self.level += 1 self.t = self.T[self.level] self.d = self.D[self.level] self.case = self.cases[self.level] return None def decrement_level(self): """ Decrease the level of self. Explanation =========== This makes the working differential extension smaller. self.level is given relative to the end of the list (-1, -2, etc.), so we don't need do worry about it when building the extension. """ if self.level <= -len(self.T): raise ValueError("The level of the differential extension cannot " "be decremented any further.") self.level -= 1 self.t = self.T[self.level] self.d = self.D[self.level] self.case = self.cases[self.level] return None def update_sets(seq, atoms, func): s = set(seq) s = atoms.intersection(s) new = atoms - s s.update(list(filter(func, new))) return list(s) class DecrementLevel: """ A context manager for decrementing the level of a DifferentialExtension. """ __slots__ = ('DE',) def __init__(self, DE): self.DE = DE return def __enter__(self): self.DE.decrement_level() def __exit__(self, exc_type, exc_value, traceback): self.DE.increment_level() class NonElementaryIntegralException(Exception): """ Exception used by subroutines within the Risch algorithm to indicate to one another that the function being integrated does not have an elementary integral in the given differential field. """ # TODO: Rewrite algorithms below to use this (?) # TODO: Pass through information about why the integral was nonelementary, # and store that in the resulting NonElementaryIntegral somehow. pass def gcdex_diophantine(a, b, c): """ Extended Euclidean Algorithm, Diophantine version. Explanation =========== Given ``a``, ``b`` in K[x] and ``c`` in (a, b), the ideal generated by ``a`` and ``b``, return (s, t) such that s*a + t*b == c and either s == 0 or s.degree() < b.degree(). """ # Extended Euclidean Algorithm (Diophantine Version) pg. 13 # TODO: This should go in densetools.py. # XXX: Bettter name? s, g = a.half_gcdex(b) s *= c.exquo(g) # Inexact division means c is not in (a, b) if s and s.degree() >= b.degree(): _, s = s.div(b) t = (c - s*a).exquo(b) return (s, t) def frac_in(f, t, *, cancel=False, **kwargs): """ Returns the tuple (fa, fd), where fa and fd are Polys in t. Explanation =========== This is a common idiom in the Risch Algorithm functions, so we abstract it out here. ``f`` should be a basic expression, a Poly, or a tuple (fa, fd), where fa and fd are either basic expressions or Polys, and f == fa/fd. **kwargs are applied to Poly. """ if type(f) is tuple: fa, fd = f f = fa.as_expr()/fd.as_expr() fa, fd = f.as_expr().as_numer_denom() fa, fd = fa.as_poly(t, **kwargs), fd.as_poly(t, **kwargs) if cancel: fa, fd = fa.cancel(fd, include=True) if fa is None or fd is None: raise ValueError("Could not turn %s into a fraction in %s." % (f, t)) return (fa, fd) def as_poly_1t(p, t, z): """ (Hackish) way to convert an element ``p`` of K[t, 1/t] to K[t, z]. In other words, ``z == 1/t`` will be a dummy variable that Poly can handle better. See issue 5131. Examples ======== >>> from sympy import random_poly >>> from sympy.integrals.risch import as_poly_1t >>> from sympy.abc import x, z >>> p1 = random_poly(x, 10, -10, 10) >>> p2 = random_poly(x, 10, -10, 10) >>> p = p1 + p2.subs(x, 1/x) >>> as_poly_1t(p, x, z).as_expr().subs(z, 1/x) == p True """ # TODO: Use this on the final result. That way, we can avoid answers like # (...)*exp(-x). pa, pd = frac_in(p, t, cancel=True) if not pd.is_monomial: # XXX: Is there a better Poly exception that we could raise here? # Either way, if you see this (from the Risch Algorithm) it indicates # a bug. raise PolynomialError("%s is not an element of K[%s, 1/%s]." % (p, t, t)) d = pd.degree(t) one_t_part = pa.slice(0, d + 1) r = pd.degree() - pa.degree() t_part = pa - one_t_part try: t_part = t_part.to_field().exquo(pd) except DomainError as e: # issue 4950 raise NotImplementedError(e) # Compute the negative degree parts. one_t_part = Poly.from_list(reversed(one_t_part.rep.rep), *one_t_part.gens, domain=one_t_part.domain) if 0 < r < oo: one_t_part *= Poly(t**r, t) one_t_part = one_t_part.replace(t, z) # z will be 1/t if pd.nth(d): one_t_part *= Poly(1/pd.nth(d), z, expand=False) ans = t_part.as_poly(t, z, expand=False) + one_t_part.as_poly(t, z, expand=False) return ans def derivation(p, DE, coefficientD=False, basic=False): """ Computes Dp. Explanation =========== Given the derivation D with D = d/dx and p is a polynomial in t over K(x), return Dp. If coefficientD is True, it computes the derivation kD (kappaD), which is defined as kD(sum(ai*Xi**i, (i, 0, n))) == sum(Dai*Xi**i, (i, 1, n)) (Definition 3.2.2, page 80). X in this case is T[-1], so coefficientD computes the derivative just with respect to T[:-1], with T[-1] treated as a constant. If ``basic=True``, the returns a Basic expression. Elements of D can still be instances of Poly. """ if basic: r = 0 else: r = Poly(0, DE.t) t = DE.t if coefficientD: if DE.level <= -len(DE.T): # 'base' case, the answer is 0. return r DE.decrement_level() D = DE.D[:len(DE.D) + DE.level + 1] T = DE.T[:len(DE.T) + DE.level + 1] for d, v in zip(D, T): pv = p.as_poly(v) if pv is None or basic: pv = p.as_expr() if basic: r += d.as_expr()*pv.diff(v) else: r += (d.as_expr()*pv.diff(v).as_expr()).as_poly(t) if basic: r = cancel(r) if coefficientD: DE.increment_level() return r def get_case(d, t): """ Returns the type of the derivation d. Returns one of {'exp', 'tan', 'base', 'primitive', 'other_linear', 'other_nonlinear'}. """ if not d.expr.has(t): if d.is_one: return 'base' return 'primitive' if d.rem(Poly(t, t)).is_zero: return 'exp' if d.rem(Poly(1 + t**2, t)).is_zero: return 'tan' if d.degree(t) > 1: return 'other_nonlinear' return 'other_linear' def splitfactor(p, DE, coefficientD=False, z=None): """ Splitting factorization. Explanation =========== Given a derivation D on k[t] and ``p`` in k[t], return (p_n, p_s) in k[t] x k[t] such that p = p_n*p_s, p_s is special, and each square factor of p_n is normal. Page. 100 """ kinv = [1/x for x in DE.T[:DE.level]] if z: kinv.append(z) One = Poly(1, DE.t, domain=p.get_domain()) Dp = derivation(p, DE, coefficientD=coefficientD) # XXX: Is this right? if p.is_zero: return (p, One) if not p.expr.has(DE.t): s = p.as_poly(*kinv).gcd(Dp.as_poly(*kinv)).as_poly(DE.t) n = p.exquo(s) return (n, s) if not Dp.is_zero: h = p.gcd(Dp).to_field() g = p.gcd(p.diff(DE.t)).to_field() s = h.exquo(g) if s.degree(DE.t) == 0: return (p, One) q_split = splitfactor(p.exquo(s), DE, coefficientD=coefficientD) return (q_split[0], q_split[1]*s) else: return (p, One) def splitfactor_sqf(p, DE, coefficientD=False, z=None, basic=False): """ Splitting Square-free Factorization. Explanation =========== Given a derivation D on k[t] and ``p`` in k[t], returns (N1, ..., Nm) and (S1, ..., Sm) in k[t]^m such that p = (N1*N2**2*...*Nm**m)*(S1*S2**2*...*Sm**m) is a splitting factorization of ``p`` and the Ni and Si are square-free and coprime. """ # TODO: This algorithm appears to be faster in every case # TODO: Verify this and splitfactor() for multiple extensions kkinv = [1/x for x in DE.T[:DE.level]] + DE.T[:DE.level] if z: kkinv = [z] S = [] N = [] p_sqf = p.sqf_list_include() if p.is_zero: return (((p, 1),), ()) for pi, i in p_sqf: Si = pi.as_poly(*kkinv).gcd(derivation(pi, DE, coefficientD=coefficientD,basic=basic).as_poly(*kkinv)).as_poly(DE.t) pi = Poly(pi, DE.t) Si = Poly(Si, DE.t) Ni = pi.exquo(Si) if not Si.is_one: S.append((Si, i)) if not Ni.is_one: N.append((Ni, i)) return (tuple(N), tuple(S)) def canonical_representation(a, d, DE): """ Canonical Representation. Explanation =========== Given a derivation D on k[t] and f = a/d in k(t), return (f_p, f_s, f_n) in k[t] x k(t) x k(t) such that f = f_p + f_s + f_n is the canonical representation of f (f_p is a polynomial, f_s is reduced (has a special denominator), and f_n is simple (has a normal denominator). """ # Make d monic l = Poly(1/d.LC(), DE.t) a, d = a.mul(l), d.mul(l) q, r = a.div(d) dn, ds = splitfactor(d, DE) b, c = gcdex_diophantine(dn.as_poly(DE.t), ds.as_poly(DE.t), r.as_poly(DE.t)) b, c = b.as_poly(DE.t), c.as_poly(DE.t) return (q, (b, ds), (c, dn)) def hermite_reduce(a, d, DE): """ Hermite Reduction - Mack's Linear Version. Given a derivation D on k(t) and f = a/d in k(t), returns g, h, r in k(t) such that f = Dg + h + r, h is simple, and r is reduced. """ # Make d monic l = Poly(1/d.LC(), DE.t) a, d = a.mul(l), d.mul(l) fp, fs, fn = canonical_representation(a, d, DE) a, d = fn l = Poly(1/d.LC(), DE.t) a, d = a.mul(l), d.mul(l) ga = Poly(0, DE.t) gd = Poly(1, DE.t) dd = derivation(d, DE) dm = gcd(d, dd).as_poly(DE.t) ds, r = d.div(dm) while dm.degree(DE.t)>0: ddm = derivation(dm, DE) dm2 = gcd(dm, ddm) dms, r = dm.div(dm2) ds_ddm = ds.mul(ddm) ds_ddm_dm, r = ds_ddm.div(dm) b, c = gcdex_diophantine(-ds_ddm_dm.as_poly(DE.t), dms.as_poly(DE.t), a.as_poly(DE.t)) b, c = b.as_poly(DE.t), c.as_poly(DE.t) db = derivation(b, DE).as_poly(DE.t) ds_dms, r = ds.div(dms) a = c.as_poly(DE.t) - db.mul(ds_dms).as_poly(DE.t) ga = ga*dm + b*gd gd = gd*dm ga, gd = ga.cancel(gd, include=True) dm = dm2 d = ds q, r = a.div(d) ga, gd = ga.cancel(gd, include=True) r, d = r.cancel(d, include=True) rra = q*fs[1] + fp*fs[1] + fs[0] rrd = fs[1] rra, rrd = rra.cancel(rrd, include=True) return ((ga, gd), (r, d), (rra, rrd)) def polynomial_reduce(p, DE): """ Polynomial Reduction. Explanation =========== Given a derivation D on k(t) and p in k[t] where t is a nonlinear monomial over k, return q, r in k[t] such that p = Dq + r, and deg(r) < deg_t(Dt). """ q = Poly(0, DE.t) while p.degree(DE.t) >= DE.d.degree(DE.t): m = p.degree(DE.t) - DE.d.degree(DE.t) + 1 q0 = Poly(DE.t**m, DE.t).mul(Poly(p.as_poly(DE.t).LC()/ (m*DE.d.LC()), DE.t)) q += q0 p = p - derivation(q0, DE) return (q, p) def laurent_series(a, d, F, n, DE): """ Contribution of ``F`` to the full partial fraction decomposition of A/D. Explanation =========== Given a field K of characteristic 0 and ``A``,``D``,``F`` in K[x] with D monic, nonzero, coprime with A, and ``F`` the factor of multiplicity n in the square- free factorization of D, return the principal parts of the Laurent series of A/D at all the zeros of ``F``. """ if F.degree()==0: return 0 Z = _symbols('z', n) Z.insert(0, z) delta_a = Poly(0, DE.t) delta_d = Poly(1, DE.t) E = d.quo(F**n) ha, hd = (a, E*Poly(z**n, DE.t)) dF = derivation(F,DE) B, G = gcdex_diophantine(E, F, Poly(1,DE.t)) C, G = gcdex_diophantine(dF, F, Poly(1,DE.t)) # initialization F_store = F V, DE_D_list, H_list= [], [], [] for j in range(0, n): # jth derivative of z would be substituted with dfnth/(j+1) where dfnth =(d^n)f/(dx)^n F_store = derivation(F_store, DE) v = (F_store.as_expr())/(j + 1) V.append(v) DE_D_list.append(Poly(Z[j + 1],Z[j])) DE_new = DifferentialExtension(extension = {'D': DE_D_list}) #a differential indeterminate for j in range(0, n): zEha = Poly(z**(n + j), DE.t)*E**(j + 1)*ha zEhd = hd Pa, Pd = cancel((zEha, zEhd))[1], cancel((zEha, zEhd))[2] Q = Pa.quo(Pd) for i in range(0, j + 1): Q = Q.subs(Z[i], V[i]) Dha = (hd*derivation(ha, DE, basic=True).as_poly(DE.t) + ha*derivation(hd, DE, basic=True).as_poly(DE.t) + hd*derivation(ha, DE_new, basic=True).as_poly(DE.t) + ha*derivation(hd, DE_new, basic=True).as_poly(DE.t)) Dhd = Poly(j + 1, DE.t)*hd**2 ha, hd = Dha, Dhd Ff, Fr = F.div(gcd(F, Q)) F_stara, F_stard = frac_in(Ff, DE.t) if F_stara.degree(DE.t) - F_stard.degree(DE.t) > 0: QBC = Poly(Q, DE.t)*B**(1 + j)*C**(n + j) H = QBC H_list.append(H) H = (QBC*F_stard).rem(F_stara) alphas = real_roots(F_stara) for alpha in list(alphas): delta_a = delta_a*Poly((DE.t - alpha)**(n - j), DE.t) + Poly(H.eval(alpha), DE.t) delta_d = delta_d*Poly((DE.t - alpha)**(n - j), DE.t) return (delta_a, delta_d, H_list) def recognize_derivative(a, d, DE, z=None): """ Compute the squarefree factorization of the denominator of f and for each Di the polynomial H in K[x] (see Theorem 2.7.1), using the LaurentSeries algorithm. Write Di = GiEi where Gj = gcd(Hn, Di) and gcd(Ei,Hn) = 1. Since the residues of f at the roots of Gj are all 0, and the residue of f at a root alpha of Ei is Hi(a) != 0, f is the derivative of a rational function if and only if Ei = 1 for each i, which is equivalent to Di | H[-1] for each i. """ flag =True a, d = a.cancel(d, include=True) q, r = a.div(d) Np, Sp = splitfactor_sqf(d, DE, coefficientD=True, z=z) j = 1 for (s, i) in Sp: delta_a, delta_d, H = laurent_series(r, d, s, j, DE) g = gcd(d, H[-1]).as_poly() if g is not d: flag = False break j = j + 1 return flag def recognize_log_derivative(a, d, DE, z=None): """ There exists a v in K(x)* such that f = dv/v where f a rational function if and only if f can be written as f = A/D where D is squarefree,deg(A) < deg(D), gcd(A, D) = 1, and all the roots of the Rothstein-Trager resultant are integers. In that case, any of the Rothstein-Trager, Lazard-Rioboo-Trager or Czichowski algorithm produces u in K(x) such that du/dx = uf. """ z = z or Dummy('z') a, d = a.cancel(d, include=True) p, a = a.div(d) pz = Poly(z, DE.t) Dd = derivation(d, DE) q = a - pz*Dd r, R = d.resultant(q, includePRS=True) r = Poly(r, z) Np, Sp = splitfactor_sqf(r, DE, coefficientD=True, z=z) for s, i in Sp: # TODO also consider the complex roots # incase we have complex roots it should turn the flag false a = real_roots(s.as_poly(z)) if any(not j.is_Integer for j in a): return False return True def residue_reduce(a, d, DE, z=None, invert=True): """ Lazard-Rioboo-Rothstein-Trager resultant reduction. Explanation =========== Given a derivation ``D`` on k(t) and f in k(t) simple, return g elementary over k(t) and a Boolean b in {True, False} such that f - Dg in k[t] if b == True or f + h and f + h - Dg do not have an elementary integral over k(t) for any h in k<t> (reduced) if b == False. Returns (G, b), where G is a tuple of tuples of the form (s_i, S_i), such that g = Add(*[RootSum(s_i, lambda z: z*log(S_i(z, t))) for S_i, s_i in G]). f - Dg is the remaining integral, which is elementary only if b == True, and hence the integral of f is elementary only if b == True. f - Dg is not calculated in this function because that would require explicitly calculating the RootSum. Use residue_reduce_derivation(). """ # TODO: Use log_to_atan() from rationaltools.py # If r = residue_reduce(...), then the logarithmic part is given by: # sum([RootSum(a[0].as_poly(z), lambda i: i*log(a[1].as_expr()).subs(z, # i)).subs(t, log(x)) for a in r[0]]) z = z or Dummy('z') a, d = a.cancel(d, include=True) a, d = a.to_field().mul_ground(1/d.LC()), d.to_field().mul_ground(1/d.LC()) kkinv = [1/x for x in DE.T[:DE.level]] + DE.T[:DE.level] if a.is_zero: return ([], True) p, a = a.div(d) pz = Poly(z, DE.t) Dd = derivation(d, DE) q = a - pz*Dd if Dd.degree(DE.t) <= d.degree(DE.t): r, R = d.resultant(q, includePRS=True) else: r, R = q.resultant(d, includePRS=True) R_map, H = {}, [] for i in R: R_map[i.degree()] = i r = Poly(r, z) Np, Sp = splitfactor_sqf(r, DE, coefficientD=True, z=z) for s, i in Sp: if i == d.degree(DE.t): s = Poly(s, z).monic() H.append((s, d)) else: h = R_map.get(i) if h is None: continue h_lc = Poly(h.as_poly(DE.t).LC(), DE.t, field=True) h_lc_sqf = h_lc.sqf_list_include(all=True) for a, j in h_lc_sqf: h = Poly(h, DE.t, field=True).exquo(Poly(gcd(a, s**j, *kkinv), DE.t)) s = Poly(s, z).monic() if invert: h_lc = Poly(h.as_poly(DE.t).LC(), DE.t, field=True, expand=False) inv, coeffs = h_lc.as_poly(z, field=True).invert(s), [S.One] for coeff in h.coeffs()[1:]: L = reduced(inv*coeff.as_poly(inv.gens), [s])[1] coeffs.append(L.as_expr()) h = Poly(dict(list(zip(h.monoms(), coeffs))), DE.t) H.append((s, h)) b = all([not cancel(i.as_expr()).has(DE.t, z) for i, _ in Np]) return (H, b) def residue_reduce_to_basic(H, DE, z): """ Converts the tuple returned by residue_reduce() into a Basic expression. """ # TODO: check what Lambda does with RootOf i = Dummy('i') s = list(zip(reversed(DE.T), reversed([f(DE.x) for f in DE.Tfuncs]))) return sum(RootSum(a[0].as_poly(z), Lambda(i, i*log(a[1].as_expr()).subs( {z: i}).subs(s))) for a in H) def residue_reduce_derivation(H, DE, z): """ Computes the derivation of an expression returned by residue_reduce(). In general, this is a rational function in t, so this returns an as_expr() result. """ # TODO: verify that this is correct for multiple extensions i = Dummy('i') return S(sum(RootSum(a[0].as_poly(z), Lambda(i, i*derivation(a[1], DE).as_expr().subs(z, i)/a[1].as_expr().subs(z, i))) for a in H)) def integrate_primitive_polynomial(p, DE): """ Integration of primitive polynomials. Explanation =========== Given a primitive monomial t over k, and ``p`` in k[t], return q in k[t], r in k, and a bool b in {True, False} such that r = p - Dq is in k if b is True, or r = p - Dq does not have an elementary integral over k(t) if b is False. """ from sympy.integrals.prde import limited_integrate Zero = Poly(0, DE.t) q = Poly(0, DE.t) if not p.expr.has(DE.t): return (Zero, p, True) while True: if not p.expr.has(DE.t): return (q, p, True) Dta, Dtb = frac_in(DE.d, DE.T[DE.level - 1]) with DecrementLevel(DE): # We had better be integrating the lowest extension (x) # with ratint(). a = p.LC() aa, ad = frac_in(a, DE.t) try: rv = limited_integrate(aa, ad, [(Dta, Dtb)], DE) if rv is None: raise NonElementaryIntegralException (ba, bd), c = rv except NonElementaryIntegralException: return (q, p, False) m = p.degree(DE.t) q0 = c[0].as_poly(DE.t)*Poly(DE.t**(m + 1)/(m + 1), DE.t) + \ (ba.as_expr()/bd.as_expr()).as_poly(DE.t)*Poly(DE.t**m, DE.t) p = p - derivation(q0, DE) q = q + q0 def integrate_primitive(a, d, DE, z=None): """ Integration of primitive functions. Explanation =========== Given a primitive monomial t over k and f in k(t), return g elementary over k(t), i in k(t), and b in {True, False} such that i = f - Dg is in k if b is True or i = f - Dg does not have an elementary integral over k(t) if b is False. This function returns a Basic expression for the first argument. If b is True, the second argument is Basic expression in k to recursively integrate. If b is False, the second argument is an unevaluated Integral, which has been proven to be nonelementary. """ # XXX: a and d must be canceled, or this might return incorrect results z = z or Dummy("z") s = list(zip(reversed(DE.T), reversed([f(DE.x) for f in DE.Tfuncs]))) g1, h, r = hermite_reduce(a, d, DE) g2, b = residue_reduce(h[0], h[1], DE, z=z) if not b: i = cancel(a.as_expr()/d.as_expr() - (g1[1]*derivation(g1[0], DE) - g1[0]*derivation(g1[1], DE)).as_expr()/(g1[1]**2).as_expr() - residue_reduce_derivation(g2, DE, z)) i = NonElementaryIntegral(cancel(i).subs(s), DE.x) return ((g1[0].as_expr()/g1[1].as_expr()).subs(s) + residue_reduce_to_basic(g2, DE, z), i, b) # h - Dg2 + r p = cancel(h[0].as_expr()/h[1].as_expr() - residue_reduce_derivation(g2, DE, z) + r[0].as_expr()/r[1].as_expr()) p = p.as_poly(DE.t) q, i, b = integrate_primitive_polynomial(p, DE) ret = ((g1[0].as_expr()/g1[1].as_expr() + q.as_expr()).subs(s) + residue_reduce_to_basic(g2, DE, z)) if not b: # TODO: This does not do the right thing when b is False i = NonElementaryIntegral(cancel(i.as_expr()).subs(s), DE.x) else: i = cancel(i.as_expr()) return (ret, i, b) def integrate_hyperexponential_polynomial(p, DE, z): """ Integration of hyperexponential polynomials. Explanation =========== Given a hyperexponential monomial t over k and ``p`` in k[t, 1/t], return q in k[t, 1/t] and a bool b in {True, False} such that p - Dq in k if b is True, or p - Dq does not have an elementary integral over k(t) if b is False. """ from sympy.integrals.rde import rischDE t1 = DE.t dtt = DE.d.exquo(Poly(DE.t, DE.t)) qa = Poly(0, DE.t) qd = Poly(1, DE.t) b = True if p.is_zero: return(qa, qd, b) with DecrementLevel(DE): for i in range(-p.degree(z), p.degree(t1) + 1): if not i: continue elif i < 0: # If you get AttributeError: 'NoneType' object has no attribute 'nth' # then this should really not have expand=False # But it shouldn't happen because p is already a Poly in t and z a = p.as_poly(z, expand=False).nth(-i) else: # If you get AttributeError: 'NoneType' object has no attribute 'nth' # then this should really not have expand=False a = p.as_poly(t1, expand=False).nth(i) aa, ad = frac_in(a, DE.t, field=True) aa, ad = aa.cancel(ad, include=True) iDt = Poly(i, t1)*dtt iDta, iDtd = frac_in(iDt, DE.t, field=True) try: va, vd = rischDE(iDta, iDtd, Poly(aa, DE.t), Poly(ad, DE.t), DE) va, vd = frac_in((va, vd), t1, cancel=True) except NonElementaryIntegralException: b = False else: qa = qa*vd + va*Poly(t1**i)*qd qd *= vd return (qa, qd, b) def integrate_hyperexponential(a, d, DE, z=None, conds='piecewise'): """ Integration of hyperexponential functions. Explanation =========== Given a hyperexponential monomial t over k and f in k(t), return g elementary over k(t), i in k(t), and a bool b in {True, False} such that i = f - Dg is in k if b is True or i = f - Dg does not have an elementary integral over k(t) if b is False. This function returns a Basic expression for the first argument. If b is True, the second argument is Basic expression in k to recursively integrate. If b is False, the second argument is an unevaluated Integral, which has been proven to be nonelementary. """ # XXX: a and d must be canceled, or this might return incorrect results z = z or Dummy("z") s = list(zip(reversed(DE.T), reversed([f(DE.x) for f in DE.Tfuncs]))) g1, h, r = hermite_reduce(a, d, DE) g2, b = residue_reduce(h[0], h[1], DE, z=z) if not b: i = cancel(a.as_expr()/d.as_expr() - (g1[1]*derivation(g1[0], DE) - g1[0]*derivation(g1[1], DE)).as_expr()/(g1[1]**2).as_expr() - residue_reduce_derivation(g2, DE, z)) i = NonElementaryIntegral(cancel(i.subs(s)), DE.x) return ((g1[0].as_expr()/g1[1].as_expr()).subs(s) + residue_reduce_to_basic(g2, DE, z), i, b) # p should be a polynomial in t and 1/t, because Sirr == k[t, 1/t] # h - Dg2 + r p = cancel(h[0].as_expr()/h[1].as_expr() - residue_reduce_derivation(g2, DE, z) + r[0].as_expr()/r[1].as_expr()) pp = as_poly_1t(p, DE.t, z) qa, qd, b = integrate_hyperexponential_polynomial(pp, DE, z) i = pp.nth(0, 0) ret = ((g1[0].as_expr()/g1[1].as_expr()).subs(s) \ + residue_reduce_to_basic(g2, DE, z)) qas = qa.as_expr().subs(s) qds = qd.as_expr().subs(s) if conds == 'piecewise' and DE.x not in qds.free_symbols: # We have to be careful if the exponent is S.Zero! # XXX: Does qd = 0 always necessarily correspond to the exponential # equaling 1? ret += Piecewise( (qas/qds, Ne(qds, 0)), (integrate((p - i).subs(DE.t, 1).subs(s), DE.x), True) ) else: ret += qas/qds if not b: i = p - (qd*derivation(qa, DE) - qa*derivation(qd, DE)).as_expr()/\ (qd**2).as_expr() i = NonElementaryIntegral(cancel(i).subs(s), DE.x) return (ret, i, b) def integrate_hypertangent_polynomial(p, DE): """ Integration of hypertangent polynomials. Explanation =========== Given a differential field k such that sqrt(-1) is not in k, a hypertangent monomial t over k, and p in k[t], return q in k[t] and c in k such that p - Dq - c*D(t**2 + 1)/(t**1 + 1) is in k and p - Dq does not have an elementary integral over k(t) if Dc != 0. """ # XXX: Make sure that sqrt(-1) is not in k. q, r = polynomial_reduce(p, DE) a = DE.d.exquo(Poly(DE.t**2 + 1, DE.t)) c = Poly(r.nth(1)/(2*a.as_expr()), DE.t) return (q, c) def integrate_nonlinear_no_specials(a, d, DE, z=None): """ Integration of nonlinear monomials with no specials. Explanation =========== Given a nonlinear monomial t over k such that Sirr ({p in k[t] | p is special, monic, and irreducible}) is empty, and f in k(t), returns g elementary over k(t) and a Boolean b in {True, False} such that f - Dg is in k if b == True, or f - Dg does not have an elementary integral over k(t) if b == False. This function is applicable to all nonlinear extensions, but in the case where it returns b == False, it will only have proven that the integral of f - Dg is nonelementary if Sirr is empty. This function returns a Basic expression. """ # TODO: Integral from k? # TODO: split out nonelementary integral # XXX: a and d must be canceled, or this might not return correct results z = z or Dummy("z") s = list(zip(reversed(DE.T), reversed([f(DE.x) for f in DE.Tfuncs]))) g1, h, r = hermite_reduce(a, d, DE) g2, b = residue_reduce(h[0], h[1], DE, z=z) if not b: return ((g1[0].as_expr()/g1[1].as_expr()).subs(s) + residue_reduce_to_basic(g2, DE, z), b) # Because f has no specials, this should be a polynomial in t, or else # there is a bug. p = cancel(h[0].as_expr()/h[1].as_expr() - residue_reduce_derivation(g2, DE, z).as_expr() + r[0].as_expr()/r[1].as_expr()).as_poly(DE.t) q1, q2 = polynomial_reduce(p, DE) if q2.expr.has(DE.t): b = False else: b = True ret = (cancel(g1[0].as_expr()/g1[1].as_expr() + q1.as_expr()).subs(s) + residue_reduce_to_basic(g2, DE, z)) return (ret, b) class NonElementaryIntegral(Integral): """ Represents a nonelementary Integral. Explanation =========== If the result of integrate() is an instance of this class, it is guaranteed to be nonelementary. Note that integrate() by default will try to find any closed-form solution, even in terms of special functions which may themselves not be elementary. To make integrate() only give elementary solutions, or, in the cases where it can prove the integral to be nonelementary, instances of this class, use integrate(risch=True). In this case, integrate() may raise NotImplementedError if it cannot make such a determination. integrate() uses the deterministic Risch algorithm to integrate elementary functions or prove that they have no elementary integral. In some cases, this algorithm can split an integral into an elementary and nonelementary part, so that the result of integrate will be the sum of an elementary expression and a NonElementaryIntegral. Examples ======== >>> from sympy import integrate, exp, log, Integral >>> from sympy.abc import x >>> a = integrate(exp(-x**2), x, risch=True) >>> print(a) Integral(exp(-x**2), x) >>> type(a) <class 'sympy.integrals.risch.NonElementaryIntegral'> >>> expr = (2*log(x)**2 - log(x) - x**2)/(log(x)**3 - x**2*log(x)) >>> b = integrate(expr, x, risch=True) >>> print(b) -log(-x + log(x))/2 + log(x + log(x))/2 + Integral(1/log(x), x) >>> type(b.atoms(Integral).pop()) <class 'sympy.integrals.risch.NonElementaryIntegral'> """ # TODO: This is useful in and of itself, because isinstance(result, # NonElementaryIntegral) will tell if the integral has been proven to be # elementary. But should we do more? Perhaps a no-op .doit() if # elementary=True? Or maybe some information on why the integral is # nonelementary. pass def risch_integrate(f, x, extension=None, handle_first='log', separate_integral=False, rewrite_complex=None, conds='piecewise'): r""" The Risch Integration Algorithm. Explanation =========== Only transcendental functions are supported. Currently, only exponentials and logarithms are supported, but support for trigonometric functions is forthcoming. If this function returns an unevaluated Integral in the result, it means that it has proven that integral to be nonelementary. Any errors will result in raising NotImplementedError. The unevaluated Integral will be an instance of NonElementaryIntegral, a subclass of Integral. handle_first may be either 'exp' or 'log'. This changes the order in which the extension is built, and may result in a different (but equivalent) solution (for an example of this, see issue 5109). It is also possible that the integral may be computed with one but not the other, because not all cases have been implemented yet. It defaults to 'log' so that the outer extension is exponential when possible, because more of the exponential case has been implemented. If ``separate_integral`` is ``True``, the result is returned as a tuple (ans, i), where the integral is ans + i, ans is elementary, and i is either a NonElementaryIntegral or 0. This useful if you want to try further integrating the NonElementaryIntegral part using other algorithms to possibly get a solution in terms of special functions. It is False by default. Examples ======== >>> from sympy.integrals.risch import risch_integrate >>> from sympy import exp, log, pprint >>> from sympy.abc import x First, we try integrating exp(-x**2). Except for a constant factor of 2/sqrt(pi), this is the famous error function. >>> pprint(risch_integrate(exp(-x**2), x)) / | | 2 | -x | e dx | / The unevaluated Integral in the result means that risch_integrate() has proven that exp(-x**2) does not have an elementary anti-derivative. In many cases, risch_integrate() can split out the elementary anti-derivative part from the nonelementary anti-derivative part. For example, >>> pprint(risch_integrate((2*log(x)**2 - log(x) - x**2)/(log(x)**3 - ... x**2*log(x)), x)) / | log(-x + log(x)) log(x + log(x)) | 1 - ---------------- + --------------- + | ------ dx 2 2 | log(x) | / This means that it has proven that the integral of 1/log(x) is nonelementary. This function is also known as the logarithmic integral, and is often denoted as Li(x). risch_integrate() currently only accepts purely transcendental functions with exponentials and logarithms, though note that this can include nested exponentials and logarithms, as well as exponentials with bases other than E. >>> pprint(risch_integrate(exp(x)*exp(exp(x)), x)) / x\ \e / e >>> pprint(risch_integrate(exp(exp(x)), x)) / | | / x\ | \e / | e dx | / >>> pprint(risch_integrate(x*x**x*log(x) + x**x + x*x**x, x)) x x*x >>> pprint(risch_integrate(x**x, x)) / | | x | x dx | / >>> pprint(risch_integrate(-1/(x*log(x)*log(log(x))**2), x)) 1 ----------- log(log(x)) """ f = S(f) DE = extension or DifferentialExtension(f, x, handle_first=handle_first, dummy=True, rewrite_complex=rewrite_complex) fa, fd = DE.fa, DE.fd result = S.Zero for case in reversed(DE.cases): if not fa.expr.has(DE.t) and not fd.expr.has(DE.t) and not case == 'base': DE.decrement_level() fa, fd = frac_in((fa, fd), DE.t) continue fa, fd = fa.cancel(fd, include=True) if case == 'exp': ans, i, b = integrate_hyperexponential(fa, fd, DE, conds=conds) elif case == 'primitive': ans, i, b = integrate_primitive(fa, fd, DE) elif case == 'base': # XXX: We can't call ratint() directly here because it doesn't # handle polynomials correctly. ans = integrate(fa.as_expr()/fd.as_expr(), DE.x, risch=False) b = False i = S.Zero else: raise NotImplementedError("Only exponential and logarithmic " "extensions are currently supported.") result += ans if b: DE.decrement_level() fa, fd = frac_in(i, DE.t) else: result = result.subs(DE.backsubs) if not i.is_zero: i = NonElementaryIntegral(i.function.subs(DE.backsubs),i.limits) if not separate_integral: result += i return result else: if isinstance(i, NonElementaryIntegral): return (result, i) else: return (result, 0)
0b2c852a298b9590d382cc491b1d1ae29785c9bc294d3048e34b4170066fe877
"""This module implements tools for integrating rational functions. """ from sympy import S, Symbol, symbols, I, log, atan, \ roots, RootSum, Lambda, cancel, Dummy from sympy.polys import Poly, resultant, ZZ def ratint(f, x, **flags): """ Performs indefinite integration of rational functions. Explanation =========== Given a field :math:`K` and a rational function :math:`f = p/q`, where :math:`p` and :math:`q` are polynomials in :math:`K[x]`, returns a function :math:`g` such that :math:`f = g'`. Examples ======== >>> from sympy.integrals.rationaltools import ratint >>> from sympy.abc import x >>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x) (12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1) References ========== .. [1] M. Bronstein, Symbolic Integration I: Transcendental Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70 See Also ======== sympy.integrals.integrals.Integral.doit sympy.integrals.rationaltools.ratint_logpart sympy.integrals.rationaltools.ratint_ratpart """ if type(f) is not tuple: p, q = f.as_numer_denom() else: p, q = f p, q = Poly(p, x, composite=False, field=True), Poly(q, x, composite=False, field=True) coeff, p, q = p.cancel(q) poly, p = p.div(q) result = poly.integrate(x).as_expr() if p.is_zero: return coeff*result g, h = ratint_ratpart(p, q, x) P, Q = h.as_numer_denom() P = Poly(P, x) Q = Poly(Q, x) q, r = P.div(Q) result += g + q.integrate(x).as_expr() if not r.is_zero: symbol = flags.get('symbol', 't') if not isinstance(symbol, Symbol): t = Dummy(symbol) else: t = symbol.as_dummy() L = ratint_logpart(r, Q, x, t) real = flags.get('real') if real is None: if type(f) is not tuple: atoms = f.atoms() else: p, q = f atoms = p.atoms() | q.atoms() for elt in atoms - {x}: if not elt.is_extended_real: real = False break else: real = True eps = S.Zero if not real: for h, q in L: _, h = h.primitive() eps += RootSum( q, Lambda(t, t*log(h.as_expr())), quadratic=True) else: for h, q in L: _, h = h.primitive() R = log_to_real(h, q, x, t) if R is not None: eps += R else: eps += RootSum( q, Lambda(t, t*log(h.as_expr())), quadratic=True) result += eps return coeff*result def ratint_ratpart(f, g, x): """ Horowitz-Ostrogradsky algorithm. Explanation =========== Given a field K and polynomials f and g in K[x], such that f and g are coprime and deg(f) < deg(g), returns fractions A and B in K(x), such that f/g = A' + B and B has square-free denominator. Examples ======== >>> from sympy.integrals.rationaltools import ratint_ratpart >>> from sympy.abc import x, y >>> from sympy import Poly >>> ratint_ratpart(Poly(1, x, domain='ZZ'), ... Poly(x + 1, x, domain='ZZ'), x) (0, 1/(x + 1)) >>> ratint_ratpart(Poly(1, x, domain='EX'), ... Poly(x**2 + y**2, x, domain='EX'), x) (0, 1/(x**2 + y**2)) >>> ratint_ratpart(Poly(36, x, domain='ZZ'), ... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x) ((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2)) See Also ======== ratint, ratint_logpart """ from sympy import solve f = Poly(f, x) g = Poly(g, x) u, v, _ = g.cofactors(g.diff()) n = u.degree() m = v.degree() A_coeffs = [ Dummy('a' + str(n - i)) for i in range(0, n) ] B_coeffs = [ Dummy('b' + str(m - i)) for i in range(0, m) ] C_coeffs = A_coeffs + B_coeffs A = Poly(A_coeffs, x, domain=ZZ[C_coeffs]) B = Poly(B_coeffs, x, domain=ZZ[C_coeffs]) H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u result = solve(H.coeffs(), C_coeffs) A = A.as_expr().subs(result) B = B.as_expr().subs(result) rat_part = cancel(A/u.as_expr(), x) log_part = cancel(B/v.as_expr(), x) return rat_part, log_part def ratint_logpart(f, g, x, t=None): r""" Lazard-Rioboo-Trager algorithm. Explanation =========== Given a field K and polynomials f and g in K[x], such that f and g are coprime, deg(f) < deg(g) and g is square-free, returns a list of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i in K[t, x] and q_i in K[t], and:: ___ ___ d f d \ ` \ ` -- - = -- ) ) a log(s_i(a, x)) dx g dx /__, /__, i=1..n a | q_i(a) = 0 Examples ======== >>> from sympy.integrals.rationaltools import ratint_logpart >>> from sympy.abc import x >>> from sympy import Poly >>> ratint_logpart(Poly(1, x, domain='ZZ'), ... Poly(x**2 + x + 1, x, domain='ZZ'), x) [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'), ...Poly(3*_t**2 + 1, _t, domain='ZZ'))] >>> ratint_logpart(Poly(12, x, domain='ZZ'), ... Poly(x**2 - x - 2, x, domain='ZZ'), x) [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'), ...Poly(-_t**2 + 16, _t, domain='ZZ'))] See Also ======== ratint, ratint_ratpart """ f, g = Poly(f, x), Poly(g, x) t = t or Dummy('t') a, b = g, f - g.diff()*Poly(t, x) res, R = resultant(a, b, includePRS=True) res = Poly(res, t, composite=False) assert res, "BUG: resultant(%s, %s) can't be zero" % (a, b) R_map, H = {}, [] for r in R: R_map[r.degree()] = r def _include_sign(c, sqf): if c.is_extended_real and (c < 0) == True: h, k = sqf[0] c_poly = c.as_poly(h.gens) sqf[0] = h*c_poly, k C, res_sqf = res.sqf_list() _include_sign(C, res_sqf) for q, i in res_sqf: _, q = q.primitive() if g.degree() == i: H.append((g, q)) else: h = R_map[i] h_lc = Poly(h.LC(), t, field=True) c, h_lc_sqf = h_lc.sqf_list(all=True) _include_sign(c, h_lc_sqf) for a, j in h_lc_sqf: h = h.quo(Poly(a.gcd(q)**j, x)) inv, coeffs = h_lc.invert(q), [S.One] for coeff in h.coeffs()[1:]: coeff = coeff.as_poly(inv.gens) T = (inv*coeff).rem(q) coeffs.append(T.as_expr()) h = Poly(dict(list(zip(h.monoms(), coeffs))), x) H.append((h, q)) return H def log_to_atan(f, g): """ Convert complex logarithms to real arctangents. Explanation =========== Given a real field K and polynomials f and g in K[x], with g != 0, returns a sum h of arctangents of polynomials in K[x], such that: dh d f + I g -- = -- I log( ------- ) dx dx f - I g Examples ======== >>> from sympy.integrals.rationaltools import log_to_atan >>> from sympy.abc import x >>> from sympy import Poly, sqrt, S >>> log_to_atan(Poly(x, x, domain='ZZ'), Poly(1, x, domain='ZZ')) 2*atan(x) >>> log_to_atan(Poly(x + S(1)/2, x, domain='QQ'), ... Poly(sqrt(3)/2, x, domain='EX')) 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3) See Also ======== log_to_real """ if f.degree() < g.degree(): f, g = -g, f f = f.to_field() g = g.to_field() p, q = f.div(g) if q.is_zero: return 2*atan(p.as_expr()) else: s, t, h = g.gcdex(-f) u = (f*s + g*t).quo(h) A = 2*atan(u.as_expr()) return A + log_to_atan(s, t) def log_to_real(h, q, x, t): r""" Convert complex logarithms to real functions. Explanation =========== Given real field K and polynomials h in K[t,x] and q in K[t], returns real function f such that: ___ df d \ ` -- = -- ) a log(h(a, x)) dx dx /__, a | q(a) = 0 Examples ======== >>> from sympy.integrals.rationaltools import log_to_real >>> from sympy.abc import x, y >>> from sympy import Poly, S >>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'), ... Poly(3*y**2 + 1, y, domain='ZZ'), x, y) 2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3 >>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'), ... Poly(-2*y + 1, y, domain='ZZ'), x, y) log(x**2 - 1)/2 See Also ======== log_to_atan """ from sympy import collect u, v = symbols('u,v', cls=Dummy) H = h.as_expr().subs({t: u + I*v}).expand() Q = q.as_expr().subs({t: u + I*v}).expand() H_map = collect(H, I, evaluate=False) Q_map = collect(Q, I, evaluate=False) a, b = H_map.get(S.One, S.Zero), H_map.get(I, S.Zero) c, d = Q_map.get(S.One, S.Zero), Q_map.get(I, S.Zero) R = Poly(resultant(c, d, v), u) R_u = roots(R, filter='R') if len(R_u) != R.count_roots(): return None result = S.Zero for r_u in R_u.keys(): C = Poly(c.subs({u: r_u}), v) R_v = roots(C, filter='R') if len(R_v) != C.count_roots(): return None R_v_paired = [] # take one from each pair of conjugate roots for r_v in R_v: if r_v not in R_v_paired and -r_v not in R_v_paired: if r_v.is_negative or r_v.could_extract_minus_sign(): R_v_paired.append(-r_v) elif not r_v.is_zero: R_v_paired.append(r_v) for r_v in R_v_paired: D = d.subs({u: r_u, v: r_v}) if D.evalf(chop=True) != 0: continue A = Poly(a.subs({u: r_u, v: r_v}), x) B = Poly(b.subs({u: r_u, v: r_v}), x) AB = (A**2 + B**2).as_expr() result += r_u*log(AB) + r_v*log_to_atan(A, B) R_q = roots(q, filter='R') if len(R_q) != q.count_roots(): return None for r in R_q.keys(): result += r*log(h.as_expr().subs(t, r)) return result
54d17634486ef93d8cf7374c343ac0625cb71b256b49e1c39ca8de3083ce6645
from sympy.core import Mul from sympy.functions import DiracDelta, Heaviside from sympy.core.compatibility import default_sort_key from sympy.core.singleton import S def change_mul(node, x): """change_mul(node, x) Rearranges the operands of a product, bringing to front any simple DiracDelta expression. Explanation =========== If no simple DiracDelta expression was found, then all the DiracDelta expressions are simplified (using DiracDelta.expand(diracdelta=True, wrt=x)). Return: (dirac, new node) Where: o dirac is either a simple DiracDelta expression or None (if no simple expression was found); o new node is either a simplified DiracDelta expressions or None (if it could not be simplified). Examples ======== >>> from sympy import DiracDelta, cos >>> from sympy.integrals.deltafunctions import change_mul >>> from sympy.abc import x, y >>> change_mul(x*y*DiracDelta(x)*cos(x), x) (DiracDelta(x), x*y*cos(x)) >>> change_mul(x*y*DiracDelta(x**2 - 1)*cos(x), x) (None, x*y*cos(x)*DiracDelta(x - 1)/2 + x*y*cos(x)*DiracDelta(x + 1)/2) >>> change_mul(x*y*DiracDelta(cos(x))*cos(x), x) (None, None) See Also ======== sympy.functions.special.delta_functions.DiracDelta deltaintegrate """ new_args = [] dirac = None #Sorting is needed so that we consistently collapse the same delta; #However, we must preserve the ordering of non-commutative terms c, nc = node.args_cnc() sorted_args = sorted(c, key=default_sort_key) sorted_args.extend(nc) for arg in sorted_args: if arg.is_Pow and isinstance(arg.base, DiracDelta): new_args.append(arg.func(arg.base, arg.exp - 1)) arg = arg.base if dirac is None and (isinstance(arg, DiracDelta) and arg.is_simple(x)): dirac = arg else: new_args.append(arg) if not dirac: # there was no simple dirac new_args = [] for arg in sorted_args: if isinstance(arg, DiracDelta): new_args.append(arg.expand(diracdelta=True, wrt=x)) elif arg.is_Pow and isinstance(arg.base, DiracDelta): new_args.append(arg.func(arg.base.expand(diracdelta=True, wrt=x), arg.exp)) else: new_args.append(arg) if new_args != sorted_args: nnode = Mul(*new_args).expand() else: # if the node didn't change there is nothing to do nnode = None return (None, nnode) return (dirac, Mul(*new_args)) def deltaintegrate(f, x): """ deltaintegrate(f, x) Explanation =========== The idea for integration is the following: - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it. If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta. 2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all. - If the node is a multiplication node having a DiracDelta term: First we expand it. If the expansion did work, then we try to integrate the expansion. If not, we try to extract a simple DiracDelta term, then we have two cases: 1) We have a simple DiracDelta term, so we return the integral. 2) We didn't have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression. Examples ======== >>> from sympy.abc import x, y, z >>> from sympy.integrals.deltafunctions import deltaintegrate >>> from sympy import sin, cos, DiracDelta >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x) sin(1)*cos(1)*Heaviside(x - 1) >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y) z**2*DiracDelta(x - z)*Heaviside(y - z) See Also ======== sympy.functions.special.delta_functions.DiracDelta sympy.integrals.integrals.Integral """ if not f.has(DiracDelta): return None from sympy.integrals import Integral, integrate from sympy.solvers import solve # g(x) = DiracDelta(h(x)) if f.func == DiracDelta: h = f.expand(diracdelta=True, wrt=x) if h == f: # can't simplify the expression #FIXME: the second term tells whether is DeltaDirac or Derivative #For integrating derivatives of DiracDelta we need the chain rule if f.is_simple(x): if (len(f.args) <= 1 or f.args[1] == 0): return Heaviside(f.args[0]) else: return (DiracDelta(f.args[0], f.args[1] - 1) / f.args[0].as_poly().LC()) else: # let's try to integrate the simplified expression fh = integrate(h, x) return fh elif f.is_Mul or f.is_Pow: # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e g = f.expand() if f != g: # the expansion worked fh = integrate(g, x) if fh is not None and not isinstance(fh, Integral): return fh else: # no expansion performed, try to extract a simple DiracDelta term deltaterm, rest_mult = change_mul(f, x) if not deltaterm: if rest_mult: fh = integrate(rest_mult, x) return fh else: deltaterm = deltaterm.expand(diracdelta=True, wrt=x) if deltaterm.is_Mul: # Take out any extracted factors deltaterm, rest_mult_2 = change_mul(deltaterm, x) rest_mult = rest_mult*rest_mult_2 point = solve(deltaterm.args[0], x)[0] # Return the largest hyperreal term left after # repeated integration by parts. For example, # # integrate(y*DiracDelta(x, 1),x) == y*DiracDelta(x,0), not 0 # # This is so Integral(y*DiracDelta(x).diff(x),x).doit() # will return y*DiracDelta(x) instead of 0 or DiracDelta(x), # both of which are correct everywhere the value is defined # but give wrong answers for nested integration. n = (0 if len(deltaterm.args)==1 else deltaterm.args[1]) m = 0 while n >= 0: r = (-1)**n*rest_mult.diff(x, n).subs(x, point) if r.is_zero: n -= 1 m += 1 else: if m == 0: return r*Heaviside(x - point) else: return r*DiracDelta(x,m-1) # In some very weak sense, x=0 is still a singularity, # but we hope will not be of any practical consequence. return S.Zero return None
c9508fc95b61b529cdc32577c82e5be2bfa728f6e6f62cae486cafd67a3036df
""" Algorithms for solving Parametric Risch Differential Equations. The methods used for solving Parametric Risch Differential Equations parallel those for solving Risch Differential Equations. See the outline in the docstring of rde.py for more information. The Parametric Risch Differential Equation problem is, given f, g1, ..., gm in K(t), to determine if there exist y in K(t) and c1, ..., cm in Const(K) such that Dy + f*y == Sum(ci*gi, (i, 1, m)), and to find such y and ci if they exist. For the algorithms here G is a list of tuples of factions of the terms on the right hand side of the equation (i.e., gi in k(t)), and Q is a list of terms on the right hand side of the equation (i.e., qi in k[t]). See the docstring of each function for more information. """ from sympy.core import Dummy, ilcm, Add, Mul, Pow, S from sympy.core.compatibility import reduce from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer, bound_degree) from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation, residue_reduce, splitfactor, residue_reduce_derivation, DecrementLevel, recognize_log_derivative) from sympy.matrices import zeros, eye from sympy.polys import Poly, lcm, cancel, sqf_list from sympy.polys.polymatrix import PolyMatrix as Matrix from sympy.solvers import solve def prde_normal_denom(fa, fd, G, DE): """ Parametric Risch Differential Equation - Normal part of the denominator. Explanation =========== Given a derivation D on k[t] and f, g1, ..., gm in k(t) with f weakly normalized with respect to t, return the tuple (a, b, G, h) such that a, h in k[t], b in k<t>, G = [g1, ..., gm] in k(t)^m, and for any solution c1, ..., cm in Const(k) and y in k(t) of Dy + f*y == Sum(ci*gi, (i, 1, m)), q == y*h in k<t> satisfies a*Dq + b*q == Sum(ci*Gi, (i, 1, m)). """ dn, ds = splitfactor(fd, DE) Gas, Gds = list(zip(*G)) gd = reduce(lambda i, j: i.lcm(j), Gds, Poly(1, DE.t)) en, es = splitfactor(gd, DE) p = dn.gcd(en) h = en.gcd(en.diff(DE.t)).quo(p.gcd(p.diff(DE.t))) a = dn*h c = a*h ba = a*fa - dn*derivation(h, DE)*fd ba, bd = ba.cancel(fd, include=True) G = [(c*A).cancel(D, include=True) for A, D in G] return (a, (ba, bd), G, h) def real_imag(ba, bd, gen): """ Helper function, to get the real and imaginary part of a rational function evaluated at sqrt(-1) without actually evaluating it at sqrt(-1). Explanation =========== Separates the even and odd power terms by checking the degree of terms wrt mod 4. Returns a tuple (ba[0], ba[1], bd) where ba[0] is real part of the numerator ba[1] is the imaginary part and bd is the denominator of the rational function. """ bd = bd.as_poly(gen).as_dict() ba = ba.as_poly(gen).as_dict() denom_real = [value if key[0] % 4 == 0 else -value if key[0] % 4 == 2 else 0 for key, value in bd.items()] denom_imag = [value if key[0] % 4 == 1 else -value if key[0] % 4 == 3 else 0 for key, value in bd.items()] bd_real = sum(r for r in denom_real) bd_imag = sum(r for r in denom_imag) num_real = [value if key[0] % 4 == 0 else -value if key[0] % 4 == 2 else 0 for key, value in ba.items()] num_imag = [value if key[0] % 4 == 1 else -value if key[0] % 4 == 3 else 0 for key, value in ba.items()] ba_real = sum(r for r in num_real) ba_imag = sum(r for r in num_imag) ba = ((ba_real*bd_real + ba_imag*bd_imag).as_poly(gen), (ba_imag*bd_real - ba_real*bd_imag).as_poly(gen)) bd = (bd_real*bd_real + bd_imag*bd_imag).as_poly(gen) return (ba[0], ba[1], bd) def prde_special_denom(a, ba, bd, G, DE, case='auto'): """ Parametric Risch Differential Equation - Special part of the denominator. Explanation =========== Case is one of {'exp', 'tan', 'primitive'} for the hyperexponential, hypertangent, and primitive cases, respectively. For the hyperexponential (resp. hypertangent) case, given a derivation D on k[t] and a in k[t], b in k<t>, and g1, ..., gm in k(t) with Dt/t in k (resp. Dt/(t**2 + 1) in k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp. gcd(a, t**2 + 1) == 1), return the tuple (A, B, GG, h) such that A, B, h in k[t], GG = [gg1, ..., ggm] in k(t)^m, and for any solution c1, ..., cm in Const(k) and q in k<t> of a*Dq + b*q == Sum(ci*gi, (i, 1, m)), r == q*h in k[t] satisfies A*Dr + B*r == Sum(ci*ggi, (i, 1, m)). For case == 'primitive', k<t> == k[t], so it returns (a, b, G, 1) in this case. """ # TODO: Merge this with the very similar special_denom() in rde.py if case == 'auto': case = DE.case if case == 'exp': p = Poly(DE.t, DE.t) elif case == 'tan': p = Poly(DE.t**2 + 1, DE.t) elif case in ['primitive', 'base']: B = ba.quo(bd) return (a, B, G, Poly(1, DE.t)) else: raise ValueError("case must be one of {'exp', 'tan', 'primitive', " "'base'}, not %s." % case) nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t) nc = min([order_at(Ga, p, DE.t) - order_at(Gd, p, DE.t) for Ga, Gd in G]) n = min(0, nc - min(0, nb)) if not nb: # Possible cancellation. if case == 'exp': dcoeff = DE.d.quo(Poly(DE.t, DE.t)) with DecrementLevel(DE): # We are guaranteed to not have problems, # because case != 'base'. alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t) etaa, etad = frac_in(dcoeff, DE.t) A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) if A is not None: Q, m, z = A if Q == 1: n = min(n, m) elif case == 'tan': dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t)) with DecrementLevel(DE): # We are guaranteed to not have problems, # because case != 'base'. betaa, alphaa, alphad = real_imag(ba, bd*a, DE.t) betad = alphad etaa, etad = frac_in(dcoeff, DE.t) if recognize_log_derivative(Poly(2, DE.t)*betaa, betad, DE): A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) B = parametric_log_deriv(betaa, betad, etaa, etad, DE) if A is not None and B is not None: Q, s, z = A # TODO: Add test if Q == 1: n = min(n, s/2) N = max(0, -nb) pN = p**N pn = p**-n # This is 1/h A = a*pN B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN G = [(Ga*pN*pn).cancel(Gd, include=True) for Ga, Gd in G] h = pn # (a*p**N, (b + n*a*Dp/p)*p**N, g1*p**(N - n), ..., gm*p**(N - n), p**-n) return (A, B, G, h) def prde_linear_constraints(a, b, G, DE): """ Parametric Risch Differential Equation - Generate linear constraints on the constants. Explanation =========== Given a derivation D on k[t], a, b, in k[t] with gcd(a, b) == 1, and G = [g1, ..., gm] in k(t)^m, return Q = [q1, ..., qm] in k[t]^m and a matrix M with entries in k(t) such that for any solution c1, ..., cm in Const(k) and p in k[t] of a*Dp + b*p == Sum(ci*gi, (i, 1, m)), (c1, ..., cm) is a solution of Mx == 0, and p and the ci satisfy a*Dp + b*p == Sum(ci*qi, (i, 1, m)). Because M has entries in k(t), and because Matrix doesn't play well with Poly, M will be a Matrix of Basic expressions. """ m = len(G) Gns, Gds = list(zip(*G)) d = reduce(lambda i, j: i.lcm(j), Gds) d = Poly(d, field=True) Q = [(ga*(d).quo(gd)).div(d) for ga, gd in G] if not all([ri.is_zero for _, ri in Q]): N = max([ri.degree(DE.t) for _, ri in Q]) M = Matrix(N + 1, m, lambda i, j: Q[j][1].nth(i)) else: M = Matrix(0, m, []) # No constraints, return the empty matrix. qs, _ = list(zip(*Q)) return (qs, M) def poly_linear_constraints(p, d): """ Given p = [p1, ..., pm] in k[t]^m and d in k[t], return q = [q1, ..., qm] in k[t]^m and a matrix M with entries in k such that Sum(ci*pi, (i, 1, m)), for c1, ..., cm in k, is divisible by d if and only if (c1, ..., cm) is a solution of Mx = 0, in which case the quotient is Sum(ci*qi, (i, 1, m)). """ m = len(p) q, r = zip(*[pi.div(d) for pi in p]) if not all([ri.is_zero for ri in r]): n = max([ri.degree() for ri in r]) M = Matrix(n + 1, m, lambda i, j: r[j].nth(i)) else: M = Matrix(0, m, []) # No constraints. return q, M def constant_system(A, u, DE): """ Generate a system for the constant solutions. Explanation =========== Given a differential field (K, D) with constant field C = Const(K), a Matrix A, and a vector (Matrix) u with coefficients in K, returns the tuple (B, v, s), where B is a Matrix with coefficients in C and v is a vector (Matrix) such that either v has coefficients in C, in which case s is True and the solutions in C of Ax == u are exactly all the solutions of Bx == v, or v has a non-constant coefficient, in which case s is False Ax == u has no constant solution. This algorithm is used both in solving parametric problems and in determining if an element a of K is a derivative of an element of K or the logarithmic derivative of a K-radical using the structure theorem approach. Because Poly does not play well with Matrix yet, this algorithm assumes that all matrix entries are Basic expressions. """ if not A: return A, u Au = A.row_join(u) Au = Au.rref(simplify=cancel, normalize_last=False)[0] # Warning: This will NOT return correct results if cancel() cannot reduce # an identically zero expression to 0. The danger is that we might # incorrectly prove that an integral is nonelementary (such as # risch_integrate(exp((sin(x)**2 + cos(x)**2 - 1)*x**2), x). # But this is a limitation in computer algebra in general, and implicit # in the correctness of the Risch Algorithm is the computability of the # constant field (actually, this same correctness problem exists in any # algorithm that uses rref()). # # We therefore limit ourselves to constant fields that are computable # via the cancel() function, in order to prevent a speed bottleneck from # calling some more complex simplification function (rational function # coefficients will fall into this class). Furthermore, (I believe) this # problem will only crop up if the integral explicitly contains an # expression in the constant field that is identically zero, but cannot # be reduced to such by cancel(). Therefore, a careful user can avoid this # problem entirely by being careful with the sorts of expressions that # appear in his integrand in the variables other than the integration # variable (the structure theorems should be able to completely decide these # problems in the integration variable). Au = Au.applyfunc(cancel) A, u = Au[:, :-1], Au[:, -1] for j in range(A.cols): for i in range(A.rows): if A[i, j].has(*DE.T): # This assumes that const(F(t0, ..., tn) == const(K) == F Ri = A[i, :] # Rm+1; m = A.rows Rm1 = Ri.applyfunc(lambda x: derivation(x, DE, basic=True)/ derivation(A[i, j], DE, basic=True)) Rm1 = Rm1.applyfunc(cancel) um1 = cancel(derivation(u[i], DE, basic=True)/ derivation(A[i, j], DE, basic=True)) for s in range(A.rows): # A[s, :] = A[s, :] - A[s, i]*A[:, m+1] Asj = A[s, j] A.row_op(s, lambda r, jj: cancel(r - Asj*Rm1[jj])) # u[s] = u[s] - A[s, j]*u[m+1 u.row_op(s, lambda r, jj: cancel(r - Asj*um1)) A = A.col_join(Rm1) u = u.col_join(Matrix([um1])) return (A, u) def prde_spde(a, b, Q, n, DE): """ Special Polynomial Differential Equation algorithm: Parametric Version. Explanation =========== Given a derivation D on k[t], an integer n, and a, b, q1, ..., qm in k[t] with deg(a) > 0 and gcd(a, b) == 1, return (A, B, Q, R, n1), with Qq = [q1, ..., qm] and R = [r1, ..., rm], such that for any solution c1, ..., cm in Const(k) and q in k[t] of degree at most n of a*Dq + b*q == Sum(ci*gi, (i, 1, m)), p = (q - Sum(ci*ri, (i, 1, m)))/a has degree at most n1 and satisfies A*Dp + B*p == Sum(ci*qi, (i, 1, m)) """ R, Z = list(zip(*[gcdex_diophantine(b, a, qi) for qi in Q])) A = a B = b + derivation(a, DE) Qq = [zi - derivation(ri, DE) for ri, zi in zip(R, Z)] R = list(R) n1 = n - a.degree(DE.t) return (A, B, Qq, R, n1) def prde_no_cancel_b_large(b, Q, n, DE): """ Parametric Poly Risch Differential Equation - No cancellation: deg(b) large enough. Explanation =========== Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), returns h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and Dq + b*q == Sum(ci*qi, (i, 1, m)), then q = Sum(dj*hj, (j, 1, r)), where d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0. """ db = b.degree(DE.t) m = len(Q) H = [Poly(0, DE.t)]*m for N in range(n, -1, -1): # [n, ..., 0] for i in range(m): si = Q[i].nth(N + db)/b.LC() sitn = Poly(si*DE.t**N, DE.t) H[i] = H[i] + sitn Q[i] = Q[i] - derivation(sitn, DE) - b*sitn if all(qi.is_zero for qi in Q): dc = -1 M = zeros(0, 2) else: dc = max([qi.degree(DE.t) for qi in Q]) M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i)) A, u = constant_system(M, zeros(dc + 1, 1), DE) c = eye(m) A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c)) return (H, A) def prde_no_cancel_b_small(b, Q, n, DE): """ Parametric Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, returns h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and Dq + b*q == Sum(ci*qi, (i, 1, m)) then q = Sum(dj*hj, (j, 1, r)) where d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0. """ m = len(Q) H = [Poly(0, DE.t)]*m for N in range(n, 0, -1): # [n, ..., 1] for i in range(m): si = Q[i].nth(N + DE.d.degree(DE.t) - 1)/(N*DE.d.LC()) sitn = Poly(si*DE.t**N, DE.t) H[i] = H[i] + sitn Q[i] = Q[i] - derivation(sitn, DE) - b*sitn if b.degree(DE.t) > 0: for i in range(m): si = Poly(Q[i].nth(b.degree(DE.t))/b.LC(), DE.t) H[i] = H[i] + si Q[i] = Q[i] - derivation(si, DE) - b*si if all(qi.is_zero for qi in Q): dc = -1 M = Matrix() else: dc = max([qi.degree(DE.t) for qi in Q]) M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i)) A, u = constant_system(M, zeros(dc + 1, 1), DE) c = eye(m) A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c)) return (H, A) # else: b is in k, deg(qi) < deg(Dt) t = DE.t if DE.case != 'base': with DecrementLevel(DE): t0 = DE.t # k = k0(t0) ba, bd = frac_in(b, t0, field=True) Q0 = [frac_in(qi.TC(), t0, field=True) for qi in Q] f, B = param_rischDE(ba, bd, Q0, DE) # f = [f1, ..., fr] in k^r and B is a matrix with # m + r columns and entries in Const(k) = Const(k0) # such that Dy0 + b*y0 = Sum(ci*qi, (i, 1, m)) has # a solution y0 in k with c1, ..., cm in Const(k) # if and only y0 = Sum(dj*fj, (j, 1, r)) where # d1, ..., dr ar in Const(k) and # B*Matrix([c1, ..., cm, d1, ..., dr]) == 0. # Transform fractions (fa, fd) in f into constant # polynomials fa/fd in k[t]. # (Is there a better way?) f = [Poly(fa.as_expr()/fd.as_expr(), t, field=True) for fa, fd in f] else: # Base case. Dy == 0 for all y in k and b == 0. # Dy + b*y = Sum(ci*qi) is solvable if and only if # Sum(ci*qi) == 0 in which case the solutions are # y = d1*f1 for f1 = 1 and any d1 in Const(k) = k. f = [Poly(1, t, field=True)] # r = 1 B = Matrix([[qi.TC() for qi in Q] + [S.Zero]]) # The condition for solvability is # B*Matrix([c1, ..., cm, d1]) == 0 # There are no constraints on d1. # Coefficients of t^j (j > 0) in Sum(ci*qi) must be zero. d = max([qi.degree(DE.t) for qi in Q]) if d > 0: M = Matrix(d, m, lambda i, j: Q[j].nth(i + 1)) A, _ = constant_system(M, zeros(d, 1), DE) else: # No constraints on the hj. A = Matrix(0, m, []) # Solutions of the original equation are # y = Sum(dj*fj, (j, 1, r) + Sum(ei*hi, (i, 1, m)), # where ei == ci (i = 1, ..., m), when # A*Matrix([c1, ..., cm]) == 0 and # B*Matrix([c1, ..., cm, d1, ..., dr]) == 0 # Build combined constraint matrix with m + r + m columns. r = len(f) I = eye(m) A = A.row_join(zeros(A.rows, r + m)) B = B.row_join(zeros(B.rows, m)) C = I.row_join(zeros(m, r)).row_join(-I) return f + H, A.col_join(B).col_join(C) def prde_cancel_liouvillian(b, Q, n, DE): """ Pg, 237. """ H = [] # Why use DecrementLevel? Below line answers that: # Assuming that we can solve such problems over 'k' (not k[t]) if DE.case == 'primitive': with DecrementLevel(DE): ba, bd = frac_in(b, DE.t, field=True) for i in range(n, -1, -1): if DE.case == 'exp': # this re-checking can be avoided with DecrementLevel(DE): ba, bd = frac_in(b + (i*(derivation(DE.t, DE)/DE.t)).as_poly(b.gens), DE.t, field=True) with DecrementLevel(DE): Qy = [frac_in(q.nth(i), DE.t, field=True) for q in Q] fi, Ai = param_rischDE(ba, bd, Qy, DE) fi = [Poly(fa.as_expr()/fd.as_expr(), DE.t, field=True) for fa, fd in fi] ri = len(fi) if i == n: M = Ai else: M = Ai.col_join(M.row_join(zeros(M.rows, ri))) Fi, hi = [None]*ri, [None]*ri # from eq. on top of p.238 (unnumbered) for j in range(ri): hji = fi[j] * (DE.t**i).as_poly(fi[j].gens) hi[j] = hji # building up Sum(djn*(D(fjn*t^n) - b*fjnt^n)) Fi[j] = -(derivation(hji, DE) - b*hji) H += hi # in the next loop instead of Q it has # to be Q + Fi taking its place Q = Q + Fi return (H, M) def param_poly_rischDE(a, b, q, n, DE): """Polynomial solutions of a parametric Risch differential equation. Explanation =========== Given a derivation D in k[t], a, b in k[t] relatively prime, and q = [q1, ..., qm] in k[t]^m, return h = [h1, ..., hr] in k[t]^r and a matrix A with m + r columns and entries in Const(k) such that a*Dp + b*p = Sum(ci*qi, (i, 1, m)) has a solution p of degree <= n in k[t] with c1, ..., cm in Const(k) if and only if p = Sum(dj*hj, (j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm, d1, ..., dr) is a solution of Ax == 0. """ m = len(q) if n < 0: # Only the trivial zero solution is possible. # Find relations between the qi. if all([qi.is_zero for qi in q]): return [], zeros(1, m) # No constraints. N = max([qi.degree(DE.t) for qi in q]) M = Matrix(N + 1, m, lambda i, j: q[j].nth(i)) A, _ = constant_system(M, zeros(M.rows, 1), DE) return [], A if a.is_ground: # Normalization: a = 1. a = a.LC() b, q = b.quo_ground(a), [qi.quo_ground(a) for qi in q] if not b.is_zero and (DE.case == 'base' or b.degree() > max(0, DE.d.degree() - 1)): return prde_no_cancel_b_large(b, q, n, DE) elif ((b.is_zero or b.degree() < DE.d.degree() - 1) and (DE.case == 'base' or DE.d.degree() >= 2)): return prde_no_cancel_b_small(b, q, n, DE) elif (DE.d.degree() >= 2 and b.degree() == DE.d.degree() - 1 and n > -b.as_poly().LC()/DE.d.as_poly().LC()): raise NotImplementedError("prde_no_cancel_b_equal() is " "not yet implemented.") else: # Liouvillian cases if DE.case == 'primitive' or DE.case == 'exp': return prde_cancel_liouvillian(b, q, n, DE) else: raise NotImplementedError("non-linear and hypertangent " "cases have not yet been implemented") # else: deg(a) > 0 # Iterate SPDE as long as possible cumulating coefficient # and terms for the recovery of original solutions. alpha, beta = a.one, [a.zero]*m while n >= 0: # and a, b relatively prime a, b, q, r, n = prde_spde(a, b, q, n, DE) beta = [betai + alpha*ri for betai, ri in zip(beta, r)] alpha *= a # Solutions p of a*Dp + b*p = Sum(ci*qi) correspond to # solutions alpha*p + Sum(ci*betai) of the initial equation. d = a.gcd(b) if not d.is_ground: break # a*Dp + b*p = Sum(ci*qi) may have a polynomial solution # only if the sum is divisible by d. qq, M = poly_linear_constraints(q, d) # qq = [qq1, ..., qqm] where qqi = qi.quo(d). # M is a matrix with m columns an entries in k. # Sum(fi*qi, (i, 1, m)), where f1, ..., fm are elements of k, is # divisible by d if and only if M*Matrix([f1, ..., fm]) == 0, # in which case the quotient is Sum(fi*qqi). A, _ = constant_system(M, zeros(M.rows, 1), DE) # A is a matrix with m columns and entries in Const(k). # Sum(ci*qqi) is Sum(ci*qi).quo(d), and the remainder is zero # for c1, ..., cm in Const(k) if and only if # A*Matrix([c1, ...,cm]) == 0. V = A.nullspace() # V = [v1, ..., vu] where each vj is a column matrix with # entries aj1, ..., ajm in Const(k). # Sum(aji*qi) is divisible by d with exact quotient Sum(aji*qqi). # Sum(ci*qi) is divisible by d if and only if ci = Sum(dj*aji) # (i = 1, ..., m) for some d1, ..., du in Const(k). # In that case, solutions of # a*Dp + b*p = Sum(ci*qi) = Sum(dj*Sum(aji*qi)) # are the same as those of # (a/d)*Dp + (b/d)*p = Sum(dj*rj) # where rj = Sum(aji*qqi). if not V: # No non-trivial solution. return [], eye(m) # Could return A, but this has # the minimum number of rows. Mqq = Matrix([qq]) # A single row. r = [(Mqq*vj)[0] for vj in V] # [r1, ..., ru] # Solutions of (a/d)*Dp + (b/d)*p = Sum(dj*rj) correspond to # solutions alpha*p + Sum(Sum(dj*aji)*betai) of the initial # equation. These are equal to alpha*p + Sum(dj*fj) where # fj = Sum(aji*betai). Mbeta = Matrix([beta]) f = [(Mbeta*vj)[0] for vj in V] # [f1, ..., fu] # # Solve the reduced equation recursively. # g, B = param_poly_rischDE(a.quo(d), b.quo(d), r, n, DE) # g = [g1, ..., gv] in k[t]^v and and B is a matrix with u + v # columns and entries in Const(k) such that # (a/d)*Dp + (b/d)*p = Sum(dj*rj) has a solution p of degree <= n # in k[t] if and only if p = Sum(ek*gk) where e1, ..., ev are in # Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0. # The solutions of the original equation are then # Sum(dj*fj, (j, 1, u)) + alpha*Sum(ek*gk, (k, 1, v)). # Collect solution components. h = f + [alpha*gk for gk in g] # Build combined relation matrix. A = -eye(m) for vj in V: A = A.row_join(vj) A = A.row_join(zeros(m, len(g))) A = A.col_join(zeros(B.rows, m).row_join(B)) return h, A def param_rischDE(fa, fd, G, DE): """ Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)). Explanation =========== Given a derivation D in k(t), f in k(t), and G = [G1, ..., Gm] in k(t)^m, return h = [h1, ..., hr] in k(t)^r and a matrix A with m + r columns and entries in Const(k) such that Dy + f*y = Sum(ci*Gi, (i, 1, m)) has a solution y in k(t) with c1, ..., cm in Const(k) if and only if y = Sum(dj*hj, (j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm, d1, ..., dr) is a solution of Ax == 0. Elements of k(t) are tuples (a, d) with a and d in k[t]. """ m = len(G) q, (fa, fd) = weak_normalizer(fa, fd, DE) # Solutions of the weakly normalized equation Dz + f*z = q*Sum(ci*Gi) # correspond to solutions y = z/q of the original equation. gamma = q G = [(q*ga).cancel(gd, include=True) for ga, gd in G] a, (ba, bd), G, hn = prde_normal_denom(fa, fd, G, DE) # Solutions q in k<t> of a*Dq + b*q = Sum(ci*Gi) correspond # to solutions z = q/hn of the weakly normalized equation. gamma *= hn A, B, G, hs = prde_special_denom(a, ba, bd, G, DE) # Solutions p in k[t] of A*Dp + B*p = Sum(ci*Gi) correspond # to solutions q = p/hs of the previous equation. gamma *= hs g = A.gcd(B) a, b, g = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for gia, gid in G] # a*Dp + b*p = Sum(ci*gi) may have a polynomial solution # only if the sum is in k[t]. q, M = prde_linear_constraints(a, b, g, DE) # q = [q1, ..., qm] where qi in k[t] is the polynomial component # of the partial fraction expansion of gi. # M is a matrix with m columns and entries in k. # Sum(fi*gi, (i, 1, m)), where f1, ..., fm are elements of k, # is a polynomial if and only if M*Matrix([f1, ..., fm]) == 0, # in which case the sum is equal to Sum(fi*qi). M, _ = constant_system(M, zeros(M.rows, 1), DE) # M is a matrix with m columns and entries in Const(k). # Sum(ci*gi) is in k[t] for c1, ..., cm in Const(k) # if and only if M*Matrix([c1, ..., cm]) == 0, # in which case the sum is Sum(ci*qi). ## Reduce number of constants at this point V = M.nullspace() # V = [v1, ..., vu] where each vj is a column matrix with # entries aj1, ..., ajm in Const(k). # Sum(aji*gi) is in k[t] and equal to Sum(aji*qi) (j = 1, ..., u). # Sum(ci*gi) is in k[t] if and only is ci = Sum(dj*aji) # (i = 1, ..., m) for some d1, ..., du in Const(k). # In that case, # Sum(ci*gi) = Sum(ci*qi) = Sum(dj*Sum(aji*qi)) = Sum(dj*rj) # where rj = Sum(aji*qi) (j = 1, ..., u) in k[t]. if not V: # No non-trivial solution return [], eye(m) Mq = Matrix([q]) # A single row. r = [(Mq*vj)[0] for vj in V] # [r1, ..., ru] # Solutions of a*Dp + b*p = Sum(dj*rj) correspond to solutions # y = p/gamma of the initial equation with ci = Sum(dj*aji). try: # We try n=5. At least for prde_spde, it will always # terminate no matter what n is. n = bound_degree(a, b, r, DE, parametric=True) except NotImplementedError: # A temporary bound is set. Eventually, it will be removed. # the currently added test case takes large time # even with n=5, and much longer with large n's. n = 5 h, B = param_poly_rischDE(a, b, r, n, DE) # h = [h1, ..., hv] in k[t]^v and and B is a matrix with u + v # columns and entries in Const(k) such that # a*Dp + b*p = Sum(dj*rj) has a solution p of degree <= n # in k[t] if and only if p = Sum(ek*hk) where e1, ..., ev are in # Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0. # The solutions of the original equation for ci = Sum(dj*aji) # (i = 1, ..., m) are then y = Sum(ek*hk, (k, 1, v))/gamma. ## Build combined relation matrix with m + u + v columns. A = -eye(m) for vj in V: A = A.row_join(vj) A = A.row_join(zeros(m, len(h))) A = A.col_join(zeros(B.rows, m).row_join(B)) ## Eliminate d1, ..., du. W = A.nullspace() # W = [w1, ..., wt] where each wl is a column matrix with # entries blk (k = 1, ..., m + u + v) in Const(k). # The vectors (bl1, ..., blm) generate the space of those # constant families (c1, ..., cm) for which a solution of # the equation Dy + f*y == Sum(ci*Gi) exists. They generate # the space and form a basis except possibly when Dy + f*y == 0 # is solvable in k(t}. The corresponding solutions are # y = Sum(blk'*hk, (k, 1, v))/gamma, where k' = k + m + u. v = len(h) M = Matrix([wl[:m] + wl[-v:] for wl in W]) # excise dj's. N = M.nullspace() # N = [n1, ..., ns] where the ni in Const(k)^(m + v) are column # vectors generating the space of linear relations between # c1, ..., cm, e1, ..., ev. C = Matrix([ni[:] for ni in N]) # rows n1, ..., ns. return [hk.cancel(gamma, include=True) for hk in h], C def limited_integrate_reduce(fa, fd, G, DE): """ Simpler version of step 1 & 2 for the limited integration problem. Explanation =========== Given a derivation D on k(t) and f, g1, ..., gn in k(t), return (a, b, h, N, g, V) such that a, b, h in k[t], N is a non-negative integer, g in k(t), V == [v1, ..., vm] in k(t)^m, and for any solution v in k(t), c1, ..., cm in C of f == Dv + Sum(ci*wi, (i, 1, m)), p = v*h is in k<t>, and p and the ci satisfy a*Dp + b*p == g + Sum(ci*vi, (i, 1, m)). Furthermore, if S1irr == Sirr, then p is in k[t], and if t is nonlinear or Liouvillian over k, then deg(p) <= N. So that the special part is always computed, this function calls the more general prde_special_denom() automatically if it cannot determine that S1irr == Sirr. Furthermore, it will automatically call bound_degree() when t is linear and non-Liouvillian, which for the transcendental case, implies that Dt == a*t + b with for some a, b in k*. """ dn, ds = splitfactor(fd, DE) E = [splitfactor(gd, DE) for _, gd in G] En, Es = list(zip(*E)) c = reduce(lambda i, j: i.lcm(j), (dn,) + En) # lcm(dn, en1, ..., enm) hn = c.gcd(c.diff(DE.t)) a = hn b = -derivation(hn, DE) N = 0 # These are the cases where we know that S1irr = Sirr, but there could be # others, and this algorithm will need to be extended to handle them. if DE.case in ['base', 'primitive', 'exp', 'tan']: hs = reduce(lambda i, j: i.lcm(j), (ds,) + Es) # lcm(ds, es1, ..., esm) a = hn*hs b -= (hn*derivation(hs, DE)).quo(hs) mu = min(order_at_oo(fa, fd, DE.t), min([order_at_oo(ga, gd, DE.t) for ga, gd in G])) # So far, all the above are also nonlinear or Liouvillian, but if this # changes, then this will need to be updated to call bound_degree() # as per the docstring of this function (DE.case == 'other_linear'). N = hn.degree(DE.t) + hs.degree(DE.t) + max(0, 1 - DE.d.degree(DE.t) - mu) else: # TODO: implement this raise NotImplementedError V = [(-a*hn*ga).cancel(gd, include=True) for ga, gd in G] return (a, b, a, N, (a*hn*fa).cancel(fd, include=True), V) def limited_integrate(fa, fd, G, DE): """ Solves the limited integration problem: f = Dv + Sum(ci*wi, (i, 1, n)) """ fa, fd = fa*Poly(1/fd.LC(), DE.t), fd.monic() # interpreting limited integration problem as a # parametric Risch DE problem Fa = Poly(0, DE.t) Fd = Poly(1, DE.t) G = [(fa, fd)] + G h, A = param_rischDE(Fa, Fd, G, DE) V = A.nullspace() V = [v for v in V if v[0] != 0] if not V: return None else: # we can take any vector from V, we take V[0] c0 = V[0][0] # v = [-1, c1, ..., cm, d1, ..., dr] v = V[0]/(-c0) r = len(h) m = len(v) - r - 1 C = list(v[1: m + 1]) y = -sum([v[m + 1 + i]*h[i][0].as_expr()/h[i][1].as_expr() \ for i in range(r)]) y_num, y_den = y.as_numer_denom() Ya, Yd = Poly(y_num, DE.t), Poly(y_den, DE.t) Y = Ya*Poly(1/Yd.LC(), DE.t), Yd.monic() return Y, C def parametric_log_deriv_heu(fa, fd, wa, wd, DE, c1=None): """ Parametric logarithmic derivative heuristic. Explanation =========== Given a derivation D on k[t], f in k(t), and a hyperexponential monomial theta over k(t), raises either NotImplementedError, in which case the heuristic failed, or returns None, in which case it has proven that no solution exists, or returns a solution (n, m, v) of the equation n*f == Dv/v + m*Dtheta/theta, with v in k(t)* and n, m in ZZ with n != 0. If this heuristic fails, the structure theorem approach will need to be used. The argument w == Dtheta/theta """ # TODO: finish writing this and write tests c1 = c1 or Dummy('c1') p, a = fa.div(fd) q, b = wa.div(wd) B = max(0, derivation(DE.t, DE).degree(DE.t) - 1) C = max(p.degree(DE.t), q.degree(DE.t)) if q.degree(DE.t) > B: eqs = [p.nth(i) - c1*q.nth(i) for i in range(B + 1, C + 1)] s = solve(eqs, c1) if not s or not s[c1].is_Rational: # deg(q) > B, no solution for c. return None M, N = s[c1].as_numer_denom() M_poly = M.as_poly(q.gens) N_poly = N.as_poly(q.gens) nfmwa = N_poly*fa*wd - M_poly*wa*fd nfmwd = fd*wd Qv = is_log_deriv_k_t_radical_in_field(nfmwa, nfmwd, DE, 'auto') if Qv is None: # (N*f - M*w) is not the logarithmic derivative of a k(t)-radical. return None Q, v = Qv if Q.is_zero or v.is_zero: return None return (Q*N, Q*M, v) if p.degree(DE.t) > B: return None c = lcm(fd.as_poly(DE.t).LC(), wd.as_poly(DE.t).LC()) l = fd.monic().lcm(wd.monic())*Poly(c, DE.t) ln, ls = splitfactor(l, DE) z = ls*ln.gcd(ln.diff(DE.t)) if not z.has(DE.t): # TODO: We treat this as 'no solution', until the structure # theorem version of parametric_log_deriv is implemented. return None u1, r1 = (fa*l.quo(fd)).div(z) # (l*f).div(z) u2, r2 = (wa*l.quo(wd)).div(z) # (l*w).div(z) eqs = [r1.nth(i) - c1*r2.nth(i) for i in range(z.degree(DE.t))] s = solve(eqs, c1) if not s or not s[c1].is_Rational: # deg(q) <= B, no solution for c. return None M, N = s[c1].as_numer_denom() nfmwa = N.as_poly(DE.t)*fa*wd - M.as_poly(DE.t)*wa*fd nfmwd = fd*wd Qv = is_log_deriv_k_t_radical_in_field(nfmwa, nfmwd, DE) if Qv is None: # (N*f - M*w) is not the logarithmic derivative of a k(t)-radical. return None Q, v = Qv if Q.is_zero or v.is_zero: return None return (Q*N, Q*M, v) def parametric_log_deriv(fa, fd, wa, wd, DE): # TODO: Write the full algorithm using the structure theorems. # try: A = parametric_log_deriv_heu(fa, fd, wa, wd, DE) # except NotImplementedError: # Heuristic failed, we have to use the full method. # TODO: This could be implemented more efficiently. # It isn't too worrisome, because the heuristic handles most difficult # cases. return A def is_deriv_k(fa, fd, DE): r""" Checks if Df/f is the derivative of an element of k(t). Explanation =========== a in k(t) is the derivative of an element of k(t) if there exists b in k(t) such that a = Db. Either returns (ans, u), such that Df/f == Du, or None, which means that Df/f is not the derivative of an element of k(t). ans is a list of tuples such that Add(*[i*j for i, j in ans]) == u. This is useful for seeing exactly which elements of k(t) produce u. This function uses the structure theorem approach, which says that for any f in K, Df/f is the derivative of a element of K if and only if there are ri in QQ such that:: --- --- Dt \ r * Dt + \ r * i Df / i i / i --- = --. --- --- t f i in L i in E i K/C(x) K/C(x) Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of hyperexponential monomials of K over C(x)). If K is an elementary extension over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the transcendence degree of K over C(x). Furthermore, because Const_D(K) == Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x) and L_K/C(x) are disjoint. The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed recursively using this same function. Therefore, it is required to pass them as indices to D (or T). E_args are the arguments of the hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] == exp(E_args[i])). This is needed to compute the final answer u such that Df/f == Du. log(f) will be the same as u up to a additive constant. This is because they will both behave the same as monomials. For example, both log(x) and log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant. Therefore, the term const is returned. const is such that log(const) + f == u. This is calculated by dividing the arguments of one logarithm from the other. Therefore, it is necessary to pass the arguments of the logarithmic terms in L_args. To handle the case where we are given Df/f, not f, use is_deriv_k_in_field(). See also ======== is_log_deriv_k_t_radical_in_field, is_log_deriv_k_t_radical """ # Compute Df/f dfa, dfd = (fd*derivation(fa, DE) - fa*derivation(fd, DE)), fd*fa dfa, dfd = dfa.cancel(dfd, include=True) # Our assumption here is that each monomial is recursively transcendental if len(DE.exts) != len(DE.D): if [i for i in DE.cases if i == 'tan'] or \ ({i for i in DE.cases if i == 'primitive'} - set(DE.indices('log'))): raise NotImplementedError("Real version of the structure " "theorems with hypertangent support is not yet implemented.") # TODO: What should really be done in this case? raise NotImplementedError("Nonelementary extensions not supported " "in the structure theorems.") E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.indices('exp')] L_part = [DE.D[i].as_expr() for i in DE.indices('log')] lhs = Matrix([E_part + L_part]) rhs = Matrix([dfa.as_expr()/dfd.as_expr()]) A, u = constant_system(lhs, rhs, DE) if not all(derivation(i, DE, basic=True).is_zero for i in u) or not A: # If the elements of u are not all constant # Note: See comment in constant_system # Also note: derivation(basic=True) calls cancel() return None else: if not all(i.is_Rational for i in u): raise NotImplementedError("Cannot work with non-rational " "coefficients in this case.") else: terms = ([DE.extargs[i] for i in DE.indices('exp')] + [DE.T[i] for i in DE.indices('log')]) ans = list(zip(terms, u)) result = Add(*[Mul(i, j) for i, j in ans]) argterms = ([DE.T[i] for i in DE.indices('exp')] + [DE.extargs[i] for i in DE.indices('log')]) l = [] ld = [] for i, j in zip(argterms, u): # We need to get around things like sqrt(x**2) != x # and also sqrt(x**2 + 2*x + 1) != x + 1 # Issue 10798: i need not be a polynomial i, d = i.as_numer_denom() icoeff, iterms = sqf_list(i) l.append(Mul(*([Pow(icoeff, j)] + [Pow(b, e*j) for b, e in iterms]))) dcoeff, dterms = sqf_list(d) ld.append(Mul(*([Pow(dcoeff, j)] + [Pow(b, e*j) for b, e in dterms]))) const = cancel(fa.as_expr()/fd.as_expr()/Mul(*l)*Mul(*ld)) return (ans, result, const) def is_log_deriv_k_t_radical(fa, fd, DE, Df=True): r""" Checks if Df is the logarithmic derivative of a k(t)-radical. Explanation =========== b in k(t) can be written as the logarithmic derivative of a k(t) radical if there exist n in ZZ and u in k(t) with n, u != 0 such that n*b == Du/u. Either returns (ans, u, n, const) or None, which means that Df cannot be written as the logarithmic derivative of a k(t)-radical. ans is a list of tuples such that Mul(*[i**j for i, j in ans]) == u. This is useful for seeing exactly what elements of k(t) produce u. This function uses the structure theorem approach, which says that for any f in K, Df is the logarithmic derivative of a K-radical if and only if there are ri in QQ such that:: --- --- Dt \ r * Dt + \ r * i / i i / i --- = Df. --- --- t i in L i in E i K/C(x) K/C(x) Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of hyperexponential monomials of K over C(x)). If K is an elementary extension over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the transcendence degree of K over C(x). Furthermore, because Const_D(K) == Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x) and L_K/C(x) are disjoint. The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed recursively using this same function. Therefore, it is required to pass them as indices to D (or T). L_args are the arguments of the logarithms indexed by L_K (i.e., if i is in L_K, then T[i] == log(L_args[i])). This is needed to compute the final answer u such that n*f == Du/u. exp(f) will be the same as u up to a multiplicative constant. This is because they will both behave the same as monomials. For example, both exp(x) and exp(x + 1) == E*exp(x) satisfy Dt == t. Therefore, the term const is returned. const is such that exp(const)*f == u. This is calculated by subtracting the arguments of one exponential from the other. Therefore, it is necessary to pass the arguments of the exponential terms in E_args. To handle the case where we are given Df, not f, use is_log_deriv_k_t_radical_in_field(). See also ======== is_log_deriv_k_t_radical_in_field, is_deriv_k """ if Df: dfa, dfd = (fd*derivation(fa, DE) - fa*derivation(fd, DE)).cancel(fd**2, include=True) else: dfa, dfd = fa, fd # Our assumption here is that each monomial is recursively transcendental if len(DE.exts) != len(DE.D): if [i for i in DE.cases if i == 'tan'] or \ ({i for i in DE.cases if i == 'primitive'} - set(DE.indices('log'))): raise NotImplementedError("Real version of the structure " "theorems with hypertangent support is not yet implemented.") # TODO: What should really be done in this case? raise NotImplementedError("Nonelementary extensions not supported " "in the structure theorems.") E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.indices('exp')] L_part = [DE.D[i].as_expr() for i in DE.indices('log')] lhs = Matrix([E_part + L_part]) rhs = Matrix([dfa.as_expr()/dfd.as_expr()]) A, u = constant_system(lhs, rhs, DE) if not all(derivation(i, DE, basic=True).is_zero for i in u) or not A: # If the elements of u are not all constant # Note: See comment in constant_system # Also note: derivation(basic=True) calls cancel() return None else: if not all(i.is_Rational for i in u): # TODO: But maybe we can tell if they're not rational, like # log(2)/log(3). Also, there should be an option to continue # anyway, even if the result might potentially be wrong. raise NotImplementedError("Cannot work with non-rational " "coefficients in this case.") else: n = reduce(ilcm, [i.as_numer_denom()[1] for i in u]) u *= n terms = ([DE.T[i] for i in DE.indices('exp')] + [DE.extargs[i] for i in DE.indices('log')]) ans = list(zip(terms, u)) result = Mul(*[Pow(i, j) for i, j in ans]) # exp(f) will be the same as result up to a multiplicative # constant. We now find the log of that constant. argterms = ([DE.extargs[i] for i in DE.indices('exp')] + [DE.T[i] for i in DE.indices('log')]) const = cancel(fa.as_expr()/fd.as_expr() - Add(*[Mul(i, j/n) for i, j in zip(argterms, u)])) return (ans, result, n, const) def is_log_deriv_k_t_radical_in_field(fa, fd, DE, case='auto', z=None): """ Checks if f can be written as the logarithmic derivative of a k(t)-radical. Explanation =========== It differs from is_log_deriv_k_t_radical(fa, fd, DE, Df=False) for any given fa, fd, DE in that it finds the solution in the given field not in some (possibly unspecified extension) and "in_field" with the function name is used to indicate that. f in k(t) can be written as the logarithmic derivative of a k(t) radical if there exist n in ZZ and u in k(t) with n, u != 0 such that n*f == Du/u. Either returns (n, u) or None, which means that f cannot be written as the logarithmic derivative of a k(t)-radical. case is one of {'primitive', 'exp', 'tan', 'auto'} for the primitive, hyperexponential, and hypertangent cases, respectively. If case is 'auto', it will attempt to determine the type of the derivation automatically. See also ======== is_log_deriv_k_t_radical, is_deriv_k """ fa, fd = fa.cancel(fd, include=True) # f must be simple n, s = splitfactor(fd, DE) if not s.is_one: pass z = z or Dummy('z') H, b = residue_reduce(fa, fd, DE, z=z) if not b: # I will have to verify, but I believe that the answer should be # None in this case. This should never happen for the # functions given when solving the parametric logarithmic # derivative problem when integration elementary functions (see # Bronstein's book, page 255), so most likely this indicates a bug. return None roots = [(i, i.real_roots()) for i, _ in H] if not all(len(j) == i.degree() and all(k.is_Rational for k in j) for i, j in roots): # If f is the logarithmic derivative of a k(t)-radical, then all the # roots of the resultant must be rational numbers. return None # [(a, i), ...], where i*log(a) is a term in the log-part of the integral # of f respolys, residues = list(zip(*roots)) or [[], []] # Note: this might be empty, but everything below should work find in that # case (it should be the same as if it were [[1, 1]]) residueterms = [(H[j][1].subs(z, i), i) for j in range(len(H)) for i in residues[j]] # TODO: finish writing this and write tests p = cancel(fa.as_expr()/fd.as_expr() - residue_reduce_derivation(H, DE, z)) p = p.as_poly(DE.t) if p is None: # f - Dg will be in k[t] if f is the logarithmic derivative of a k(t)-radical return None if p.degree(DE.t) >= max(1, DE.d.degree(DE.t)): return None if case == 'auto': case = DE.case if case == 'exp': wa, wd = derivation(DE.t, DE).cancel(Poly(DE.t, DE.t), include=True) with DecrementLevel(DE): pa, pd = frac_in(p, DE.t, cancel=True) wa, wd = frac_in((wa, wd), DE.t) A = parametric_log_deriv(pa, pd, wa, wd, DE) if A is None: return None n, e, u = A u *= DE.t**e elif case == 'primitive': with DecrementLevel(DE): pa, pd = frac_in(p, DE.t) A = is_log_deriv_k_t_radical_in_field(pa, pd, DE, case='auto') if A is None: return None n, u = A elif case == 'base': # TODO: we can use more efficient residue reduction from ratint() if not fd.is_sqf or fa.degree() >= fd.degree(): # f is the logarithmic derivative in the base case if and only if # f = fa/fd, fd is square-free, deg(fa) < deg(fd), and # gcd(fa, fd) == 1. The last condition is handled by cancel() above. return None # Note: if residueterms = [], returns (1, 1) # f had better be 0 in that case. n = reduce(ilcm, [i.as_numer_denom()[1] for _, i in residueterms], S.One) u = Mul(*[Pow(i, j*n) for i, j in residueterms]) return (n, u) elif case == 'tan': raise NotImplementedError("The hypertangent case is " "not yet implemented for is_log_deriv_k_t_radical_in_field()") elif case in ['other_linear', 'other_nonlinear']: # XXX: If these are supported by the structure theorems, change to NotImplementedError. raise ValueError("The %s case is not supported in this function." % case) else: raise ValueError("case must be one of {'primitive', 'exp', 'tan', " "'base', 'auto'}, not %s" % case) common_denom = reduce(ilcm, [i.as_numer_denom()[1] for i in [j for _, j in residueterms]] + [n], S.One) residueterms = [(i, j*common_denom) for i, j in residueterms] m = common_denom//n if common_denom != n*m: # Verify exact division raise ValueError("Inexact division") u = cancel(u**m*Mul(*[Pow(i, j) for i, j in residueterms])) return (common_denom, u)
74361de19f6a29952f10389225a12b972b615302c92cbdfbcca7ba3d601d0a8c
from sympy.core import cacheit, Dummy, Ne, Integer, Rational, S, Wild from sympy.functions import binomial, sin, cos, Piecewise # TODO sin(a*x)*cos(b*x) -> sin((a+b)x) + sin((a-b)x) ? # creating, each time, Wild's and sin/cos/Mul is expensive. Also, our match & # subs are very slow when not cached, and if we create Wild each time, we # effectively block caching. # # so we cache the pattern # need to use a function instead of lamda since hash of lambda changes on # each call to _pat_sincos def _integer_instance(n): return isinstance(n , Integer) @cacheit def _pat_sincos(x): a = Wild('a', exclude=[x]) n, m = [Wild(s, exclude=[x], properties=[_integer_instance]) for s in 'nm'] pat = sin(a*x)**n * cos(a*x)**m return pat, a, n, m _u = Dummy('u') def trigintegrate(f, x, conds='piecewise'): """ Integrate f = Mul(trig) over x. Examples ======== >>> from sympy import sin, cos, tan, sec >>> from sympy.integrals.trigonometry import trigintegrate >>> from sympy.abc import x >>> trigintegrate(sin(x)*cos(x), x) sin(x)**2/2 >>> trigintegrate(sin(x)**2, x) x/2 - sin(x)*cos(x)/2 >>> trigintegrate(tan(x)*sec(x), x) 1/cos(x) >>> trigintegrate(sin(x)*tan(x), x) -log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x) References ========== .. [1] http://en.wikibooks.org/wiki/Calculus/Integration_techniques See Also ======== sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral """ from sympy.integrals.integrals import integrate pat, a, n, m = _pat_sincos(x) f = f.rewrite('sincos') M = f.match(pat) if M is None: return n, m = M[n], M[m] if n.is_zero and m.is_zero: return x zz = x if n.is_zero else S.Zero a = M[a] if n.is_odd or m.is_odd: u = _u n_, m_ = n.is_odd, m.is_odd # take smallest n or m -- to choose simplest substitution if n_ and m_: # Make sure to choose the positive one # otherwise an incorrect integral can occur. if n < 0 and m > 0: m_ = True n_ = False elif m < 0 and n > 0: n_ = True m_ = False # Both are negative so choose the smallest n or m # in absolute value for simplest substitution. elif (n < 0 and m < 0): n_ = n > m m_ = not (n > m) # Both n and m are odd and positive else: n_ = (n < m) # NB: careful here, one of the m_ = not (n < m) # conditions *must* be true # n m u=C (n-1)/2 m # S(x) * C(x) dx --> -(1-u^2) * u du if n_: ff = -(1 - u**2)**((n - 1)/2) * u**m uu = cos(a*x) # n m u=S n (m-1)/2 # S(x) * C(x) dx --> u * (1-u^2) du elif m_: ff = u**n * (1 - u**2)**((m - 1)/2) uu = sin(a*x) fi = integrate(ff, u) # XXX cyclic deps fx = fi.subs(u, uu) if conds == 'piecewise': return Piecewise((fx / a, Ne(a, 0)), (zz, True)) return fx / a # n & m are both even # # 2k 2m 2l 2l # we transform S (x) * C (x) into terms with only S (x) or C (x) # # example: # 100 4 100 2 2 100 4 2 # S (x) * C (x) = S (x) * (1-S (x)) = S (x) * (1 + S (x) - 2*S (x)) # # 104 102 100 # = S (x) - 2*S (x) + S (x) # 2k # then S is integrated with recursive formula # take largest n or m -- to choose simplest substitution n_ = (abs(n) > abs(m)) m_ = (abs(m) > abs(n)) res = S.Zero if n_: # 2k 2 k i 2i # C = (1 - S ) = sum(i, (-) * B(k, i) * S ) if m > 0: for i in range(0, m//2 + 1): res += ((-1)**i * binomial(m//2, i) * _sin_pow_integrate(n + 2*i, x)) elif m == 0: res = _sin_pow_integrate(n, x) else: # m < 0 , |n| > |m| # / # | # | m n # | cos (x) sin (x) dx = # | # | #/ # / # | # -1 m+1 n-1 n - 1 | m+2 n-2 # ________ cos (x) sin (x) + _______ | cos (x) sin (x) dx # | # m + 1 m + 1 | # / res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) + Rational(n - 1, m + 1) * trigintegrate(cos(x)**(m + 2)*sin(x)**(n - 2), x)) elif m_: # 2k 2 k i 2i # S = (1 - C ) = sum(i, (-) * B(k, i) * C ) if n > 0: # / / # | | # | m n | -m n # | cos (x)*sin (x) dx or | cos (x) * sin (x) dx # | | # / / # # |m| > |n| ; m, n >0 ; m, n belong to Z - {0} # n 2 # sin (x) term is expanded here in terms of cos (x), # and then integrated. # for i in range(0, n//2 + 1): res += ((-1)**i * binomial(n//2, i) * _cos_pow_integrate(m + 2*i, x)) elif n == 0: # / # | # | 1 # | _ _ _ # | m # | cos (x) # / # res = _cos_pow_integrate(m, x) else: # n < 0 , |m| > |n| # / # | # | m n # | cos (x) sin (x) dx = # | # | #/ # / # | # 1 m-1 n+1 m - 1 | m-2 n+2 # _______ cos (x) sin (x) + _______ | cos (x) sin (x) dx # | # n + 1 n + 1 | # / res = (Rational(1, n + 1) * cos(x)**(m - 1)*sin(x)**(n + 1) + Rational(m - 1, n + 1) * trigintegrate(cos(x)**(m - 2)*sin(x)**(n + 2), x)) else: if m == n: ##Substitute sin(2x)/2 for sin(x)cos(x) and then Integrate. res = integrate((sin(2*x)*S.Half)**m, x) elif (m == -n): if n < 0: # Same as the scheme described above. # the function argument to integrate in the end will # be 1 , this cannot be integrated by trigintegrate. # Hence use sympy.integrals.integrate. res = (Rational(1, n + 1) * cos(x)**(m - 1) * sin(x)**(n + 1) + Rational(m - 1, n + 1) * integrate(cos(x)**(m - 2) * sin(x)**(n + 2), x)) else: res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) + Rational(n - 1, m + 1) * integrate(cos(x)**(m + 2)*sin(x)**(n - 2), x)) if conds == 'piecewise': return Piecewise((res.subs(x, a*x) / a, Ne(a, 0)), (zz, True)) return res.subs(x, a*x) / a def _sin_pow_integrate(n, x): if n > 0: if n == 1: #Recursion break return -cos(x) # n > 0 # / / # | | # | n -1 n-1 n - 1 | n-2 # | sin (x) dx = ______ cos (x) sin (x) + _______ | sin (x) dx # | | # | n n | #/ / # # return (Rational(-1, n) * cos(x) * sin(x)**(n - 1) + Rational(n - 1, n) * _sin_pow_integrate(n - 2, x)) if n < 0: if n == -1: ##Make sure this does not come back here again. ##Recursion breaks here or at n==0. return trigintegrate(1/sin(x), x) # n < 0 # / / # | | # | n 1 n+1 n + 2 | n+2 # | sin (x) dx = _______ cos (x) sin (x) + _______ | sin (x) dx # | | # | n + 1 n + 1 | #/ / # return (Rational(1, n + 1) * cos(x) * sin(x)**(n + 1) + Rational(n + 2, n + 1) * _sin_pow_integrate(n + 2, x)) else: #n == 0 #Recursion break. return x def _cos_pow_integrate(n, x): if n > 0: if n == 1: #Recursion break. return sin(x) # n > 0 # / / # | | # | n 1 n-1 n - 1 | n-2 # | sin (x) dx = ______ sin (x) cos (x) + _______ | cos (x) dx # | | # | n n | #/ / # return (Rational(1, n) * sin(x) * cos(x)**(n - 1) + Rational(n - 1, n) * _cos_pow_integrate(n - 2, x)) if n < 0: if n == -1: ##Recursion break return trigintegrate(1/cos(x), x) # n < 0 # / / # | | # | n -1 n+1 n + 2 | n+2 # | cos (x) dx = _______ sin (x) cos (x) + _______ | cos (x) dx # | | # | n + 1 n + 1 | #/ / # return (Rational(-1, n + 1) * sin(x) * cos(x)**(n + 1) + Rational(n + 2, n + 1) * _cos_pow_integrate(n + 2, x)) else: # n == 0 #Recursion Break. return x
a3b3e253920a50a8a9a7b8f11d6161e67fece52181c5355bd7319c8f3c31bbe8
""" Integrate functions by rewriting them as Meijer G-functions. There are three user-visible functions that can be used by other parts of the sympy library to solve various integration problems: - meijerint_indefinite - meijerint_definite - meijerint_inversion They can be used to compute, respectively, indefinite integrals, definite integrals over intervals of the real line, and inverse laplace-type integrals (from c-I*oo to c+I*oo). See the respective docstrings for details. The main references for this are: [L] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 [R] Kelly B. Roach. Meijer G Function Representations. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pages 205-211, New York, 1997. ACM. [P] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990). Integrals and Series: More Special Functions, Vol. 3,. Gordon and Breach Science Publisher """ from typing import Dict, Tuple from sympy.core import oo, S, pi, Expr from sympy.core.exprtools import factor_terms from sympy.core.function import expand, expand_mul, expand_power_base from sympy.core.add import Add from sympy.core.mul import Mul from sympy.core.numbers import Rational from sympy.core.cache import cacheit from sympy.core.symbol import Dummy, Wild from sympy.simplify import hyperexpand, powdenest, collect from sympy.simplify.fu import sincos_to_sum from sympy.logic.boolalg import And, Or, BooleanAtom from sympy.functions.special.delta_functions import DiracDelta, Heaviside from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold from sympy.functions.elementary.hyperbolic import \ _rewrite_hyperbolics_as_exp, HyperbolicFunction from sympy.functions.elementary.trigonometric import cos, sin from sympy.functions.special.hyper import meijerg from sympy.utilities.iterables import multiset_partitions, ordered from sympy.utilities.misc import debug as _debug from sympy.utilities import default_sort_key # keep this at top for easy reference z = Dummy('z') def _has(res, *f): # return True if res has f; in the case of Piecewise # only return True if *all* pieces have f res = piecewise_fold(res) if getattr(res, 'is_Piecewise', False): return all(_has(i, *f) for i in res.args) return res.has(*f) def _create_lookup_table(table): """ Add formulae for the function -> meijerg lookup table. """ def wild(n): return Wild(n, exclude=[z]) p, q, a, b, c = list(map(wild, 'pqabc')) n = Wild('n', properties=[lambda x: x.is_Integer and x > 0]) t = p*z**q def add(formula, an, ap, bm, bq, arg=t, fac=S.One, cond=True, hint=True): table.setdefault(_mytype(formula, z), []).append((formula, [(fac, meijerg(an, ap, bm, bq, arg))], cond, hint)) def addi(formula, inst, cond, hint=True): table.setdefault( _mytype(formula, z), []).append((formula, inst, cond, hint)) def constant(a): return [(a, meijerg([1], [], [], [0], z)), (a, meijerg([], [1], [0], [], z))] table[()] = [(a, constant(a), True, True)] # [P], Section 8. from sympy import unpolarify, Function, Not class IsNonPositiveInteger(Function): @classmethod def eval(cls, arg): arg = unpolarify(arg) if arg.is_Integer is True: return arg <= 0 # Section 8.4.2 from sympy import (gamma, pi, cos, exp, re, sin, sinc, sqrt, sinh, cosh, factorial, log, erf, erfc, erfi, polar_lift) # TODO this needs more polar_lift (c/f entry for exp) add(Heaviside(t - b)*(t - b)**(a - 1), [a], [], [], [0], t/b, gamma(a)*b**(a - 1), And(b > 0)) add(Heaviside(b - t)*(b - t)**(a - 1), [], [a], [0], [], t/b, gamma(a)*b**(a - 1), And(b > 0)) add(Heaviside(z - (b/p)**(1/q))*(t - b)**(a - 1), [a], [], [], [0], t/b, gamma(a)*b**(a - 1), And(b > 0)) add(Heaviside((b/p)**(1/q) - z)*(b - t)**(a - 1), [], [a], [0], [], t/b, gamma(a)*b**(a - 1), And(b > 0)) add((b + t)**(-a), [1 - a], [], [0], [], t/b, b**(-a)/gamma(a), hint=Not(IsNonPositiveInteger(a))) add(abs(b - t)**(-a), [1 - a], [(1 - a)/2], [0], [(1 - a)/2], t/b, 2*sin(pi*a/2)*gamma(1 - a)*abs(b)**(-a), re(a) < 1) add((t**a - b**a)/(t - b), [0, a], [], [0, a], [], t/b, b**(a - 1)*sin(a*pi)/pi) # 12 def A1(r, sign, nu): return pi**Rational(-1, 2)*(-sign*nu/2)**(1 - 2*r) def tmpadd(r, sgn): # XXX the a**2 is bad for matching add((sqrt(a**2 + t) + sgn*a)**b/(a**2 + t)**r, [(1 + b)/2, 1 - 2*r + b/2], [], [(b - sgn*b)/2], [(b + sgn*b)/2], t/a**2, a**(b - 2*r)*A1(r, sgn, b)) tmpadd(0, 1) tmpadd(0, -1) tmpadd(S.Half, 1) tmpadd(S.Half, -1) # 13 def tmpadd(r, sgn): add((sqrt(a + p*z**q) + sgn*sqrt(p)*z**(q/2))**b/(a + p*z**q)**r, [1 - r + sgn*b/2], [1 - r - sgn*b/2], [0, S.Half], [], p*z**q/a, a**(b/2 - r)*A1(r, sgn, b)) tmpadd(0, 1) tmpadd(0, -1) tmpadd(S.Half, 1) tmpadd(S.Half, -1) # (those after look obscure) # Section 8.4.3 add(exp(polar_lift(-1)*t), [], [], [0], []) # TODO can do sin^n, sinh^n by expansion ... where? # 8.4.4 (hyperbolic functions) add(sinh(t), [], [1], [S.Half], [1, 0], t**2/4, pi**Rational(3, 2)) add(cosh(t), [], [S.Half], [0], [S.Half, S.Half], t**2/4, pi**Rational(3, 2)) # Section 8.4.5 # TODO can do t + a. but can also do by expansion... (XXX not really) add(sin(t), [], [], [S.Half], [0], t**2/4, sqrt(pi)) add(cos(t), [], [], [0], [S.Half], t**2/4, sqrt(pi)) # Section 8.4.6 (sinc function) add(sinc(t), [], [], [0], [Rational(-1, 2)], t**2/4, sqrt(pi)/2) # Section 8.5.5 def make_log1(subs): N = subs[n] return [((-1)**N*factorial(N), meijerg([], [1]*(N + 1), [0]*(N + 1), [], t))] def make_log2(subs): N = subs[n] return [(factorial(N), meijerg([1]*(N + 1), [], [], [0]*(N + 1), t))] # TODO these only hold for positive p, and can be made more general # but who uses log(x)*Heaviside(a-x) anyway ... # TODO also it would be nice to derive them recursively ... addi(log(t)**n*Heaviside(1 - t), make_log1, True) addi(log(t)**n*Heaviside(t - 1), make_log2, True) def make_log3(subs): return make_log1(subs) + make_log2(subs) addi(log(t)**n, make_log3, True) addi(log(t + a), constant(log(a)) + [(S.One, meijerg([1, 1], [], [1], [0], t/a))], True) addi(log(abs(t - a)), constant(log(abs(a))) + [(pi, meijerg([1, 1], [S.Half], [1], [0, S.Half], t/a))], True) # TODO log(x)/(x+a) and log(x)/(x-1) can also be done. should they # be derivable? # TODO further formulae in this section seem obscure # Sections 8.4.9-10 # TODO # Section 8.4.11 from sympy import Ei, I, expint, Si, Ci, Shi, Chi, fresnels, fresnelc addi(Ei(t), constant(-I*pi) + [(S.NegativeOne, meijerg([], [1], [0, 0], [], t*polar_lift(-1)))], True) # Section 8.4.12 add(Si(t), [1], [], [S.Half], [0, 0], t**2/4, sqrt(pi)/2) add(Ci(t), [], [1], [0, 0], [S.Half], t**2/4, -sqrt(pi)/2) # Section 8.4.13 add(Shi(t), [S.Half], [], [0], [Rational(-1, 2), Rational(-1, 2)], polar_lift(-1)*t**2/4, t*sqrt(pi)/4) add(Chi(t), [], [S.Half, 1], [0, 0], [S.Half, S.Half], t**2/4, - pi**S('3/2')/2) # generalized exponential integral add(expint(a, t), [], [a], [a - 1, 0], [], t) # Section 8.4.14 add(erf(t), [1], [], [S.Half], [0], t**2, 1/sqrt(pi)) # TODO exp(-x)*erf(I*x) does not work add(erfc(t), [], [1], [0, S.Half], [], t**2, 1/sqrt(pi)) # This formula for erfi(z) yields a wrong(?) minus sign #add(erfi(t), [1], [], [S.Half], [0], -t**2, I/sqrt(pi)) add(erfi(t), [S.Half], [], [0], [Rational(-1, 2)], -t**2, t/sqrt(pi)) # Fresnel Integrals add(fresnels(t), [1], [], [Rational(3, 4)], [0, Rational(1, 4)], pi**2*t**4/16, S.Half) add(fresnelc(t), [1], [], [Rational(1, 4)], [0, Rational(3, 4)], pi**2*t**4/16, S.Half) ##### bessel-type functions ##### from sympy import besselj, bessely, besseli, besselk # Section 8.4.19 add(besselj(a, t), [], [], [a/2], [-a/2], t**2/4) # all of the following are derivable #add(sin(t)*besselj(a, t), [Rational(1, 4), Rational(3, 4)], [], [(1+a)/2], # [-a/2, a/2, (1-a)/2], t**2, 1/sqrt(2)) #add(cos(t)*besselj(a, t), [Rational(1, 4), Rational(3, 4)], [], [a/2], # [-a/2, (1+a)/2, (1-a)/2], t**2, 1/sqrt(2)) #add(besselj(a, t)**2, [S.Half], [], [a], [-a, 0], t**2, 1/sqrt(pi)) #add(besselj(a, t)*besselj(b, t), [0, S.Half], [], [(a + b)/2], # [-(a+b)/2, (a - b)/2, (b - a)/2], t**2, 1/sqrt(pi)) # Section 8.4.20 add(bessely(a, t), [], [-(a + 1)/2], [a/2, -a/2], [-(a + 1)/2], t**2/4) # TODO all of the following should be derivable #add(sin(t)*bessely(a, t), [Rational(1, 4), Rational(3, 4)], [(1 - a - 1)/2], # [(1 + a)/2, (1 - a)/2], [(1 - a - 1)/2, (1 - 1 - a)/2, (1 - 1 + a)/2], # t**2, 1/sqrt(2)) #add(cos(t)*bessely(a, t), [Rational(1, 4), Rational(3, 4)], [(0 - a - 1)/2], # [(0 + a)/2, (0 - a)/2], [(0 - a - 1)/2, (1 - 0 - a)/2, (1 - 0 + a)/2], # t**2, 1/sqrt(2)) #add(besselj(a, t)*bessely(b, t), [0, S.Half], [(a - b - 1)/2], # [(a + b)/2, (a - b)/2], [(a - b - 1)/2, -(a + b)/2, (b - a)/2], # t**2, 1/sqrt(pi)) #addi(bessely(a, t)**2, # [(2/sqrt(pi), meijerg([], [S.Half, S.Half - a], [0, a, -a], # [S.Half - a], t**2)), # (1/sqrt(pi), meijerg([S.Half], [], [a], [-a, 0], t**2))], # True) #addi(bessely(a, t)*bessely(b, t), # [(2/sqrt(pi), meijerg([], [0, S.Half, (1 - a - b)/2], # [(a + b)/2, (a - b)/2, (b - a)/2, -(a + b)/2], # [(1 - a - b)/2], t**2)), # (1/sqrt(pi), meijerg([0, S.Half], [], [(a + b)/2], # [-(a + b)/2, (a - b)/2, (b - a)/2], t**2))], # True) # Section 8.4.21 ? # Section 8.4.22 add(besseli(a, t), [], [(1 + a)/2], [a/2], [-a/2, (1 + a)/2], t**2/4, pi) # TODO many more formulas. should all be derivable # Section 8.4.23 add(besselk(a, t), [], [], [a/2, -a/2], [], t**2/4, S.Half) # TODO many more formulas. should all be derivable # Complete elliptic integrals K(z) and E(z) from sympy import elliptic_k, elliptic_e add(elliptic_k(t), [S.Half, S.Half], [], [0], [0], -t, S.Half) add(elliptic_e(t), [S.Half, 3*S.Half], [], [0], [0], -t, Rational(-1, 2)/2) #################################################################### # First some helper functions. #################################################################### from sympy.utilities.timeutils import timethis timeit = timethis('meijerg') def _mytype(f, x): """ Create a hashable entity describing the type of f. """ if x not in f.free_symbols: return () elif f.is_Function: return (type(f),) else: types = [_mytype(a, x) for a in f.args] res = [] for t in types: res += list(t) res.sort() return tuple(res) class _CoeffExpValueError(ValueError): """ Exception raised by _get_coeff_exp, for internal use only. """ pass def _get_coeff_exp(expr, x): """ When expr is known to be of the form c*x**b, with c and/or b possibly 1, return c, b. Examples ======== >>> from sympy.abc import x, a, b >>> from sympy.integrals.meijerint import _get_coeff_exp >>> _get_coeff_exp(a*x**b, x) (a, b) >>> _get_coeff_exp(x, x) (1, 1) >>> _get_coeff_exp(2*x, x) (2, 1) >>> _get_coeff_exp(x**3, x) (1, 3) """ from sympy import powsimp (c, m) = expand_power_base(powsimp(expr)).as_coeff_mul(x) if not m: return c, S.Zero [m] = m if m.is_Pow: if m.base != x: raise _CoeffExpValueError('expr not of form a*x**b') return c, m.exp elif m == x: return c, S.One else: raise _CoeffExpValueError('expr not of form a*x**b: %s' % expr) def _exponents(expr, x): """ Find the exponents of ``x`` (not including zero) in ``expr``. Examples ======== >>> from sympy.integrals.meijerint import _exponents >>> from sympy.abc import x, y >>> from sympy import sin >>> _exponents(x, x) {1} >>> _exponents(x**2, x) {2} >>> _exponents(x**2 + x, x) {1, 2} >>> _exponents(x**3*sin(x + x**y) + 1/x, x) {-1, 1, 3, y} """ def _exponents_(expr, x, res): if expr == x: res.update([1]) return if expr.is_Pow and expr.base == x: res.update([expr.exp]) return for arg in expr.args: _exponents_(arg, x, res) res = set() _exponents_(expr, x, res) return res def _functions(expr, x): """ Find the types of functions in expr, to estimate the complexity. """ from sympy import Function return {e.func for e in expr.atoms(Function) if x in e.free_symbols} def _find_splitting_points(expr, x): """ Find numbers a such that a linear substitution x -> x + a would (hopefully) simplify expr. Examples ======== >>> from sympy.integrals.meijerint import _find_splitting_points as fsp >>> from sympy import sin >>> from sympy.abc import x >>> fsp(x, x) {0} >>> fsp((x-1)**3, x) {1} >>> fsp(sin(x+3)*x, x) {-3, 0} """ p, q = [Wild(n, exclude=[x]) for n in 'pq'] def compute_innermost(expr, res): if not isinstance(expr, Expr): return m = expr.match(p*x + q) if m and m[p] != 0: res.add(-m[q]/m[p]) return if expr.is_Atom: return for arg in expr.args: compute_innermost(arg, res) innermost = set() compute_innermost(expr, innermost) return innermost def _split_mul(f, x): """ Split expression ``f`` into fac, po, g, where fac is a constant factor, po = x**s for some s independent of s, and g is "the rest". Examples ======== >>> from sympy.integrals.meijerint import _split_mul >>> from sympy import sin >>> from sympy.abc import s, x >>> _split_mul((3*x)**s*sin(x**2)*x, x) (3**s, x*x**s, sin(x**2)) """ from sympy import polarify, unpolarify fac = S.One po = S.One g = S.One f = expand_power_base(f) args = Mul.make_args(f) for a in args: if a == x: po *= x elif x not in a.free_symbols: fac *= a else: if a.is_Pow and x not in a.exp.free_symbols: c, t = a.base.as_coeff_mul(x) if t != (x,): c, t = expand_mul(a.base).as_coeff_mul(x) if t == (x,): po *= x**a.exp fac *= unpolarify(polarify(c**a.exp, subs=False)) continue g *= a return fac, po, g def _mul_args(f): """ Return a list ``L`` such that ``Mul(*L) == f``. If ``f`` is not a ``Mul`` or ``Pow``, ``L=[f]``. If ``f=g**n`` for an integer ``n``, ``L=[g]*n``. If ``f`` is a ``Mul``, ``L`` comes from applying ``_mul_args`` to all factors of ``f``. """ args = Mul.make_args(f) gs = [] for g in args: if g.is_Pow and g.exp.is_Integer: n = g.exp base = g.base if n < 0: n = -n base = 1/base gs += [base]*n else: gs.append(g) return gs def _mul_as_two_parts(f): """ Find all the ways to split ``f`` into a product of two terms. Return None on failure. Explanation =========== Although the order is canonical from multiset_partitions, this is not necessarily the best order to process the terms. For example, if the case of len(gs) == 2 is removed and multiset is allowed to sort the terms, some tests fail. Examples ======== >>> from sympy.integrals.meijerint import _mul_as_two_parts >>> from sympy import sin, exp, ordered >>> from sympy.abc import x >>> list(ordered(_mul_as_two_parts(x*sin(x)*exp(x)))) [(x, exp(x)*sin(x)), (x*exp(x), sin(x)), (x*sin(x), exp(x))] """ gs = _mul_args(f) if len(gs) < 2: return None if len(gs) == 2: return [tuple(gs)] return [(Mul(*x), Mul(*y)) for (x, y) in multiset_partitions(gs, 2)] def _inflate_g(g, n): """ Return C, h such that h is a G function of argument z**n and g = C*h. """ # TODO should this be a method of meijerg? # See: [L, page 150, equation (5)] def inflate(params, n): """ (a1, .., ak) -> (a1/n, (a1+1)/n, ..., (ak + n-1)/n) """ res = [] for a in params: for i in range(n): res.append((a + i)/n) return res v = S(len(g.ap) - len(g.bq)) C = n**(1 + g.nu + v/2) C /= (2*pi)**((n - 1)*g.delta) return C, meijerg(inflate(g.an, n), inflate(g.aother, n), inflate(g.bm, n), inflate(g.bother, n), g.argument**n * n**(n*v)) def _flip_g(g): """ Turn the G function into one of inverse argument (i.e. G(1/x) -> G'(x)) """ # See [L], section 5.2 def tr(l): return [1 - a for a in l] return meijerg(tr(g.bm), tr(g.bother), tr(g.an), tr(g.aother), 1/g.argument) def _inflate_fox_h(g, a): r""" Let d denote the integrand in the definition of the G function ``g``. Consider the function H which is defined in the same way, but with integrand d/Gamma(a*s) (contour conventions as usual). If ``a`` is rational, the function H can be written as C*G, for a constant C and a G-function G. This function returns C, G. """ if a < 0: return _inflate_fox_h(_flip_g(g), -a) p = S(a.p) q = S(a.q) # We use the substitution s->qs, i.e. inflate g by q. We are left with an # extra factor of Gamma(p*s), for which we use Gauss' multiplication # theorem. D, g = _inflate_g(g, q) z = g.argument D /= (2*pi)**((1 - p)/2)*p**Rational(-1, 2) z /= p**p bs = [(n + 1)/p for n in range(p)] return D, meijerg(g.an, g.aother, g.bm, list(g.bother) + bs, z) _dummies = {} # type: Dict[Tuple[str, str], Dummy] def _dummy(name, token, expr, **kwargs): """ Return a dummy. This will return the same dummy if the same token+name is requested more than once, and it is not already in expr. This is for being cache-friendly. """ d = _dummy_(name, token, **kwargs) if d in expr.free_symbols: return Dummy(name, **kwargs) return d def _dummy_(name, token, **kwargs): """ Return a dummy associated to name and token. Same effect as declaring it globally. """ global _dummies if not (name, token) in _dummies: _dummies[(name, token)] = Dummy(name, **kwargs) return _dummies[(name, token)] def _is_analytic(f, x): """ Check if f(x), when expressed using G functions on the positive reals, will in fact agree with the G functions almost everywhere """ from sympy import Heaviside, Abs return not any(x in expr.free_symbols for expr in f.atoms(Heaviside, Abs)) def _condsimp(cond): """ Do naive simplifications on ``cond``. Explanation =========== Note that this routine is completely ad-hoc, simplification rules being added as need arises rather than following any logical pattern. Examples ======== >>> from sympy.integrals.meijerint import _condsimp as simp >>> from sympy import Or, Eq, And >>> from sympy.abc import x, y, z >>> simp(Or(x < y, z, Eq(x, y))) z | (x <= y) >>> simp(Or(x <= y, And(x < y, z))) x <= y """ from sympy import ( symbols, Wild, Eq, unbranched_argument, exp_polar, pi, I, arg, periodic_argument, oo, polar_lift) from sympy.logic.boolalg import BooleanFunction if not isinstance(cond, BooleanFunction): return cond cond = cond.func(*list(map(_condsimp, cond.args))) change = True p, q, r = symbols('p q r', cls=Wild) rules = [ (Or(p < q, Eq(p, q)), p <= q), # The next two obviously are instances of a general pattern, but it is # easier to spell out the few cases we care about. (And(abs(arg(p)) <= pi, abs(arg(p) - 2*pi) <= pi), Eq(arg(p) - pi, 0)), (And(abs(2*arg(p) + pi) <= pi, abs(2*arg(p) - pi) <= pi), Eq(arg(p), 0)), (And(abs(unbranched_argument(p)) <= pi, abs(unbranched_argument(exp_polar(-2*pi*I)*p)) <= pi), Eq(unbranched_argument(exp_polar(-I*pi)*p), 0)), (And(abs(unbranched_argument(p)) <= pi/2, abs(unbranched_argument(exp_polar(-pi*I)*p)) <= pi/2), Eq(unbranched_argument(exp_polar(-I*pi/2)*p), 0)), (Or(p <= q, And(p < q, r)), p <= q) ] while change: change = False for fro, to in rules: if fro.func != cond.func: continue for n, arg1 in enumerate(cond.args): if r in fro.args[0].free_symbols: m = arg1.match(fro.args[1]) num = 1 else: num = 0 m = arg1.match(fro.args[0]) if not m: continue otherargs = [x.subs(m) for x in fro.args[:num] + fro.args[num + 1:]] otherlist = [n] for arg2 in otherargs: for k, arg3 in enumerate(cond.args): if k in otherlist: continue if arg2 == arg3: otherlist += [k] break if isinstance(arg3, And) and arg2.args[1] == r and \ isinstance(arg2, And) and arg2.args[0] in arg3.args: otherlist += [k] break if isinstance(arg3, And) and arg2.args[0] == r and \ isinstance(arg2, And) and arg2.args[1] in arg3.args: otherlist += [k] break if len(otherlist) != len(otherargs) + 1: continue newargs = [arg_ for (k, arg_) in enumerate(cond.args) if k not in otherlist] + [to.subs(m)] cond = cond.func(*newargs) change = True break # final tweak def repl_eq(orig): if orig.lhs == 0: expr = orig.rhs elif orig.rhs == 0: expr = orig.lhs else: return orig m = expr.match(arg(p)**q) if not m: m = expr.match(unbranched_argument(polar_lift(p)**q)) if not m: if isinstance(expr, periodic_argument) and not expr.args[0].is_polar \ and expr.args[1] is oo: return (expr.args[0] > 0) return orig return (m[p] > 0) return cond.replace( lambda expr: expr.is_Relational and expr.rel_op == '==', repl_eq) def _eval_cond(cond): """ Re-evaluate the conditions. """ if isinstance(cond, bool): return cond return _condsimp(cond.doit()) #################################################################### # Now the "backbone" functions to do actual integration. #################################################################### def _my_principal_branch(expr, period, full_pb=False): """ Bring expr nearer to its principal branch by removing superfluous factors. This function does *not* guarantee to yield the principal branch, to avoid introducing opaque principal_branch() objects, unless full_pb=True. """ from sympy import principal_branch res = principal_branch(expr, period) if not full_pb: res = res.replace(principal_branch, lambda x, y: x) return res def _rewrite_saxena_1(fac, po, g, x): """ Rewrite the integral fac*po*g dx, from zero to infinity, as integral fac*G, where G has argument a*x. Note po=x**s. Return fac, G. """ _, s = _get_coeff_exp(po, x) a, b = _get_coeff_exp(g.argument, x) period = g.get_period() a = _my_principal_branch(a, period) # We substitute t = x**b. C = fac/(abs(b)*a**((s + 1)/b - 1)) # Absorb a factor of (at)**((1 + s)/b - 1). def tr(l): return [a + (1 + s)/b - 1 for a in l] return C, meijerg(tr(g.an), tr(g.aother), tr(g.bm), tr(g.bother), a*x) def _check_antecedents_1(g, x, helper=False): r""" Return a condition under which the mellin transform of g exists. Any power of x has already been absorbed into the G function, so this is just $\int_0^\infty g\, dx$. See [L, section 5.6.1]. (Note that s=1.) If ``helper`` is True, only check if the MT exists at infinity, i.e. if $\int_1^\infty g\, dx$ exists. """ # NOTE if you update these conditions, please update the documentation as well from sympy import Eq, Not, ceiling, Ne, re, unbranched_argument as arg delta = g.delta eta, _ = _get_coeff_exp(g.argument, x) m, n, p, q = S([len(g.bm), len(g.an), len(g.ap), len(g.bq)]) if p > q: def tr(l): return [1 - x for x in l] return _check_antecedents_1(meijerg(tr(g.bm), tr(g.bother), tr(g.an), tr(g.aother), x/eta), x) tmp = [] for b in g.bm: tmp += [-re(b) < 1] for a in g.an: tmp += [1 < 1 - re(a)] cond_3 = And(*tmp) for b in g.bother: tmp += [-re(b) < 1] for a in g.aother: tmp += [1 < 1 - re(a)] cond_3_star = And(*tmp) cond_4 = (-re(g.nu) + (q + 1 - p)/2 > q - p) def debug(*msg): _debug(*msg) debug('Checking antecedents for 1 function:') debug(' delta=%s, eta=%s, m=%s, n=%s, p=%s, q=%s' % (delta, eta, m, n, p, q)) debug(' ap = %s, %s' % (list(g.an), list(g.aother))) debug(' bq = %s, %s' % (list(g.bm), list(g.bother))) debug(' cond_3=%s, cond_3*=%s, cond_4=%s' % (cond_3, cond_3_star, cond_4)) conds = [] # case 1 case1 = [] tmp1 = [1 <= n, p < q, 1 <= m] tmp2 = [1 <= p, 1 <= m, Eq(q, p + 1), Not(And(Eq(n, 0), Eq(m, p + 1)))] tmp3 = [1 <= p, Eq(q, p)] for k in range(ceiling(delta/2) + 1): tmp3 += [Ne(abs(arg(eta)), (delta - 2*k)*pi)] tmp = [delta > 0, abs(arg(eta)) < delta*pi] extra = [Ne(eta, 0), cond_3] if helper: extra = [] for t in [tmp1, tmp2, tmp3]: case1 += [And(*(t + tmp + extra))] conds += case1 debug(' case 1:', case1) # case 2 extra = [cond_3] if helper: extra = [] case2 = [And(Eq(n, 0), p + 1 <= m, m <= q, abs(arg(eta)) < delta*pi, *extra)] conds += case2 debug(' case 2:', case2) # case 3 extra = [cond_3, cond_4] if helper: extra = [] case3 = [And(p < q, 1 <= m, delta > 0, Eq(abs(arg(eta)), delta*pi), *extra)] case3 += [And(p <= q - 2, Eq(delta, 0), Eq(abs(arg(eta)), 0), *extra)] conds += case3 debug(' case 3:', case3) # TODO altered cases 4-7 # extra case from wofram functions site: # (reproduced verbatim from Prudnikov, section 2.24.2) # http://functions.wolfram.com/HypergeometricFunctions/MeijerG/21/02/01/ case_extra = [] case_extra += [Eq(p, q), Eq(delta, 0), Eq(arg(eta), 0), Ne(eta, 0)] if not helper: case_extra += [cond_3] s = [] for a, b in zip(g.ap, g.bq): s += [b - a] case_extra += [re(Add(*s)) < 0] case_extra = And(*case_extra) conds += [case_extra] debug(' extra case:', [case_extra]) case_extra_2 = [And(delta > 0, abs(arg(eta)) < delta*pi)] if not helper: case_extra_2 += [cond_3] case_extra_2 = And(*case_extra_2) conds += [case_extra_2] debug(' second extra case:', [case_extra_2]) # TODO This leaves only one case from the three listed by Prudnikov. # Investigate if these indeed cover everything; if so, remove the rest. return Or(*conds) def _int0oo_1(g, x): r""" Evaluate $\int_0^\infty g\, dx$ using G functions, assuming the necessary conditions are fulfilled. Examples ======== >>> from sympy.abc import a, b, c, d, x, y >>> from sympy import meijerg >>> from sympy.integrals.meijerint import _int0oo_1 >>> _int0oo_1(meijerg([a], [b], [c], [d], x*y), x) gamma(-a)*gamma(c + 1)/(y*gamma(-d)*gamma(b + 1)) """ # See [L, section 5.6.1]. Note that s=1. from sympy import gamma, gammasimp, unpolarify eta, _ = _get_coeff_exp(g.argument, x) res = 1/eta # XXX TODO we should reduce order first for b in g.bm: res *= gamma(b + 1) for a in g.an: res *= gamma(1 - a - 1) for b in g.bother: res /= gamma(1 - b - 1) for a in g.aother: res /= gamma(a + 1) return gammasimp(unpolarify(res)) def _rewrite_saxena(fac, po, g1, g2, x, full_pb=False): """ Rewrite the integral ``fac*po*g1*g2`` from 0 to oo in terms of G functions with argument ``c*x``. Explanation =========== Return C, f1, f2 such that integral C f1 f2 from 0 to infinity equals integral fac ``po``, ``g1``, ``g2`` from 0 to infinity. Examples ======== >>> from sympy.integrals.meijerint import _rewrite_saxena >>> from sympy.abc import s, t, m >>> from sympy import meijerg >>> g1 = meijerg([], [], [0], [], s*t) >>> g2 = meijerg([], [], [m/2], [-m/2], t**2/4) >>> r = _rewrite_saxena(1, t**0, g1, g2, t) >>> r[0] s/(4*sqrt(pi)) >>> r[1] meijerg(((), ()), ((-1/2, 0), ()), s**2*t/4) >>> r[2] meijerg(((), ()), ((m/2,), (-m/2,)), t/4) """ from sympy.core.numbers import ilcm def pb(g): a, b = _get_coeff_exp(g.argument, x) per = g.get_period() return meijerg(g.an, g.aother, g.bm, g.bother, _my_principal_branch(a, per, full_pb)*x**b) _, s = _get_coeff_exp(po, x) _, b1 = _get_coeff_exp(g1.argument, x) _, b2 = _get_coeff_exp(g2.argument, x) if (b1 < 0) == True: b1 = -b1 g1 = _flip_g(g1) if (b2 < 0) == True: b2 = -b2 g2 = _flip_g(g2) if not b1.is_Rational or not b2.is_Rational: return m1, n1 = b1.p, b1.q m2, n2 = b2.p, b2.q tau = ilcm(m1*n2, m2*n1) r1 = tau//(m1*n2) r2 = tau//(m2*n1) C1, g1 = _inflate_g(g1, r1) C2, g2 = _inflate_g(g2, r2) g1 = pb(g1) g2 = pb(g2) fac *= C1*C2 a1, b = _get_coeff_exp(g1.argument, x) a2, _ = _get_coeff_exp(g2.argument, x) # arbitrarily tack on the x**s part to g1 # TODO should we try both? exp = (s + 1)/b - 1 fac = fac/(abs(b) * a1**exp) def tr(l): return [a + exp for a in l] g1 = meijerg(tr(g1.an), tr(g1.aother), tr(g1.bm), tr(g1.bother), a1*x) g2 = meijerg(g2.an, g2.aother, g2.bm, g2.bother, a2*x) return powdenest(fac, polar=True), g1, g2 def _check_antecedents(g1, g2, x): """ Return a condition under which the integral theorem applies. """ from sympy import re, Eq, Ne, cos, I, exp, sin, sign, unpolarify from sympy import arg as arg_, unbranched_argument as arg # Yes, this is madness. # XXX TODO this is a testing *nightmare* # NOTE if you update these conditions, please update the documentation as well # The following conditions are found in # [P], Section 2.24.1 # # They are also reproduced (verbatim!) at # http://functions.wolfram.com/HypergeometricFunctions/MeijerG/21/02/03/ # # Note: k=l=r=alpha=1 sigma, _ = _get_coeff_exp(g1.argument, x) omega, _ = _get_coeff_exp(g2.argument, x) s, t, u, v = S([len(g1.bm), len(g1.an), len(g1.ap), len(g1.bq)]) m, n, p, q = S([len(g2.bm), len(g2.an), len(g2.ap), len(g2.bq)]) bstar = s + t - (u + v)/2 cstar = m + n - (p + q)/2 rho = g1.nu + (u - v)/2 + 1 mu = g2.nu + (p - q)/2 + 1 phi = q - p - (v - u) eta = 1 - (v - u) - mu - rho psi = (pi*(q - m - n) + abs(arg(omega)))/(q - p) theta = (pi*(v - s - t) + abs(arg(sigma)))/(v - u) _debug('Checking antecedents:') _debug(' sigma=%s, s=%s, t=%s, u=%s, v=%s, b*=%s, rho=%s' % (sigma, s, t, u, v, bstar, rho)) _debug(' omega=%s, m=%s, n=%s, p=%s, q=%s, c*=%s, mu=%s,' % (omega, m, n, p, q, cstar, mu)) _debug(' phi=%s, eta=%s, psi=%s, theta=%s' % (phi, eta, psi, theta)) def _c1(): for g in [g1, g2]: for i in g.an: for j in g.bm: diff = i - j if diff.is_integer and diff.is_positive: return False return True c1 = _c1() c2 = And(*[re(1 + i + j) > 0 for i in g1.bm for j in g2.bm]) c3 = And(*[re(1 + i + j) < 1 + 1 for i in g1.an for j in g2.an]) c4 = And(*[(p - q)*re(1 + i - 1) - re(mu) > Rational(-3, 2) for i in g1.an]) c5 = And(*[(p - q)*re(1 + i) - re(mu) > Rational(-3, 2) for i in g1.bm]) c6 = And(*[(u - v)*re(1 + i - 1) - re(rho) > Rational(-3, 2) for i in g2.an]) c7 = And(*[(u - v)*re(1 + i) - re(rho) > Rational(-3, 2) for i in g2.bm]) c8 = (abs(phi) + 2*re((rho - 1)*(q - p) + (v - u)*(q - p) + (mu - 1)*(v - u)) > 0) c9 = (abs(phi) - 2*re((rho - 1)*(q - p) + (v - u)*(q - p) + (mu - 1)*(v - u)) > 0) c10 = (abs(arg(sigma)) < bstar*pi) c11 = Eq(abs(arg(sigma)), bstar*pi) c12 = (abs(arg(omega)) < cstar*pi) c13 = Eq(abs(arg(omega)), cstar*pi) # The following condition is *not* implemented as stated on the wolfram # function site. In the book of Prudnikov there is an additional part # (the And involving re()). However, I only have this book in russian, and # I don't read any russian. The following condition is what other people # have told me it means. # Worryingly, it is different from the condition implemented in REDUCE. # The REDUCE implementation: # https://reduce-algebra.svn.sourceforge.net/svnroot/reduce-algebra/trunk/packages/defint/definta.red # (search for tst14) # The Wolfram alpha version: # http://functions.wolfram.com/HypergeometricFunctions/MeijerG/21/02/03/03/0014/ z0 = exp(-(bstar + cstar)*pi*I) zos = unpolarify(z0*omega/sigma) zso = unpolarify(z0*sigma/omega) if zos == 1/zso: c14 = And(Eq(phi, 0), bstar + cstar <= 1, Or(Ne(zos, 1), re(mu + rho + v - u) < 1, re(mu + rho + q - p) < 1)) else: def _cond(z): '''Returns True if abs(arg(1-z)) < pi, avoiding arg(0). Explanation =========== If ``z`` is 1 then arg is NaN. This raises a TypeError on `NaN < pi`. Previously this gave `False` so this behavior has been hardcoded here but someone should check if this NaN is more serious! This NaN is triggered by test_meijerint() in test_meijerint.py: `meijerint_definite(exp(x), x, 0, I)` ''' return z != 1 and abs(arg_(1 - z)) < pi c14 = And(Eq(phi, 0), bstar - 1 + cstar <= 0, Or(And(Ne(zos, 1), _cond(zos)), And(re(mu + rho + v - u) < 1, Eq(zos, 1)))) c14_alt = And(Eq(phi, 0), cstar - 1 + bstar <= 0, Or(And(Ne(zso, 1), _cond(zso)), And(re(mu + rho + q - p) < 1, Eq(zso, 1)))) # Since r=k=l=1, in our case there is c14_alt which is the same as calling # us with (g1, g2) = (g2, g1). The conditions below enumerate all cases # (i.e. we don't have to try arguments reversed by hand), and indeed try # all symmetric cases. (i.e. whenever there is a condition involving c14, # there is also a dual condition which is exactly what we would get when g1, # g2 were interchanged, *but c14 was unaltered*). # Hence the following seems correct: c14 = Or(c14, c14_alt) ''' When `c15` is NaN (e.g. from `psi` being NaN as happens during 'test_issue_4992' and/or `theta` is NaN as in 'test_issue_6253', both in `test_integrals.py`) the comparison to 0 formerly gave False whereas now an error is raised. To keep the old behavior, the value of NaN is replaced with False but perhaps a closer look at this condition should be made: XXX how should conditions leading to c15=NaN be handled? ''' try: lambda_c = (q - p)*abs(omega)**(1/(q - p))*cos(psi) \ + (v - u)*abs(sigma)**(1/(v - u))*cos(theta) # the TypeError might be raised here, e.g. if lambda_c is NaN if _eval_cond(lambda_c > 0) != False: c15 = (lambda_c > 0) else: def lambda_s0(c1, c2): return c1*(q - p)*abs(omega)**(1/(q - p))*sin(psi) \ + c2*(v - u)*abs(sigma)**(1/(v - u))*sin(theta) lambda_s = Piecewise( ((lambda_s0(+1, +1)*lambda_s0(-1, -1)), And(Eq(arg(sigma), 0), Eq(arg(omega), 0))), (lambda_s0(sign(arg(omega)), +1)*lambda_s0(sign(arg(omega)), -1), And(Eq(arg(sigma), 0), Ne(arg(omega), 0))), (lambda_s0(+1, sign(arg(sigma)))*lambda_s0(-1, sign(arg(sigma))), And(Ne(arg(sigma), 0), Eq(arg(omega), 0))), (lambda_s0(sign(arg(omega)), sign(arg(sigma))), True)) tmp = [lambda_c > 0, And(Eq(lambda_c, 0), Ne(lambda_s, 0), re(eta) > -1), And(Eq(lambda_c, 0), Eq(lambda_s, 0), re(eta) > 0)] c15 = Or(*tmp) except TypeError: c15 = False for cond, i in [(c1, 1), (c2, 2), (c3, 3), (c4, 4), (c5, 5), (c6, 6), (c7, 7), (c8, 8), (c9, 9), (c10, 10), (c11, 11), (c12, 12), (c13, 13), (c14, 14), (c15, 15)]: _debug(' c%s:' % i, cond) # We will return Or(*conds) conds = [] def pr(count): _debug(' case %s:' % count, conds[-1]) conds += [And(m*n*s*t != 0, bstar.is_positive is True, cstar.is_positive is True, c1, c2, c3, c10, c12)] # 1 pr(1) conds += [And(Eq(u, v), Eq(bstar, 0), cstar.is_positive is True, sigma.is_positive is True, re(rho) < 1, c1, c2, c3, c12)] # 2 pr(2) conds += [And(Eq(p, q), Eq(cstar, 0), bstar.is_positive is True, omega.is_positive is True, re(mu) < 1, c1, c2, c3, c10)] # 3 pr(3) conds += [And(Eq(p, q), Eq(u, v), Eq(bstar, 0), Eq(cstar, 0), sigma.is_positive is True, omega.is_positive is True, re(mu) < 1, re(rho) < 1, Ne(sigma, omega), c1, c2, c3)] # 4 pr(4) conds += [And(Eq(p, q), Eq(u, v), Eq(bstar, 0), Eq(cstar, 0), sigma.is_positive is True, omega.is_positive is True, re(mu + rho) < 1, Ne(omega, sigma), c1, c2, c3)] # 5 pr(5) conds += [And(p > q, s.is_positive is True, bstar.is_positive is True, cstar >= 0, c1, c2, c3, c5, c10, c13)] # 6 pr(6) conds += [And(p < q, t.is_positive is True, bstar.is_positive is True, cstar >= 0, c1, c2, c3, c4, c10, c13)] # 7 pr(7) conds += [And(u > v, m.is_positive is True, cstar.is_positive is True, bstar >= 0, c1, c2, c3, c7, c11, c12)] # 8 pr(8) conds += [And(u < v, n.is_positive is True, cstar.is_positive is True, bstar >= 0, c1, c2, c3, c6, c11, c12)] # 9 pr(9) conds += [And(p > q, Eq(u, v), Eq(bstar, 0), cstar >= 0, sigma.is_positive is True, re(rho) < 1, c1, c2, c3, c5, c13)] # 10 pr(10) conds += [And(p < q, Eq(u, v), Eq(bstar, 0), cstar >= 0, sigma.is_positive is True, re(rho) < 1, c1, c2, c3, c4, c13)] # 11 pr(11) conds += [And(Eq(p, q), u > v, bstar >= 0, Eq(cstar, 0), omega.is_positive is True, re(mu) < 1, c1, c2, c3, c7, c11)] # 12 pr(12) conds += [And(Eq(p, q), u < v, bstar >= 0, Eq(cstar, 0), omega.is_positive is True, re(mu) < 1, c1, c2, c3, c6, c11)] # 13 pr(13) conds += [And(p < q, u > v, bstar >= 0, cstar >= 0, c1, c2, c3, c4, c7, c11, c13)] # 14 pr(14) conds += [And(p > q, u < v, bstar >= 0, cstar >= 0, c1, c2, c3, c5, c6, c11, c13)] # 15 pr(15) conds += [And(p > q, u > v, bstar >= 0, cstar >= 0, c1, c2, c3, c5, c7, c8, c11, c13, c14)] # 16 pr(16) conds += [And(p < q, u < v, bstar >= 0, cstar >= 0, c1, c2, c3, c4, c6, c9, c11, c13, c14)] # 17 pr(17) conds += [And(Eq(t, 0), s.is_positive is True, bstar.is_positive is True, phi.is_positive is True, c1, c2, c10)] # 18 pr(18) conds += [And(Eq(s, 0), t.is_positive is True, bstar.is_positive is True, phi.is_negative is True, c1, c3, c10)] # 19 pr(19) conds += [And(Eq(n, 0), m.is_positive is True, cstar.is_positive is True, phi.is_negative is True, c1, c2, c12)] # 20 pr(20) conds += [And(Eq(m, 0), n.is_positive is True, cstar.is_positive is True, phi.is_positive is True, c1, c3, c12)] # 21 pr(21) conds += [And(Eq(s*t, 0), bstar.is_positive is True, cstar.is_positive is True, c1, c2, c3, c10, c12)] # 22 pr(22) conds += [And(Eq(m*n, 0), bstar.is_positive is True, cstar.is_positive is True, c1, c2, c3, c10, c12)] # 23 pr(23) # The following case is from [Luke1969]. As far as I can tell, it is *not* # covered by Prudnikov's. # Let G1 and G2 be the two G-functions. Suppose the integral exists from # 0 to a > 0 (this is easy the easy part), that G1 is exponential decay at # infinity, and that the mellin transform of G2 exists. # Then the integral exists. mt1_exists = _check_antecedents_1(g1, x, helper=True) mt2_exists = _check_antecedents_1(g2, x, helper=True) conds += [And(mt2_exists, Eq(t, 0), u < s, bstar.is_positive is True, c10, c1, c2, c3)] pr('E1') conds += [And(mt2_exists, Eq(s, 0), v < t, bstar.is_positive is True, c10, c1, c2, c3)] pr('E2') conds += [And(mt1_exists, Eq(n, 0), p < m, cstar.is_positive is True, c12, c1, c2, c3)] pr('E3') conds += [And(mt1_exists, Eq(m, 0), q < n, cstar.is_positive is True, c12, c1, c2, c3)] pr('E4') # Let's short-circuit if this worked ... # the rest is corner-cases and terrible to read. r = Or(*conds) if _eval_cond(r) != False: return r conds += [And(m + n > p, Eq(t, 0), Eq(phi, 0), s.is_positive is True, bstar.is_positive is True, cstar.is_negative is True, abs(arg(omega)) < (m + n - p + 1)*pi, c1, c2, c10, c14, c15)] # 24 pr(24) conds += [And(m + n > q, Eq(s, 0), Eq(phi, 0), t.is_positive is True, bstar.is_positive is True, cstar.is_negative is True, abs(arg(omega)) < (m + n - q + 1)*pi, c1, c3, c10, c14, c15)] # 25 pr(25) conds += [And(Eq(p, q - 1), Eq(t, 0), Eq(phi, 0), s.is_positive is True, bstar.is_positive is True, cstar >= 0, cstar*pi < abs(arg(omega)), c1, c2, c10, c14, c15)] # 26 pr(26) conds += [And(Eq(p, q + 1), Eq(s, 0), Eq(phi, 0), t.is_positive is True, bstar.is_positive is True, cstar >= 0, cstar*pi < abs(arg(omega)), c1, c3, c10, c14, c15)] # 27 pr(27) conds += [And(p < q - 1, Eq(t, 0), Eq(phi, 0), s.is_positive is True, bstar.is_positive is True, cstar >= 0, cstar*pi < abs(arg(omega)), abs(arg(omega)) < (m + n - p + 1)*pi, c1, c2, c10, c14, c15)] # 28 pr(28) conds += [And( p > q + 1, Eq(s, 0), Eq(phi, 0), t.is_positive is True, bstar.is_positive is True, cstar >= 0, cstar*pi < abs(arg(omega)), abs(arg(omega)) < (m + n - q + 1)*pi, c1, c3, c10, c14, c15)] # 29 pr(29) conds += [And(Eq(n, 0), Eq(phi, 0), s + t > 0, m.is_positive is True, cstar.is_positive is True, bstar.is_negative is True, abs(arg(sigma)) < (s + t - u + 1)*pi, c1, c2, c12, c14, c15)] # 30 pr(30) conds += [And(Eq(m, 0), Eq(phi, 0), s + t > v, n.is_positive is True, cstar.is_positive is True, bstar.is_negative is True, abs(arg(sigma)) < (s + t - v + 1)*pi, c1, c3, c12, c14, c15)] # 31 pr(31) conds += [And(Eq(n, 0), Eq(phi, 0), Eq(u, v - 1), m.is_positive is True, cstar.is_positive is True, bstar >= 0, bstar*pi < abs(arg(sigma)), abs(arg(sigma)) < (bstar + 1)*pi, c1, c2, c12, c14, c15)] # 32 pr(32) conds += [And(Eq(m, 0), Eq(phi, 0), Eq(u, v + 1), n.is_positive is True, cstar.is_positive is True, bstar >= 0, bstar*pi < abs(arg(sigma)), abs(arg(sigma)) < (bstar + 1)*pi, c1, c3, c12, c14, c15)] # 33 pr(33) conds += [And( Eq(n, 0), Eq(phi, 0), u < v - 1, m.is_positive is True, cstar.is_positive is True, bstar >= 0, bstar*pi < abs(arg(sigma)), abs(arg(sigma)) < (s + t - u + 1)*pi, c1, c2, c12, c14, c15)] # 34 pr(34) conds += [And( Eq(m, 0), Eq(phi, 0), u > v + 1, n.is_positive is True, cstar.is_positive is True, bstar >= 0, bstar*pi < abs(arg(sigma)), abs(arg(sigma)) < (s + t - v + 1)*pi, c1, c3, c12, c14, c15)] # 35 pr(35) return Or(*conds) # NOTE An alternative, but as far as I can tell weaker, set of conditions # can be found in [L, section 5.6.2]. def _int0oo(g1, g2, x): """ Express integral from zero to infinity g1*g2 using a G function, assuming the necessary conditions are fulfilled. Examples ======== >>> from sympy.integrals.meijerint import _int0oo >>> from sympy.abc import s, t, m >>> from sympy import meijerg, S >>> g1 = meijerg([], [], [-S(1)/2, 0], [], s**2*t/4) >>> g2 = meijerg([], [], [m/2], [-m/2], t/4) >>> _int0oo(g1, g2, t) 4*meijerg(((1/2, 0), ()), ((m/2,), (-m/2,)), s**(-2))/s**2 """ # See: [L, section 5.6.2, equation (1)] eta, _ = _get_coeff_exp(g1.argument, x) omega, _ = _get_coeff_exp(g2.argument, x) def neg(l): return [-x for x in l] a1 = neg(g1.bm) + list(g2.an) a2 = list(g2.aother) + neg(g1.bother) b1 = neg(g1.an) + list(g2.bm) b2 = list(g2.bother) + neg(g1.aother) return meijerg(a1, a2, b1, b2, omega/eta)/eta def _rewrite_inversion(fac, po, g, x): """ Absorb ``po`` == x**s into g. """ _, s = _get_coeff_exp(po, x) a, b = _get_coeff_exp(g.argument, x) def tr(l): return [t + s/b for t in l] return (powdenest(fac/a**(s/b), polar=True), meijerg(tr(g.an), tr(g.aother), tr(g.bm), tr(g.bother), g.argument)) def _check_antecedents_inversion(g, x): """ Check antecedents for the laplace inversion integral. """ from sympy import re, im, Or, And, Eq, exp, I, Add, nan, Ne _debug('Checking antecedents for inversion:') z = g.argument _, e = _get_coeff_exp(z, x) if e < 0: _debug(' Flipping G.') # We want to assume that argument gets large as |x| -> oo return _check_antecedents_inversion(_flip_g(g), x) def statement_half(a, b, c, z, plus): coeff, exponent = _get_coeff_exp(z, x) a *= exponent b *= coeff**c c *= exponent conds = [] wp = b*exp(I*re(c)*pi/2) wm = b*exp(-I*re(c)*pi/2) if plus: w = wp else: w = wm conds += [And(Or(Eq(b, 0), re(c) <= 0), re(a) <= -1)] conds += [And(Ne(b, 0), Eq(im(c), 0), re(c) > 0, re(w) < 0)] conds += [And(Ne(b, 0), Eq(im(c), 0), re(c) > 0, re(w) <= 0, re(a) <= -1)] return Or(*conds) def statement(a, b, c, z): """ Provide a convergence statement for z**a * exp(b*z**c), c/f sphinx docs. """ return And(statement_half(a, b, c, z, True), statement_half(a, b, c, z, False)) # Notations from [L], section 5.7-10 m, n, p, q = S([len(g.bm), len(g.an), len(g.ap), len(g.bq)]) tau = m + n - p nu = q - m - n rho = (tau - nu)/2 sigma = q - p if sigma == 1: epsilon = S.Half elif sigma > 1: epsilon = 1 else: epsilon = nan theta = ((1 - sigma)/2 + Add(*g.bq) - Add(*g.ap))/sigma delta = g.delta _debug(' m=%s, n=%s, p=%s, q=%s, tau=%s, nu=%s, rho=%s, sigma=%s' % ( m, n, p, q, tau, nu, rho, sigma)) _debug(' epsilon=%s, theta=%s, delta=%s' % (epsilon, theta, delta)) # First check if the computation is valid. if not (g.delta >= e/2 or (p >= 1 and p >= q)): _debug(' Computation not valid for these parameters.') return False # Now check if the inversion integral exists. # Test "condition A" for a in g.an: for b in g.bm: if (a - b).is_integer and a > b: _debug(' Not a valid G function.') return False # There are two cases. If p >= q, we can directly use a slater expansion # like [L], 5.2 (11). Note in particular that the asymptotics of such an # expansion even hold when some of the parameters differ by integers, i.e. # the formula itself would not be valid! (b/c G functions are cts. in their # parameters) # When p < q, we need to use the theorems of [L], 5.10. if p >= q: _debug(' Using asymptotic Slater expansion.') return And(*[statement(a - 1, 0, 0, z) for a in g.an]) def E(z): return And(*[statement(a - 1, 0, 0, z) for a in g.an]) def H(z): return statement(theta, -sigma, 1/sigma, z) def Hp(z): return statement_half(theta, -sigma, 1/sigma, z, True) def Hm(z): return statement_half(theta, -sigma, 1/sigma, z, False) # [L], section 5.10 conds = [] # Theorem 1 -- p < q from test above conds += [And(1 <= n, 1 <= m, rho*pi - delta >= pi/2, delta > 0, E(z*exp(I*pi*(nu + 1))))] # Theorem 2, statements (2) and (3) conds += [And(p + 1 <= m, m + 1 <= q, delta > 0, delta < pi/2, n == 0, (m - p + 1)*pi - delta >= pi/2, Hp(z*exp(I*pi*(q - m))), Hm(z*exp(-I*pi*(q - m))))] # Theorem 2, statement (5) -- p < q from test above conds += [And(m == q, n == 0, delta > 0, (sigma + epsilon)*pi - delta >= pi/2, H(z))] # Theorem 3, statements (6) and (7) conds += [And(Or(And(p <= q - 2, 1 <= tau, tau <= sigma/2), And(p + 1 <= m + n, m + n <= (p + q)/2)), delta > 0, delta < pi/2, (tau + 1)*pi - delta >= pi/2, Hp(z*exp(I*pi*nu)), Hm(z*exp(-I*pi*nu)))] # Theorem 4, statements (10) and (11) -- p < q from test above conds += [And(1 <= m, rho > 0, delta > 0, delta + rho*pi < pi/2, (tau + epsilon)*pi - delta >= pi/2, Hp(z*exp(I*pi*nu)), Hm(z*exp(-I*pi*nu)))] # Trivial case conds += [m == 0] # TODO # Theorem 5 is quite general # Theorem 6 contains special cases for q=p+1 return Or(*conds) def _int_inversion(g, x, t): """ Compute the laplace inversion integral, assuming the formula applies. """ b, a = _get_coeff_exp(g.argument, x) C, g = _inflate_fox_h(meijerg(g.an, g.aother, g.bm, g.bother, b/t**a), -a) return C/t*g #################################################################### # Finally, the real meat. #################################################################### _lookup_table = None @cacheit @timeit def _rewrite_single(f, x, recursive=True): """ Try to rewrite f as a sum of single G functions of the form C*x**s*G(a*x**b), where b is a rational number and C is independent of x. We guarantee that result.argument.as_coeff_mul(x) returns (a, (x**b,)) or (a, ()). Returns a list of tuples (C, s, G) and a condition cond. Returns None on failure. """ from sympy import polarify, unpolarify, oo, zoo, Tuple global _lookup_table if not _lookup_table: _lookup_table = {} _create_lookup_table(_lookup_table) if isinstance(f, meijerg): from sympy import factor coeff, m = factor(f.argument, x).as_coeff_mul(x) if len(m) > 1: return None m = m[0] if m.is_Pow: if m.base != x or not m.exp.is_Rational: return None elif m != x: return None return [(1, 0, meijerg(f.an, f.aother, f.bm, f.bother, coeff*m))], True f_ = f f = f.subs(x, z) t = _mytype(f, z) if t in _lookup_table: l = _lookup_table[t] for formula, terms, cond, hint in l: subs = f.match(formula, old=True) if subs: subs_ = {} for fro, to in subs.items(): subs_[fro] = unpolarify(polarify(to, lift=True), exponents_only=True) subs = subs_ if not isinstance(hint, bool): hint = hint.subs(subs) if hint == False: continue if not isinstance(cond, (bool, BooleanAtom)): cond = unpolarify(cond.subs(subs)) if _eval_cond(cond) == False: continue if not isinstance(terms, list): terms = terms(subs) res = [] for fac, g in terms: r1 = _get_coeff_exp(unpolarify(fac.subs(subs).subs(z, x), exponents_only=True), x) try: g = g.subs(subs).subs(z, x) except ValueError: continue # NOTE these substitutions can in principle introduce oo, # zoo and other absurdities. It shouldn't matter, # but better be safe. if Tuple(*(r1 + (g,))).has(oo, zoo, -oo): continue g = meijerg(g.an, g.aother, g.bm, g.bother, unpolarify(g.argument, exponents_only=True)) res.append(r1 + (g,)) if res: return res, cond # try recursive mellin transform if not recursive: return None _debug('Trying recursive Mellin transform method.') from sympy.integrals.transforms import (mellin_transform, inverse_mellin_transform, IntegralTransformError, MellinTransformStripError) from sympy import oo, nan, zoo, simplify, cancel def my_imt(F, s, x, strip): """ Calling simplify() all the time is slow and not helpful, since most of the time it only factors things in a way that has to be un-done anyway. But sometimes it can remove apparent poles. """ # XXX should this be in inverse_mellin_transform? try: return inverse_mellin_transform(F, s, x, strip, as_meijerg=True, needeval=True) except MellinTransformStripError: return inverse_mellin_transform( simplify(cancel(expand(F))), s, x, strip, as_meijerg=True, needeval=True) f = f_ s = _dummy('s', 'rewrite-single', f) # to avoid infinite recursion, we have to force the two g functions case def my_integrator(f, x): from sympy import Integral, hyperexpand r = _meijerint_definite_4(f, x, only_double=True) if r is not None: res, cond = r res = _my_unpolarify(hyperexpand(res, rewrite='nonrepsmall')) return Piecewise((res, cond), (Integral(f, (x, 0, oo)), True)) return Integral(f, (x, 0, oo)) try: F, strip, _ = mellin_transform(f, x, s, integrator=my_integrator, simplify=False, needeval=True) g = my_imt(F, s, x, strip) except IntegralTransformError: g = None if g is None: # We try to find an expression by analytic continuation. # (also if the dummy is already in the expression, there is no point in # putting in another one) a = _dummy_('a', 'rewrite-single') if a not in f.free_symbols and _is_analytic(f, x): try: F, strip, _ = mellin_transform(f.subs(x, a*x), x, s, integrator=my_integrator, needeval=True, simplify=False) g = my_imt(F, s, x, strip).subs(a, 1) except IntegralTransformError: g = None if g is None or g.has(oo, nan, zoo): _debug('Recursive Mellin transform failed.') return None args = Add.make_args(g) res = [] for f in args: c, m = f.as_coeff_mul(x) if len(m) > 1: raise NotImplementedError('Unexpected form...') g = m[0] a, b = _get_coeff_exp(g.argument, x) res += [(c, 0, meijerg(g.an, g.aother, g.bm, g.bother, unpolarify(polarify( a, lift=True), exponents_only=True) *x**b))] _debug('Recursive Mellin transform worked:', g) return res, True def _rewrite1(f, x, recursive=True): """ Try to rewrite ``f`` using a (sum of) single G functions with argument a*x**b. Return fac, po, g such that f = fac*po*g, fac is independent of ``x``. and po = x**s. Here g is a result from _rewrite_single. Return None on failure. """ fac, po, g = _split_mul(f, x) g = _rewrite_single(g, x, recursive) if g: return fac, po, g[0], g[1] def _rewrite2(f, x): """ Try to rewrite ``f`` as a product of two G functions of arguments a*x**b. Return fac, po, g1, g2 such that f = fac*po*g1*g2, where fac is independent of x and po is x**s. Here g1 and g2 are results of _rewrite_single. Returns None on failure. """ fac, po, g = _split_mul(f, x) if any(_rewrite_single(expr, x, False) is None for expr in _mul_args(g)): return None l = _mul_as_two_parts(g) if not l: return None l = list(ordered(l, [ lambda p: max(len(_exponents(p[0], x)), len(_exponents(p[1], x))), lambda p: max(len(_functions(p[0], x)), len(_functions(p[1], x))), lambda p: max(len(_find_splitting_points(p[0], x)), len(_find_splitting_points(p[1], x)))])) for recursive in [False, True]: for fac1, fac2 in l: g1 = _rewrite_single(fac1, x, recursive) g2 = _rewrite_single(fac2, x, recursive) if g1 and g2: cond = And(g1[1], g2[1]) if cond != False: return fac, po, g1[0], g2[0], cond def meijerint_indefinite(f, x): """ Compute an indefinite integral of ``f`` by rewriting it as a G function. Examples ======== >>> from sympy.integrals.meijerint import meijerint_indefinite >>> from sympy import sin >>> from sympy.abc import x >>> meijerint_indefinite(sin(x), x) -cos(x) """ from sympy import hyper, meijerg results = [] for a in sorted(_find_splitting_points(f, x) | {S.Zero}, key=default_sort_key): res = _meijerint_indefinite_1(f.subs(x, x + a), x) if not res: continue res = res.subs(x, x - a) if _has(res, hyper, meijerg): results.append(res) else: return res if f.has(HyperbolicFunction): _debug('Try rewriting hyperbolics in terms of exp.') rv = meijerint_indefinite( _rewrite_hyperbolics_as_exp(f), x) if rv: if not type(rv) is list: return collect(factor_terms(rv), rv.atoms(exp)) results.extend(rv) if results: return next(ordered(results)) def _meijerint_indefinite_1(f, x): """ Helper that does not attempt any substitution. """ from sympy import Integral, piecewise_fold, nan, zoo _debug('Trying to compute the indefinite integral of', f, 'wrt', x) gs = _rewrite1(f, x) if gs is None: # Note: the code that calls us will do expand() and try again return None fac, po, gl, cond = gs _debug(' could rewrite:', gs) res = S.Zero for C, s, g in gl: a, b = _get_coeff_exp(g.argument, x) _, c = _get_coeff_exp(po, x) c += s # we do a substitution t=a*x**b, get integrand fac*t**rho*g fac_ = fac * C / (b*a**((1 + c)/b)) rho = (c + 1)/b - 1 # we now use t**rho*G(params, t) = G(params + rho, t) # [L, page 150, equation (4)] # and integral G(params, t) dt = G(1, params+1, 0, t) # (or a similar expression with 1 and 0 exchanged ... pick the one # which yields a well-defined function) # [R, section 5] # (Note that this dummy will immediately go away again, so we # can safely pass S.One for ``expr``.) t = _dummy('t', 'meijerint-indefinite', S.One) def tr(p): return [a + rho + 1 for a in p] if any(b.is_integer and (b <= 0) == True for b in tr(g.bm)): r = -meijerg( tr(g.an), tr(g.aother) + [1], tr(g.bm) + [0], tr(g.bother), t) else: r = meijerg( tr(g.an) + [1], tr(g.aother), tr(g.bm), tr(g.bother) + [0], t) # The antiderivative is most often expected to be defined # in the neighborhood of x = 0. if b.is_extended_nonnegative and not f.subs(x, 0).has(nan, zoo): place = 0 # Assume we can expand at zero else: place = None r = hyperexpand(r.subs(t, a*x**b), place=place) # now substitute back # Note: we really do want the powers of x to combine. res += powdenest(fac_*r, polar=True) def _clean(res): """This multiplies out superfluous powers of x we created, and chops off constants: >> _clean(x*(exp(x)/x - 1/x) + 3) exp(x) cancel is used before mul_expand since it is possible for an expression to have an additive constant that doesn't become isolated with simple expansion. Such a situation was identified in issue 6369: Examples ======== >>> from sympy import sqrt, cancel >>> from sympy.abc import x >>> a = sqrt(2*x + 1) >>> bad = (3*x*a**5 + 2*x - a**5 + 1)/a**2 >>> bad.expand().as_independent(x)[0] 0 >>> cancel(bad).expand().as_independent(x)[0] 1 """ from sympy import cancel res = expand_mul(cancel(res), deep=False) return Add._from_args(res.as_coeff_add(x)[1]) res = piecewise_fold(res) if res.is_Piecewise: newargs = [] for expr, cond in res.args: expr = _my_unpolarify(_clean(expr)) newargs += [(expr, cond)] res = Piecewise(*newargs) else: res = _my_unpolarify(_clean(res)) return Piecewise((res, _my_unpolarify(cond)), (Integral(f, x), True)) @timeit def meijerint_definite(f, x, a, b): """ Integrate ``f`` over the interval [``a``, ``b``], by rewriting it as a product of two G functions, or as a single G function. Return res, cond, where cond are convergence conditions. Examples ======== >>> from sympy.integrals.meijerint import meijerint_definite >>> from sympy import exp, oo >>> from sympy.abc import x >>> meijerint_definite(exp(-x**2), x, -oo, oo) (sqrt(pi), True) This function is implemented as a succession of functions meijerint_definite, _meijerint_definite_2, _meijerint_definite_3, _meijerint_definite_4. Each function in the list calls the next one (presumably) several times. This means that calling meijerint_definite can be very costly. """ # This consists of three steps: # 1) Change the integration limits to 0, oo # 2) Rewrite in terms of G functions # 3) Evaluate the integral # # There are usually several ways of doing this, and we want to try all. # This function does (1), calls _meijerint_definite_2 for step (2). from sympy import arg, exp, I, And, DiracDelta, SingularityFunction _debug('Integrating', f, 'wrt %s from %s to %s.' % (x, a, b)) if f.has(DiracDelta): _debug('Integrand has DiracDelta terms - giving up.') return None if f.has(SingularityFunction): _debug('Integrand has Singularity Function terms - giving up.') return None f_, x_, a_, b_ = f, x, a, b # Let's use a dummy in case any of the boundaries has x. d = Dummy('x') f = f.subs(x, d) x = d if a == b: return (S.Zero, True) results = [] if a is -oo and b is not oo: return meijerint_definite(f.subs(x, -x), x, -b, -a) elif a is -oo: # Integrating -oo to oo. We need to find a place to split the integral. _debug(' Integrating -oo to +oo.') innermost = _find_splitting_points(f, x) _debug(' Sensible splitting points:', innermost) for c in sorted(innermost, key=default_sort_key, reverse=True) + [S.Zero]: _debug(' Trying to split at', c) if not c.is_extended_real: _debug(' Non-real splitting point.') continue res1 = _meijerint_definite_2(f.subs(x, x + c), x) if res1 is None: _debug(' But could not compute first integral.') continue res2 = _meijerint_definite_2(f.subs(x, c - x), x) if res2 is None: _debug(' But could not compute second integral.') continue res1, cond1 = res1 res2, cond2 = res2 cond = _condsimp(And(cond1, cond2)) if cond == False: _debug(' But combined condition is always false.') continue res = res1 + res2 return res, cond elif a is oo: res = meijerint_definite(f, x, b, oo) return -res[0], res[1] elif (a, b) == (0, oo): # This is a common case - try it directly first. res = _meijerint_definite_2(f, x) if res: if _has(res[0], meijerg): results.append(res) else: return res else: if b is oo: for split in _find_splitting_points(f, x): if (a - split >= 0) == True: _debug('Trying x -> x + %s' % split) res = _meijerint_definite_2(f.subs(x, x + split) *Heaviside(x + split - a), x) if res: if _has(res[0], meijerg): results.append(res) else: return res f = f.subs(x, x + a) b = b - a a = 0 if b != oo: phi = exp(I*arg(b)) b = abs(b) f = f.subs(x, phi*x) f *= Heaviside(b - x)*phi b = oo _debug('Changed limits to', a, b) _debug('Changed function to', f) res = _meijerint_definite_2(f, x) if res: if _has(res[0], meijerg): results.append(res) else: return res if f_.has(HyperbolicFunction): _debug('Try rewriting hyperbolics in terms of exp.') rv = meijerint_definite( _rewrite_hyperbolics_as_exp(f_), x_, a_, b_) if rv: if not type(rv) is list: rv = (collect(factor_terms(rv[0]), rv[0].atoms(exp)),) + rv[1:] return rv results.extend(rv) if results: return next(ordered(results)) def _guess_expansion(f, x): """ Try to guess sensible rewritings for integrand f(x). """ from sympy import expand_trig from sympy.functions.elementary.trigonometric import TrigonometricFunction res = [(f, 'original integrand')] orig = res[-1][0] saw = {orig} expanded = expand_mul(orig) if expanded not in saw: res += [(expanded, 'expand_mul')] saw.add(expanded) expanded = expand(orig) if expanded not in saw: res += [(expanded, 'expand')] saw.add(expanded) if orig.has(TrigonometricFunction, HyperbolicFunction): expanded = expand_mul(expand_trig(orig)) if expanded not in saw: res += [(expanded, 'expand_trig, expand_mul')] saw.add(expanded) if orig.has(cos, sin): reduced = sincos_to_sum(orig) if reduced not in saw: res += [(reduced, 'trig power reduction')] saw.add(reduced) return res def _meijerint_definite_2(f, x): """ Try to integrate f dx from zero to infinity. The body of this function computes various 'simplifications' f1, f2, ... of f (e.g. by calling expand_mul(), trigexpand() - see _guess_expansion) and calls _meijerint_definite_3 with each of these in succession. If _meijerint_definite_3 succeeds with any of the simplified functions, returns this result. """ # This function does preparation for (2), calls # _meijerint_definite_3 for (2) and (3) combined. # use a positive dummy - we integrate from 0 to oo # XXX if a nonnegative symbol is used there will be test failures dummy = _dummy('x', 'meijerint-definite2', f, positive=True) f = f.subs(x, dummy) x = dummy if f == 0: return S.Zero, True for g, explanation in _guess_expansion(f, x): _debug('Trying', explanation) res = _meijerint_definite_3(g, x) if res: return res def _meijerint_definite_3(f, x): """ Try to integrate f dx from zero to infinity. This function calls _meijerint_definite_4 to try to compute the integral. If this fails, it tries using linearity. """ res = _meijerint_definite_4(f, x) if res and res[1] != False: return res if f.is_Add: _debug('Expanding and evaluating all terms.') ress = [_meijerint_definite_4(g, x) for g in f.args] if all(r is not None for r in ress): conds = [] res = S.Zero for r, c in ress: res += r conds += [c] c = And(*conds) if c != False: return res, c def _my_unpolarify(f): from sympy import unpolarify return _eval_cond(unpolarify(f)) @timeit def _meijerint_definite_4(f, x, only_double=False): """ Try to integrate f dx from zero to infinity. Explanation =========== This function tries to apply the integration theorems found in literature, i.e. it tries to rewrite f as either one or a product of two G-functions. The parameter ``only_double`` is used internally in the recursive algorithm to disable trying to rewrite f as a single G-function. """ # This function does (2) and (3) _debug('Integrating', f) # Try single G function. if not only_double: gs = _rewrite1(f, x, recursive=False) if gs is not None: fac, po, g, cond = gs _debug('Could rewrite as single G function:', fac, po, g) res = S.Zero for C, s, f in g: if C == 0: continue C, f = _rewrite_saxena_1(fac*C, po*x**s, f, x) res += C*_int0oo_1(f, x) cond = And(cond, _check_antecedents_1(f, x)) if cond == False: break cond = _my_unpolarify(cond) if cond == False: _debug('But cond is always False.') else: _debug('Result before branch substitutions is:', res) return _my_unpolarify(hyperexpand(res)), cond # Try two G functions. gs = _rewrite2(f, x) if gs is not None: for full_pb in [False, True]: fac, po, g1, g2, cond = gs _debug('Could rewrite as two G functions:', fac, po, g1, g2) res = S.Zero for C1, s1, f1 in g1: for C2, s2, f2 in g2: r = _rewrite_saxena(fac*C1*C2, po*x**(s1 + s2), f1, f2, x, full_pb) if r is None: _debug('Non-rational exponents.') return C, f1_, f2_ = r _debug('Saxena subst for yielded:', C, f1_, f2_) cond = And(cond, _check_antecedents(f1_, f2_, x)) if cond == False: break res += C*_int0oo(f1_, f2_, x) else: continue break cond = _my_unpolarify(cond) if cond == False: _debug('But cond is always False (full_pb=%s).' % full_pb) else: _debug('Result before branch substitutions is:', res) if only_double: return res, cond return _my_unpolarify(hyperexpand(res)), cond def meijerint_inversion(f, x, t): r""" Compute the inverse laplace transform $\int_{c+i\infty}^{c-i\infty} f(x) e^{tx}\, dx$, for real c larger than the real part of all singularities of ``f``. Note that ``t`` is always assumed real and positive. Return None if the integral does not exist or could not be evaluated. Examples ======== >>> from sympy.abc import x, t >>> from sympy.integrals.meijerint import meijerint_inversion >>> meijerint_inversion(1/x, x, t) Heaviside(t) """ from sympy import exp, expand, log, Add, Mul, Heaviside f_ = f t_ = t t = Dummy('t', polar=True) # We don't want sqrt(t**2) = abs(t) etc f = f.subs(t_, t) _debug('Laplace-inverting', f) if not _is_analytic(f, x): _debug('But expression is not analytic.') return None # Exponentials correspond to shifts; we filter them out and then # shift the result later. If we are given an Add this will not # work, but the calling code will take care of that. shift = S.Zero if f.is_Mul: args = list(f.args) elif isinstance(f, exp): args = [f] else: args = None if args: newargs = [] exponentials = [] while args: arg = args.pop() if isinstance(arg, exp): arg2 = expand(arg) if arg2.is_Mul: args += arg2.args continue try: a, b = _get_coeff_exp(arg.args[0], x) except _CoeffExpValueError: b = 0 if b == 1: exponentials.append(a) else: newargs.append(arg) elif arg.is_Pow: arg2 = expand(arg) if arg2.is_Mul: args += arg2.args continue if x not in arg.base.free_symbols: try: a, b = _get_coeff_exp(arg.exp, x) except _CoeffExpValueError: b = 0 if b == 1: exponentials.append(a*log(arg.base)) newargs.append(arg) else: newargs.append(arg) shift = Add(*exponentials) f = Mul(*newargs) if x not in f.free_symbols: _debug('Expression consists of constant and exp shift:', f, shift) from sympy import Eq, im cond = Eq(im(shift), 0) if cond == False: _debug('but shift is nonreal, cannot be a Laplace transform') return None res = f*DiracDelta(t + shift) _debug('Result is a delta function, possibly conditional:', res, cond) # cond is True or Eq return Piecewise((res.subs(t, t_), cond)) gs = _rewrite1(f, x) if gs is not None: fac, po, g, cond = gs _debug('Could rewrite as single G function:', fac, po, g) res = S.Zero for C, s, f in g: C, f = _rewrite_inversion(fac*C, po*x**s, f, x) res += C*_int_inversion(f, x, t) cond = And(cond, _check_antecedents_inversion(f, x)) if cond == False: break cond = _my_unpolarify(cond) if cond == False: _debug('But cond is always False.') else: _debug('Result before branch substitution:', res) res = _my_unpolarify(hyperexpand(res)) if not res.has(Heaviside): res *= Heaviside(t) res = res.subs(t, t + shift) if not isinstance(cond, bool): cond = cond.subs(t, t + shift) from sympy import InverseLaplaceTransform return Piecewise((res.subs(t, t_), cond), (InverseLaplaceTransform(f_.subs(t, t_), x, t_, None), True))
3d1e65958beb399e2237be3191c70ac8df3f84ee001cf03472689fde84cc1ada
""" SymPy core decorators. The purpose of this module is to expose decorators without any other dependencies, so that they can be easily imported anywhere in sympy/core. """ from functools import wraps from .sympify import SympifyError, sympify from sympy.core.compatibility import get_function_code def deprecated(**decorator_kwargs): """This is a decorator which can be used to mark functions as deprecated. It will result in a warning being emitted when the function is used.""" from sympy.utilities.exceptions import SymPyDeprecationWarning def _warn_deprecation(wrapped, stacklevel): decorator_kwargs.setdefault('feature', wrapped.__name__) SymPyDeprecationWarning(**decorator_kwargs).warn(stacklevel=stacklevel) def deprecated_decorator(wrapped): if hasattr(wrapped, '__mro__'): # wrapped is actually a class class wrapper(wrapped): __doc__ = wrapped.__doc__ __name__ = wrapped.__name__ __module__ = wrapped.__module__ _sympy_deprecated_func = wrapped def __init__(self, *args, **kwargs): _warn_deprecation(wrapped, 4) super().__init__(*args, **kwargs) else: @wraps(wrapped) def wrapper(*args, **kwargs): _warn_deprecation(wrapped, 3) return wrapped(*args, **kwargs) wrapper._sympy_deprecated_func = wrapped return wrapper return deprecated_decorator def _sympifyit(arg, retval=None): """ decorator to smartly _sympify function arguments Explanation =========== @_sympifyit('other', NotImplemented) def add(self, other): ... In add, other can be thought of as already being a SymPy object. If it is not, the code is likely to catch an exception, then other will be explicitly _sympified, and the whole code restarted. if _sympify(arg) fails, NotImplemented will be returned See also ======== __sympifyit """ def deco(func): return __sympifyit(func, arg, retval) return deco def __sympifyit(func, arg, retval=None): """decorator to _sympify `arg` argument for function `func` don't use directly -- use _sympifyit instead """ # we support f(a,b) only if not get_function_code(func).co_argcount: raise LookupError("func not found") # only b is _sympified assert get_function_code(func).co_varnames[1] == arg if retval is None: @wraps(func) def __sympifyit_wrapper(a, b): return func(a, sympify(b, strict=True)) else: @wraps(func) def __sympifyit_wrapper(a, b): try: # If an external class has _op_priority, it knows how to deal # with sympy objects. Otherwise, it must be converted. if not hasattr(b, '_op_priority'): b = sympify(b, strict=True) return func(a, b) except SympifyError: return retval return __sympifyit_wrapper def call_highest_priority(method_name): """A decorator for binary special methods to handle _op_priority. Explanation =========== Binary special methods in Expr and its subclasses use a special attribute '_op_priority' to determine whose special method will be called to handle the operation. In general, the object having the highest value of '_op_priority' will handle the operation. Expr and subclasses that define custom binary special methods (__mul__, etc.) should decorate those methods with this decorator to add the priority logic. The ``method_name`` argument is the name of the method of the other class that will be called. Use this decorator in the following manner:: # Call other.__rmul__ if other._op_priority > self._op_priority @call_highest_priority('__rmul__') def __mul__(self, other): ... # Call other.__mul__ if other._op_priority > self._op_priority @call_highest_priority('__mul__') def __rmul__(self, other): ... """ def priority_decorator(func): @wraps(func) def binary_op_wrapper(self, other): if hasattr(other, '_op_priority'): if other._op_priority > self._op_priority: f = getattr(other, method_name, None) if f is not None: return f(self) return func(self, other) return binary_op_wrapper return priority_decorator def sympify_method_args(cls): '''Decorator for a class with methods that sympify arguments. Explanation =========== The sympify_method_args decorator is to be used with the sympify_return decorator for automatic sympification of method arguments. This is intended for the common idiom of writing a class like : Examples ======== >>> from sympy.core.basic import Basic >>> from sympy.core.sympify import _sympify, SympifyError >>> class MyTuple(Basic): ... def __add__(self, other): ... try: ... other = _sympify(other) ... except SympifyError: ... return NotImplemented ... if not isinstance(other, MyTuple): ... return NotImplemented ... return MyTuple(*(self.args + other.args)) >>> MyTuple(1, 2) + MyTuple(3, 4) MyTuple(1, 2, 3, 4) In the above it is important that we return NotImplemented when other is not sympifiable and also when the sympified result is not of the expected type. This allows the MyTuple class to be used cooperatively with other classes that overload __add__ and want to do something else in combination with instance of Tuple. Using this decorator the above can be written as >>> from sympy.core.decorators import sympify_method_args, sympify_return >>> @sympify_method_args ... class MyTuple(Basic): ... @sympify_return([('other', 'MyTuple')], NotImplemented) ... def __add__(self, other): ... return MyTuple(*(self.args + other.args)) >>> MyTuple(1, 2) + MyTuple(3, 4) MyTuple(1, 2, 3, 4) The idea here is that the decorators take care of the boiler-plate code for making this happen in each method that potentially needs to accept unsympified arguments. Then the body of e.g. the __add__ method can be written without needing to worry about calling _sympify or checking the type of the resulting object. The parameters for sympify_return are a list of tuples of the form (parameter_name, expected_type) and the value to return (e.g. NotImplemented). The expected_type parameter can be a type e.g. Tuple or a string 'Tuple'. Using a string is useful for specifying a Type within its class body (as in the above example). Notes: Currently sympify_return only works for methods that take a single argument (not including self). Specifying an expected_type as a string only works for the class in which the method is defined. ''' # Extract the wrapped methods from each of the wrapper objects created by # the sympify_return decorator. Doing this here allows us to provide the # cls argument which is used for forward string referencing. for attrname, obj in cls.__dict__.items(): if isinstance(obj, _SympifyWrapper): setattr(cls, attrname, obj.make_wrapped(cls)) return cls def sympify_return(*args): '''Function/method decorator to sympify arguments automatically See the docstring of sympify_method_args for explanation. ''' # Store a wrapper object for the decorated method def wrapper(func): return _SympifyWrapper(func, args) return wrapper class _SympifyWrapper: '''Internal class used by sympify_return and sympify_method_args''' def __init__(self, func, args): self.func = func self.args = args def make_wrapped(self, cls): func = self.func parameters, retval = self.args # XXX: Handle more than one parameter? [(parameter, expectedcls)] = parameters # Handle forward references to the current class using strings if expectedcls == cls.__name__: expectedcls = cls # Raise RuntimeError since this is a failure at import time and should # not be recoverable. nargs = get_function_code(func).co_argcount # we support f(a, b) only if nargs != 2: raise RuntimeError('sympify_return can only be used with 2 argument functions') # only b is _sympified if get_function_code(func).co_varnames[1] != parameter: raise RuntimeError('parameter name mismatch "%s" in %s' % (parameter, func.__name__)) @wraps(func) def _func(self, other): # XXX: The check for _op_priority here should be removed. It is # needed to stop mutable matrices from being sympified to # immutable matrices which breaks things in quantum... if not hasattr(other, '_op_priority'): try: other = sympify(other, strict=True) except SympifyError: return retval if not isinstance(other, expectedcls): return retval return func(self, other) return _func
64a5020f0a72b5970ffd271ac6fbdf497415e3cec8cedb2343593f43901c622e
from math import log as _log from .sympify import _sympify from .cache import cacheit from .singleton import S from .expr import Expr from .evalf import PrecisionExhausted from .function import (_coeff_isneg, expand_complex, expand_multinomial, expand_mul) from .logic import fuzzy_bool, fuzzy_not, fuzzy_and from .compatibility import as_int, HAS_GMPY, gmpy from .parameters import global_parameters from sympy.utilities.iterables import sift from sympy.utilities.exceptions import SymPyDeprecationWarning from sympy.multipledispatch import Dispatcher from mpmath.libmp import sqrtrem as mpmath_sqrtrem from math import sqrt as _sqrt def isqrt(n): """Return the largest integer less than or equal to sqrt(n).""" if n < 0: raise ValueError("n must be nonnegative") n = int(n) # Fast path: with IEEE 754 binary64 floats and a correctly-rounded # math.sqrt, int(math.sqrt(n)) works for any integer n satisfying 0 <= n < # 4503599761588224 = 2**52 + 2**27. But Python doesn't guarantee either # IEEE 754 format floats *or* correct rounding of math.sqrt, so check the # answer and fall back to the slow method if necessary. if n < 4503599761588224: s = int(_sqrt(n)) if 0 <= n - s*s <= 2*s: return s return integer_nthroot(n, 2)[0] def integer_nthroot(y, n): """ Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y). Examples ======== >>> from sympy import integer_nthroot >>> integer_nthroot(16, 2) (4, True) >>> integer_nthroot(26, 2) (5, False) To simply determine if a number is a perfect square, the is_square function should be used: >>> from sympy.ntheory.primetest import is_square >>> is_square(26) False See Also ======== sympy.ntheory.primetest.is_square integer_log """ y, n = as_int(y), as_int(n) if y < 0: raise ValueError("y must be nonnegative") if n < 1: raise ValueError("n must be positive") if HAS_GMPY and n < 2**63: # Currently it works only for n < 2**63, else it produces TypeError # sympy issue: https://github.com/sympy/sympy/issues/18374 # gmpy2 issue: https://github.com/aleaxit/gmpy/issues/257 if HAS_GMPY >= 2: x, t = gmpy.iroot(y, n) else: x, t = gmpy.root(y, n) return as_int(x), bool(t) return _integer_nthroot_python(y, n) def _integer_nthroot_python(y, n): if y in (0, 1): return y, True if n == 1: return y, True if n == 2: x, rem = mpmath_sqrtrem(y) return int(x), not rem if n > y: return 1, False # Get initial estimate for Newton's method. Care must be taken to # avoid overflow try: guess = int(y**(1./n) + 0.5) except OverflowError: exp = _log(y, 2)/n if exp > 53: shift = int(exp - 53) guess = int(2.0**(exp - shift) + 1) << shift else: guess = int(2.0**exp) if guess > 2**50: # Newton iteration xprev, x = -1, guess while 1: t = x**(n - 1) xprev, x = x, ((n - 1)*x + y//t)//n if abs(x - xprev) < 2: break else: x = guess # Compensate t = x**n while t < y: x += 1 t = x**n while t > y: x -= 1 t = x**n return int(x), t == y # int converts long to int if possible def integer_log(y, x): r""" Returns ``(e, bool)`` where e is the largest nonnegative integer such that :math:`|y| \geq |x^e|` and ``bool`` is True if $y = x^e$. Examples ======== >>> from sympy import integer_log >>> integer_log(125, 5) (3, True) >>> integer_log(17, 9) (1, False) >>> integer_log(4, -2) (2, True) >>> integer_log(-125,-5) (3, True) See Also ======== integer_nthroot sympy.ntheory.primetest.is_square sympy.ntheory.factor_.multiplicity sympy.ntheory.factor_.perfect_power """ if x == 1: raise ValueError('x cannot take value as 1') if y == 0: raise ValueError('y cannot take value as 0') if x in (-2, 2): x = int(x) y = as_int(y) e = y.bit_length() - 1 return e, x**e == y if x < 0: n, b = integer_log(y if y > 0 else -y, -x) return n, b and bool(n % 2 if y < 0 else not n % 2) x = as_int(x) y = as_int(y) r = e = 0 while y >= x: d = x m = 1 while y >= d: y, rem = divmod(y, d) r = r or rem e += m if y > d: d *= d m *= 2 return e, r == 0 and y == 1 class Pow(Expr): """ Defines the expression x**y as "x raised to a power y" Singleton definitions involving (0, 1, -1, oo, -oo, I, -I): +--------------+---------+-----------------------------------------------+ | expr | value | reason | +==============+=========+===============================================+ | z**0 | 1 | Although arguments over 0**0 exist, see [2]. | +--------------+---------+-----------------------------------------------+ | z**1 | z | | +--------------+---------+-----------------------------------------------+ | (-oo)**(-1) | 0 | | +--------------+---------+-----------------------------------------------+ | (-1)**-1 | -1 | | +--------------+---------+-----------------------------------------------+ | S.Zero**-1 | zoo | This is not strictly true, as 0**-1 may be | | | | undefined, but is convenient in some contexts | | | | where the base is assumed to be positive. | +--------------+---------+-----------------------------------------------+ | 1**-1 | 1 | | +--------------+---------+-----------------------------------------------+ | oo**-1 | 0 | | +--------------+---------+-----------------------------------------------+ | 0**oo | 0 | Because for all complex numbers z near | | | | 0, z**oo -> 0. | +--------------+---------+-----------------------------------------------+ | 0**-oo | zoo | This is not strictly true, as 0**oo may be | | | | oscillating between positive and negative | | | | values or rotating in the complex plane. | | | | It is convenient, however, when the base | | | | is positive. | +--------------+---------+-----------------------------------------------+ | 1**oo | nan | Because there are various cases where | | 1**-oo | | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), | | | | but lim( x(t)**y(t), t) != 1. See [3]. | +--------------+---------+-----------------------------------------------+ | b**zoo | nan | Because b**z has no limit as z -> zoo | +--------------+---------+-----------------------------------------------+ | (-1)**oo | nan | Because of oscillations in the limit. | | (-1)**(-oo) | | | +--------------+---------+-----------------------------------------------+ | oo**oo | oo | | +--------------+---------+-----------------------------------------------+ | oo**-oo | 0 | | +--------------+---------+-----------------------------------------------+ | (-oo)**oo | nan | | | (-oo)**-oo | | | +--------------+---------+-----------------------------------------------+ | oo**I | nan | oo**e could probably be best thought of as | | (-oo)**I | | the limit of x**e for real x as x tends to | | | | oo. If e is I, then the limit does not exist | | | | and nan is used to indicate that. | +--------------+---------+-----------------------------------------------+ | oo**(1+I) | zoo | If the real part of e is positive, then the | | (-oo)**(1+I) | | limit of abs(x**e) is oo. So the limit value | | | | is zoo. | +--------------+---------+-----------------------------------------------+ | oo**(-1+I) | 0 | If the real part of e is negative, then the | | -oo**(-1+I) | | limit is 0. | +--------------+---------+-----------------------------------------------+ Because symbolic computations are more flexible that floating point calculations and we prefer to never return an incorrect answer, we choose not to conform to all IEEE 754 conventions. This helps us avoid extra test-case code in the calculation of limits. See Also ======== sympy.core.numbers.Infinity sympy.core.numbers.NegativeInfinity sympy.core.numbers.NaN References ========== .. [1] https://en.wikipedia.org/wiki/Exponentiation .. [2] https://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero .. [3] https://en.wikipedia.org/wiki/Indeterminate_forms """ is_Pow = True __slots__ = ('is_commutative',) @cacheit def __new__(cls, b, e, evaluate=None): if evaluate is None: evaluate = global_parameters.evaluate from sympy.functions.elementary.exponential import exp_polar b = _sympify(b) e = _sympify(e) # XXX: This can be removed when non-Expr args are disallowed rather # than deprecated. from sympy.core.relational import Relational if isinstance(b, Relational) or isinstance(e, Relational): raise TypeError('Relational can not be used in Pow') # XXX: This should raise TypeError once deprecation period is over: if not (isinstance(b, Expr) and isinstance(e, Expr)): SymPyDeprecationWarning( feature="Pow with non-Expr args", useinstead="Expr args", issue=19445, deprecated_since_version="1.7" ).warn() if evaluate: if b is S.Zero and e is S.NegativeInfinity: return S.ComplexInfinity if e is S.ComplexInfinity: return S.NaN if e is S.Zero: return S.One elif e is S.One: return b elif e == -1 and not b: return S.ComplexInfinity # Only perform autosimplification if exponent or base is a Symbol or number elif (b.is_Symbol or b.is_number) and (e.is_Symbol or e.is_number) and\ e.is_integer and _coeff_isneg(b): if e.is_even: b = -b elif e.is_odd: return -Pow(-b, e) if S.NaN in (b, e): # XXX S.NaN**x -> S.NaN under assumption that x != 0 return S.NaN elif b is S.One: if abs(e).is_infinite: return S.NaN return S.One else: # recognize base as E if not e.is_Atom and b is not S.Exp1 and not isinstance(b, exp_polar): from sympy import numer, denom, log, sign, im, factor_terms c, ex = factor_terms(e, sign=False).as_coeff_Mul() den = denom(ex) if isinstance(den, log) and den.args[0] == b: return S.Exp1**(c*numer(ex)) elif den.is_Add: s = sign(im(b)) if s.is_Number and s and den == \ log(-factor_terms(b, sign=False)) + s*S.ImaginaryUnit*S.Pi: return S.Exp1**(c*numer(ex)) obj = b._eval_power(e) if obj is not None: return obj obj = Expr.__new__(cls, b, e) obj = cls._exec_constructor_postprocessors(obj) if not isinstance(obj, Pow): return obj obj.is_commutative = (b.is_commutative and e.is_commutative) return obj @property def base(self): return self._args[0] @property def exp(self): return self._args[1] @classmethod def class_key(cls): return 3, 2, cls.__name__ def _eval_refine(self, assumptions): from sympy.assumptions.ask import ask, Q b, e = self.as_base_exp() if ask(Q.integer(e), assumptions) and _coeff_isneg(b): if ask(Q.even(e), assumptions): return Pow(-b, e) elif ask(Q.odd(e), assumptions): return -Pow(-b, e) def _eval_power(self, other): from sympy import arg, exp, floor, im, log, re, sign b, e = self.as_base_exp() if b is S.NaN: return (b**e)**other # let __new__ handle it s = None if other.is_integer: s = 1 elif b.is_polar: # e.g. exp_polar, besselj, var('p', polar=True)... s = 1 elif e.is_extended_real is not None: # helper functions =========================== def _half(e): """Return True if the exponent has a literal 2 as the denominator, else None.""" if getattr(e, 'q', None) == 2: return True n, d = e.as_numer_denom() if n.is_integer and d == 2: return True def _n2(e): """Return ``e`` evaluated to a Number with 2 significant digits, else None.""" try: rv = e.evalf(2, strict=True) if rv.is_Number: return rv except PrecisionExhausted: pass # =================================================== if e.is_extended_real: # we need _half(other) with constant floor or # floor(S.Half - e*arg(b)/2/pi) == 0 # handle -1 as special case if e == -1: # floor arg. is 1/2 + arg(b)/2/pi if _half(other): if b.is_negative is True: return S.NegativeOne**other*Pow(-b, e*other) elif b.is_negative is False: return Pow(b, -other) elif e.is_even: if b.is_extended_real: b = abs(b) if b.is_imaginary: b = abs(im(b))*S.ImaginaryUnit if (abs(e) < 1) == True or e == 1: s = 1 # floor = 0 elif b.is_extended_nonnegative: s = 1 # floor = 0 elif re(b).is_extended_nonnegative and (abs(e) < 2) == True: s = 1 # floor = 0 elif fuzzy_not(im(b).is_zero) and abs(e) == 2: s = 1 # floor = 0 elif _half(other): s = exp(2*S.Pi*S.ImaginaryUnit*other*floor( S.Half - e*arg(b)/(2*S.Pi))) if s.is_extended_real and _n2(sign(s) - s) == 0: s = sign(s) else: s = None else: # e.is_extended_real is False requires: # _half(other) with constant floor or # floor(S.Half - im(e*log(b))/2/pi) == 0 try: s = exp(2*S.ImaginaryUnit*S.Pi*other* floor(S.Half - im(e*log(b))/2/S.Pi)) # be careful to test that s is -1 or 1 b/c sign(I) == I: # so check that s is real if s.is_extended_real and _n2(sign(s) - s) == 0: s = sign(s) else: s = None except PrecisionExhausted: s = None if s is not None: return s*Pow(b, e*other) def _eval_Mod(self, q): r"""A dispatched function to compute `b^e \bmod q`, dispatched by ``Mod``. Notes ===== Algorithms: 1. For unevaluated integer power, use built-in ``pow`` function with 3 arguments, if powers are not too large wrt base. 2. For very large powers, use totient reduction if e >= lg(m). Bound on m, is for safe factorization memory wise ie m^(1/4). For pollard-rho to be faster than built-in pow lg(e) > m^(1/4) check is added. 3. For any unevaluated power found in `b` or `e`, the step 2 will be recursed down to the base and the exponent such that the `b \bmod q` becomes the new base and ``\phi(q) + e \bmod \phi(q)`` becomes the new exponent, and then the computation for the reduced expression can be done. """ from sympy.ntheory import totient from .mod import Mod base, exp = self.base, self.exp if exp.is_integer and exp.is_positive: if q.is_integer and base % q == 0: return S.Zero if base.is_Integer and exp.is_Integer and q.is_Integer: b, e, m = int(base), int(exp), int(q) mb = m.bit_length() if mb <= 80 and e >= mb and e.bit_length()**4 >= m: phi = totient(m) return Integer(pow(b, phi + e%phi, m)) return Integer(pow(b, e, m)) if isinstance(base, Pow) and base.is_integer and base.is_number: base = Mod(base, q) return Mod(Pow(base, exp, evaluate=False), q) if isinstance(exp, Pow) and exp.is_integer and exp.is_number: bit_length = int(q).bit_length() # XXX Mod-Pow actually attempts to do a hanging evaluation # if this dispatched function returns None. # May need some fixes in the dispatcher itself. if bit_length <= 80: phi = totient(q) exp = phi + Mod(exp, phi) return Mod(Pow(base, exp, evaluate=False), q) def _eval_is_even(self): if self.exp.is_integer and self.exp.is_positive: return self.base.is_even def _eval_is_negative(self): ext_neg = Pow._eval_is_extended_negative(self) if ext_neg is True: return self.is_finite return ext_neg def _eval_is_positive(self): ext_pos = Pow._eval_is_extended_positive(self) if ext_pos is True: return self.is_finite return ext_pos def _eval_is_extended_positive(self): from sympy import log if self.base == self.exp: if self.base.is_extended_nonnegative: return True elif self.base.is_positive: if self.exp.is_real: return True elif self.base.is_extended_negative: if self.exp.is_even: return True if self.exp.is_odd: return False elif self.base.is_zero: if self.exp.is_extended_real: return self.exp.is_zero elif self.base.is_extended_nonpositive: if self.exp.is_odd: return False elif self.base.is_imaginary: if self.exp.is_integer: m = self.exp % 4 if m.is_zero: return True if m.is_integer and m.is_zero is False: return False if self.exp.is_imaginary: return log(self.base).is_imaginary def _eval_is_extended_negative(self): if self.exp is S(1)/2: if self.base.is_complex or self.base.is_extended_real: return False if self.base.is_extended_negative: if self.exp.is_odd and self.base.is_finite: return True if self.exp.is_even: return False elif self.base.is_extended_positive: if self.exp.is_extended_real: return False elif self.base.is_zero: if self.exp.is_extended_real: return False elif self.base.is_extended_nonnegative: if self.exp.is_extended_nonnegative: return False elif self.base.is_extended_nonpositive: if self.exp.is_even: return False elif self.base.is_extended_real: if self.exp.is_even: return False def _eval_is_zero(self): if self.base.is_zero: if self.exp.is_extended_positive: return True elif self.exp.is_extended_nonpositive: return False elif self.base.is_zero is False: if self.base.is_finite and self.exp.is_finite: return False elif self.exp.is_negative: return self.base.is_infinite elif self.exp.is_nonnegative: return False elif self.exp.is_infinite and self.exp.is_extended_real: if (1 - abs(self.base)).is_extended_positive: return self.exp.is_extended_positive elif (1 - abs(self.base)).is_extended_negative: return self.exp.is_extended_negative else: # when self.base.is_zero is None if self.base.is_finite and self.exp.is_negative: return False def _eval_is_integer(self): b, e = self.args if b.is_rational: if b.is_integer is False and e.is_positive: return False # rat**nonneg if b.is_integer and e.is_integer: if b is S.NegativeOne: return True if e.is_nonnegative or e.is_positive: return True if b.is_integer and e.is_negative and (e.is_finite or e.is_integer): if fuzzy_not((b - 1).is_zero) and fuzzy_not((b + 1).is_zero): return False if b.is_Number and e.is_Number: check = self.func(*self.args) return check.is_Integer if e.is_negative and b.is_positive and (b - 1).is_positive: return False if e.is_negative and b.is_negative and (b + 1).is_negative: return False def _eval_is_extended_real(self): from sympy import arg, exp, log, Mul real_b = self.base.is_extended_real if real_b is None: if self.base.func == exp and self.base.args[0].is_imaginary: return self.exp.is_imaginary return real_e = self.exp.is_extended_real if real_e is None: return if real_b and real_e: if self.base.is_extended_positive: return True elif self.base.is_extended_nonnegative and self.exp.is_extended_nonnegative: return True elif self.exp.is_integer and self.base.is_extended_nonzero: return True elif self.exp.is_integer and self.exp.is_nonnegative: return True elif self.base.is_extended_negative: if self.exp.is_Rational: return False if real_e and self.exp.is_extended_negative and self.base.is_zero is False: return Pow(self.base, -self.exp).is_extended_real im_b = self.base.is_imaginary im_e = self.exp.is_imaginary if im_b: if self.exp.is_integer: if self.exp.is_even: return True elif self.exp.is_odd: return False elif im_e and log(self.base).is_imaginary: return True elif self.exp.is_Add: c, a = self.exp.as_coeff_Add() if c and c.is_Integer: return Mul( self.base**c, self.base**a, evaluate=False).is_extended_real elif self.base in (-S.ImaginaryUnit, S.ImaginaryUnit): if (self.exp/2).is_integer is False: return False if real_b and im_e: if self.base is S.NegativeOne: return True c = self.exp.coeff(S.ImaginaryUnit) if c: if self.base.is_rational and c.is_rational: if self.base.is_nonzero and (self.base - 1).is_nonzero and c.is_nonzero: return False ok = (c*log(self.base)/S.Pi).is_integer if ok is not None: return ok if real_b is False: # we already know it's not imag i = arg(self.base)*self.exp/S.Pi if i.is_complex: # finite return i.is_integer def _eval_is_complex(self): if all(a.is_complex for a in self.args) and self._eval_is_finite(): return True def _eval_is_imaginary(self): from sympy import arg, log if self.base.is_imaginary: if self.exp.is_integer: odd = self.exp.is_odd if odd is not None: return odd return if self.exp.is_imaginary: imlog = log(self.base).is_imaginary if imlog is not None: return False # I**i -> real; (2*I)**i -> complex ==> not imaginary if self.base.is_extended_real and self.exp.is_extended_real: if self.base.is_positive: return False else: rat = self.exp.is_rational if not rat: return rat if self.exp.is_integer: return False else: half = (2*self.exp).is_integer if half: return self.base.is_negative return half if self.base.is_extended_real is False: # we already know it's not imag i = arg(self.base)*self.exp/S.Pi isodd = (2*i).is_odd if isodd is not None: return isodd if self.exp.is_negative: return (1/self).is_imaginary def _eval_is_odd(self): if self.exp.is_integer: if self.exp.is_positive: return self.base.is_odd elif self.exp.is_nonnegative and self.base.is_odd: return True elif self.base is S.NegativeOne: return True def _eval_is_finite(self): if self.exp.is_negative: if self.base.is_zero: return False if self.base.is_infinite or self.base.is_nonzero: return True c1 = self.base.is_finite if c1 is None: return c2 = self.exp.is_finite if c2 is None: return if c1 and c2: if self.exp.is_nonnegative or fuzzy_not(self.base.is_zero): return True def _eval_is_prime(self): ''' An integer raised to the n(>=2)-th power cannot be a prime. ''' if self.base.is_integer and self.exp.is_integer and (self.exp - 1).is_positive: return False def _eval_is_composite(self): """ A power is composite if both base and exponent are greater than 1 """ if (self.base.is_integer and self.exp.is_integer and ((self.base - 1).is_positive and (self.exp - 1).is_positive or (self.base + 1).is_negative and self.exp.is_positive and self.exp.is_even)): return True def _eval_is_polar(self): return self.base.is_polar def _eval_subs(self, old, new): from sympy import exp, log, Symbol def _check(ct1, ct2, old): """Return (bool, pow, remainder_pow) where, if bool is True, then the exponent of Pow `old` will combine with `pow` so the substitution is valid, otherwise bool will be False. For noncommutative objects, `pow` will be an integer, and a factor `Pow(old.base, remainder_pow)` needs to be included. If there is no such factor, None is returned. For commutative objects, remainder_pow is always None. cti are the coefficient and terms of an exponent of self or old In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y) will give y**2 since (b**x)**2 == b**(2*x); if that equality does not hold then the substitution should not occur so `bool` will be False. """ coeff1, terms1 = ct1 coeff2, terms2 = ct2 if terms1 == terms2: if old.is_commutative: # Allow fractional powers for commutative objects pow = coeff1/coeff2 try: as_int(pow, strict=False) combines = True except ValueError: combines = isinstance(Pow._eval_power( Pow(*old.as_base_exp(), evaluate=False), pow), (Pow, exp, Symbol)) return combines, pow, None else: # With noncommutative symbols, substitute only integer powers if not isinstance(terms1, tuple): terms1 = (terms1,) if not all(term.is_integer for term in terms1): return False, None, None try: # Round pow toward zero pow, remainder = divmod(as_int(coeff1), as_int(coeff2)) if pow < 0 and remainder != 0: pow += 1 remainder -= as_int(coeff2) if remainder == 0: remainder_pow = None else: remainder_pow = Mul(remainder, *terms1) return True, pow, remainder_pow except ValueError: # Can't substitute pass return False, None, None if old == self.base: return new**self.exp._subs(old, new) # issue 10829: (4**x - 3*y + 2).subs(2**x, y) -> y**2 - 3*y + 2 if isinstance(old, self.func) and self.exp == old.exp: l = log(self.base, old.base) if l.is_Number: return Pow(new, l) if isinstance(old, self.func) and self.base == old.base: if self.exp.is_Add is False: ct1 = self.exp.as_independent(Symbol, as_Add=False) ct2 = old.exp.as_independent(Symbol, as_Add=False) ok, pow, remainder_pow = _check(ct1, ct2, old) if ok: # issue 5180: (x**(6*y)).subs(x**(3*y),z)->z**2 result = self.func(new, pow) if remainder_pow is not None: result = Mul(result, Pow(old.base, remainder_pow)) return result else: # b**(6*x + a).subs(b**(3*x), y) -> y**2 * b**a # exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2)) oarg = old.exp new_l = [] o_al = [] ct2 = oarg.as_coeff_mul() for a in self.exp.args: newa = a._subs(old, new) ct1 = newa.as_coeff_mul() ok, pow, remainder_pow = _check(ct1, ct2, old) if ok: new_l.append(new**pow) if remainder_pow is not None: o_al.append(remainder_pow) continue elif not old.is_commutative and not newa.is_integer: # If any term in the exponent is non-integer, # we do not do any substitutions in the noncommutative case return o_al.append(newa) if new_l: expo = Add(*o_al) new_l.append(Pow(self.base, expo, evaluate=False) if expo != 1 else self.base) return Mul(*new_l) if isinstance(old, exp) and self.exp.is_extended_real and self.base.is_positive: ct1 = old.args[0].as_independent(Symbol, as_Add=False) ct2 = (self.exp*log(self.base)).as_independent( Symbol, as_Add=False) ok, pow, remainder_pow = _check(ct1, ct2, old) if ok: result = self.func(new, pow) # (2**x).subs(exp(x*log(2)), z) -> z if remainder_pow is not None: result = Mul(result, Pow(old.base, remainder_pow)) return result def as_base_exp(self): """Return base and exp of self. Explnation ========== If base is 1/Integer, then return Integer, -exp. If this extra processing is not needed, the base and exp properties will give the raw arguments Examples ======== >>> from sympy import Pow, S >>> p = Pow(S.Half, 2, evaluate=False) >>> p.as_base_exp() (2, -2) >>> p.args (1/2, 2) """ b, e = self.args if b.is_Rational and b.p == 1 and b.q != 1: return Integer(b.q), -e return b, e def _eval_adjoint(self): from sympy.functions.elementary.complexes import adjoint i, p = self.exp.is_integer, self.base.is_positive if i: return adjoint(self.base)**self.exp if p: return self.base**adjoint(self.exp) if i is False and p is False: expanded = expand_complex(self) if expanded != self: return adjoint(expanded) def _eval_conjugate(self): from sympy.functions.elementary.complexes import conjugate as c i, p = self.exp.is_integer, self.base.is_positive if i: return c(self.base)**self.exp if p: return self.base**c(self.exp) if i is False and p is False: expanded = expand_complex(self) if expanded != self: return c(expanded) if self.is_extended_real: return self def _eval_transpose(self): from sympy.functions.elementary.complexes import transpose i, p = self.exp.is_integer, (self.base.is_complex or self.base.is_infinite) if p: return self.base**self.exp if i: return transpose(self.base)**self.exp if i is False and p is False: expanded = expand_complex(self) if expanded != self: return transpose(expanded) def _eval_expand_power_exp(self, **hints): """a**(n + m) -> a**n*a**m""" b = self.base e = self.exp if e.is_Add and e.is_commutative: expr = [] for x in e.args: expr.append(self.func(self.base, x)) return Mul(*expr) return self.func(b, e) def _eval_expand_power_base(self, **hints): """(a*b)**n -> a**n * b**n""" force = hints.get('force', False) b = self.base e = self.exp if not b.is_Mul: return self cargs, nc = b.args_cnc(split_1=False) # expand each term - this is top-level-only # expansion but we have to watch out for things # that don't have an _eval_expand method if nc: nc = [i._eval_expand_power_base(**hints) if hasattr(i, '_eval_expand_power_base') else i for i in nc] if e.is_Integer: if e.is_positive: rv = Mul(*nc*e) else: rv = Mul(*[i**-1 for i in nc[::-1]]*-e) if cargs: rv *= Mul(*cargs)**e return rv if not cargs: return self.func(Mul(*nc), e, evaluate=False) nc = [Mul(*nc)] # sift the commutative bases other, maybe_real = sift(cargs, lambda x: x.is_extended_real is False, binary=True) def pred(x): if x is S.ImaginaryUnit: return S.ImaginaryUnit polar = x.is_polar if polar: return True if polar is None: return fuzzy_bool(x.is_extended_nonnegative) sifted = sift(maybe_real, pred) nonneg = sifted[True] other += sifted[None] neg = sifted[False] imag = sifted[S.ImaginaryUnit] if imag: I = S.ImaginaryUnit i = len(imag) % 4 if i == 0: pass elif i == 1: other.append(I) elif i == 2: if neg: nonn = -neg.pop() if nonn is not S.One: nonneg.append(nonn) else: neg.append(S.NegativeOne) else: if neg: nonn = -neg.pop() if nonn is not S.One: nonneg.append(nonn) else: neg.append(S.NegativeOne) other.append(I) del imag # bring out the bases that can be separated from the base if force or e.is_integer: # treat all commutatives the same and put nc in other cargs = nonneg + neg + other other = nc else: # this is just like what is happening automatically, except # that now we are doing it for an arbitrary exponent for which # no automatic expansion is done assert not e.is_Integer # handle negatives by making them all positive and putting # the residual -1 in other if len(neg) > 1: o = S.One if not other and neg[0].is_Number: o *= neg.pop(0) if len(neg) % 2: o = -o for n in neg: nonneg.append(-n) if o is not S.One: other.append(o) elif neg and other: if neg[0].is_Number and neg[0] is not S.NegativeOne: other.append(S.NegativeOne) nonneg.append(-neg[0]) else: other.extend(neg) else: other.extend(neg) del neg cargs = nonneg other += nc rv = S.One if cargs: if e.is_Rational: npow, cargs = sift(cargs, lambda x: x.is_Pow and x.exp.is_Rational and x.base.is_number, binary=True) rv = Mul(*[self.func(b.func(*b.args), e) for b in npow]) rv *= Mul(*[self.func(b, e, evaluate=False) for b in cargs]) if other: rv *= self.func(Mul(*other), e, evaluate=False) return rv def _eval_expand_multinomial(self, **hints): """(a + b + ..)**n -> a**n + n*a**(n-1)*b + .., n is nonzero integer""" base, exp = self.args result = self if exp.is_Rational and exp.p > 0 and base.is_Add: if not exp.is_Integer: n = Integer(exp.p // exp.q) if not n: return result else: radical, result = self.func(base, exp - n), [] expanded_base_n = self.func(base, n) if expanded_base_n.is_Pow: expanded_base_n = \ expanded_base_n._eval_expand_multinomial() for term in Add.make_args(expanded_base_n): result.append(term*radical) return Add(*result) n = int(exp) if base.is_commutative: order_terms, other_terms = [], [] for b in base.args: if b.is_Order: order_terms.append(b) else: other_terms.append(b) if order_terms: # (f(x) + O(x^n))^m -> f(x)^m + m*f(x)^{m-1} *O(x^n) f = Add(*other_terms) o = Add(*order_terms) if n == 2: return expand_multinomial(f**n, deep=False) + n*f*o else: g = expand_multinomial(f**(n - 1), deep=False) return expand_mul(f*g, deep=False) + n*g*o if base.is_number: # Efficiently expand expressions of the form (a + b*I)**n # where 'a' and 'b' are real numbers and 'n' is integer. a, b = base.as_real_imag() if a.is_Rational and b.is_Rational: if not a.is_Integer: if not b.is_Integer: k = self.func(a.q * b.q, n) a, b = a.p*b.q, a.q*b.p else: k = self.func(a.q, n) a, b = a.p, a.q*b elif not b.is_Integer: k = self.func(b.q, n) a, b = a*b.q, b.p else: k = 1 a, b, c, d = int(a), int(b), 1, 0 while n: if n & 1: c, d = a*c - b*d, b*c + a*d n -= 1 a, b = a*a - b*b, 2*a*b n //= 2 I = S.ImaginaryUnit if k == 1: return c + I*d else: return Integer(c)/k + I*d/k p = other_terms # (x + y)**3 -> x**3 + 3*x**2*y + 3*x*y**2 + y**3 # in this particular example: # p = [x,y]; n = 3 # so now it's easy to get the correct result -- we get the # coefficients first: from sympy import multinomial_coefficients from sympy.polys.polyutils import basic_from_dict expansion_dict = multinomial_coefficients(len(p), n) # in our example: {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3} # and now construct the expression. return basic_from_dict(expansion_dict, *p) else: if n == 2: return Add(*[f*g for f in base.args for g in base.args]) else: multi = (base**(n - 1))._eval_expand_multinomial() if multi.is_Add: return Add(*[f*g for f in base.args for g in multi.args]) else: # XXX can this ever happen if base was an Add? return Add(*[f*multi for f in base.args]) elif (exp.is_Rational and exp.p < 0 and base.is_Add and abs(exp.p) > exp.q): return 1 / self.func(base, -exp)._eval_expand_multinomial() elif exp.is_Add and base.is_Number: # a + b a b # n --> n n , where n, a, b are Numbers coeff, tail = S.One, S.Zero for term in exp.args: if term.is_Number: coeff *= self.func(base, term) else: tail += term return coeff * self.func(base, tail) else: return result def as_real_imag(self, deep=True, **hints): from sympy import atan2, cos, im, re, sin from sympy.polys.polytools import poly if self.exp.is_Integer: exp = self.exp re_e, im_e = self.base.as_real_imag(deep=deep) if not im_e: return self, S.Zero a, b = symbols('a b', cls=Dummy) if exp >= 0: if re_e.is_Number and im_e.is_Number: # We can be more efficient in this case expr = expand_multinomial(self.base**exp) if expr != self: return expr.as_real_imag() expr = poly( (a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp else: mag = re_e**2 + im_e**2 re_e, im_e = re_e/mag, -im_e/mag if re_e.is_Number and im_e.is_Number: # We can be more efficient in this case expr = expand_multinomial((re_e + im_e*S.ImaginaryUnit)**-exp) if expr != self: return expr.as_real_imag() expr = poly((a + b)**-exp) # Terms with even b powers will be real r = [i for i in expr.terms() if not i[0][1] % 2] re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) # Terms with odd b powers will be imaginary r = [i for i in expr.terms() if i[0][1] % 4 == 1] im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) r = [i for i in expr.terms() if i[0][1] % 4 == 3] im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) return (re_part.subs({a: re_e, b: S.ImaginaryUnit*im_e}), im_part1.subs({a: re_e, b: im_e}) + im_part3.subs({a: re_e, b: -im_e})) elif self.exp.is_Rational: re_e, im_e = self.base.as_real_imag(deep=deep) if im_e.is_zero and self.exp is S.Half: if re_e.is_extended_nonnegative: return self, S.Zero if re_e.is_extended_nonpositive: return S.Zero, (-self.base)**self.exp # XXX: This is not totally correct since for x**(p/q) with # x being imaginary there are actually q roots, but # only a single one is returned from here. r = self.func(self.func(re_e, 2) + self.func(im_e, 2), S.Half) t = atan2(im_e, re_e) rp, tp = self.func(r, self.exp), t*self.exp return (rp*cos(tp), rp*sin(tp)) else: if deep: hints['complex'] = False expanded = self.expand(deep, **hints) if hints.get('ignore') == expanded: return None else: return (re(expanded), im(expanded)) else: return (re(self), im(self)) def _eval_derivative(self, s): from sympy import log dbase = self.base.diff(s) dexp = self.exp.diff(s) return self * (dexp * log(self.base) + dbase * self.exp/self.base) def _eval_evalf(self, prec): base, exp = self.as_base_exp() base = base._evalf(prec) if not exp.is_Integer: exp = exp._evalf(prec) if exp.is_negative and base.is_number and base.is_extended_real is False: base = base.conjugate() / (base * base.conjugate())._evalf(prec) exp = -exp return self.func(base, exp).expand() return self.func(base, exp) def _eval_is_polynomial(self, syms): if self.exp.has(*syms): return False if self.base.has(*syms): return bool(self.base._eval_is_polynomial(syms) and self.exp.is_Integer and (self.exp >= 0)) else: return True def _eval_is_rational(self): # The evaluation of self.func below can be very expensive in the case # of integer**integer if the exponent is large. We should try to exit # before that if possible: if (self.exp.is_integer and self.base.is_rational and fuzzy_not(fuzzy_and([self.exp.is_negative, self.base.is_zero]))): return True p = self.func(*self.as_base_exp()) # in case it's unevaluated if not p.is_Pow: return p.is_rational b, e = p.as_base_exp() if e.is_Rational and b.is_Rational: # we didn't check that e is not an Integer # because Rational**Integer autosimplifies return False if e.is_integer: if b.is_rational: if fuzzy_not(b.is_zero) or e.is_nonnegative: return True if b == e: # always rational, even for 0**0 return True elif b.is_irrational: return e.is_zero def _eval_is_algebraic(self): def _is_one(expr): try: return (expr - 1).is_zero except ValueError: # when the operation is not allowed return False if self.base.is_zero or _is_one(self.base): return True elif self.exp.is_rational: if self.base.is_algebraic is False: return self.exp.is_zero if self.base.is_zero is False: if self.exp.is_nonzero: return self.base.is_algebraic elif self.base.is_algebraic: return True if self.exp.is_positive: return self.base.is_algebraic elif self.base.is_algebraic and self.exp.is_algebraic: if ((fuzzy_not(self.base.is_zero) and fuzzy_not(_is_one(self.base))) or self.base.is_integer is False or self.base.is_irrational): return self.exp.is_rational def _eval_is_rational_function(self, syms): if self.exp.has(*syms): return False if self.base.has(*syms): return self.base._eval_is_rational_function(syms) and \ self.exp.is_Integer else: return True def _eval_is_meromorphic(self, x, a): # f**g is meromorphic if g is an integer and f is meromorphic. # E**(log(f)*g) is meromorphic if log(f)*g is meromorphic # and finite. base_merom = self.base._eval_is_meromorphic(x, a) exp_integer = self.exp.is_Integer if exp_integer: return base_merom exp_merom = self.exp._eval_is_meromorphic(x, a) if base_merom is False: # f**g = E**(log(f)*g) may be meromorphic if the # singularities of log(f) and g cancel each other, # for example, if g = 1/log(f). Hence, return False if exp_merom else None elif base_merom is None: return None b = self.base.subs(x, a) # b is extended complex as base is meromorphic. # log(base) is finite and meromorphic when b != 0, zoo. b_zero = b.is_zero if b_zero: log_defined = False else: log_defined = fuzzy_and((b.is_finite, fuzzy_not(b_zero))) if log_defined is False: # zero or pole of base return exp_integer # False or None elif log_defined is None: return None if not exp_merom: return exp_merom # False or None return self.exp.subs(x, a).is_finite def _eval_is_algebraic_expr(self, syms): if self.exp.has(*syms): return False if self.base.has(*syms): return self.base._eval_is_algebraic_expr(syms) and \ self.exp.is_Rational else: return True def _eval_rewrite_as_exp(self, base, expo, **kwargs): from sympy import exp, log, I, arg if base.is_zero or base.has(exp) or expo.has(exp): return base**expo if base.has(Symbol): # delay evaluation if expo is non symbolic # (as exp(x*log(5)) automatically reduces to x**5) return exp(log(base)*expo, evaluate=expo.has(Symbol)) else: return exp((log(abs(base)) + I*arg(base))*expo) def as_numer_denom(self): if not self.is_commutative: return self, S.One base, exp = self.as_base_exp() n, d = base.as_numer_denom() # this should be the same as ExpBase.as_numer_denom wrt # exponent handling neg_exp = exp.is_negative if not neg_exp and not (-exp).is_negative: neg_exp = _coeff_isneg(exp) int_exp = exp.is_integer # the denominator cannot be separated from the numerator if # its sign is unknown unless the exponent is an integer, e.g. # sqrt(a/b) != sqrt(a)/sqrt(b) when a=1 and b=-1. But if the # denominator is negative the numerator and denominator can # be negated and the denominator (now positive) separated. if not (d.is_extended_real or int_exp): n = base d = S.One dnonpos = d.is_nonpositive if dnonpos: n, d = -n, -d elif dnonpos is None and not int_exp: n = base d = S.One if neg_exp: n, d = d, n exp = -exp if exp.is_infinite: if n is S.One and d is not S.One: return n, self.func(d, exp) if n is not S.One and d is S.One: return self.func(n, exp), d return self.func(n, exp), self.func(d, exp) def matches(self, expr, repl_dict={}, old=False): expr = _sympify(expr) repl_dict = repl_dict.copy() # special case, pattern = 1 and expr.exp can match to 0 if expr is S.One: d = self.exp.matches(S.Zero, repl_dict) if d is not None: return d # make sure the expression to be matched is an Expr if not isinstance(expr, Expr): return None b, e = expr.as_base_exp() # special case number sb, se = self.as_base_exp() if sb.is_Symbol and se.is_Integer and expr: if e.is_rational: return sb.matches(b**(e/se), repl_dict) return sb.matches(expr**(1/se), repl_dict) d = repl_dict.copy() d = self.base.matches(b, d) if d is None: return None d = self.exp.xreplace(d).matches(e, d) if d is None: return Expr.matches(self, expr, repl_dict) return d def _eval_nseries(self, x, n, logx, cdir=0): # NOTE! This function is an important part of the gruntz algorithm # for computing limits. It has to return a generalized power # series with coefficients in C(log, log(x)). In more detail: # It has to return an expression # c_0*x**e_0 + c_1*x**e_1 + ... (finitely many terms) # where e_i are numbers (not necessarily integers) and c_i are # expressions involving only numbers, the log function, and log(x). # The series expansion of b**e is computed as follows: # 1) We express b as f*(1 + g) where f is the leading term of b. # g has order O(x**d) where d is strictly positive. # 2) Then b**e = (f**e)*((1 + g)**e). # (1 + g)**e is computed using binomial series. from sympy import im, I, ceiling, polygamma, limit, logcombine, EulerGamma, exp, nan, zoo, log, factorial, ff, PoleError, O, powdenest, Wild from itertools import product self = powdenest(self, force=True).trigsimp() b, e = self.as_base_exp() if e.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN): raise PoleError() if e.has(x): return exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir) if logx is not None and b.has(log): c, ex = symbols('c, ex', cls=Wild, exclude=[x]) b = b.replace(log(c*x**ex), log(c) + ex*logx) self = b**e b = b.removeO() try: if b.has(polygamma, EulerGamma) and logx is not None: raise ValueError() _, m = b.leadterm(x) except (ValueError, NotImplementedError): b = b._eval_nseries(x, n=max(2, n), logx=logx, cdir=cdir).removeO() if b.has(nan, zoo): raise NotImplementedError() _, m = b.leadterm(x) if e.has(log): e = logcombine(e).cancel() if not (m.is_zero or e.is_number and e.is_real): return exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir) f = b.as_leading_term(x) g = (b/f - S.One).cancel() maxpow = n - m*e if maxpow < S.Zero: return O(x**(m*e), x) if g.is_zero: return f**e def coeff_exp(term, x): coeff, exp = S.One, S.Zero for factor in Mul.make_args(term): if factor.has(x): base, exp = factor.as_base_exp() if base != x: try: return term.leadterm(x) except ValueError: return term, S.Zero else: coeff *= factor return coeff, exp def mul(d1, d2): res = {} for e1, e2 in product(d1, d2): ex = e1 + e2 if ex < maxpow: res[ex] = res.get(ex, S.Zero) + d1[e1]*d2[e2] return res try: _, d = g.leadterm(x) except (ValueError, NotImplementedError): if limit(g/x**maxpow, x, 0) == 0: # g has higher order zero return f**e + e*f**e*g # first term of binomial series else: raise NotImplementedError() if not d.is_positive: g = (b - f).simplify()/f _, d = g.leadterm(x) if not d.is_positive: raise NotImplementedError() gpoly = g._eval_nseries(x, n=ceiling(maxpow), logx=logx, cdir=cdir).removeO() gterms = {} for term in Add.make_args(gpoly): co1, e1 = coeff_exp(term, x) gterms[e1] = gterms.get(e1, S.Zero) + co1 k = S.One terms = {S.Zero: S.One} tk = gterms while k*d < maxpow: coeff = ff(e, k)/factorial(k) for ex in tk: terms[ex] = terms.get(ex, S.Zero) + coeff*tk[ex] tk = mul(tk, gterms) k += S.One if (not e.is_integer and m.is_zero and f.is_real and f.is_negative and im((b - f).dir(x, cdir)) < 0): inco, inex = coeff_exp(f**e*exp(-2*e*S.Pi*I), x) else: inco, inex = coeff_exp(f**e, x) res = S.Zero for e1 in terms: ex = e1 + inex res += terms[e1]*inco*x**(ex) for i in (1, 2, 3): if (res - self).subs(x, i) is not S.Zero: res += O(x**n, x) break return res def _eval_as_leading_term(self, x, cdir=0): from sympy import exp, I, im, log e = self.exp b = self.base if e.has(x): return exp(e * log(b)).as_leading_term(x, cdir=cdir) f = b.as_leading_term(x, cdir=cdir) if (not e.is_integer and f.is_constant() and f.is_real and f.is_negative and im((b - f).dir(x, cdir)) < 0): return self.func(f, e)*exp(-2*e*S.Pi*I) return self.func(f, e) @cacheit def _taylor_term(self, n, x, *previous_terms): # of (1 + x)**e from sympy import binomial return binomial(self.exp, n) * self.func(x, n) def _sage_(self): return self.args[0]._sage_()**self.args[1]._sage_() def as_content_primitive(self, radical=False, clear=True): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import sqrt >>> sqrt(4 + 4*sqrt(2)).as_content_primitive() (2, sqrt(1 + sqrt(2))) >>> sqrt(3 + 3*sqrt(2)).as_content_primitive() (1, sqrt(3)*sqrt(1 + sqrt(2))) >>> from sympy import expand_power_base, powsimp, Mul >>> from sympy.abc import x, y >>> ((2*x + 2)**2).as_content_primitive() (4, (x + 1)**2) >>> (4**((1 + y)/2)).as_content_primitive() (2, 4**(y/2)) >>> (3**((1 + y)/2)).as_content_primitive() (1, 3**((y + 1)/2)) >>> (3**((5 + y)/2)).as_content_primitive() (9, 3**((y + 1)/2)) >>> eq = 3**(2 + 2*x) >>> powsimp(eq) == eq True >>> eq.as_content_primitive() (9, 3**(2*x)) >>> powsimp(Mul(*_)) 3**(2*x + 2) >>> eq = (2 + 2*x)**y >>> s = expand_power_base(eq); s.is_Mul, s (False, (2*x + 2)**y) >>> eq.as_content_primitive() (1, (2*(x + 1))**y) >>> s = expand_power_base(_[1]); s.is_Mul, s (True, 2**y*(x + 1)**y) See docstring of Expr.as_content_primitive for more examples. """ b, e = self.as_base_exp() b = _keep_coeff(*b.as_content_primitive(radical=radical, clear=clear)) ce, pe = e.as_content_primitive(radical=radical, clear=clear) if b.is_Rational: #e #= ce*pe #= ce*(h + t) #= ce*h + ce*t #=> self #= b**(ce*h)*b**(ce*t) #= b**(cehp/cehq)*b**(ce*t) #= b**(iceh + r/cehq)*b**(ce*t) #= b**(iceh)*b**(r/cehq)*b**(ce*t) #= b**(iceh)*b**(ce*t + r/cehq) h, t = pe.as_coeff_Add() if h.is_Rational: ceh = ce*h c = self.func(b, ceh) r = S.Zero if not c.is_Rational: iceh, r = divmod(ceh.p, ceh.q) c = self.func(b, iceh) return c, self.func(b, _keep_coeff(ce, t + r/ce/ceh.q)) e = _keep_coeff(ce, pe) # b**e = (h*t)**e = h**e*t**e = c*m*t**e if e.is_Rational and b.is_Mul: h, t = b.as_content_primitive(radical=radical, clear=clear) # h is positive c, m = self.func(h, e).as_coeff_Mul() # so c is positive m, me = m.as_base_exp() if m is S.One or me == e: # probably always true # return the following, not return c, m*Pow(t, e) # which would change Pow into Mul; we let sympy # decide what to do by using the unevaluated Mul, e.g # should it stay as sqrt(2 + 2*sqrt(5)) or become # sqrt(2)*sqrt(1 + sqrt(5)) return c, self.func(_keep_coeff(m, t), e) return S.One, self.func(b, e) def is_constant(self, *wrt, **flags): expr = self if flags.get('simplify', True): expr = expr.simplify() b, e = expr.as_base_exp() bz = b.equals(0) if bz: # recalculate with assumptions in case it's unevaluated new = b**e if new != expr: return new.is_constant() econ = e.is_constant(*wrt) bcon = b.is_constant(*wrt) if bcon: if econ: return True bz = b.equals(0) if bz is False: return False elif bcon is None: return None return e.equals(0) def _eval_difference_delta(self, n, step): b, e = self.args if e.has(n) and not b.has(n): new_e = e.subs(n, n + step) return (b**(new_e - e) - 1) * self power = Dispatcher('power') power.add((object, object), Pow) from .add import Add from .numbers import Integer from .mul import Mul, _keep_coeff from .symbol import Symbol, Dummy, symbols
55bba8aa9d194051f74deda63e84e21b17e44ecd7e3298cb8b4ce208a47f38d1
"""Thread-safe global parameters""" from .cache import clear_cache from contextlib import contextmanager from threading import local class _global_parameters(local): """ Thread-local global parameters. Explanation =========== This class generates thread-local container for SymPy's global parameters. Every global parameters must be passed as keyword argument when generating its instance. A variable, `global_parameters` is provided as default instance for this class. WARNING! Although the global parameters are thread-local, SymPy's cache is not by now. This may lead to undesired result in multi-threading operations. Examples ======== >>> from sympy.abc import x >>> from sympy.core.cache import clear_cache >>> from sympy.core.parameters import global_parameters as gp >>> gp.evaluate True >>> x+x 2*x >>> log = [] >>> def f(): ... clear_cache() ... gp.evaluate = False ... log.append(x+x) ... clear_cache() >>> import threading >>> thread = threading.Thread(target=f) >>> thread.start() >>> thread.join() >>> print(log) [x + x] >>> gp.evaluate True >>> x+x 2*x References ========== .. [1] https://docs.python.org/3/library/threading.html """ def __init__(self, **kwargs): self.__dict__.update(kwargs) def __setattr__(self, name, value): if getattr(self, name) != value: clear_cache() return super().__setattr__(name, value) global_parameters = _global_parameters(evaluate=True, distribute=True) @contextmanager def evaluate(x): """ Control automatic evaluation Explanation =========== This context manager controls whether or not all SymPy functions evaluate by default. Note that much of SymPy expects evaluated expressions. This functionality is experimental and is unlikely to function as intended on large expressions. Examples ======== >>> from sympy.abc import x >>> from sympy.core.parameters import evaluate >>> print(x + x) 2*x >>> with evaluate(False): ... print(x + x) x + x """ old = global_parameters.evaluate try: global_parameters.evaluate = x yield finally: global_parameters.evaluate = old @contextmanager def distribute(x): """ Control automatic distribution of Number over Add Explanation =========== This context manager controls whether or not Mul distribute Number over Add. Plan is to avoid distributing Number over Add in all of sympy. Once that is done, this contextmanager will be removed. Examples ======== >>> from sympy.abc import x >>> from sympy.core.parameters import distribute >>> print(2*(x + 1)) 2*x + 2 >>> with distribute(False): ... print(2*(x + 1)) 2*(x + 1) """ old = global_parameters.distribute try: global_parameters.distribute = x yield finally: global_parameters.distribute = old
1439299d856bc66e580fcc2c24998910a93c111d7471b8bdd79ef0444b359e51
"""Tools for manipulating of large commutative expressions. """ from sympy.core.add import Add from sympy.core.compatibility import iterable, is_sequence, SYMPY_INTS from sympy.core.mul import Mul, _keep_coeff from sympy.core.power import Pow from sympy.core.basic import Basic, preorder_traversal from sympy.core.expr import Expr from sympy.core.sympify import sympify from sympy.core.numbers import Rational, Integer, Number, I from sympy.core.singleton import S from sympy.core.symbol import Dummy from sympy.core.coreerrors import NonCommutativeExpression from sympy.core.containers import Tuple, Dict from sympy.utilities import default_sort_key from sympy.utilities.iterables import (common_prefix, common_suffix, variations, ordered) from collections import defaultdict _eps = Dummy(positive=True) def _isnumber(i): return isinstance(i, (SYMPY_INTS, float)) or i.is_Number def _monotonic_sign(self): """Return the value closest to 0 that ``self`` may have if all symbols are signed and the result is uniformly the same sign for all values of symbols. If a symbol is only signed but not known to be an integer or the result is 0 then a symbol representative of the sign of self will be returned. Otherwise, None is returned if a) the sign could be positive or negative or b) self is not in one of the following forms: - L(x, y, ...) + A: a function linear in all symbols x, y, ... with an additive constant; if A is zero then the function can be a monomial whose sign is monotonic over the range of the variables, e.g. (x + 1)**3 if x is nonnegative. - A/L(x, y, ...) + B: the inverse of a function linear in all symbols x, y, ... that does not have a sign change from positive to negative for any set of values for the variables. - M(x, y, ...) + A: a monomial M whose factors are all signed and a constant, A. - A/M(x, y, ...) + B: the inverse of a monomial and constants A and B. - P(x): a univariate polynomial Examples ======== >>> from sympy.core.exprtools import _monotonic_sign as F >>> from sympy import Dummy >>> nn = Dummy(integer=True, nonnegative=True) >>> p = Dummy(integer=True, positive=True) >>> p2 = Dummy(integer=True, positive=True) >>> F(nn + 1) 1 >>> F(p - 1) _nneg >>> F(nn*p + 1) 1 >>> F(p2*p + 1) 2 >>> F(nn - 1) # could be negative, zero or positive """ if not self.is_extended_real: return if (-self).is_Symbol: rv = _monotonic_sign(-self) return rv if rv is None else -rv if not self.is_Add and self.as_numer_denom()[1].is_number: s = self if s.is_prime: if s.is_odd: return S(3) else: return S(2) elif s.is_composite: if s.is_odd: return S(9) else: return S(4) elif s.is_positive: if s.is_even: if s.is_prime is False: return S(4) else: return S(2) elif s.is_integer: return S.One else: return _eps elif s.is_extended_negative: if s.is_even: return S(-2) elif s.is_integer: return S.NegativeOne else: return -_eps if s.is_zero or s.is_extended_nonpositive or s.is_extended_nonnegative: return S.Zero return None # univariate polynomial free = self.free_symbols if len(free) == 1: if self.is_polynomial(): from sympy.polys.polytools import real_roots from sympy.polys.polyroots import roots from sympy.polys.polyerrors import PolynomialError x = free.pop() x0 = _monotonic_sign(x) if x0 == _eps or x0 == -_eps: x0 = S.Zero if x0 is not None: d = self.diff(x) if d.is_number: currentroots = [] else: try: currentroots = real_roots(d) except (PolynomialError, NotImplementedError): currentroots = [r for r in roots(d, x) if r.is_extended_real] y = self.subs(x, x0) if x.is_nonnegative and all(r <= x0 for r in currentroots): if y.is_nonnegative and d.is_positive: if y: return y if y.is_positive else Dummy('pos', positive=True) else: return Dummy('nneg', nonnegative=True) if y.is_nonpositive and d.is_negative: if y: return y if y.is_negative else Dummy('neg', negative=True) else: return Dummy('npos', nonpositive=True) elif x.is_nonpositive and all(r >= x0 for r in currentroots): if y.is_nonnegative and d.is_negative: if y: return Dummy('pos', positive=True) else: return Dummy('nneg', nonnegative=True) if y.is_nonpositive and d.is_positive: if y: return Dummy('neg', negative=True) else: return Dummy('npos', nonpositive=True) else: n, d = self.as_numer_denom() den = None if n.is_number: den = _monotonic_sign(d) elif not d.is_number: if _monotonic_sign(n) is not None: den = _monotonic_sign(d) if den is not None and (den.is_positive or den.is_negative): v = n*den if v.is_positive: return Dummy('pos', positive=True) elif v.is_nonnegative: return Dummy('nneg', nonnegative=True) elif v.is_negative: return Dummy('neg', negative=True) elif v.is_nonpositive: return Dummy('npos', nonpositive=True) return None # multivariate c, a = self.as_coeff_Add() v = None if not a.is_polynomial(): # F/A or A/F where A is a number and F is a signed, rational monomial n, d = a.as_numer_denom() if not (n.is_number or d.is_number): return if ( a.is_Mul or a.is_Pow) and \ a.is_rational and \ all(p.exp.is_Integer for p in a.atoms(Pow) if p.is_Pow) and \ (a.is_positive or a.is_negative): v = S.One for ai in Mul.make_args(a): if ai.is_number: v *= ai continue reps = {} for x in ai.free_symbols: reps[x] = _monotonic_sign(x) if reps[x] is None: return v *= ai.subs(reps) elif c: # signed linear expression if not any(p for p in a.atoms(Pow) if not p.is_number) and (a.is_nonpositive or a.is_nonnegative): free = list(a.free_symbols) p = {} for i in free: v = _monotonic_sign(i) if v is None: return p[i] = v or (_eps if i.is_nonnegative else -_eps) v = a.xreplace(p) if v is not None: rv = v + c if v.is_nonnegative and rv.is_positive: return rv.subs(_eps, 0) if v.is_nonpositive and rv.is_negative: return rv.subs(_eps, 0) def decompose_power(expr): """ Decompose power into symbolic base and integer exponent. Explanation =========== This is strictly only valid if the exponent from which the integer is extracted is itself an integer or the base is positive. These conditions are assumed and not checked here. Examples ======== >>> from sympy.core.exprtools import decompose_power >>> from sympy.abc import x, y >>> decompose_power(x) (x, 1) >>> decompose_power(x**2) (x, 2) >>> decompose_power(x**(2*y)) (x**y, 2) >>> decompose_power(x**(2*y/3)) (x**(y/3), 2) """ base, exp = expr.as_base_exp() if exp.is_Number: if exp.is_Rational: if not exp.is_Integer: base = Pow(base, Rational(1, exp.q)) exp = exp.p else: base, exp = expr, 1 else: exp, tail = exp.as_coeff_Mul(rational=True) if exp is S.NegativeOne: base, exp = Pow(base, tail), -1 elif exp is not S.One: tail = _keep_coeff(Rational(1, exp.q), tail) base, exp = Pow(base, tail), exp.p else: base, exp = expr, 1 return base, exp def decompose_power_rat(expr): """ Decompose power into symbolic base and rational exponent. """ base, exp = expr.as_base_exp() if exp.is_Number: if not exp.is_Rational: base, exp = expr, 1 else: exp, tail = exp.as_coeff_Mul(rational=True) if exp is S.NegativeOne: base, exp = Pow(base, tail), -1 elif exp is not S.One: tail = _keep_coeff(Rational(1, exp.q), tail) base, exp = Pow(base, tail), exp.p else: base, exp = expr, 1 return base, exp class Factors: """Efficient representation of ``f_1*f_2*...*f_n``.""" __slots__ = ('factors', 'gens') def __init__(self, factors=None): # Factors """Initialize Factors from dict or expr. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x >>> from sympy import I >>> e = 2*x**3 >>> Factors(e) Factors({2: 1, x: 3}) >>> Factors(e.as_powers_dict()) Factors({2: 1, x: 3}) >>> f = _ >>> f.factors # underlying dictionary {2: 1, x: 3} >>> f.gens # base of each factor frozenset({2, x}) >>> Factors(0) Factors({0: 1}) >>> Factors(I) Factors({I: 1}) Notes ===== Although a dictionary can be passed, only minimal checking is performed: powers of -1 and I are made canonical. """ if isinstance(factors, (SYMPY_INTS, float)): factors = S(factors) if isinstance(factors, Factors): factors = factors.factors.copy() elif factors is None or factors is S.One: factors = {} elif factors is S.Zero or factors == 0: factors = {S.Zero: S.One} elif isinstance(factors, Number): n = factors factors = {} if n < 0: factors[S.NegativeOne] = S.One n = -n if n is not S.One: if n.is_Float or n.is_Integer or n is S.Infinity: factors[n] = S.One elif n.is_Rational: # since we're processing Numbers, the denominator is # stored with a negative exponent; all other factors # are left . if n.p != 1: factors[Integer(n.p)] = S.One factors[Integer(n.q)] = S.NegativeOne else: raise ValueError('Expected Float|Rational|Integer, not %s' % n) elif isinstance(factors, Basic) and not factors.args: factors = {factors: S.One} elif isinstance(factors, Expr): c, nc = factors.args_cnc() i = c.count(I) for _ in range(i): c.remove(I) factors = dict(Mul._from_args(c).as_powers_dict()) # Handle all rational Coefficients for f in list(factors.keys()): if isinstance(f, Rational) and not isinstance(f, Integer): p, q = Integer(f.p), Integer(f.q) factors[p] = (factors[p] if p in factors else S.Zero) + factors[f] factors[q] = (factors[q] if q in factors else S.Zero) - factors[f] factors.pop(f) if i: factors[I] = S.One*i if nc: factors[Mul(*nc, evaluate=False)] = S.One else: factors = factors.copy() # /!\ should be dict-like # tidy up -/+1 and I exponents if Rational handle = [] for k in factors: if k is I or k in (-1, 1): handle.append(k) if handle: i1 = S.One for k in handle: if not _isnumber(factors[k]): continue i1 *= k**factors.pop(k) if i1 is not S.One: for a in i1.args if i1.is_Mul else [i1]: # at worst, -1.0*I*(-1)**e if a is S.NegativeOne: factors[a] = S.One elif a is I: factors[I] = S.One elif a.is_Pow: if S.NegativeOne not in factors: factors[S.NegativeOne] = S.Zero factors[S.NegativeOne] += a.exp elif a == 1: factors[a] = S.One elif a == -1: factors[-a] = S.One factors[S.NegativeOne] = S.One else: raise ValueError('unexpected factor in i1: %s' % a) self.factors = factors keys = getattr(factors, 'keys', None) if keys is None: raise TypeError('expecting Expr or dictionary') self.gens = frozenset(keys()) def __hash__(self): # Factors keys = tuple(ordered(self.factors.keys())) values = [self.factors[k] for k in keys] return hash((keys, values)) def __repr__(self): # Factors return "Factors({%s})" % ', '.join( ['%s: %s' % (k, v) for k, v in ordered(self.factors.items())]) @property def is_zero(self): # Factors """ >>> from sympy.core.exprtools import Factors >>> Factors(0).is_zero True """ f = self.factors return len(f) == 1 and S.Zero in f @property def is_one(self): # Factors """ >>> from sympy.core.exprtools import Factors >>> Factors(1).is_one True """ return not self.factors def as_expr(self): # Factors """Return the underlying expression. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y >>> Factors((x*y**2).as_powers_dict()).as_expr() x*y**2 """ args = [] for factor, exp in self.factors.items(): if exp != 1: if isinstance(exp, Integer): b, e = factor.as_base_exp() e = _keep_coeff(exp, e) args.append(b**e) else: args.append(factor**exp) else: args.append(factor) return Mul(*args) def mul(self, other): # Factors """Return Factors of ``self * other``. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> a = Factors((x*y**2).as_powers_dict()) >>> b = Factors((x*y/z).as_powers_dict()) >>> a.mul(b) Factors({x: 2, y: 3, z: -1}) >>> a*b Factors({x: 2, y: 3, z: -1}) """ if not isinstance(other, Factors): other = Factors(other) if any(f.is_zero for f in (self, other)): return Factors(S.Zero) factors = dict(self.factors) for factor, exp in other.factors.items(): if factor in factors: exp = factors[factor] + exp if not exp: del factors[factor] continue factors[factor] = exp return Factors(factors) def normal(self, other): """Return ``self`` and ``other`` with ``gcd`` removed from each. The only differences between this and method ``div`` is that this is 1) optimized for the case when there are few factors in common and 2) this does not raise an error if ``other`` is zero. See Also ======== div """ if not isinstance(other, Factors): other = Factors(other) if other.is_zero: return (Factors(), Factors(S.Zero)) if self.is_zero: return (Factors(S.Zero), Factors()) self_factors = dict(self.factors) other_factors = dict(other.factors) for factor, self_exp in self.factors.items(): try: other_exp = other.factors[factor] except KeyError: continue exp = self_exp - other_exp if not exp: del self_factors[factor] del other_factors[factor] elif _isnumber(exp): if exp > 0: self_factors[factor] = exp del other_factors[factor] else: del self_factors[factor] other_factors[factor] = -exp else: r = self_exp.extract_additively(other_exp) if r is not None: if r: self_factors[factor] = r del other_factors[factor] else: # should be handled already del self_factors[factor] del other_factors[factor] else: sc, sa = self_exp.as_coeff_Add() if sc: oc, oa = other_exp.as_coeff_Add() diff = sc - oc if diff > 0: self_factors[factor] -= oc other_exp = oa elif diff < 0: self_factors[factor] -= sc other_factors[factor] -= sc other_exp = oa - diff else: self_factors[factor] = sa other_exp = oa if other_exp: other_factors[factor] = other_exp else: del other_factors[factor] return Factors(self_factors), Factors(other_factors) def div(self, other): # Factors """Return ``self`` and ``other`` with ``gcd`` removed from each. This is optimized for the case when there are many factors in common. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> from sympy import S >>> a = Factors((x*y**2).as_powers_dict()) >>> a.div(a) (Factors({}), Factors({})) >>> a.div(x*z) (Factors({y: 2}), Factors({z: 1})) The ``/`` operator only gives ``quo``: >>> a/x Factors({y: 2}) Factors treats its factors as though they are all in the numerator, so if you violate this assumption the results will be correct but will not strictly correspond to the numerator and denominator of the ratio: >>> a.div(x/z) (Factors({y: 2}), Factors({z: -1})) Factors is also naive about bases: it does not attempt any denesting of Rational-base terms, for example the following does not become 2**(2*x)/2. >>> Factors(2**(2*x + 2)).div(S(8)) (Factors({2: 2*x + 2}), Factors({8: 1})) factor_terms can clean up such Rational-bases powers: >>> from sympy.core.exprtools import factor_terms >>> n, d = Factors(2**(2*x + 2)).div(S(8)) >>> n.as_expr()/d.as_expr() 2**(2*x + 2)/8 >>> factor_terms(_) 2**(2*x)/2 """ quo, rem = dict(self.factors), {} if not isinstance(other, Factors): other = Factors(other) if other.is_zero: raise ZeroDivisionError if self.is_zero: return (Factors(S.Zero), Factors()) for factor, exp in other.factors.items(): if factor in quo: d = quo[factor] - exp if _isnumber(d): if d <= 0: del quo[factor] if d >= 0: if d: quo[factor] = d continue exp = -d else: r = quo[factor].extract_additively(exp) if r is not None: if r: quo[factor] = r else: # should be handled already del quo[factor] else: other_exp = exp sc, sa = quo[factor].as_coeff_Add() if sc: oc, oa = other_exp.as_coeff_Add() diff = sc - oc if diff > 0: quo[factor] -= oc other_exp = oa elif diff < 0: quo[factor] -= sc other_exp = oa - diff else: quo[factor] = sa other_exp = oa if other_exp: rem[factor] = other_exp else: assert factor not in rem continue rem[factor] = exp return Factors(quo), Factors(rem) def quo(self, other): # Factors """Return numerator Factor of ``self / other``. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> a = Factors((x*y**2).as_powers_dict()) >>> b = Factors((x*y/z).as_powers_dict()) >>> a.quo(b) # same as a/b Factors({y: 1}) """ return self.div(other)[0] def rem(self, other): # Factors """Return denominator Factors of ``self / other``. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> a = Factors((x*y**2).as_powers_dict()) >>> b = Factors((x*y/z).as_powers_dict()) >>> a.rem(b) Factors({z: -1}) >>> a.rem(a) Factors({}) """ return self.div(other)[1] def pow(self, other): # Factors """Return self raised to a non-negative integer power. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y >>> a = Factors((x*y**2).as_powers_dict()) >>> a**2 Factors({x: 2, y: 4}) """ if isinstance(other, Factors): other = other.as_expr() if other.is_Integer: other = int(other) if isinstance(other, SYMPY_INTS) and other >= 0: factors = {} if other: for factor, exp in self.factors.items(): factors[factor] = exp*other return Factors(factors) else: raise ValueError("expected non-negative integer, got %s" % other) def gcd(self, other): # Factors """Return Factors of ``gcd(self, other)``. The keys are the intersection of factors with the minimum exponent for each factor. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> a = Factors((x*y**2).as_powers_dict()) >>> b = Factors((x*y/z).as_powers_dict()) >>> a.gcd(b) Factors({x: 1, y: 1}) """ if not isinstance(other, Factors): other = Factors(other) if other.is_zero: return Factors(self.factors) factors = {} for factor, exp in self.factors.items(): factor, exp = sympify(factor), sympify(exp) if factor in other.factors: lt = (exp - other.factors[factor]).is_negative if lt == True: factors[factor] = exp elif lt == False: factors[factor] = other.factors[factor] return Factors(factors) def lcm(self, other): # Factors """Return Factors of ``lcm(self, other)`` which are the union of factors with the maximum exponent for each factor. Examples ======== >>> from sympy.core.exprtools import Factors >>> from sympy.abc import x, y, z >>> a = Factors((x*y**2).as_powers_dict()) >>> b = Factors((x*y/z).as_powers_dict()) >>> a.lcm(b) Factors({x: 1, y: 2, z: -1}) """ if not isinstance(other, Factors): other = Factors(other) if any(f.is_zero for f in (self, other)): return Factors(S.Zero) factors = dict(self.factors) for factor, exp in other.factors.items(): if factor in factors: exp = max(exp, factors[factor]) factors[factor] = exp return Factors(factors) def __mul__(self, other): # Factors return self.mul(other) def __divmod__(self, other): # Factors return self.div(other) def __truediv__(self, other): # Factors return self.quo(other) def __mod__(self, other): # Factors return self.rem(other) def __pow__(self, other): # Factors return self.pow(other) def __eq__(self, other): # Factors if not isinstance(other, Factors): other = Factors(other) return self.factors == other.factors def __ne__(self, other): # Factors return not self == other class Term: """Efficient representation of ``coeff*(numer/denom)``. """ __slots__ = ('coeff', 'numer', 'denom') def __init__(self, term, numer=None, denom=None): # Term if numer is None and denom is None: if not term.is_commutative: raise NonCommutativeExpression( 'commutative expression expected') coeff, factors = term.as_coeff_mul() numer, denom = defaultdict(int), defaultdict(int) for factor in factors: base, exp = decompose_power(factor) if base.is_Add: cont, base = base.primitive() coeff *= cont**exp if exp > 0: numer[base] += exp else: denom[base] += -exp numer = Factors(numer) denom = Factors(denom) else: coeff = term if numer is None: numer = Factors() if denom is None: denom = Factors() self.coeff = coeff self.numer = numer self.denom = denom def __hash__(self): # Term return hash((self.coeff, self.numer, self.denom)) def __repr__(self): # Term return "Term(%s, %s, %s)" % (self.coeff, self.numer, self.denom) def as_expr(self): # Term return self.coeff*(self.numer.as_expr()/self.denom.as_expr()) def mul(self, other): # Term coeff = self.coeff*other.coeff numer = self.numer.mul(other.numer) denom = self.denom.mul(other.denom) numer, denom = numer.normal(denom) return Term(coeff, numer, denom) def inv(self): # Term return Term(1/self.coeff, self.denom, self.numer) def quo(self, other): # Term return self.mul(other.inv()) def pow(self, other): # Term if other < 0: return self.inv().pow(-other) else: return Term(self.coeff ** other, self.numer.pow(other), self.denom.pow(other)) def gcd(self, other): # Term return Term(self.coeff.gcd(other.coeff), self.numer.gcd(other.numer), self.denom.gcd(other.denom)) def lcm(self, other): # Term return Term(self.coeff.lcm(other.coeff), self.numer.lcm(other.numer), self.denom.lcm(other.denom)) def __mul__(self, other): # Term if isinstance(other, Term): return self.mul(other) else: return NotImplemented def __truediv__(self, other): # Term if isinstance(other, Term): return self.quo(other) else: return NotImplemented def __pow__(self, other): # Term if isinstance(other, SYMPY_INTS): return self.pow(other) else: return NotImplemented def __eq__(self, other): # Term return (self.coeff == other.coeff and self.numer == other.numer and self.denom == other.denom) def __ne__(self, other): # Term return not self == other def _gcd_terms(terms, isprimitive=False, fraction=True): """Helper function for :func:`gcd_terms`. Parameters ========== isprimitive : boolean, optional If ``isprimitive`` is True then the call to primitive for an Add will be skipped. This is useful when the content has already been extrated. fraction : boolean, optional If ``fraction`` is True then the expression will appear over a common denominator, the lcm of all term denominators. """ if isinstance(terms, Basic) and not isinstance(terms, Tuple): terms = Add.make_args(terms) terms = list(map(Term, [t for t in terms if t])) # there is some simplification that may happen if we leave this # here rather than duplicate it before the mapping of Term onto # the terms if len(terms) == 0: return S.Zero, S.Zero, S.One if len(terms) == 1: cont = terms[0].coeff numer = terms[0].numer.as_expr() denom = terms[0].denom.as_expr() else: cont = terms[0] for term in terms[1:]: cont = cont.gcd(term) for i, term in enumerate(terms): terms[i] = term.quo(cont) if fraction: denom = terms[0].denom for term in terms[1:]: denom = denom.lcm(term.denom) numers = [] for term in terms: numer = term.numer.mul(denom.quo(term.denom)) numers.append(term.coeff*numer.as_expr()) else: numers = [t.as_expr() for t in terms] denom = Term(S.One).numer cont = cont.as_expr() numer = Add(*numers) denom = denom.as_expr() if not isprimitive and numer.is_Add: _cont, numer = numer.primitive() cont *= _cont return cont, numer, denom def gcd_terms(terms, isprimitive=False, clear=True, fraction=True): """Compute the GCD of ``terms`` and put them together. Parameters ========== terms : Expr Can be an expression or a non-Basic sequence of expressions which will be handled as though they are terms from a sum. isprimitive : bool, optional If ``isprimitive`` is True the _gcd_terms will not run the primitive method on the terms. clear : bool, optional It controls the removal of integers from the denominator of an Add expression. When True (default), all numerical denominator will be cleared; when False the denominators will be cleared only if all terms had numerical denominators other than 1. fraction : bool, optional When True (default), will put the expression over a common denominator. Examples ======== >>> from sympy.core import gcd_terms >>> from sympy.abc import x, y >>> gcd_terms((x + 1)**2*y + (x + 1)*y**2) y*(x + 1)*(x + y + 1) >>> gcd_terms(x/2 + 1) (x + 2)/2 >>> gcd_terms(x/2 + 1, clear=False) x/2 + 1 >>> gcd_terms(x/2 + y/2, clear=False) (x + y)/2 >>> gcd_terms(x/2 + 1/x) (x**2 + 2)/(2*x) >>> gcd_terms(x/2 + 1/x, fraction=False) (x + 2/x)/2 >>> gcd_terms(x/2 + 1/x, fraction=False, clear=False) x/2 + 1/x >>> gcd_terms(x/2/y + 1/x/y) (x**2 + 2)/(2*x*y) >>> gcd_terms(x/2/y + 1/x/y, clear=False) (x**2/2 + 1)/(x*y) >>> gcd_terms(x/2/y + 1/x/y, clear=False, fraction=False) (x/2 + 1/x)/y The ``clear`` flag was ignored in this case because the returned expression was a rational expression, not a simple sum. See Also ======== factor_terms, sympy.polys.polytools.terms_gcd """ def mask(terms): """replace nc portions of each term with a unique Dummy symbols and return the replacements to restore them""" args = [(a, []) if a.is_commutative else a.args_cnc() for a in terms] reps = [] for i, (c, nc) in enumerate(args): if nc: nc = Mul(*nc) d = Dummy() reps.append((d, nc)) c.append(d) args[i] = Mul(*c) else: args[i] = c return args, dict(reps) isadd = isinstance(terms, Add) addlike = isadd or not isinstance(terms, Basic) and \ is_sequence(terms, include=set) and \ not isinstance(terms, Dict) if addlike: if isadd: # i.e. an Add terms = list(terms.args) else: terms = sympify(terms) terms, reps = mask(terms) cont, numer, denom = _gcd_terms(terms, isprimitive, fraction) numer = numer.xreplace(reps) coeff, factors = cont.as_coeff_Mul() if not clear: c, _coeff = coeff.as_coeff_Mul() if not c.is_Integer and not clear and numer.is_Add: n, d = c.as_numer_denom() _numer = numer/d if any(a.as_coeff_Mul()[0].is_Integer for a in _numer.args): numer = _numer coeff = n*_coeff return _keep_coeff(coeff, factors*numer/denom, clear=clear) if not isinstance(terms, Basic): return terms if terms.is_Atom: return terms if terms.is_Mul: c, args = terms.as_coeff_mul() return _keep_coeff(c, Mul(*[gcd_terms(i, isprimitive, clear, fraction) for i in args]), clear=clear) def handle(a): # don't treat internal args like terms of an Add if not isinstance(a, Expr): if isinstance(a, Basic): return a.func(*[handle(i) for i in a.args]) return type(a)([handle(i) for i in a]) return gcd_terms(a, isprimitive, clear, fraction) if isinstance(terms, Dict): return Dict(*[(k, handle(v)) for k, v in terms.args]) return terms.func(*[handle(i) for i in terms.args]) def _factor_sum_int(expr, **kwargs): """Return Sum or Integral object with factors that are not in the wrt variables removed. In cases where there are additive terms in the function of the object that are independent, the object will be separated into two objects. Examples ======== >>> from sympy import Sum, factor_terms >>> from sympy.abc import x, y >>> factor_terms(Sum(x + y, (x, 1, 3))) y*Sum(1, (x, 1, 3)) + Sum(x, (x, 1, 3)) >>> factor_terms(Sum(x*y, (x, 1, 3))) y*Sum(x, (x, 1, 3)) Notes ===== If a function in the summand or integrand is replaced with a symbol, then this simplification should not be done or else an incorrect result will be obtained when the symbol is replaced with an expression that depends on the variables of summation/integration: >>> eq = Sum(y, (x, 1, 3)) >>> factor_terms(eq).subs(y, x).doit() 3*x >>> eq.subs(y, x).doit() 6 """ result = expr.function if result == 0: return S.Zero limits = expr.limits # get the wrt variables wrt = {i.args[0] for i in limits} # factor out any common terms that are independent of wrt f = factor_terms(result, **kwargs) i, d = f.as_independent(*wrt) if isinstance(f, Add): return i * expr.func(1, *limits) + expr.func(d, *limits) else: return i * expr.func(d, *limits) def factor_terms(expr, radical=False, clear=False, fraction=False, sign=True): """Remove common factors from terms in all arguments without changing the underlying structure of the expr. No expansion or simplification (and no processing of non-commutatives) is performed. Parameters ========== radical: bool, optional If radical=True then a radical common to all terms will be factored out of any Add sub-expressions of the expr. clear : bool, optional If clear=False (default) then coefficients will not be separated from a single Add if they can be distributed to leave one or more terms with integer coefficients. fraction : bool, optional If fraction=True (default is False) then a common denominator will be constructed for the expression. sign : bool, optional If sign=True (default) then even if the only factor in common is a -1, it will be factored out of the expression. Examples ======== >>> from sympy import factor_terms, Symbol >>> from sympy.abc import x, y >>> factor_terms(x + x*(2 + 4*y)**3) x*(8*(2*y + 1)**3 + 1) >>> A = Symbol('A', commutative=False) >>> factor_terms(x*A + x*A + x*y*A) x*(y*A + 2*A) When ``clear`` is False, a rational will only be factored out of an Add expression if all terms of the Add have coefficients that are fractions: >>> factor_terms(x/2 + 1, clear=False) x/2 + 1 >>> factor_terms(x/2 + 1, clear=True) (x + 2)/2 If a -1 is all that can be factored out, to *not* factor it out, the flag ``sign`` must be False: >>> factor_terms(-x - y) -(x + y) >>> factor_terms(-x - y, sign=False) -x - y >>> factor_terms(-2*x - 2*y, sign=False) -2*(x + y) See Also ======== gcd_terms, sympy.polys.polytools.terms_gcd """ def do(expr): from sympy.concrete.summations import Sum from sympy.integrals.integrals import Integral is_iterable = iterable(expr) if not isinstance(expr, Basic) or expr.is_Atom: if is_iterable: return type(expr)([do(i) for i in expr]) return expr if expr.is_Pow or expr.is_Function or \ is_iterable or not hasattr(expr, 'args_cnc'): args = expr.args newargs = tuple([do(i) for i in args]) if newargs == args: return expr return expr.func(*newargs) if isinstance(expr, (Sum, Integral)): return _factor_sum_int(expr, radical=radical, clear=clear, fraction=fraction, sign=sign) cont, p = expr.as_content_primitive(radical=radical, clear=clear) if p.is_Add: list_args = [do(a) for a in Add.make_args(p)] # get a common negative (if there) which gcd_terms does not remove if all(a.as_coeff_Mul()[0].extract_multiplicatively(-1) is not None for a in list_args): cont = -cont list_args = [-a for a in list_args] # watch out for exp(-(x+2)) which gcd_terms will change to exp(-x-2) special = {} for i, a in enumerate(list_args): b, e = a.as_base_exp() if e.is_Mul and e != Mul(*e.args): list_args[i] = Dummy() special[list_args[i]] = a # rebuild p not worrying about the order which gcd_terms will fix p = Add._from_args(list_args) p = gcd_terms(p, isprimitive=True, clear=clear, fraction=fraction).xreplace(special) elif p.args: p = p.func( *[do(a) for a in p.args]) rv = _keep_coeff(cont, p, clear=clear, sign=sign) return rv expr = sympify(expr) return do(expr) def _mask_nc(eq, name=None): """ Return ``eq`` with non-commutative objects replaced with Dummy symbols. A dictionary that can be used to restore the original values is returned: if it is None, the expression is noncommutative and cannot be made commutative. The third value returned is a list of any non-commutative symbols that appear in the returned equation. Explanation =========== All non-commutative objects other than Symbols are replaced with a non-commutative Symbol. Identical objects will be identified by identical symbols. If there is only 1 non-commutative object in an expression it will be replaced with a commutative symbol. Otherwise, the non-commutative entities are retained and the calling routine should handle replacements in this case since some care must be taken to keep track of the ordering of symbols when they occur within Muls. Parameters ========== name : str ``name``, if given, is the name that will be used with numbered Dummy variables that will replace the non-commutative objects and is mainly used for doctesting purposes. Examples ======== >>> from sympy.physics.secondquant import Commutator, NO, F, Fd >>> from sympy import symbols >>> from sympy.core.exprtools import _mask_nc >>> from sympy.abc import x, y >>> A, B, C = symbols('A,B,C', commutative=False) One nc-symbol: >>> _mask_nc(A**2 - x**2, 'd') (_d0**2 - x**2, {_d0: A}, []) Multiple nc-symbols: >>> _mask_nc(A**2 - B**2, 'd') (A**2 - B**2, {}, [A, B]) An nc-object with nc-symbols but no others outside of it: >>> _mask_nc(1 + x*Commutator(A, B), 'd') (_d0*x + 1, {_d0: Commutator(A, B)}, []) >>> _mask_nc(NO(Fd(x)*F(y)), 'd') (_d0, {_d0: NO(CreateFermion(x)*AnnihilateFermion(y))}, []) Multiple nc-objects: >>> eq = x*Commutator(A, B) + x*Commutator(A, C)*Commutator(A, B) >>> _mask_nc(eq, 'd') (x*_d0 + x*_d1*_d0, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1]) Multiple nc-objects and nc-symbols: >>> eq = A*Commutator(A, B) + B*Commutator(A, C) >>> _mask_nc(eq, 'd') (A*_d0 + B*_d1, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1, A, B]) """ name = name or 'mask' # Make Dummy() append sequential numbers to the name def numbered_names(): i = 0 while True: yield name + str(i) i += 1 names = numbered_names() def Dummy(*args, **kwargs): from sympy import Dummy return Dummy(next(names), *args, **kwargs) expr = eq if expr.is_commutative: return eq, {}, [] # identify nc-objects; symbols and other rep = [] nc_obj = set() nc_syms = set() pot = preorder_traversal(expr, keys=default_sort_key) for i, a in enumerate(pot): if any(a == r[0] for r in rep): pot.skip() elif not a.is_commutative: if a.is_symbol: nc_syms.add(a) pot.skip() elif not (a.is_Add or a.is_Mul or a.is_Pow): nc_obj.add(a) pot.skip() # If there is only one nc symbol or object, it can be factored regularly # but polys is going to complain, so replace it with a Dummy. if len(nc_obj) == 1 and not nc_syms: rep.append((nc_obj.pop(), Dummy())) elif len(nc_syms) == 1 and not nc_obj: rep.append((nc_syms.pop(), Dummy())) # Any remaining nc-objects will be replaced with an nc-Dummy and # identified as an nc-Symbol to watch out for nc_obj = sorted(nc_obj, key=default_sort_key) for n in nc_obj: nc = Dummy(commutative=False) rep.append((n, nc)) nc_syms.add(nc) expr = expr.subs(rep) nc_syms = list(nc_syms) nc_syms.sort(key=default_sort_key) return expr, {v: k for k, v in rep}, nc_syms def factor_nc(expr): """Return the factored form of ``expr`` while handling non-commutative expressions. Examples ======== >>> from sympy.core.exprtools import factor_nc >>> from sympy import Symbol >>> from sympy.abc import x >>> A = Symbol('A', commutative=False) >>> B = Symbol('B', commutative=False) >>> factor_nc((x**2 + 2*A*x + A**2).expand()) (x + A)**2 >>> factor_nc(((x + A)*(x + B)).expand()) (x + A)*(x + B) """ from sympy.simplify.simplify import powsimp from sympy.polys import gcd, factor def _pemexpand(expr): "Expand with the minimal set of hints necessary to check the result." return expr.expand(deep=True, mul=True, power_exp=True, power_base=False, basic=False, multinomial=True, log=False) expr = sympify(expr) if not isinstance(expr, Expr) or not expr.args: return expr if not expr.is_Add: return expr.func(*[factor_nc(a) for a in expr.args]) expr, rep, nc_symbols = _mask_nc(expr) if rep: return factor(expr).subs(rep) else: args = [a.args_cnc() for a in Add.make_args(expr)] c = g = l = r = S.One hit = False # find any commutative gcd term for i, a in enumerate(args): if i == 0: c = Mul._from_args(a[0]) elif a[0]: c = gcd(c, Mul._from_args(a[0])) else: c = S.One if c is not S.One: hit = True c, g = c.as_coeff_Mul() if g is not S.One: for i, (cc, _) in enumerate(args): cc = list(Mul.make_args(Mul._from_args(list(cc))/g)) args[i][0] = cc for i, (cc, _) in enumerate(args): cc[0] = cc[0]/c args[i][0] = cc # find any noncommutative common prefix for i, a in enumerate(args): if i == 0: n = a[1][:] else: n = common_prefix(n, a[1]) if not n: # is there a power that can be extracted? if not args[0][1]: break b, e = args[0][1][0].as_base_exp() ok = False if e.is_Integer: for t in args: if not t[1]: break bt, et = t[1][0].as_base_exp() if et.is_Integer and bt == b: e = min(e, et) else: break else: ok = hit = True l = b**e il = b**-e for _ in args: _[1][0] = il*_[1][0] break if not ok: break else: hit = True lenn = len(n) l = Mul(*n) for _ in args: _[1] = _[1][lenn:] # find any noncommutative common suffix for i, a in enumerate(args): if i == 0: n = a[1][:] else: n = common_suffix(n, a[1]) if not n: # is there a power that can be extracted? if not args[0][1]: break b, e = args[0][1][-1].as_base_exp() ok = False if e.is_Integer: for t in args: if not t[1]: break bt, et = t[1][-1].as_base_exp() if et.is_Integer and bt == b: e = min(e, et) else: break else: ok = hit = True r = b**e il = b**-e for _ in args: _[1][-1] = _[1][-1]*il break if not ok: break else: hit = True lenn = len(n) r = Mul(*n) for _ in args: _[1] = _[1][:len(_[1]) - lenn] if hit: mid = Add(*[Mul(*cc)*Mul(*nc) for cc, nc in args]) else: mid = expr # sort the symbols so the Dummys would appear in the same # order as the original symbols, otherwise you may introduce # a factor of -1, e.g. A**2 - B**2) -- {A:y, B:x} --> y**2 - x**2 # and the former factors into two terms, (A - B)*(A + B) while the # latter factors into 3 terms, (-1)*(x - y)*(x + y) rep1 = [(n, Dummy()) for n in sorted(nc_symbols, key=default_sort_key)] unrep1 = [(v, k) for k, v in rep1] unrep1.reverse() new_mid, r2, _ = _mask_nc(mid.subs(rep1)) new_mid = powsimp(factor(new_mid)) new_mid = new_mid.subs(r2).subs(unrep1) if new_mid.is_Pow: return _keep_coeff(c, g*l*new_mid*r) if new_mid.is_Mul: # XXX TODO there should be a way to inspect what order the terms # must be in and just select the plausible ordering without # checking permutations cfac = [] ncfac = [] for f in new_mid.args: if f.is_commutative: cfac.append(f) else: b, e = f.as_base_exp() if e.is_Integer: ncfac.extend([b]*e) else: ncfac.append(f) pre_mid = g*Mul(*cfac)*l target = _pemexpand(expr/c) for s in variations(ncfac, len(ncfac)): ok = pre_mid*Mul(*s)*r if _pemexpand(ok) == target: return _keep_coeff(c, ok) # mid was an Add that didn't factor successfully return _keep_coeff(c, g*l*mid*r)
0db8b0facd7480a59df421c1900e262ebaccaf533008e16bcd80f2b39565d180
"""Singleton mechanism""" from typing import Any, Dict, Type from .core import Registry from .assumptions import ManagedProperties from .sympify import sympify class SingletonRegistry(Registry): """ The registry for the singleton classes (accessible as ``S``). Explanation =========== This class serves as two separate things. The first thing it is is the ``SingletonRegistry``. Several classes in SymPy appear so often that they are singletonized, that is, using some metaprogramming they are made so that they can only be instantiated once (see the :class:`sympy.core.singleton.Singleton` class for details). For instance, every time you create ``Integer(0)``, this will return the same instance, :class:`sympy.core.numbers.Zero`. All singleton instances are attributes of the ``S`` object, so ``Integer(0)`` can also be accessed as ``S.Zero``. Singletonization offers two advantages: it saves memory, and it allows fast comparison. It saves memory because no matter how many times the singletonized objects appear in expressions in memory, they all point to the same single instance in memory. The fast comparison comes from the fact that you can use ``is`` to compare exact instances in Python (usually, you need to use ``==`` to compare things). ``is`` compares objects by memory address, and is very fast. Examples ======== >>> from sympy import S, Integer >>> a = Integer(0) >>> a is S.Zero True For the most part, the fact that certain objects are singletonized is an implementation detail that users shouldn't need to worry about. In SymPy library code, ``is`` comparison is often used for performance purposes The primary advantage of ``S`` for end users is the convenient access to certain instances that are otherwise difficult to type, like ``S.Half`` (instead of ``Rational(1, 2)``). When using ``is`` comparison, make sure the argument is sympified. For instance, >>> x = 0 >>> x is S.Zero False This problem is not an issue when using ``==``, which is recommended for most use-cases: >>> 0 == S.Zero True The second thing ``S`` is is a shortcut for :func:`sympy.core.sympify.sympify`. :func:`sympy.core.sympify.sympify` is the function that converts Python objects such as ``int(1)`` into SymPy objects such as ``Integer(1)``. It also converts the string form of an expression into a SymPy expression, like ``sympify("x**2")`` -> ``Symbol("x")**2``. ``S(1)`` is the same thing as ``sympify(1)`` (basically, ``S.__call__`` has been defined to call ``sympify``). This is for convenience, since ``S`` is a single letter. It's mostly useful for defining rational numbers. Consider an expression like ``x + 1/2``. If you enter this directly in Python, it will evaluate the ``1/2`` and give ``0.5`` (or just ``0`` in Python 2, because of integer division), because both arguments are ints (see also :ref:`tutorial-gotchas-final-notes`). However, in SymPy, you usually want the quotient of two integers to give an exact rational number. The way Python's evaluation works, at least one side of an operator needs to be a SymPy object for the SymPy evaluation to take over. You could write this as ``x + Rational(1, 2)``, but this is a lot more typing. A shorter version is ``x + S(1)/2``. Since ``S(1)`` returns ``Integer(1)``, the division will return a ``Rational`` type, since it will call ``Integer.__truediv__``, which knows how to return a ``Rational``. """ __slots__ = () # Also allow things like S(5) __call__ = staticmethod(sympify) def __init__(self): self._classes_to_install = {} # Dict of classes that have been registered, but that have not have been # installed as an attribute of this SingletonRegistry. # Installation automatically happens at the first attempt to access the # attribute. # The purpose of this is to allow registration during class # initialization during import, but not trigger object creation until # actual use (which should not happen until after all imports are # finished). def register(self, cls): # Make sure a duplicate class overwrites the old one if hasattr(self, cls.__name__): delattr(self, cls.__name__) self._classes_to_install[cls.__name__] = cls def __getattr__(self, name): """Python calls __getattr__ if no attribute of that name was installed yet. Explanation =========== This __getattr__ checks whether a class with the requested name was already registered but not installed; if no, raises an AttributeError. Otherwise, retrieves the class, calculates its singleton value, installs it as an attribute of the given name, and unregisters the class.""" if name not in self._classes_to_install: raise AttributeError( "Attribute '%s' was not installed on SymPy registry %s" % ( name, self)) class_to_install = self._classes_to_install[name] value_to_install = class_to_install() self.__setattr__(name, value_to_install) del self._classes_to_install[name] return value_to_install def __repr__(self): return "S" S = SingletonRegistry() class Singleton(ManagedProperties): """ Metaclass for singleton classes. Explanation =========== A singleton class has only one instance which is returned every time the class is instantiated. Additionally, this instance can be accessed through the global registry object ``S`` as ``S.<class_name>``. Examples ======== >>> from sympy import S, Basic >>> from sympy.core.singleton import Singleton >>> class MySingleton(Basic, metaclass=Singleton): ... pass >>> Basic() is Basic() False >>> MySingleton() is MySingleton() True >>> S.MySingleton is MySingleton() True Notes ===== Instance creation is delayed until the first time the value is accessed. (SymPy versions before 1.0 would create the instance during class creation time, which would be prone to import cycles.) This metaclass is a subclass of ManagedProperties because that is the metaclass of many classes that need to be Singletons (Python does not allow subclasses to have a different metaclass than the superclass, except the subclass may use a subclassed metaclass). """ _instances = {} # type: Dict[Type[Any], Any] "Maps singleton classes to their instances." def __new__(cls, *args, **kwargs): result = super().__new__(cls, *args, **kwargs) S.register(result) return result def __call__(self, *args, **kwargs): # Called when application code says SomeClass(), where SomeClass is a # class of which Singleton is the metaclas. # __call__ is invoked first, before __new__() and __init__(). if self not in Singleton._instances: Singleton._instances[self] = \ super().__call__(*args, **kwargs) # Invokes the standard constructor of SomeClass. return Singleton._instances[self] # Inject pickling support. def __getnewargs__(self): return () self.__getnewargs__ = __getnewargs__
59f95bab3e4fa782ea132e85187b7ffb3307c851fd69321fec51123d98cec9ae
""" There are three types of functions implemented in SymPy: 1) defined functions (in the sense that they can be evaluated) like exp or sin; they have a name and a body: f = exp 2) undefined function which have a name but no body. Undefined functions can be defined using a Function class as follows: f = Function('f') (the result will be a Function instance) 3) anonymous function (or lambda function) which have a body (defined with dummy variables) but have no name: f = Lambda(x, exp(x)*x) f = Lambda((x, y), exp(x)*y) The fourth type of functions are composites, like (sin + cos)(x); these work in SymPy core, but are not yet part of SymPy. Examples ======== >>> import sympy >>> f = sympy.Function("f") >>> from sympy.abc import x >>> f(x) f(x) >>> print(sympy.srepr(f(x).func)) Function('f') >>> f(x).args (x,) """ from typing import Any, Dict as tDict, Optional, Set as tSet, Tuple as tTuple, Union from .add import Add from .assumptions import ManagedProperties from .basic import Basic, _atomic from .cache import cacheit from .compatibility import iterable, is_sequence, as_int, ordered, Iterable from .decorators import _sympifyit from .expr import Expr, AtomicExpr from .numbers import Rational, Float from .operations import LatticeOp from .rules import Transform from .singleton import S from .sympify import sympify from sympy.core.containers import Tuple, Dict from sympy.core.parameters import global_parameters from sympy.core.logic import fuzzy_and, fuzzy_or, fuzzy_not, FuzzyBool from sympy.utilities import default_sort_key from sympy.utilities.exceptions import SymPyDeprecationWarning from sympy.utilities.iterables import has_dups, sift from sympy.utilities.misc import filldedent import mpmath import mpmath.libmp as mlib import inspect from collections import Counter def _coeff_isneg(a): """Return True if the leading Number is negative. Examples ======== >>> from sympy.core.function import _coeff_isneg >>> from sympy import S, Symbol, oo, pi >>> _coeff_isneg(-3*pi) True >>> _coeff_isneg(S(3)) False >>> _coeff_isneg(-oo) True >>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1 False For matrix expressions: >>> from sympy import MatrixSymbol, sqrt >>> A = MatrixSymbol("A", 3, 3) >>> _coeff_isneg(-sqrt(2)*A) True >>> _coeff_isneg(sqrt(2)*A) False """ if a.is_MatMul: a = a.args[0] if a.is_Mul: a = a.args[0] return a.is_Number and a.is_extended_negative class PoleError(Exception): pass class ArgumentIndexError(ValueError): def __str__(self): return ("Invalid operation with argument number %s for Function %s" % (self.args[1], self.args[0])) class BadSignatureError(TypeError): '''Raised when a Lambda is created with an invalid signature''' pass class BadArgumentsError(TypeError): '''Raised when a Lambda is called with an incorrect number of arguments''' pass # Python 2/3 version that does not raise a Deprecation warning def arity(cls): """Return the arity of the function if it is known, else None. Explanation =========== When default values are specified for some arguments, they are optional and the arity is reported as a tuple of possible values. Examples ======== >>> from sympy.core.function import arity >>> from sympy import log >>> arity(lambda x: x) 1 >>> arity(log) (1, 2) >>> arity(lambda *x: sum(x)) is None True """ eval_ = getattr(cls, 'eval', cls) parameters = inspect.signature(eval_).parameters.items() if [p for _, p in parameters if p.kind == p.VAR_POSITIONAL]: return p_or_k = [p for _, p in parameters if p.kind == p.POSITIONAL_OR_KEYWORD] # how many have no default and how many have a default value no, yes = map(len, sift(p_or_k, lambda p:p.default == p.empty, binary=True)) return no if not yes else tuple(range(no, no + yes + 1)) class FunctionClass(ManagedProperties): """ Base class for function classes. FunctionClass is a subclass of type. Use Function('<function name>' [ , signature ]) to create undefined function classes. """ _new = type.__new__ def __init__(cls, *args, **kwargs): # honor kwarg value or class-defined value before using # the number of arguments in the eval function (if present) nargs = kwargs.pop('nargs', cls.__dict__.get('nargs', arity(cls))) if nargs is None and 'nargs' not in cls.__dict__: for supcls in cls.__mro__: if hasattr(supcls, '_nargs'): nargs = supcls._nargs break else: continue # Canonicalize nargs here; change to set in nargs. if is_sequence(nargs): if not nargs: raise ValueError(filldedent(''' Incorrectly specified nargs as %s: if there are no arguments, it should be `nargs = 0`; if there are any number of arguments, it should be `nargs = None`''' % str(nargs))) nargs = tuple(ordered(set(nargs))) elif nargs is not None: nargs = (as_int(nargs),) cls._nargs = nargs super().__init__(*args, **kwargs) @property def __signature__(self): """ Allow Python 3's inspect.signature to give a useful signature for Function subclasses. """ # Python 3 only, but backports (like the one in IPython) still might # call this. try: from inspect import signature except ImportError: return None # TODO: Look at nargs return signature(self.eval) @property def free_symbols(self): return set() @property def xreplace(self): # Function needs args so we define a property that returns # a function that takes args...and then use that function # to return the right value return lambda rule, **_: rule.get(self, self) @property def nargs(self): """Return a set of the allowed number of arguments for the function. Examples ======== >>> from sympy.core.function import Function >>> f = Function('f') If the function can take any number of arguments, the set of whole numbers is returned: >>> Function('f').nargs Naturals0 If the function was initialized to accept one or more arguments, a corresponding set will be returned: >>> Function('f', nargs=1).nargs FiniteSet(1) >>> Function('f', nargs=(2, 1)).nargs FiniteSet(1, 2) The undefined function, after application, also has the nargs attribute; the actual number of arguments is always available by checking the ``args`` attribute: >>> f = Function('f') >>> f(1).nargs Naturals0 >>> len(f(1).args) 1 """ from sympy.sets.sets import FiniteSet # XXX it would be nice to handle this in __init__ but there are import # problems with trying to import FiniteSet there return FiniteSet(*self._nargs) if self._nargs else S.Naturals0 def __repr__(cls): return cls.__name__ class Application(Basic, metaclass=FunctionClass): """ Base class for applied functions. Explanation =========== Instances of Application represent the result of applying an application of any type to any object. """ is_Function = True @cacheit def __new__(cls, *args, **options): from sympy.sets.fancysets import Naturals0 from sympy.sets.sets import FiniteSet args = list(map(sympify, args)) evaluate = options.pop('evaluate', global_parameters.evaluate) # WildFunction (and anything else like it) may have nargs defined # and we throw that value away here options.pop('nargs', None) if options: raise ValueError("Unknown options: %s" % options) if evaluate: evaluated = cls.eval(*args) if evaluated is not None: return evaluated obj = super().__new__(cls, *args, **options) # make nargs uniform here sentinel = object() objnargs = getattr(obj, "nargs", sentinel) if objnargs is not sentinel: # things passing through here: # - functions subclassed from Function (e.g. myfunc(1).nargs) # - functions like cos(1).nargs # - AppliedUndef with given nargs like Function('f', nargs=1)(1).nargs # Canonicalize nargs here if is_sequence(objnargs): nargs = tuple(ordered(set(objnargs))) elif objnargs is not None: nargs = (as_int(objnargs),) else: nargs = None else: # things passing through here: # - WildFunction('f').nargs # - AppliedUndef with no nargs like Function('f')(1).nargs nargs = obj._nargs # note the underscore here # convert to FiniteSet obj.nargs = FiniteSet(*nargs) if nargs else Naturals0() return obj @classmethod def eval(cls, *args): """ Returns a canonical form of cls applied to arguments args. Explanation =========== The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None. Examples of eval() for the function "sign" --------------------------------------------- .. code-block:: python @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN if arg.is_zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul): coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One: return cls(coeff) * cls(terms) """ return @property def func(self): return self.__class__ def _eval_subs(self, old, new): if (old.is_Function and new.is_Function and callable(old) and callable(new) and old == self.func and len(self.args) in new.nargs): return new(*[i._subs(old, new) for i in self.args]) class Function(Application, Expr): """ Base class for applied mathematical functions. It also serves as a constructor for undefined function classes. Examples ======== First example shows how to use Function as a constructor for undefined function classes: >>> from sympy import Function, Symbol >>> x = Symbol('x') >>> f = Function('f') >>> g = Function('g')(x) >>> f f >>> f(x) f(x) >>> g g(x) >>> f(x).diff(x) Derivative(f(x), x) >>> g.diff(x) Derivative(g(x), x) Assumptions can be passed to Function, and if function is initialized with a Symbol, the function inherits the name and assumptions associated with the Symbol: >>> f_real = Function('f', real=True) >>> f_real(x).is_real True >>> f_real_inherit = Function(Symbol('f', real=True)) >>> f_real_inherit(x).is_real True Note that assumptions on a function are unrelated to the assumptions on the variable it is called on. If you want to add a relationship, subclass Function and define the appropriate ``_eval_is_assumption`` methods. In the following example Function is used as a base class for ``my_func`` that represents a mathematical function *my_func*. Suppose that it is well known, that *my_func(0)* is *1* and *my_func* at infinity goes to *0*, so we want those two simplifications to occur automatically. Suppose also that *my_func(x)* is real exactly when *x* is real. Here is an implementation that honours those requirements: >>> from sympy import Function, S, oo, I, sin >>> class my_func(Function): ... ... @classmethod ... def eval(cls, x): ... if x.is_Number: ... if x.is_zero: ... return S.One ... elif x is S.Infinity: ... return S.Zero ... ... def _eval_is_real(self): ... return self.args[0].is_real ... >>> x = S('x') >>> my_func(0) + sin(0) 1 >>> my_func(oo) 0 >>> my_func(3.54).n() # Not yet implemented for my_func. my_func(3.54) >>> my_func(I).is_real False In order for ``my_func`` to become useful, several other methods would need to be implemented. See source code of some of the already implemented functions for more complete examples. Also, if the function can take more than one argument, then ``nargs`` must be defined, e.g. if ``my_func`` can take one or two arguments then, >>> class my_func(Function): ... nargs = (1, 2) ... >>> """ @property def _diff_wrt(self): return False @cacheit def __new__(cls, *args, **options): # Handle calls like Function('f') if cls is Function: return UndefinedFunction(*args, **options) n = len(args) if n not in cls.nargs: # XXX: exception message must be in exactly this format to # make it work with NumPy's functions like vectorize(). See, # for example, https://github.com/numpy/numpy/issues/1697. # The ideal solution would be just to attach metadata to # the exception and change NumPy to take advantage of this. temp = ('%(name)s takes %(qual)s %(args)s ' 'argument%(plural)s (%(given)s given)') raise TypeError(temp % { 'name': cls, 'qual': 'exactly' if len(cls.nargs) == 1 else 'at least', 'args': min(cls.nargs), 'plural': 's'*(min(cls.nargs) != 1), 'given': n}) evaluate = options.get('evaluate', global_parameters.evaluate) result = super().__new__(cls, *args, **options) if evaluate and isinstance(result, cls) and result.args: pr2 = min(cls._should_evalf(a) for a in result.args) if pr2 > 0: pr = max(cls._should_evalf(a) for a in result.args) result = result.evalf(mlib.libmpf.prec_to_dps(pr)) return result @classmethod def _should_evalf(cls, arg): """ Decide if the function should automatically evalf(). Explanation =========== By default (in this implementation), this happens if (and only if) the ARG is a floating point number. This function is used by __new__. Returns the precision to evalf to, or -1 if it shouldn't evalf. """ from sympy.core.evalf import pure_complex if arg.is_Float: return arg._prec if not arg.is_Add: return -1 m = pure_complex(arg) if m is None or not (m[0].is_Float or m[1].is_Float): return -1 l = [i._prec for i in m if i.is_Float] l.append(-1) return max(l) @classmethod def class_key(cls): from sympy.sets.fancysets import Naturals0 funcs = { 'exp': 10, 'log': 11, 'sin': 20, 'cos': 21, 'tan': 22, 'cot': 23, 'sinh': 30, 'cosh': 31, 'tanh': 32, 'coth': 33, 'conjugate': 40, 're': 41, 'im': 42, 'arg': 43, } name = cls.__name__ try: i = funcs[name] except KeyError: i = 0 if isinstance(cls.nargs, Naturals0) else 10000 return 4, i, name def _eval_evalf(self, prec): def _get_mpmath_func(fname): """Lookup mpmath function based on name""" if isinstance(self, AppliedUndef): # Shouldn't lookup in mpmath but might have ._imp_ return None if not hasattr(mpmath, fname): from sympy.utilities.lambdify import MPMATH_TRANSLATIONS fname = MPMATH_TRANSLATIONS.get(fname, None) if fname is None: return None return getattr(mpmath, fname) func = _get_mpmath_func(self.func.__name__) # Fall-back evaluation if func is None: imp = getattr(self, '_imp_', None) if imp is None: return None try: return Float(imp(*[i.evalf(prec) for i in self.args]), prec) except (TypeError, ValueError): return None # Convert all args to mpf or mpc # Convert the arguments to *higher* precision than requested for the # final result. # XXX + 5 is a guess, it is similar to what is used in evalf.py. Should # we be more intelligent about it? try: args = [arg._to_mpmath(prec + 5) for arg in self.args] def bad(m): from mpmath import mpf, mpc # the precision of an mpf value is the last element # if that is 1 (and m[1] is not 1 which would indicate a # power of 2), then the eval failed; so check that none of # the arguments failed to compute to a finite precision. # Note: An mpc value has two parts, the re and imag tuple; # check each of those parts, too. Anything else is allowed to # pass if isinstance(m, mpf): m = m._mpf_ return m[1] !=1 and m[-1] == 1 elif isinstance(m, mpc): m, n = m._mpc_ return m[1] !=1 and m[-1] == 1 and \ n[1] !=1 and n[-1] == 1 else: return False if any(bad(a) for a in args): raise ValueError # one or more args failed to compute with significance except ValueError: return with mpmath.workprec(prec): v = func(*args) return Expr._from_mpmath(v, prec) def _eval_derivative(self, s): # f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s) i = 0 l = [] for a in self.args: i += 1 da = a.diff(s) if da.is_zero: continue try: df = self.fdiff(i) except ArgumentIndexError: df = Function.fdiff(self, i) l.append(df * da) return Add(*l) def _eval_is_commutative(self): return fuzzy_and(a.is_commutative for a in self.args) def _eval_is_meromorphic(self, x, a): if not self.args: return True if any(arg.has(x) for arg in self.args[1:]): return False arg = self.args[0] if not arg._eval_is_meromorphic(x, a): return None return fuzzy_not(type(self).is_singular(arg.subs(x, a))) _singularities = None # type: Union[FuzzyBool, tTuple[Expr, ...]] @classmethod def is_singular(cls, a): """ Tests whether the argument is an essential singularity or a branch point, or the functions is non-holomorphic. """ ss = cls._singularities if ss in (True, None, False): return ss return fuzzy_or(a.is_infinite if s is S.ComplexInfinity else (a - s).is_zero for s in ss) def as_base_exp(self): """ Returns the method as the 2-tuple (base, exponent). """ return self, S.One def _eval_aseries(self, n, args0, x, logx): """ Compute an asymptotic expansion around args0, in terms of self.args. This function is only used internally by _eval_nseries and should not be called directly; derived classes can overwrite this to implement asymptotic expansions. """ from sympy.utilities.misc import filldedent raise PoleError(filldedent(''' Asymptotic expansion of %s around %s is not implemented.''' % (type(self), args0))) def _eval_nseries(self, x, n, logx, cdir=0): """ This function does compute series for multivariate functions, but the expansion is always in terms of *one* variable. Examples ======== >>> from sympy import atan2 >>> from sympy.abc import x, y >>> atan2(x, y).series(x, n=2) atan2(0, y) + x/y + O(x**2) >>> atan2(x, y).series(y, n=2) -y/x + atan2(x, 0) + O(y**2) This function also computes asymptotic expansions, if necessary and possible: >>> from sympy import loggamma >>> loggamma(1/x)._eval_nseries(x,0,None) -1/x - log(x)/x + log(x)/2 + O(1) """ from sympy import Order from sympy.core.symbol import uniquely_named_symbol from sympy.sets.sets import FiniteSet args = self.args args0 = [t.limit(x, 0) for t in args] if any(t.is_finite is False for t in args0): from sympy import oo, zoo, nan # XXX could use t.as_leading_term(x) here but it's a little # slower a = [t.compute_leading_term(x, logx=logx) for t in args] a0 = [t.limit(x, 0) for t in a] if any([t.has(oo, -oo, zoo, nan) for t in a0]): return self._eval_aseries(n, args0, x, logx) # Careful: the argument goes to oo, but only logarithmically so. We # are supposed to do a power series expansion "around the # logarithmic term". e.g. # f(1+x+log(x)) # -> f(1+logx) + x*f'(1+logx) + O(x**2) # where 'logx' is given in the argument a = [t._eval_nseries(x, n, logx) for t in args] z = [r - r0 for (r, r0) in zip(a, a0)] p = [Dummy() for _ in z] q = [] v = None for ai, zi, pi in zip(a0, z, p): if zi.has(x): if v is not None: raise NotImplementedError q.append(ai + pi) v = pi else: q.append(ai) e1 = self.func(*q) if v is None: return e1 s = e1._eval_nseries(v, n, logx) o = s.getO() s = s.removeO() s = s.subs(v, zi).expand() + Order(o.expr.subs(v, zi), x) return s if (self.func.nargs is S.Naturals0 or (self.func.nargs == FiniteSet(1) and args0[0]) or any(c > 1 for c in self.func.nargs)): e = self e1 = e.expand() if e == e1: #for example when e = sin(x+1) or e = sin(cos(x)) #let's try the general algorithm if len(e.args) == 1: # issue 14411 e = e.func(e.args[0].cancel()) term = e.subs(x, S.Zero) if term.is_finite is False or term is S.NaN: raise PoleError("Cannot expand %s around 0" % (self)) series = term fact = S.One _x = uniquely_named_symbol('xi', self) e = e.subs(x, _x) for i in range(n - 1): i += 1 fact *= Rational(i) e = e.diff(_x) subs = e.subs(_x, S.Zero) if subs is S.NaN: # try to evaluate a limit if we have to subs = e.limit(_x, S.Zero) if subs.is_finite is False: raise PoleError("Cannot expand %s around 0" % (self)) term = subs*(x**i)/fact term = term.expand() series += term return series + Order(x**n, x) return e1.nseries(x, n=n, logx=logx) arg = self.args[0] l = [] g = None # try to predict a number of terms needed nterms = n + 2 cf = Order(arg.as_leading_term(x), x).getn() if cf != 0: nterms = (n/cf).ceiling() for i in range(nterms): g = self.taylor_term(i, arg, g) g = g.nseries(x, n=n, logx=logx) l.append(g) return Add(*l) + Order(x**n, x) def fdiff(self, argindex=1): """ Returns the first derivative of the function. """ if not (1 <= argindex <= len(self.args)): raise ArgumentIndexError(self, argindex) ix = argindex - 1 A = self.args[ix] if A._diff_wrt: if len(self.args) == 1 or not A.is_Symbol: return _derivative_dispatch(self, A) for i, v in enumerate(self.args): if i != ix and A in v.free_symbols: # it can't be in any other argument's free symbols # issue 8510 break else: return _derivative_dispatch(self, A) # See issue 4624 and issue 4719, 5600 and 8510 D = Dummy('xi_%i' % argindex, dummy_index=hash(A)) args = self.args[:ix] + (D,) + self.args[ix + 1:] return Subs(Derivative(self.func(*args), D), D, A) def _eval_as_leading_term(self, x, cdir=0): """Stub that should be overridden by new Functions to return the first non-zero term in a series if ever an x-dependent argument whose leading term vanishes as x -> 0 might be encountered. See, for example, cos._eval_as_leading_term. """ from sympy import Order args = [a.as_leading_term(x) for a in self.args] o = Order(1, x) if any(x in a.free_symbols and o.contains(a) for a in args): # Whereas x and any finite number are contained in O(1, x), # expressions like 1/x are not. If any arg simplified to a # vanishing expression as x -> 0 (like x or x**2, but not # 3, 1/x, etc...) then the _eval_as_leading_term is needed # to supply the first non-zero term of the series, # # e.g. expression leading term # ---------- ------------ # cos(1/x) cos(1/x) # cos(cos(x)) cos(1) # cos(x) 1 <- _eval_as_leading_term needed # sin(x) x <- _eval_as_leading_term needed # raise NotImplementedError( '%s has no _eval_as_leading_term routine' % self.func) else: return self.func(*args) def _sage_(self): import sage.all as sage fname = self.func.__name__ func = getattr(sage, fname, None) args = [arg._sage_() for arg in self.args] # In the case the function is not known in sage: if func is None: import sympy if getattr(sympy, fname, None) is None: # abstract function return sage.function(fname)(*args) else: # the function defined in sympy is not known in sage # this exception is caught in sage raise AttributeError return func(*args) class AppliedUndef(Function): """ Base class for expressions resulting from the application of an undefined function. """ is_number = False def __new__(cls, *args, **options): args = list(map(sympify, args)) u = [a.name for a in args if isinstance(a, UndefinedFunction)] if u: raise TypeError('Invalid argument: expecting an expression, not UndefinedFunction%s: %s' % ( 's'*(len(u) > 1), ', '.join(u))) obj = super().__new__(cls, *args, **options) return obj def _eval_as_leading_term(self, x, cdir=0): return self def _sage_(self): import sage.all as sage fname = str(self.func) args = [arg._sage_() for arg in self.args] func = sage.function(fname)(*args) return func @property def _diff_wrt(self): """ Allow derivatives wrt to undefined functions. Examples ======== >>> from sympy import Function, Symbol >>> f = Function('f') >>> x = Symbol('x') >>> f(x)._diff_wrt True >>> f(x).diff(x) Derivative(f(x), x) """ return True class UndefSageHelper: """ Helper to facilitate Sage conversion. """ def __get__(self, ins, typ): import sage.all as sage if ins is None: return lambda: sage.function(typ.__name__) else: args = [arg._sage_() for arg in ins.args] return lambda : sage.function(ins.__class__.__name__)(*args) _undef_sage_helper = UndefSageHelper() class UndefinedFunction(FunctionClass): """ The (meta)class of undefined functions. """ def __new__(mcl, name, bases=(AppliedUndef,), __dict__=None, **kwargs): from .symbol import _filter_assumptions # Allow Function('f', real=True) # and/or Function(Symbol('f', real=True)) assumptions, kwargs = _filter_assumptions(kwargs) if isinstance(name, Symbol): assumptions = name._merge(assumptions) name = name.name elif not isinstance(name, str): raise TypeError('expecting string or Symbol for name') else: commutative = assumptions.get('commutative', None) assumptions = Symbol(name, **assumptions).assumptions0 if commutative is None: assumptions.pop('commutative') __dict__ = __dict__ or {} # put the `is_*` for into __dict__ __dict__.update({'is_%s' % k: v for k, v in assumptions.items()}) # You can add other attributes, although they do have to be hashable # (but seriously, if you want to add anything other than assumptions, # just subclass Function) __dict__.update(kwargs) # add back the sanitized assumptions without the is_ prefix kwargs.update(assumptions) # Save these for __eq__ __dict__.update({'_kwargs': kwargs}) # do this for pickling __dict__['__module__'] = None obj = super().__new__(mcl, name, bases, __dict__) obj.name = name obj._sage_ = _undef_sage_helper return obj def __instancecheck__(cls, instance): return cls in type(instance).__mro__ _kwargs = {} # type: tDict[str, Optional[bool]] def __hash__(self): return hash((self.class_key(), frozenset(self._kwargs.items()))) def __eq__(self, other): return (isinstance(other, self.__class__) and self.class_key() == other.class_key() and self._kwargs == other._kwargs) def __ne__(self, other): return not self == other @property def _diff_wrt(self): return False # XXX: The type: ignore on WildFunction is because mypy complains: # # sympy/core/function.py:939: error: Cannot determine type of 'sort_key' in # base class 'Expr' # # Somehow this is because of the @cacheit decorator but it is not clear how to # fix it. class WildFunction(Function, AtomicExpr): # type: ignore """ A WildFunction function matches any function (with its arguments). Examples ======== >>> from sympy import WildFunction, Function, cos >>> from sympy.abc import x, y >>> F = WildFunction('F') >>> f = Function('f') >>> F.nargs Naturals0 >>> x.match(F) >>> F.match(F) {F_: F_} >>> f(x).match(F) {F_: f(x)} >>> cos(x).match(F) {F_: cos(x)} >>> f(x, y).match(F) {F_: f(x, y)} To match functions with a given number of arguments, set ``nargs`` to the desired value at instantiation: >>> F = WildFunction('F', nargs=2) >>> F.nargs FiniteSet(2) >>> f(x).match(F) >>> f(x, y).match(F) {F_: f(x, y)} To match functions with a range of arguments, set ``nargs`` to a tuple containing the desired number of arguments, e.g. if ``nargs = (1, 2)`` then functions with 1 or 2 arguments will be matched. >>> F = WildFunction('F', nargs=(1, 2)) >>> F.nargs FiniteSet(1, 2) >>> f(x).match(F) {F_: f(x)} >>> f(x, y).match(F) {F_: f(x, y)} >>> f(x, y, 1).match(F) """ # XXX: What is this class attribute used for? include = set() # type: tSet[Any] def __init__(cls, name, **assumptions): from sympy.sets.sets import Set, FiniteSet cls.name = name nargs = assumptions.pop('nargs', S.Naturals0) if not isinstance(nargs, Set): # Canonicalize nargs here. See also FunctionClass. if is_sequence(nargs): nargs = tuple(ordered(set(nargs))) elif nargs is not None: nargs = (as_int(nargs),) nargs = FiniteSet(*nargs) cls.nargs = nargs def matches(self, expr, repl_dict={}, old=False): if not isinstance(expr, (AppliedUndef, Function)): return None if len(expr.args) not in self.nargs: return None repl_dict = repl_dict.copy() repl_dict[self] = expr return repl_dict class Derivative(Expr): """ Carries out differentiation of the given expression with respect to symbols. Examples ======== >>> from sympy import Derivative, Function, symbols, Subs >>> from sympy.abc import x, y >>> f, g = symbols('f g', cls=Function) >>> Derivative(x**2, x, evaluate=True) 2*x Denesting of derivatives retains the ordering of variables: >>> Derivative(Derivative(f(x, y), y), x) Derivative(f(x, y), y, x) Contiguously identical symbols are merged into a tuple giving the symbol and the count: >>> Derivative(f(x), x, x, y, x) Derivative(f(x), (x, 2), y, x) If the derivative cannot be performed, and evaluate is True, the order of the variables of differentiation will be made canonical: >>> Derivative(f(x, y), y, x, evaluate=True) Derivative(f(x, y), x, y) Derivatives with respect to undefined functions can be calculated: >>> Derivative(f(x)**2, f(x), evaluate=True) 2*f(x) Such derivatives will show up when the chain rule is used to evalulate a derivative: >>> f(g(x)).diff(x) Derivative(f(g(x)), g(x))*Derivative(g(x), x) Substitution is used to represent derivatives of functions with arguments that are not symbols or functions: >>> f(2*x + 3).diff(x) == 2*Subs(f(y).diff(y), y, 2*x + 3) True Notes ===== Simplification of high-order derivatives: Because there can be a significant amount of simplification that can be done when multiple differentiations are performed, results will be automatically simplified in a fairly conservative fashion unless the keyword ``simplify`` is set to False. >>> from sympy import sqrt, diff, Function, symbols >>> from sympy.abc import x, y, z >>> f, g = symbols('f,g', cls=Function) >>> e = sqrt((x + 1)**2 + x) >>> diff(e, (x, 5), simplify=False).count_ops() 136 >>> diff(e, (x, 5)).count_ops() 30 Ordering of variables: If evaluate is set to True and the expression cannot be evaluated, the list of differentiation symbols will be sorted, that is, the expression is assumed to have continuous derivatives up to the order asked. Derivative wrt non-Symbols: For the most part, one may not differentiate wrt non-symbols. For example, we do not allow differentiation wrt `x*y` because there are multiple ways of structurally defining where x*y appears in an expression: a very strict definition would make (x*y*z).diff(x*y) == 0. Derivatives wrt defined functions (like cos(x)) are not allowed, either: >>> (x*y*z).diff(x*y) Traceback (most recent call last): ... ValueError: Can't calculate derivative wrt x*y. To make it easier to work with variational calculus, however, derivatives wrt AppliedUndef and Derivatives are allowed. For example, in the Euler-Lagrange method one may write F(t, u, v) where u = f(t) and v = f'(t). These variables can be written explicitly as functions of time:: >>> from sympy.abc import t >>> F = Function('F') >>> U = f(t) >>> V = U.diff(t) The derivative wrt f(t) can be obtained directly: >>> direct = F(t, U, V).diff(U) When differentiation wrt a non-Symbol is attempted, the non-Symbol is temporarily converted to a Symbol while the differentiation is performed and the same answer is obtained: >>> indirect = F(t, U, V).subs(U, x).diff(x).subs(x, U) >>> assert direct == indirect The implication of this non-symbol replacement is that all functions are treated as independent of other functions and the symbols are independent of the functions that contain them:: >>> x.diff(f(x)) 0 >>> g(x).diff(f(x)) 0 It also means that derivatives are assumed to depend only on the variables of differentiation, not on anything contained within the expression being differentiated:: >>> F = f(x) >>> Fx = F.diff(x) >>> Fx.diff(F) # derivative depends on x, not F 0 >>> Fxx = Fx.diff(x) >>> Fxx.diff(Fx) # derivative depends on x, not Fx 0 The last example can be made explicit by showing the replacement of Fx in Fxx with y: >>> Fxx.subs(Fx, y) Derivative(y, x) Since that in itself will evaluate to zero, differentiating wrt Fx will also be zero: >>> _.doit() 0 Replacing undefined functions with concrete expressions One must be careful to replace undefined functions with expressions that contain variables consistent with the function definition and the variables of differentiation or else insconsistent result will be obtained. Consider the following example: >>> eq = f(x)*g(y) >>> eq.subs(f(x), x*y).diff(x, y).doit() y*Derivative(g(y), y) + g(y) >>> eq.diff(x, y).subs(f(x), x*y).doit() y*Derivative(g(y), y) The results differ because `f(x)` was replaced with an expression that involved both variables of differentiation. In the abstract case, differentiation of `f(x)` by `y` is 0; in the concrete case, the presence of `y` made that derivative nonvanishing and produced the extra `g(y)` term. Defining differentiation for an object An object must define ._eval_derivative(symbol) method that returns the differentiation result. This function only needs to consider the non-trivial case where expr contains symbol and it should call the diff() method internally (not _eval_derivative); Derivative should be the only one to call _eval_derivative. Any class can allow derivatives to be taken with respect to itself (while indicating its scalar nature). See the docstring of Expr._diff_wrt. See Also ======== _sort_variable_count """ is_Derivative = True @property def _diff_wrt(self): """An expression may be differentiated wrt a Derivative if it is in elementary form. Examples ======== >>> from sympy import Function, Derivative, cos >>> from sympy.abc import x >>> f = Function('f') >>> Derivative(f(x), x)._diff_wrt True >>> Derivative(cos(x), x)._diff_wrt False >>> Derivative(x + 1, x)._diff_wrt False A Derivative might be an unevaluated form of what will not be a valid variable of differentiation if evaluated. For example, >>> Derivative(f(f(x)), x).doit() Derivative(f(x), x)*Derivative(f(f(x)), f(x)) Such an expression will present the same ambiguities as arise when dealing with any other product, like ``2*x``, so ``_diff_wrt`` is False: >>> Derivative(f(f(x)), x)._diff_wrt False """ return self.expr._diff_wrt and isinstance(self.doit(), Derivative) def __new__(cls, expr, *variables, **kwargs): from sympy.matrices.common import MatrixCommon from sympy import Integer, MatrixExpr from sympy.tensor.array import Array, NDimArray from sympy.utilities.misc import filldedent expr = sympify(expr) symbols_or_none = getattr(expr, "free_symbols", None) has_symbol_set = isinstance(symbols_or_none, set) if not has_symbol_set: raise ValueError(filldedent(''' Since there are no variables in the expression %s, it cannot be differentiated.''' % expr)) # determine value for variables if it wasn't given if not variables: variables = expr.free_symbols if len(variables) != 1: if expr.is_number: return S.Zero if len(variables) == 0: raise ValueError(filldedent(''' Since there are no variables in the expression, the variable(s) of differentiation must be supplied to differentiate %s''' % expr)) else: raise ValueError(filldedent(''' Since there is more than one variable in the expression, the variable(s) of differentiation must be supplied to differentiate %s''' % expr)) # Standardize the variables by sympifying them: variables = list(sympify(variables)) # Split the list of variables into a list of the variables we are diff # wrt, where each element of the list has the form (s, count) where # s is the entity to diff wrt and count is the order of the # derivative. variable_count = [] array_likes = (tuple, list, Tuple) for i, v in enumerate(variables): if isinstance(v, Integer): if i == 0: raise ValueError("First variable cannot be a number: %i" % v) count = v prev, prevcount = variable_count[-1] if prevcount != 1: raise TypeError("tuple {} followed by number {}".format((prev, prevcount), v)) if count == 0: variable_count.pop() else: variable_count[-1] = Tuple(prev, count) else: if isinstance(v, array_likes): if len(v) == 0: # Ignore empty tuples: Derivative(expr, ... , (), ... ) continue if isinstance(v[0], array_likes): # Derive by array: Derivative(expr, ... , [[x, y, z]], ... ) if len(v) == 1: v = Array(v[0]) count = 1 else: v, count = v v = Array(v) else: v, count = v if count == 0: continue elif isinstance(v, UndefinedFunction): raise TypeError( "cannot differentiate wrt " "UndefinedFunction: %s" % v) else: count = 1 variable_count.append(Tuple(v, count)) # light evaluation of contiguous, identical # items: (x, 1), (x, 1) -> (x, 2) merged = [] for t in variable_count: v, c = t if c.is_negative: raise ValueError( 'order of differentiation must be nonnegative') if merged and merged[-1][0] == v: c += merged[-1][1] if not c: merged.pop() else: merged[-1] = Tuple(v, c) else: merged.append(t) variable_count = merged # sanity check of variables of differentation; we waited # until the counts were computed since some variables may # have been removed because the count was 0 for v, c in variable_count: # v must have _diff_wrt True if not v._diff_wrt: __ = '' # filler to make error message neater raise ValueError(filldedent(''' Can't calculate derivative wrt %s.%s''' % (v, __))) # We make a special case for 0th derivative, because there is no # good way to unambiguously print this. if len(variable_count) == 0: return expr evaluate = kwargs.get('evaluate', False) if evaluate: if isinstance(expr, Derivative): expr = expr.canonical variable_count = [ (v.canonical if isinstance(v, Derivative) else v, c) for v, c in variable_count] # Look for a quick exit if there are symbols that don't appear in # expression at all. Note, this cannot check non-symbols like # Derivatives as those can be created by intermediate # derivatives. zero = False free = expr.free_symbols for v, c in variable_count: vfree = v.free_symbols if c.is_positive and vfree: if isinstance(v, AppliedUndef): # these match exactly since # x.diff(f(x)) == g(x).diff(f(x)) == 0 # and are not created by differentiation D = Dummy() if not expr.xreplace({v: D}).has(D): zero = True break elif isinstance(v, MatrixExpr): zero = False break elif isinstance(v, Symbol) and v not in free: zero = True break else: if not free & vfree: # e.g. v is IndexedBase or Matrix zero = True break if zero: return cls._get_zero_with_shape_like(expr) # make the order of symbols canonical #TODO: check if assumption of discontinuous derivatives exist variable_count = cls._sort_variable_count(variable_count) # denest if isinstance(expr, Derivative): variable_count = list(expr.variable_count) + variable_count expr = expr.expr return _derivative_dispatch(expr, *variable_count, **kwargs) # we return here if evaluate is False or if there is no # _eval_derivative method if not evaluate or not hasattr(expr, '_eval_derivative'): # return an unevaluated Derivative if evaluate and variable_count == [(expr, 1)] and expr.is_scalar: # special hack providing evaluation for classes # that have defined is_scalar=True but have no # _eval_derivative defined return S.One return Expr.__new__(cls, expr, *variable_count) # evaluate the derivative by calling _eval_derivative method # of expr for each variable # ------------------------------------------------------------- nderivs = 0 # how many derivatives were performed unhandled = [] for i, (v, count) in enumerate(variable_count): old_expr = expr old_v = None is_symbol = v.is_symbol or isinstance(v, (Iterable, Tuple, MatrixCommon, NDimArray)) if not is_symbol: old_v = v v = Dummy('xi') expr = expr.xreplace({old_v: v}) # Derivatives and UndefinedFunctions are independent # of all others clashing = not (isinstance(old_v, Derivative) or \ isinstance(old_v, AppliedUndef)) if not v in expr.free_symbols and not clashing: return expr.diff(v) # expr's version of 0 if not old_v.is_scalar and not hasattr( old_v, '_eval_derivative'): # special hack providing evaluation for classes # that have defined is_scalar=True but have no # _eval_derivative defined expr *= old_v.diff(old_v) obj = cls._dispatch_eval_derivative_n_times(expr, v, count) if obj is not None and obj.is_zero: return obj nderivs += count if old_v is not None: if obj is not None: # remove the dummy that was used obj = obj.subs(v, old_v) # restore expr expr = old_expr if obj is None: # we've already checked for quick-exit conditions # that give 0 so the remaining variables # are contained in the expression but the expression # did not compute a derivative so we stop taking # derivatives unhandled = variable_count[i:] break expr = obj # what we have so far can be made canonical expr = expr.replace( lambda x: isinstance(x, Derivative), lambda x: x.canonical) if unhandled: if isinstance(expr, Derivative): unhandled = list(expr.variable_count) + unhandled expr = expr.expr expr = Expr.__new__(cls, expr, *unhandled) if (nderivs > 1) == True and kwargs.get('simplify', True): from sympy.core.exprtools import factor_terms from sympy.simplify.simplify import signsimp expr = factor_terms(signsimp(expr)) return expr @property def canonical(cls): return cls.func(cls.expr, *Derivative._sort_variable_count(cls.variable_count)) @classmethod def _sort_variable_count(cls, vc): """ Sort (variable, count) pairs into canonical order while retaining order of variables that do not commute during differentiation: * symbols and functions commute with each other * derivatives commute with each other * a derivative doesn't commute with anything it contains * any other object is not allowed to commute if it has free symbols in common with another object Examples ======== >>> from sympy import Derivative, Function, symbols >>> vsort = Derivative._sort_variable_count >>> x, y, z = symbols('x y z') >>> f, g, h = symbols('f g h', cls=Function) Contiguous items are collapsed into one pair: >>> vsort([(x, 1), (x, 1)]) [(x, 2)] >>> vsort([(y, 1), (f(x), 1), (y, 1), (f(x), 1)]) [(y, 2), (f(x), 2)] Ordering is canonical. >>> def vsort0(*v): ... # docstring helper to ... # change vi -> (vi, 0), sort, and return vi vals ... return [i[0] for i in vsort([(i, 0) for i in v])] >>> vsort0(y, x) [x, y] >>> vsort0(g(y), g(x), f(y)) [f(y), g(x), g(y)] Symbols are sorted as far to the left as possible but never move to the left of a derivative having the same symbol in its variables; the same applies to AppliedUndef which are always sorted after Symbols: >>> dfx = f(x).diff(x) >>> assert vsort0(dfx, y) == [y, dfx] >>> assert vsort0(dfx, x) == [dfx, x] """ from sympy.utilities.iterables import uniq, topological_sort if not vc: return [] vc = list(vc) if len(vc) == 1: return [Tuple(*vc[0])] V = list(range(len(vc))) E = [] v = lambda i: vc[i][0] D = Dummy() def _block(d, v, wrt=False): # return True if v should not come before d else False if d == v: return wrt if d.is_Symbol: return False if isinstance(d, Derivative): # a derivative blocks if any of it's variables contain # v; the wrt flag will return True for an exact match # and will cause an AppliedUndef to block if v is in # the arguments if any(_block(k, v, wrt=True) for k in d._wrt_variables): return True return False if not wrt and isinstance(d, AppliedUndef): return False if v.is_Symbol: return v in d.free_symbols if isinstance(v, AppliedUndef): return _block(d.xreplace({v: D}), D) return d.free_symbols & v.free_symbols for i in range(len(vc)): for j in range(i): if _block(v(j), v(i)): E.append((j,i)) # this is the default ordering to use in case of ties O = dict(zip(ordered(uniq([i for i, c in vc])), range(len(vc)))) ix = topological_sort((V, E), key=lambda i: O[v(i)]) # merge counts of contiguously identical items merged = [] for v, c in [vc[i] for i in ix]: if merged and merged[-1][0] == v: merged[-1][1] += c else: merged.append([v, c]) return [Tuple(*i) for i in merged] def _eval_is_commutative(self): return self.expr.is_commutative def _eval_derivative(self, v): # If v (the variable of differentiation) is not in # self.variables, we might be able to take the derivative. if v not in self._wrt_variables: dedv = self.expr.diff(v) if isinstance(dedv, Derivative): return dedv.func(dedv.expr, *(self.variable_count + dedv.variable_count)) # dedv (d(self.expr)/dv) could have simplified things such that the # derivative wrt things in self.variables can now be done. Thus, # we set evaluate=True to see if there are any other derivatives # that can be done. The most common case is when dedv is a simple # number so that the derivative wrt anything else will vanish. return self.func(dedv, *self.variables, evaluate=True) # In this case v was in self.variables so the derivative wrt v has # already been attempted and was not computed, either because it # couldn't be or evaluate=False originally. variable_count = list(self.variable_count) variable_count.append((v, 1)) return self.func(self.expr, *variable_count, evaluate=False) def doit(self, **hints): expr = self.expr if hints.get('deep', True): expr = expr.doit(**hints) hints['evaluate'] = True rv = self.func(expr, *self.variable_count, **hints) if rv!= self and rv.has(Derivative): rv = rv.doit(**hints) return rv @_sympifyit('z0', NotImplementedError) def doit_numerically(self, z0): """ Evaluate the derivative at z numerically. When we can represent derivatives at a point, this should be folded into the normal evalf. For now, we need a special method. """ if len(self.free_symbols) != 1 or len(self.variables) != 1: raise NotImplementedError('partials and higher order derivatives') z = list(self.free_symbols)[0] def eval(x): f0 = self.expr.subs(z, Expr._from_mpmath(x, prec=mpmath.mp.prec)) f0 = f0.evalf(mlib.libmpf.prec_to_dps(mpmath.mp.prec)) return f0._to_mpmath(mpmath.mp.prec) return Expr._from_mpmath(mpmath.diff(eval, z0._to_mpmath(mpmath.mp.prec)), mpmath.mp.prec) @property def expr(self): return self._args[0] @property def _wrt_variables(self): # return the variables of differentiation without # respect to the type of count (int or symbolic) return [i[0] for i in self.variable_count] @property def variables(self): # TODO: deprecate? YES, make this 'enumerated_variables' and # name _wrt_variables as variables # TODO: support for `d^n`? rv = [] for v, count in self.variable_count: if not count.is_Integer: raise TypeError(filldedent(''' Cannot give expansion for symbolic count. If you just want a list of all variables of differentiation, use _wrt_variables.''')) rv.extend([v]*count) return tuple(rv) @property def variable_count(self): return self._args[1:] @property def derivative_count(self): return sum([count for var, count in self.variable_count], 0) @property def free_symbols(self): ret = self.expr.free_symbols # Add symbolic counts to free_symbols for var, count in self.variable_count: ret.update(count.free_symbols) return ret def _eval_subs(self, old, new): # The substitution (old, new) cannot be done inside # Derivative(expr, vars) for a variety of reasons # as handled below. if old in self._wrt_variables: # first handle the counts expr = self.func(self.expr, *[(v, c.subs(old, new)) for v, c in self.variable_count]) if expr != self: return expr._eval_subs(old, new) # quick exit case if not getattr(new, '_diff_wrt', False): # case (0): new is not a valid variable of # differentiation if isinstance(old, Symbol): # don't introduce a new symbol if the old will do return Subs(self, old, new) else: xi = Dummy('xi') return Subs(self.xreplace({old: xi}), xi, new) # If both are Derivatives with the same expr, check if old is # equivalent to self or if old is a subderivative of self. if old.is_Derivative and old.expr == self.expr: if self.canonical == old.canonical: return new # collections.Counter doesn't have __le__ def _subset(a, b): return all((a[i] <= b[i]) == True for i in a) old_vars = Counter(dict(reversed(old.variable_count))) self_vars = Counter(dict(reversed(self.variable_count))) if _subset(old_vars, self_vars): return _derivative_dispatch(new, *(self_vars - old_vars).items()).canonical args = list(self.args) newargs = list(x._subs(old, new) for x in args) if args[0] == old: # complete replacement of self.expr # we already checked that the new is valid so we know # it won't be a problem should it appear in variables return _derivative_dispatch(*newargs) if newargs[0] != args[0]: # case (1) can't change expr by introducing something that is in # the _wrt_variables if it was already in the expr # e.g. # for Derivative(f(x, g(y)), y), x cannot be replaced with # anything that has y in it; for f(g(x), g(y)).diff(g(y)) # g(x) cannot be replaced with anything that has g(y) syms = {vi: Dummy() for vi in self._wrt_variables if not vi.is_Symbol} wrt = {syms.get(vi, vi) for vi in self._wrt_variables} forbidden = args[0].xreplace(syms).free_symbols & wrt nfree = new.xreplace(syms).free_symbols ofree = old.xreplace(syms).free_symbols if (nfree - ofree) & forbidden: return Subs(self, old, new) viter = ((i, j) for ((i, _), (j, _)) in zip(newargs[1:], args[1:])) if any(i != j for i, j in viter): # a wrt-variable change # case (2) can't change vars by introducing a variable # that is contained in expr, e.g. # for Derivative(f(z, g(h(x), y)), y), y cannot be changed to # x, h(x), or g(h(x), y) for a in _atomic(self.expr, recursive=True): for i in range(1, len(newargs)): vi, _ = newargs[i] if a == vi and vi != args[i][0]: return Subs(self, old, new) # more arg-wise checks vc = newargs[1:] oldv = self._wrt_variables newe = self.expr subs = [] for i, (vi, ci) in enumerate(vc): if not vi._diff_wrt: # case (3) invalid differentiation expression so # create a replacement dummy xi = Dummy('xi_%i' % i) # replace the old valid variable with the dummy # in the expression newe = newe.xreplace({oldv[i]: xi}) # and replace the bad variable with the dummy vc[i] = (xi, ci) # and record the dummy with the new (invalid) # differentiation expression subs.append((xi, vi)) if subs: # handle any residual substitution in the expression newe = newe._subs(old, new) # return the Subs-wrapped derivative return Subs(Derivative(newe, *vc), *zip(*subs)) # everything was ok return _derivative_dispatch(*newargs) def _eval_lseries(self, x, logx, cdir=0): dx = self.variables for term in self.expr.lseries(x, logx=logx, cdir=cdir): yield self.func(term, *dx) def _eval_nseries(self, x, n, logx, cdir=0): arg = self.expr.nseries(x, n=n, logx=logx) o = arg.getO() dx = self.variables rv = [self.func(a, *dx) for a in Add.make_args(arg.removeO())] if o: rv.append(o/x) return Add(*rv) def _eval_as_leading_term(self, x, cdir=0): series_gen = self.expr.lseries(x) d = S.Zero for leading_term in series_gen: d = diff(leading_term, *self.variables) if d != 0: break return d def _sage_(self): import sage.all as sage args = [arg._sage_() for arg in self.args] return sage.derivative(*args) def as_finite_difference(self, points=1, x0=None, wrt=None): """ Expresses a Derivative instance as a finite difference. Parameters ========== points : sequence or coefficient, optional If sequence: discrete values (length >= order+1) of the independent variable used for generating the finite difference weights. If it is a coefficient, it will be used as the step-size for generating an equidistant sequence of length order+1 centered around ``x0``. Default: 1 (step-size 1) x0 : number or Symbol, optional the value of the independent variable (``wrt``) at which the derivative is to be approximated. Default: same as ``wrt``. wrt : Symbol, optional "with respect to" the variable for which the (partial) derivative is to be approximated for. If not provided it is required that the derivative is ordinary. Default: ``None``. Examples ======== >>> from sympy import symbols, Function, exp, sqrt, Symbol >>> x, h = symbols('x h') >>> f = Function('f') >>> f(x).diff(x).as_finite_difference() -f(x - 1/2) + f(x + 1/2) The default step size and number of points are 1 and ``order + 1`` respectively. We can change the step size by passing a symbol as a parameter: >>> f(x).diff(x).as_finite_difference(h) -f(-h/2 + x)/h + f(h/2 + x)/h We can also specify the discretized values to be used in a sequence: >>> f(x).diff(x).as_finite_difference([x, x+h, x+2*h]) -3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h) The algorithm is not restricted to use equidistant spacing, nor do we need to make the approximation around ``x0``, but we can get an expression estimating the derivative at an offset: >>> e, sq2 = exp(1), sqrt(2) >>> xl = [x-h, x+h, x+e*h] >>> f(x).diff(x, 1).as_finite_difference(xl, x+h*sq2) # doctest: +ELLIPSIS 2*h*((h + sqrt(2)*h)/(2*h) - (-sqrt(2)*h + h)/(2*h))*f(E*h + x)/... To approximate ``Derivative`` around ``x0`` using a non-equidistant spacing step, the algorithm supports assignment of undefined functions to ``points``: >>> dx = Function('dx') >>> f(x).diff(x).as_finite_difference(points=dx(x), x0=x-h) -f(-h + x - dx(-h + x)/2)/dx(-h + x) + f(-h + x + dx(-h + x)/2)/dx(-h + x) Partial derivatives are also supported: >>> y = Symbol('y') >>> d2fdxdy=f(x,y).diff(x,y) >>> d2fdxdy.as_finite_difference(wrt=x) -Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y) We can apply ``as_finite_difference`` to ``Derivative`` instances in compound expressions using ``replace``: >>> (1 + 42**f(x).diff(x)).replace(lambda arg: arg.is_Derivative, ... lambda arg: arg.as_finite_difference()) 42**(-f(x - 1/2) + f(x + 1/2)) + 1 See also ======== sympy.calculus.finite_diff.apply_finite_diff sympy.calculus.finite_diff.differentiate_finite sympy.calculus.finite_diff.finite_diff_weights """ from ..calculus.finite_diff import _as_finite_diff return _as_finite_diff(self, points, x0, wrt) @classmethod def _get_zero_with_shape_like(cls, expr): return S.Zero @classmethod def _dispatch_eval_derivative_n_times(cls, expr, v, count): # Evaluate the derivative `n` times. If # `_eval_derivative_n_times` is not overridden by the current # object, the default in `Basic` will call a loop over # `_eval_derivative`: return expr._eval_derivative_n_times(v, count) def _derivative_dispatch(expr, *variables, **kwargs): from sympy.matrices.common import MatrixCommon from sympy import MatrixExpr from sympy import NDimArray array_types = (MatrixCommon, MatrixExpr, NDimArray, list, tuple, Tuple) if isinstance(expr, array_types) or any(isinstance(i[0], array_types) if isinstance(i, (tuple, list, Tuple)) else isinstance(i, array_types) for i in variables): from sympy.tensor.array.array_derivatives import ArrayDerivative return ArrayDerivative(expr, *variables, **kwargs) return Derivative(expr, *variables, **kwargs) class Lambda(Expr): """ Lambda(x, expr) represents a lambda function similar to Python's 'lambda x: expr'. A function of several variables is written as Lambda((x, y, ...), expr). Examples ======== A simple example: >>> from sympy import Lambda >>> from sympy.abc import x >>> f = Lambda(x, x**2) >>> f(4) 16 For multivariate functions, use: >>> from sympy.abc import y, z, t >>> f2 = Lambda((x, y, z, t), x + y**z + t**z) >>> f2(1, 2, 3, 4) 73 It is also possible to unpack tuple arguments: >>> f = Lambda( ((x, y), z) , x + y + z) >>> f((1, 2), 3) 6 A handy shortcut for lots of arguments: >>> p = x, y, z >>> f = Lambda(p, x + y*z) >>> f(*p) x + y*z """ is_Function = True def __new__(cls, signature, expr): if iterable(signature) and not isinstance(signature, (tuple, Tuple)): SymPyDeprecationWarning( feature="non tuple iterable of argument symbols to Lambda", useinstead="tuple of argument symbols", issue=17474, deprecated_since_version="1.5").warn() signature = tuple(signature) sig = signature if iterable(signature) else (signature,) sig = sympify(sig) cls._check_signature(sig) if len(sig) == 1 and sig[0] == expr: return S.IdentityFunction return Expr.__new__(cls, sig, sympify(expr)) @classmethod def _check_signature(cls, sig): syms = set() def rcheck(args): for a in args: if a.is_symbol: if a in syms: raise BadSignatureError("Duplicate symbol %s" % a) syms.add(a) elif isinstance(a, Tuple): rcheck(a) else: raise BadSignatureError("Lambda signature should be only tuples" " and symbols, not %s" % a) if not isinstance(sig, Tuple): raise BadSignatureError("Lambda signature should be a tuple not %s" % sig) # Recurse through the signature: rcheck(sig) @property def signature(self): """The expected form of the arguments to be unpacked into variables""" return self._args[0] @property def expr(self): """The return value of the function""" return self._args[1] @property def variables(self): """The variables used in the internal representation of the function""" def _variables(args): if isinstance(args, Tuple): for arg in args: yield from _variables(arg) else: yield args return tuple(_variables(self.signature)) @property def nargs(self): from sympy.sets.sets import FiniteSet return FiniteSet(len(self.signature)) bound_symbols = variables @property def free_symbols(self): return self.expr.free_symbols - set(self.variables) def __call__(self, *args): n = len(args) if n not in self.nargs: # Lambda only ever has 1 value in nargs # XXX: exception message must be in exactly this format to # make it work with NumPy's functions like vectorize(). See, # for example, https://github.com/numpy/numpy/issues/1697. # The ideal solution would be just to attach metadata to # the exception and change NumPy to take advantage of this. ## XXX does this apply to Lambda? If not, remove this comment. temp = ('%(name)s takes exactly %(args)s ' 'argument%(plural)s (%(given)s given)') raise BadArgumentsError(temp % { 'name': self, 'args': list(self.nargs)[0], 'plural': 's'*(list(self.nargs)[0] != 1), 'given': n}) d = self._match_signature(self.signature, args) return self.expr.xreplace(d) def _match_signature(self, sig, args): symargmap = {} def rmatch(pars, args): for par, arg in zip(pars, args): if par.is_symbol: symargmap[par] = arg elif isinstance(par, Tuple): if not isinstance(arg, (tuple, Tuple)) or len(args) != len(pars): raise BadArgumentsError("Can't match %s and %s" % (args, pars)) rmatch(par, arg) rmatch(sig, args) return symargmap @property def is_identity(self): """Return ``True`` if this ``Lambda`` is an identity function. """ return self.signature == self.expr class Subs(Expr): """ Represents unevaluated substitutions of an expression. ``Subs(expr, x, x0)`` represents the expression resulting from substituting x with x0 in expr. Parameters ========== expr : Expr An expression. x : tuple, variable A variable or list of distinct variables. x0 : tuple or list of tuples A point or list of evaluation points corresponding to those variables. Notes ===== ``Subs`` objects are generally useful to represent unevaluated derivatives calculated at a point. The variables may be expressions, but they are subjected to the limitations of subs(), so it is usually a good practice to use only symbols for variables, since in that case there can be no ambiguity. There's no automatic expansion - use the method .doit() to effect all possible substitutions of the object and also of objects inside the expression. When evaluating derivatives at a point that is not a symbol, a Subs object is returned. One is also able to calculate derivatives of Subs objects - in this case the expression is always expanded (for the unevaluated form, use Derivative()). Examples ======== >>> from sympy import Subs, Function, sin, cos >>> from sympy.abc import x, y, z >>> f = Function('f') Subs are created when a particular substitution cannot be made. The x in the derivative cannot be replaced with 0 because 0 is not a valid variables of differentiation: >>> f(x).diff(x).subs(x, 0) Subs(Derivative(f(x), x), x, 0) Once f is known, the derivative and evaluation at 0 can be done: >>> _.subs(f, sin).doit() == sin(x).diff(x).subs(x, 0) == cos(0) True Subs can also be created directly with one or more variables: >>> Subs(f(x)*sin(y) + z, (x, y), (0, 1)) Subs(z + f(x)*sin(y), (x, y), (0, 1)) >>> _.doit() z + f(0)*sin(1) Notes ===== In order to allow expressions to combine before doit is done, a representation of the Subs expression is used internally to make expressions that are superficially different compare the same: >>> a, b = Subs(x, x, 0), Subs(y, y, 0) >>> a + b 2*Subs(x, x, 0) This can lead to unexpected consequences when using methods like `has` that are cached: >>> s = Subs(x, x, 0) >>> s.has(x), s.has(y) (True, False) >>> ss = s.subs(x, y) >>> ss.has(x), ss.has(y) (True, False) >>> s, ss (Subs(x, x, 0), Subs(y, y, 0)) """ def __new__(cls, expr, variables, point, **assumptions): from sympy import Symbol if not is_sequence(variables, Tuple): variables = [variables] variables = Tuple(*variables) if has_dups(variables): repeated = [str(v) for v, i in Counter(variables).items() if i > 1] __ = ', '.join(repeated) raise ValueError(filldedent(''' The following expressions appear more than once: %s ''' % __)) point = Tuple(*(point if is_sequence(point, Tuple) else [point])) if len(point) != len(variables): raise ValueError('Number of point values must be the same as ' 'the number of variables.') if not point: return sympify(expr) # denest if isinstance(expr, Subs): variables = expr.variables + variables point = expr.point + point expr = expr.expr else: expr = sympify(expr) # use symbols with names equal to the point value (with prepended _) # to give a variable-independent expression pre = "_" pts = sorted(set(point), key=default_sort_key) from sympy.printing import StrPrinter class CustomStrPrinter(StrPrinter): def _print_Dummy(self, expr): return str(expr) + str(expr.dummy_index) def mystr(expr, **settings): p = CustomStrPrinter(settings) return p.doprint(expr) while 1: s_pts = {p: Symbol(pre + mystr(p)) for p in pts} reps = [(v, s_pts[p]) for v, p in zip(variables, point)] # if any underscore-prepended symbol is already a free symbol # and is a variable with a different point value, then there # is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0)) # because the new symbol that would be created is _1 but _1 # is already mapped to 0 so __0 and __1 are used for the new # symbols if any(r in expr.free_symbols and r in variables and Symbol(pre + mystr(point[variables.index(r)])) != r for _, r in reps): pre += "_" continue break obj = Expr.__new__(cls, expr, Tuple(*variables), point) obj._expr = expr.xreplace(dict(reps)) return obj def _eval_is_commutative(self): return self.expr.is_commutative def doit(self, **hints): e, v, p = self.args # remove self mappings for i, (vi, pi) in enumerate(zip(v, p)): if vi == pi: v = v[:i] + v[i + 1:] p = p[:i] + p[i + 1:] if not v: return self.expr if isinstance(e, Derivative): # apply functions first, e.g. f -> cos undone = [] for i, vi in enumerate(v): if isinstance(vi, FunctionClass): e = e.subs(vi, p[i]) else: undone.append((vi, p[i])) if not isinstance(e, Derivative): e = e.doit() if isinstance(e, Derivative): # do Subs that aren't related to differentiation undone2 = [] D = Dummy() for vi, pi in undone: if D not in e.xreplace({vi: D}).free_symbols: e = e.subs(vi, pi) else: undone2.append((vi, pi)) undone = undone2 # differentiate wrt variables that are present wrt = [] D = Dummy() expr = e.expr free = expr.free_symbols for vi, ci in e.variable_count: if isinstance(vi, Symbol) and vi in free: expr = expr.diff((vi, ci)) elif D in expr.subs(vi, D).free_symbols: expr = expr.diff((vi, ci)) else: wrt.append((vi, ci)) # inject remaining subs rv = expr.subs(undone) # do remaining differentiation *in order given* for vc in wrt: rv = rv.diff(vc) else: # inject remaining subs rv = e.subs(undone) else: rv = e.doit(**hints).subs(list(zip(v, p))) if hints.get('deep', True) and rv != self: rv = rv.doit(**hints) return rv def evalf(self, prec=None, **options): return self.doit().evalf(prec, **options) n = evalf @property def variables(self): """The variables to be evaluated""" return self._args[1] bound_symbols = variables @property def expr(self): """The expression on which the substitution operates""" return self._args[0] @property def point(self): """The values for which the variables are to be substituted""" return self._args[2] @property def free_symbols(self): return (self.expr.free_symbols - set(self.variables) | set(self.point.free_symbols)) @property def expr_free_symbols(self): return (self.expr.expr_free_symbols - set(self.variables) | set(self.point.expr_free_symbols)) def __eq__(self, other): if not isinstance(other, Subs): return False return self._hashable_content() == other._hashable_content() def __ne__(self, other): return not(self == other) def __hash__(self): return super().__hash__() def _hashable_content(self): return (self._expr.xreplace(self.canonical_variables), ) + tuple(ordered([(v, p) for v, p in zip(self.variables, self.point) if not self.expr.has(v)])) def _eval_subs(self, old, new): # Subs doit will do the variables in order; the semantics # of subs for Subs is have the following invariant for # Subs object foo: # foo.doit().subs(reps) == foo.subs(reps).doit() pt = list(self.point) if old in self.variables: if _atomic(new) == {new} and not any( i.has(new) for i in self.args): # the substitution is neutral return self.xreplace({old: new}) # any occurrence of old before this point will get # handled by replacements from here on i = self.variables.index(old) for j in range(i, len(self.variables)): pt[j] = pt[j]._subs(old, new) return self.func(self.expr, self.variables, pt) v = [i._subs(old, new) for i in self.variables] if v != list(self.variables): return self.func(self.expr, self.variables + (old,), pt + [new]) expr = self.expr._subs(old, new) pt = [i._subs(old, new) for i in self.point] return self.func(expr, v, pt) def _eval_derivative(self, s): # Apply the chain rule of the derivative on the substitution variables: val = Add.fromiter(p.diff(s) * Subs(self.expr.diff(v), self.variables, self.point).doit() for v, p in zip(self.variables, self.point)) # Check if there are free symbols in `self.expr`: # First get the `expr_free_symbols`, which returns the free symbols # that are directly contained in an expression node (i.e. stop # searching if the node isn't an expression). At this point turn the # expressions into `free_symbols` and check if there are common free # symbols in `self.expr` and the deriving factor. fs1 = {j for i in self.expr_free_symbols for j in i.free_symbols} if len(fs1 & s.free_symbols) > 0: val += Subs(self.expr.diff(s), self.variables, self.point).doit() return val def _eval_nseries(self, x, n, logx, cdir=0): if x in self.point: # x is the variable being substituted into apos = self.point.index(x) other = self.variables[apos] else: other = x arg = self.expr.nseries(other, n=n, logx=logx) o = arg.getO() terms = Add.make_args(arg.removeO()) rv = Add(*[self.func(a, *self.args[1:]) for a in terms]) if o: rv += o.subs(other, x) return rv def _eval_as_leading_term(self, x, cdir=0): if x in self.point: ipos = self.point.index(x) xvar = self.variables[ipos] return self.expr.as_leading_term(xvar) if x in self.variables: # if `x` is a dummy variable, it means it won't exist after the # substitution has been performed: return self # The variable is independent of the substitution: return self.expr.as_leading_term(x) def diff(f, *symbols, **kwargs): """ Differentiate f with respect to symbols. Explanation =========== This is just a wrapper to unify .diff() and the Derivative class; its interface is similar to that of integrate(). You can use the same shortcuts for multiple variables as with Derivative. For example, diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative of f(x). You can pass evaluate=False to get an unevaluated Derivative class. Note that if there are 0 symbols (such as diff(f(x), x, 0), then the result will be the function (the zeroth derivative), even if evaluate=False. Examples ======== >>> from sympy import sin, cos, Function, diff >>> from sympy.abc import x, y >>> f = Function('f') >>> diff(sin(x), x) cos(x) >>> diff(f(x), x, x, x) Derivative(f(x), (x, 3)) >>> diff(f(x), x, 3) Derivative(f(x), (x, 3)) >>> diff(sin(x)*cos(y), x, 2, y, 2) sin(x)*cos(y) >>> type(diff(sin(x), x)) cos >>> type(diff(sin(x), x, evaluate=False)) <class 'sympy.core.function.Derivative'> >>> type(diff(sin(x), x, 0)) sin >>> type(diff(sin(x), x, 0, evaluate=False)) sin >>> diff(sin(x)) cos(x) >>> diff(sin(x*y)) Traceback (most recent call last): ... ValueError: specify differentiation variables to differentiate sin(x*y) Note that ``diff(sin(x))`` syntax is meant only for convenience in interactive sessions and should be avoided in library code. References ========== http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html See Also ======== Derivative idiff: computes the derivative implicitly """ if hasattr(f, 'diff'): return f.diff(*symbols, **kwargs) kwargs.setdefault('evaluate', True) return _derivative_dispatch(f, *symbols, **kwargs) def expand(e, deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints): r""" Expand an expression using methods given as hints. Explanation =========== Hints evaluated unless explicitly set to False are: ``basic``, ``log``, ``multinomial``, ``mul``, ``power_base``, and ``power_exp`` The following hints are supported but not applied unless set to True: ``complex``, ``func``, and ``trig``. In addition, the following meta-hints are supported by some or all of the other hints: ``frac``, ``numer``, ``denom``, ``modulus``, and ``force``. ``deep`` is supported by all hints. Additionally, subclasses of Expr may define their own hints or meta-hints. The ``basic`` hint is used for any special rewriting of an object that should be done automatically (along with the other hints like ``mul``) when expand is called. This is a catch-all hint to handle any sort of expansion that may not be described by the existing hint names. To use this hint an object should override the ``_eval_expand_basic`` method. Objects may also define their own expand methods, which are not run by default. See the API section below. If ``deep`` is set to ``True`` (the default), things like arguments of functions are recursively expanded. Use ``deep=False`` to only expand on the top level. If the ``force`` hint is used, assumptions about variables will be ignored in making the expansion. Hints ===== These hints are run by default mul --- Distributes multiplication over addition: >>> from sympy import cos, exp, sin >>> from sympy.abc import x, y, z >>> (y*(x + z)).expand(mul=True) x*y + y*z multinomial ----------- Expand (x + y + ...)**n where n is a positive integer. >>> ((x + y + z)**2).expand(multinomial=True) x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2 power_exp --------- Expand addition in exponents into multiplied bases. >>> exp(x + y).expand(power_exp=True) exp(x)*exp(y) >>> (2**(x + y)).expand(power_exp=True) 2**x*2**y power_base ---------- Split powers of multiplied bases. This only happens by default if assumptions allow, or if the ``force`` meta-hint is used: >>> ((x*y)**z).expand(power_base=True) (x*y)**z >>> ((x*y)**z).expand(power_base=True, force=True) x**z*y**z >>> ((2*y)**z).expand(power_base=True) 2**z*y**z Note that in some cases where this expansion always holds, SymPy performs it automatically: >>> (x*y)**2 x**2*y**2 log --- Pull out power of an argument as a coefficient and split logs products into sums of logs. Note that these only work if the arguments of the log function have the proper assumptions--the arguments must be positive and the exponents must be real--or else the ``force`` hint must be True: >>> from sympy import log, symbols >>> log(x**2*y).expand(log=True) log(x**2*y) >>> log(x**2*y).expand(log=True, force=True) 2*log(x) + log(y) >>> x, y = symbols('x,y', positive=True) >>> log(x**2*y).expand(log=True) 2*log(x) + log(y) basic ----- This hint is intended primarily as a way for custom subclasses to enable expansion by default. These hints are not run by default: complex ------- Split an expression into real and imaginary parts. >>> x, y = symbols('x,y') >>> (x + y).expand(complex=True) re(x) + re(y) + I*im(x) + I*im(y) >>> cos(x).expand(complex=True) -I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x)) Note that this is just a wrapper around ``as_real_imag()``. Most objects that wish to redefine ``_eval_expand_complex()`` should consider redefining ``as_real_imag()`` instead. func ---- Expand other functions. >>> from sympy import gamma >>> gamma(x + 1).expand(func=True) x*gamma(x) trig ---- Do trigonometric expansions. >>> cos(x + y).expand(trig=True) -sin(x)*sin(y) + cos(x)*cos(y) >>> sin(2*x).expand(trig=True) 2*sin(x)*cos(x) Note that the forms of ``sin(n*x)`` and ``cos(n*x)`` in terms of ``sin(x)`` and ``cos(x)`` are not unique, due to the identity `\sin^2(x) + \cos^2(x) = 1`. The current implementation uses the form obtained from Chebyshev polynomials, but this may change. See `this MathWorld article <http://mathworld.wolfram.com/Multiple-AngleFormulas.html>`_ for more information. Notes ===== - You can shut off unwanted methods:: >>> (exp(x + y)*(x + y)).expand() x*exp(x)*exp(y) + y*exp(x)*exp(y) >>> (exp(x + y)*(x + y)).expand(power_exp=False) x*exp(x + y) + y*exp(x + y) >>> (exp(x + y)*(x + y)).expand(mul=False) (x + y)*exp(x)*exp(y) - Use deep=False to only expand on the top level:: >>> exp(x + exp(x + y)).expand() exp(x)*exp(exp(x)*exp(y)) >>> exp(x + exp(x + y)).expand(deep=False) exp(x)*exp(exp(x + y)) - Hints are applied in an arbitrary, but consistent order (in the current implementation, they are applied in alphabetical order, except multinomial comes before mul, but this may change). Because of this, some hints may prevent expansion by other hints if they are applied first. For example, ``mul`` may distribute multiplications and prevent ``log`` and ``power_base`` from expanding them. Also, if ``mul`` is applied before ``multinomial`, the expression might not be fully distributed. The solution is to use the various ``expand_hint`` helper functions or to use ``hint=False`` to this function to finely control which hints are applied. Here are some examples:: >>> from sympy import expand, expand_mul, expand_power_base >>> x, y, z = symbols('x,y,z', positive=True) >>> expand(log(x*(y + z))) log(x) + log(y + z) Here, we see that ``log`` was applied before ``mul``. To get the mul expanded form, either of the following will work:: >>> expand_mul(log(x*(y + z))) log(x*y + x*z) >>> expand(log(x*(y + z)), log=False) log(x*y + x*z) A similar thing can happen with the ``power_base`` hint:: >>> expand((x*(y + z))**x) (x*y + x*z)**x To get the ``power_base`` expanded form, either of the following will work:: >>> expand((x*(y + z))**x, mul=False) x**x*(y + z)**x >>> expand_power_base((x*(y + z))**x) x**x*(y + z)**x >>> expand((x + y)*y/x) y + y**2/x The parts of a rational expression can be targeted:: >>> expand((x + y)*y/x/(x + 1), frac=True) (x*y + y**2)/(x**2 + x) >>> expand((x + y)*y/x/(x + 1), numer=True) (x*y + y**2)/(x*(x + 1)) >>> expand((x + y)*y/x/(x + 1), denom=True) y*(x + y)/(x**2 + x) - The ``modulus`` meta-hint can be used to reduce the coefficients of an expression post-expansion:: >>> expand((3*x + 1)**2) 9*x**2 + 6*x + 1 >>> expand((3*x + 1)**2, modulus=5) 4*x**2 + x + 1 - Either ``expand()`` the function or ``.expand()`` the method can be used. Both are equivalent:: >>> expand((x + 1)**2) x**2 + 2*x + 1 >>> ((x + 1)**2).expand() x**2 + 2*x + 1 API === Objects can define their own expand hints by defining ``_eval_expand_hint()``. The function should take the form:: def _eval_expand_hint(self, **hints): # Only apply the method to the top-level expression ... See also the example below. Objects should define ``_eval_expand_hint()`` methods only if ``hint`` applies to that specific object. The generic ``_eval_expand_hint()`` method defined in Expr will handle the no-op case. Each hint should be responsible for expanding that hint only. Furthermore, the expansion should be applied to the top-level expression only. ``expand()`` takes care of the recursion that happens when ``deep=True``. You should only call ``_eval_expand_hint()`` methods directly if you are 100% sure that the object has the method, as otherwise you are liable to get unexpected ``AttributeError``s. Note, again, that you do not need to recursively apply the hint to args of your object: this is handled automatically by ``expand()``. ``_eval_expand_hint()`` should generally not be used at all outside of an ``_eval_expand_hint()`` method. If you want to apply a specific expansion from within another method, use the public ``expand()`` function, method, or ``expand_hint()`` functions. In order for expand to work, objects must be rebuildable by their args, i.e., ``obj.func(*obj.args) == obj`` must hold. Expand methods are passed ``**hints`` so that expand hints may use 'metahints'--hints that control how different expand methods are applied. For example, the ``force=True`` hint described above that causes ``expand(log=True)`` to ignore assumptions is such a metahint. The ``deep`` meta-hint is handled exclusively by ``expand()`` and is not passed to ``_eval_expand_hint()`` methods. Note that expansion hints should generally be methods that perform some kind of 'expansion'. For hints that simply rewrite an expression, use the .rewrite() API. Examples ======== >>> from sympy import Expr, sympify >>> class MyClass(Expr): ... def __new__(cls, *args): ... args = sympify(args) ... return Expr.__new__(cls, *args) ... ... def _eval_expand_double(self, *, force=False, **hints): ... ''' ... Doubles the args of MyClass. ... ... If there more than four args, doubling is not performed, ... unless force=True is also used (False by default). ... ''' ... if not force and len(self.args) > 4: ... return self ... return self.func(*(self.args + self.args)) ... >>> a = MyClass(1, 2, MyClass(3, 4)) >>> a MyClass(1, 2, MyClass(3, 4)) >>> a.expand(double=True) MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4)) >>> a.expand(double=True, deep=False) MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4)) >>> b = MyClass(1, 2, 3, 4, 5) >>> b.expand(double=True) MyClass(1, 2, 3, 4, 5) >>> b.expand(double=True, force=True) MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5) See Also ======== expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig, expand_power_base, expand_power_exp, expand_func, sympy.simplify.hyperexpand.hyperexpand """ # don't modify this; modify the Expr.expand method hints['power_base'] = power_base hints['power_exp'] = power_exp hints['mul'] = mul hints['log'] = log hints['multinomial'] = multinomial hints['basic'] = basic return sympify(e).expand(deep=deep, modulus=modulus, **hints) # This is a special application of two hints def _mexpand(expr, recursive=False): # expand multinomials and then expand products; this may not always # be sufficient to give a fully expanded expression (see # test_issue_8247_8354 in test_arit) if expr is None: return was = None while was != expr: was, expr = expr, expand_mul(expand_multinomial(expr)) if not recursive: break return expr # These are simple wrappers around single hints. def expand_mul(expr, deep=True): """ Wrapper around expand that only uses the mul hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_mul, exp, log >>> x, y = symbols('x,y', positive=True) >>> expand_mul(exp(x+y)*(x+y)*log(x*y**2)) x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2) """ return sympify(expr).expand(deep=deep, mul=True, power_exp=False, power_base=False, basic=False, multinomial=False, log=False) def expand_multinomial(expr, deep=True): """ Wrapper around expand that only uses the multinomial hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_multinomial, exp >>> x, y = symbols('x y', positive=True) >>> expand_multinomial((x + exp(x + 1))**2) x**2 + 2*x*exp(x + 1) + exp(2*x + 2) """ return sympify(expr).expand(deep=deep, mul=False, power_exp=False, power_base=False, basic=False, multinomial=True, log=False) def expand_log(expr, deep=True, force=False, factor=False): """ Wrapper around expand that only uses the log hint. See the expand docstring for more information. Examples ======== >>> from sympy import symbols, expand_log, exp, log >>> x, y = symbols('x,y', positive=True) >>> expand_log(exp(x+y)*(x+y)*log(x*y**2)) (x + y)*(log(x) + 2*log(y))*exp(x + y) """ from sympy import Mul, log if factor is False: def _handle(x): x1 = expand_mul(expand_log(x, deep=deep, force=force, factor=True)) if x1.count(log) <= x.count(log): return x1 return x expr = expr.replace( lambda x: x.is_Mul and all(any(isinstance(i, log) and i.args[0].is_Rational for i in Mul.make_args(j)) for j in x.as_numer_denom()), lambda x: _handle(x)) return sympify(expr).expand(deep=deep, log=True, mul=False, power_exp=False, power_base=False, multinomial=False, basic=False, force=force, factor=factor) def expand_func(expr, deep=True): """ Wrapper around expand that only uses the func hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_func, gamma >>> from sympy.abc import x >>> expand_func(gamma(x + 2)) x*(x + 1)*gamma(x) """ return sympify(expr).expand(deep=deep, func=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False) def expand_trig(expr, deep=True): """ Wrapper around expand that only uses the trig hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_trig, sin >>> from sympy.abc import x, y >>> expand_trig(sin(x+y)*(x+y)) (x + y)*(sin(x)*cos(y) + sin(y)*cos(x)) """ return sympify(expr).expand(deep=deep, trig=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False) def expand_complex(expr, deep=True): """ Wrapper around expand that only uses the complex hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_complex, exp, sqrt, I >>> from sympy.abc import z >>> expand_complex(exp(z)) I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z)) >>> expand_complex(sqrt(I)) sqrt(2)/2 + sqrt(2)*I/2 See Also ======== sympy.core.expr.Expr.as_real_imag """ return sympify(expr).expand(deep=deep, complex=True, basic=False, log=False, mul=False, power_exp=False, power_base=False, multinomial=False) def expand_power_base(expr, deep=True, force=False): """ Wrapper around expand that only uses the power_base hint. A wrapper to expand(power_base=True) which separates a power with a base that is a Mul into a product of powers, without performing any other expansions, provided that assumptions about the power's base and exponent allow. deep=False (default is True) will only apply to the top-level expression. force=True (default is False) will cause the expansion to ignore assumptions about the base and exponent. When False, the expansion will only happen if the base is non-negative or the exponent is an integer. >>> from sympy.abc import x, y, z >>> from sympy import expand_power_base, sin, cos, exp >>> (x*y)**2 x**2*y**2 >>> (2*x)**y (2*x)**y >>> expand_power_base(_) 2**y*x**y >>> expand_power_base((x*y)**z) (x*y)**z >>> expand_power_base((x*y)**z, force=True) x**z*y**z >>> expand_power_base(sin((x*y)**z), deep=False) sin((x*y)**z) >>> expand_power_base(sin((x*y)**z), force=True) sin(x**z*y**z) >>> expand_power_base((2*sin(x))**y + (2*cos(x))**y) 2**y*sin(x)**y + 2**y*cos(x)**y >>> expand_power_base((2*exp(y))**x) 2**x*exp(y)**x >>> expand_power_base((2*cos(x))**y) 2**y*cos(x)**y Notice that sums are left untouched. If this is not the desired behavior, apply full ``expand()`` to the expression: >>> expand_power_base(((x+y)*z)**2) z**2*(x + y)**2 >>> (((x+y)*z)**2).expand() x**2*z**2 + 2*x*y*z**2 + y**2*z**2 >>> expand_power_base((2*y)**(1+z)) 2**(z + 1)*y**(z + 1) >>> ((2*y)**(1+z)).expand() 2*2**z*y*y**z See Also ======== expand """ return sympify(expr).expand(deep=deep, log=False, mul=False, power_exp=False, power_base=True, multinomial=False, basic=False, force=force) def expand_power_exp(expr, deep=True): """ Wrapper around expand that only uses the power_exp hint. See the expand docstring for more information. Examples ======== >>> from sympy import expand_power_exp >>> from sympy.abc import x, y >>> expand_power_exp(x**(y + 2)) x**2*x**y """ return sympify(expr).expand(deep=deep, complex=False, basic=False, log=False, mul=False, power_exp=True, power_base=False, multinomial=False) def count_ops(expr, visual=False): """ Return a representation (integer or expression) of the operations in expr. Parameters ========== expr : Expr If expr is an iterable, the sum of the op counts of the items will be returned. visual : bool, optional If ``False`` (default) then the sum of the coefficients of the visual expression will be returned. If ``True`` then the number of each type of operation is shown with the core class types (or their virtual equivalent) multiplied by the number of times they occur. Examples ======== >>> from sympy.abc import a, b, x, y >>> from sympy import sin, count_ops Although there isn't a SUB object, minus signs are interpreted as either negations or subtractions: >>> (x - y).count_ops(visual=True) SUB >>> (-x).count_ops(visual=True) NEG Here, there are two Adds and a Pow: >>> (1 + a + b**2).count_ops(visual=True) 2*ADD + POW In the following, an Add, Mul, Pow and two functions: >>> (sin(x)*x + sin(x)**2).count_ops(visual=True) ADD + MUL + POW + 2*SIN for a total of 5: >>> (sin(x)*x + sin(x)**2).count_ops(visual=False) 5 Note that "what you type" is not always what you get. The expression 1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather than two DIVs: >>> (1/x/y).count_ops(visual=True) DIV + MUL The visual option can be used to demonstrate the difference in operations for expressions in different forms. Here, the Horner representation is compared with the expanded form of a polynomial: >>> eq=x*(1 + x*(2 + x*(3 + x))) >>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True) -MUL + 3*POW The count_ops function also handles iterables: >>> count_ops([x, sin(x), None, True, x + 2], visual=False) 2 >>> count_ops([x, sin(x), None, True, x + 2], visual=True) ADD + SIN >>> count_ops({x: sin(x), x + 2: y + 1}, visual=True) 2*ADD + SIN """ from sympy import Integral, Sum, Symbol from sympy.core.relational import Relational from sympy.simplify.radsimp import fraction from sympy.logic.boolalg import BooleanFunction from sympy.utilities.misc import func_name expr = sympify(expr) if isinstance(expr, Expr) and not expr.is_Relational: ops = [] args = [expr] NEG = Symbol('NEG') DIV = Symbol('DIV') SUB = Symbol('SUB') ADD = Symbol('ADD') while args: a = args.pop() if a.is_Rational: #-1/3 = NEG + DIV if a is not S.One: if a.p < 0: ops.append(NEG) if a.q != 1: ops.append(DIV) continue elif a.is_Mul or a.is_MatMul: if _coeff_isneg(a): ops.append(NEG) if a.args[0] is S.NegativeOne: a = a.as_two_terms()[1] else: a = -a n, d = fraction(a) if n.is_Integer: ops.append(DIV) if n < 0: ops.append(NEG) args.append(d) continue # won't be -Mul but could be Add elif d is not S.One: if not d.is_Integer: args.append(d) ops.append(DIV) args.append(n) continue # could be -Mul elif a.is_Add or a.is_MatAdd: aargs = list(a.args) negs = 0 for i, ai in enumerate(aargs): if _coeff_isneg(ai): negs += 1 args.append(-ai) if i > 0: ops.append(SUB) else: args.append(ai) if i > 0: ops.append(ADD) if negs == len(aargs): # -x - y = NEG + SUB ops.append(NEG) elif _coeff_isneg(aargs[0]): # -x + y = SUB, but already recorded ADD ops.append(SUB - ADD) continue if a.is_Pow and a.exp is S.NegativeOne: ops.append(DIV) args.append(a.base) # won't be -Mul but could be Add continue if a.is_Mul or isinstance(a, LatticeOp): o = Symbol(a.func.__name__.upper()) # count the args ops.append(o*(len(a.args) - 1)) elif a.args and ( a.is_Pow or a.is_Function or isinstance(a, Derivative) or isinstance(a, Integral) or isinstance(a, Sum)): # if it's not in the list above we don't # consider a.func something to count, e.g. # Tuple, MatrixSymbol, etc... o = Symbol(a.func.__name__.upper()) ops.append(o) if not a.is_Symbol: args.extend(a.args) elif isinstance(expr, Dict): ops = [count_ops(k, visual=visual) + count_ops(v, visual=visual) for k, v in expr.items()] elif iterable(expr): ops = [count_ops(i, visual=visual) for i in expr] elif isinstance(expr, (Relational, BooleanFunction)): ops = [] for arg in expr.args: ops.append(count_ops(arg, visual=True)) o = Symbol(func_name(expr, short=True).upper()) ops.append(o) elif not isinstance(expr, Basic): ops = [] else: # it's Basic not isinstance(expr, Expr): if not isinstance(expr, Basic): raise TypeError("Invalid type of expr") else: ops = [] args = [expr] while args: a = args.pop() if a.args: o = Symbol(a.func.__name__.upper()) if a.is_Boolean: ops.append(o*(len(a.args)-1)) else: ops.append(o) args.extend(a.args) if not ops: if visual: return S.Zero return 0 ops = Add(*ops) if visual: return ops if ops.is_Number: return int(ops) return sum(int((a.args or [1])[0]) for a in Add.make_args(ops)) def nfloat(expr, n=15, exponent=False, dkeys=False): """Make all Rationals in expr Floats except those in exponents (unless the exponents flag is set to True). When processing dictionaries, don't modify the keys unless ``dkeys=True``. Examples ======== >>> from sympy.core.function import nfloat >>> from sympy.abc import x, y >>> from sympy import cos, pi, sqrt >>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y)) x**4 + 0.5*x + sqrt(y) + 1.5 >>> nfloat(x**4 + sqrt(y), exponent=True) x**4.0 + y**0.5 Container types are not modified: >>> type(nfloat((1, 2))) is tuple True """ from sympy.core.power import Pow from sympy.polys.rootoftools import RootOf from sympy import MatrixBase kw = dict(n=n, exponent=exponent, dkeys=dkeys) if isinstance(expr, MatrixBase): return expr.applyfunc(lambda e: nfloat(e, **kw)) # handling of iterable containers if iterable(expr, exclude=str): if isinstance(expr, (dict, Dict)): if dkeys: args = [tuple(map(lambda i: nfloat(i, **kw), a)) for a in expr.items()] else: args = [(k, nfloat(v, **kw)) for k, v in expr.items()] if isinstance(expr, dict): return type(expr)(args) else: return expr.func(*args) elif isinstance(expr, Basic): return expr.func(*[nfloat(a, **kw) for a in expr.args]) return type(expr)([nfloat(a, **kw) for a in expr]) rv = sympify(expr) if rv.is_Number: return Float(rv, n) elif rv.is_number: # evalf doesn't always set the precision rv = rv.n(n) if rv.is_Number: rv = Float(rv.n(n), n) else: pass # pure_complex(rv) is likely True return rv elif rv.is_Atom: return rv elif rv.is_Relational: args_nfloat = (nfloat(arg, **kw) for arg in rv.args) return rv.func(*args_nfloat) # watch out for RootOf instances that don't like to have # their exponents replaced with Dummies and also sometimes have # problems with evaluating at low precision (issue 6393) rv = rv.xreplace({ro: ro.n(n) for ro in rv.atoms(RootOf)}) if not exponent: reps = [(p, Pow(p.base, Dummy())) for p in rv.atoms(Pow)] rv = rv.xreplace(dict(reps)) rv = rv.n(n) if not exponent: rv = rv.xreplace({d.exp: p.exp for p, d in reps}) else: # Pow._eval_evalf special cases Integer exponents so if # exponent is suppose to be handled we have to do so here rv = rv.xreplace(Transform( lambda x: Pow(x.base, Float(x.exp, n)), lambda x: x.is_Pow and x.exp.is_Integer)) return rv.xreplace(Transform( lambda x: x.func(*nfloat(x.args, n, exponent)), lambda x: isinstance(x, Function))) from sympy.core.symbol import Dummy, Symbol
009b99ced998299ff25084cb641eab8598c3cba44e042b5a8dc62a3dae74b237
""" Replacement rules. """ class Transform: """ Immutable mapping that can be used as a generic transformation rule. Parameters ========== transform : callable Computes the value corresponding to any key. filter : callable, optional If supplied, specifies which objects are in the mapping. Examples ======== >>> from sympy.core.rules import Transform >>> from sympy.abc import x This Transform will return, as a value, one more than the key: >>> add1 = Transform(lambda x: x + 1) >>> add1[1] 2 >>> add1[x] x + 1 By default, all values are considered to be in the dictionary. If a filter is supplied, only the objects for which it returns True are considered as being in the dictionary: >>> add1_odd = Transform(lambda x: x + 1, lambda x: x%2 == 1) >>> 2 in add1_odd False >>> add1_odd.get(2, 0) 0 >>> 3 in add1_odd True >>> add1_odd[3] 4 >>> add1_odd.get(3, 0) 4 """ def __init__(self, transform, filter=lambda x: True): self._transform = transform self._filter = filter def __contains__(self, item): return self._filter(item) def __getitem__(self, key): if self._filter(key): return self._transform(key) else: raise KeyError(key) def get(self, item, default=None): if item in self: return self[item] else: return default
adbdb15511d692dc09bcfdb5739f0ea524d7a19af465ae5f2d0282f8cd38888a
from collections import defaultdict from functools import cmp_to_key from .basic import Basic from .compatibility import reduce, is_sequence from .parameters import global_parameters from .logic import _fuzzy_group, fuzzy_or, fuzzy_not from .singleton import S from .operations import AssocOp, AssocOpDispatcher from .cache import cacheit from .numbers import ilcm, igcd from .expr import Expr # Key for sorting commutative args in canonical order _args_sortkey = cmp_to_key(Basic.compare) def _addsort(args): # in-place sorting of args args.sort(key=_args_sortkey) def _unevaluated_Add(*args): """Return a well-formed unevaluated Add: Numbers are collected and put in slot 0 and args are sorted. Use this when args have changed but you still want to return an unevaluated Add. Examples ======== >>> from sympy.core.add import _unevaluated_Add as uAdd >>> from sympy import S, Add >>> from sympy.abc import x, y >>> a = uAdd(*[S(1.0), x, S(2)]) >>> a.args[0] 3.00000000000000 >>> a.args[1] x Beyond the Number being in slot 0, there is no other assurance of order for the arguments since they are hash sorted. So, for testing purposes, output produced by this in some other function can only be tested against the output of this function or as one of several options: >>> opts = (Add(x, y, evaluate=False), Add(y, x, evaluate=False)) >>> a = uAdd(x, y) >>> assert a in opts and a == uAdd(x, y) >>> uAdd(x + 1, x + 2) x + x + 3 """ args = list(args) newargs = [] co = S.Zero while args: a = args.pop() if a.is_Add: # this will keep nesting from building up # so that x + (x + 1) -> x + x + 1 (3 args) args.extend(a.args) elif a.is_Number: co += a else: newargs.append(a) _addsort(newargs) if co: newargs.insert(0, co) return Add._from_args(newargs) class Add(Expr, AssocOp): __slots__ = () is_Add = True _args_type = Expr @classmethod def flatten(cls, seq): """ Takes the sequence "seq" of nested Adds and returns a flatten list. Returns: (commutative_part, noncommutative_part, order_symbols) Applies associativity, all terms are commutable with respect to addition. NB: the removal of 0 is already handled by AssocOp.__new__ See also ======== sympy.core.mul.Mul.flatten """ from sympy.calculus.util import AccumBounds from sympy.matrices.expressions import MatrixExpr from sympy.tensor.tensor import TensExpr rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a if a.is_Rational: if b.is_Mul: rv = [a, b], [], None if rv: if all(s.is_commutative for s in rv[0]): return rv return [], rv[0], None terms = {} # term -> coeff # e.g. x**2 -> 5 for ... + 5*x**2 + ... coeff = S.Zero # coefficient (Number or zoo) to always be in slot 0 # e.g. 3 + ... order_factors = [] extra = [] for o in seq: # O(x) if o.is_Order: if o.expr.is_zero: continue for o1 in order_factors: if o1.contains(o): o = None break if o is None: continue order_factors = [o] + [ o1 for o1 in order_factors if not o.contains(o1)] continue # 3 or NaN elif o.is_Number: if (o is S.NaN or coeff is S.ComplexInfinity and o.is_finite is False) and not extra: # we know for sure the result will be nan return [S.NaN], [], None if coeff.is_Number or isinstance(coeff, AccumBounds): coeff += o if coeff is S.NaN and not extra: # we know for sure the result will be nan return [S.NaN], [], None continue elif isinstance(o, AccumBounds): coeff = o.__add__(coeff) continue elif isinstance(o, MatrixExpr): # can't add 0 to Matrix so make sure coeff is not 0 extra.append(o) continue elif isinstance(o, TensExpr): coeff = o.__add__(coeff) if coeff else o continue elif o is S.ComplexInfinity: if coeff.is_finite is False and not extra: # we know for sure the result will be nan return [S.NaN], [], None coeff = S.ComplexInfinity continue # Add([...]) elif o.is_Add: # NB: here we assume Add is always commutative seq.extend(o.args) # TODO zerocopy? continue # Mul([...]) elif o.is_Mul: c, s = o.as_coeff_Mul() # check for unevaluated Pow, e.g. 2**3 or 2**(-1/2) elif o.is_Pow: b, e = o.as_base_exp() if b.is_Number and (e.is_Integer or (e.is_Rational and e.is_negative)): seq.append(b**e) continue c, s = S.One, o else: # everything else c = S.One s = o # now we have: # o = c*s, where # # c is a Number # s is an expression with number factor extracted # let's collect terms with the same s, so e.g. # 2*x**2 + 3*x**2 -> 5*x**2 if s in terms: terms[s] += c if terms[s] is S.NaN and not extra: # we know for sure the result will be nan return [S.NaN], [], None else: terms[s] = c # now let's construct new args: # [2*x**2, x**3, 7*x**4, pi, ...] newseq = [] noncommutative = False for s, c in terms.items(): # 0*s if c.is_zero: continue # 1*s elif c is S.One: newseq.append(s) # c*s else: if s.is_Mul: # Mul, already keeps its arguments in perfect order. # so we can simply put c in slot0 and go the fast way. cs = s._new_rawargs(*((c,) + s.args)) newseq.append(cs) elif s.is_Add: # we just re-create the unevaluated Mul newseq.append(Mul(c, s, evaluate=False)) else: # alternatively we have to call all Mul's machinery (slow) newseq.append(Mul(c, s)) noncommutative = noncommutative or not s.is_commutative # oo, -oo if coeff is S.Infinity: newseq = [f for f in newseq if not (f.is_extended_nonnegative or f.is_real)] elif coeff is S.NegativeInfinity: newseq = [f for f in newseq if not (f.is_extended_nonpositive or f.is_real)] if coeff is S.ComplexInfinity: # zoo might be # infinite_real + finite_im # finite_real + infinite_im # infinite_real + infinite_im # addition of a finite real or imaginary number won't be able to # change the zoo nature; adding an infinite qualtity would result # in a NaN condition if it had sign opposite of the infinite # portion of zoo, e.g., infinite_real - infinite_real. newseq = [c for c in newseq if not (c.is_finite and c.is_extended_real is not None)] # process O(x) if order_factors: newseq2 = [] for t in newseq: for o in order_factors: # x + O(x) -> O(x) if o.contains(t): t = None break # x + O(x**2) -> x + O(x**2) if t is not None: newseq2.append(t) newseq = newseq2 + order_factors # 1 + O(1) -> O(1) for o in order_factors: if o.contains(coeff): coeff = S.Zero break # order args canonically _addsort(newseq) # current code expects coeff to be first if coeff is not S.Zero: newseq.insert(0, coeff) if extra: newseq += extra noncommutative = True # we are done if noncommutative: return [], newseq, None else: return newseq, [], None @classmethod def class_key(cls): """Nice order of classes""" return 3, 1, cls.__name__ def as_coefficients_dict(a): """Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term. Examples ======== >>> from sympy.abc import a, x >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} """ d = defaultdict(list) for ai in a.args: c, m = ai.as_coeff_Mul() d[m].append(c) for k, v in d.items(): if len(v) == 1: d[k] = v[0] else: d[k] = Add(*v) di = defaultdict(int) di.update(d) return di @cacheit def as_coeff_add(self, *deps): """ Returns a tuple (coeff, args) where self is treated as an Add and coeff is the Number term and args is a tuple of all other terms. Examples ======== >>> from sympy.abc import x >>> (7 + 3*x).as_coeff_add() (7, (3*x,)) >>> (7*x).as_coeff_add() (0, (7*x,)) """ if deps: from sympy.utilities.iterables import sift l1, l2 = sift(self.args, lambda x: x.has(*deps), binary=True) return self._new_rawargs(*l2), tuple(l1) coeff, notrat = self.args[0].as_coeff_add() if coeff is not S.Zero: return coeff, notrat + self.args[1:] return S.Zero, self.args def as_coeff_Add(self, rational=False, deps=None): """ Efficiently extract the coefficient of a summation. """ coeff, args = self.args[0], self.args[1:] if coeff.is_Number and not rational or coeff.is_Rational: return coeff, self._new_rawargs(*args) return S.Zero, self # Note, we intentionally do not implement Add.as_coeff_mul(). Rather, we # let Expr.as_coeff_mul() just always return (S.One, self) for an Add. See # issue 5524. def _eval_power(self, e): if e.is_Rational and self.is_number: from sympy.core.evalf import pure_complex from sympy.core.mul import _unevaluated_Mul from sympy.core.exprtools import factor_terms from sympy.core.function import expand_multinomial from sympy.functions.elementary.complexes import sign from sympy.functions.elementary.miscellaneous import sqrt ri = pure_complex(self) if ri: r, i = ri if e.q == 2: D = sqrt(r**2 + i**2) if D.is_Rational: # (r, i, D) is a Pythagorean triple root = sqrt(factor_terms((D - r)/2))**e.p return root*expand_multinomial(( # principle value (D + r)/abs(i) + sign(i)*S.ImaginaryUnit)**e.p) elif e == -1: return _unevaluated_Mul( r - i*S.ImaginaryUnit, 1/(r**2 + i**2)) elif e.is_Number and abs(e) != 1: # handle the Float case: (2.0 + 4*x)**e -> 4**e*(0.5 + x)**e c, m = zip(*[i.as_coeff_Mul() for i in self.args]) if any(i.is_Float for i in c): # XXX should this always be done? big = -1 for i in c: if abs(i) >= big: big = abs(i) if big > 0 and big != 1: from sympy.functions.elementary.complexes import sign bigs = (big, -big) c = [sign(i) if i in bigs else i/big for i in c] addpow = Add(*[c*m for c, m in zip(c, m)])**e return big**e*addpow @cacheit def _eval_derivative(self, s): return self.func(*[a.diff(s) for a in self.args]) def _eval_nseries(self, x, n, logx, cdir=0): terms = [t.nseries(x, n=n, logx=logx, cdir=cdir) for t in self.args] return self.func(*terms) def _matches_simple(self, expr, repl_dict): # handle (w+3).matches('x+5') -> {w: x+2} coeff, terms = self.as_coeff_add() if len(terms) == 1: return terms[0].matches(expr - coeff, repl_dict) return def matches(self, expr, repl_dict={}, old=False): return self._matches_commutative(expr, repl_dict, old) @staticmethod def _combine_inverse(lhs, rhs): """ Returns lhs - rhs, but treats oo like a symbol so oo - oo returns 0, instead of a nan. """ from sympy.simplify.simplify import signsimp from sympy.core.symbol import Dummy inf = (S.Infinity, S.NegativeInfinity) if lhs.has(*inf) or rhs.has(*inf): oo = Dummy('oo') reps = { S.Infinity: oo, S.NegativeInfinity: -oo} ireps = {v: k for k, v in reps.items()} eq = signsimp(lhs.xreplace(reps) - rhs.xreplace(reps)) if eq.has(oo): eq = eq.replace( lambda x: x.is_Pow and x.base is oo, lambda x: x.base) return eq.xreplace(ireps) else: return signsimp(lhs - rhs) @cacheit def as_two_terms(self): """Return head and tail of self. This is the most efficient way to get the head and tail of an expression. - if you want only the head, use self.args[0]; - if you want to process the arguments of the tail then use self.as_coef_add() which gives the head and a tuple containing the arguments of the tail when treated as an Add. - if you want the coefficient when self is treated as a Mul then use self.as_coeff_mul()[0] >>> from sympy.abc import x, y >>> (3*x - 2*y + 5).as_two_terms() (5, 3*x - 2*y) """ return self.args[0], self._new_rawargs(*self.args[1:]) def as_numer_denom(self): """ Decomposes an expression to its numerator part and its denominator part. Examples ======== >>> from sympy.abc import x, y, z >>> (x*y/z).as_numer_denom() (x*y, z) >>> (x*(y + 1)/y**7).as_numer_denom() (x*(y + 1), y**7) See Also ======== sympy.core.expr.Expr.as_numer_denom """ # clear rational denominator content, expr = self.primitive() ncon, dcon = content.as_numer_denom() # collect numerators and denominators of the terms nd = defaultdict(list) for f in expr.args: ni, di = f.as_numer_denom() nd[di].append(ni) # check for quick exit if len(nd) == 1: d, n = nd.popitem() return self.func( *[_keep_coeff(ncon, ni) for ni in n]), _keep_coeff(dcon, d) # sum up the terms having a common denominator for d, n in nd.items(): if len(n) == 1: nd[d] = n[0] else: nd[d] = self.func(*n) # assemble single numerator and denominator denoms, numers = [list(i) for i in zip(*iter(nd.items()))] n, d = self.func(*[Mul(*(denoms[:i] + [numers[i]] + denoms[i + 1:])) for i in range(len(numers))]), Mul(*denoms) return _keep_coeff(ncon, n), _keep_coeff(dcon, d) def _eval_is_polynomial(self, syms): return all(term._eval_is_polynomial(syms) for term in self.args) def _eval_is_rational_function(self, syms): return all(term._eval_is_rational_function(syms) for term in self.args) def _eval_is_meromorphic(self, x, a): return _fuzzy_group((arg.is_meromorphic(x, a) for arg in self.args), quick_exit=True) def _eval_is_algebraic_expr(self, syms): return all(term._eval_is_algebraic_expr(syms) for term in self.args) # assumption methods _eval_is_real = lambda self: _fuzzy_group( (a.is_real for a in self.args), quick_exit=True) _eval_is_extended_real = lambda self: _fuzzy_group( (a.is_extended_real for a in self.args), quick_exit=True) _eval_is_complex = lambda self: _fuzzy_group( (a.is_complex for a in self.args), quick_exit=True) _eval_is_antihermitian = lambda self: _fuzzy_group( (a.is_antihermitian for a in self.args), quick_exit=True) _eval_is_finite = lambda self: _fuzzy_group( (a.is_finite for a in self.args), quick_exit=True) _eval_is_hermitian = lambda self: _fuzzy_group( (a.is_hermitian for a in self.args), quick_exit=True) _eval_is_integer = lambda self: _fuzzy_group( (a.is_integer for a in self.args), quick_exit=True) _eval_is_rational = lambda self: _fuzzy_group( (a.is_rational for a in self.args), quick_exit=True) _eval_is_algebraic = lambda self: _fuzzy_group( (a.is_algebraic for a in self.args), quick_exit=True) _eval_is_commutative = lambda self: _fuzzy_group( a.is_commutative for a in self.args) def _eval_is_infinite(self): sawinf = False for a in self.args: ainf = a.is_infinite if ainf is None: return None elif ainf is True: # infinite+infinite might not be infinite if sawinf is True: return None sawinf = True return sawinf def _eval_is_imaginary(self): nz = [] im_I = [] for a in self.args: if a.is_extended_real: if a.is_zero: pass elif a.is_zero is False: nz.append(a) else: return elif a.is_imaginary: im_I.append(a*S.ImaginaryUnit) elif (S.ImaginaryUnit*a).is_extended_real: im_I.append(a*S.ImaginaryUnit) else: return b = self.func(*nz) if b.is_zero: return fuzzy_not(self.func(*im_I).is_zero) elif b.is_zero is False: return False def _eval_is_zero(self): if self.is_commutative is False: # issue 10528: there is no way to know if a nc symbol # is zero or not return nz = [] z = 0 im_or_z = False im = False for a in self.args: if a.is_extended_real: if a.is_zero: z += 1 elif a.is_zero is False: nz.append(a) else: return elif a.is_imaginary: im = True elif (S.ImaginaryUnit*a).is_extended_real: im_or_z = True else: return if z == len(self.args): return True if len(nz) == 0 or len(nz) == len(self.args): return None b = self.func(*nz) if b.is_zero: if not im_or_z and not im: return True if im and not im_or_z: return False if b.is_zero is False: return False def _eval_is_odd(self): l = [f for f in self.args if not (f.is_even is True)] if not l: return False if l[0].is_odd: return self._new_rawargs(*l[1:]).is_even def _eval_is_irrational(self): for t in self.args: a = t.is_irrational if a: others = list(self.args) others.remove(t) if all(x.is_rational is True for x in others): return True return None if a is None: return return False def _eval_is_extended_positive(self): from sympy.core.exprtools import _monotonic_sign if self.is_number: return super()._eval_is_extended_positive() c, a = self.as_coeff_Add() if not c.is_zero: v = _monotonic_sign(a) if v is not None: s = v + c if s != self and s.is_extended_positive and a.is_extended_nonnegative: return True if len(self.free_symbols) == 1: v = _monotonic_sign(self) if v is not None and v != self and v.is_extended_positive: return True pos = nonneg = nonpos = unknown_sign = False saw_INF = set() args = [a for a in self.args if not a.is_zero] if not args: return False for a in args: ispos = a.is_extended_positive infinite = a.is_infinite if infinite: saw_INF.add(fuzzy_or((ispos, a.is_extended_nonnegative))) if True in saw_INF and False in saw_INF: return if ispos: pos = True continue elif a.is_extended_nonnegative: nonneg = True continue elif a.is_extended_nonpositive: nonpos = True continue if infinite is None: return unknown_sign = True if saw_INF: if len(saw_INF) > 1: return return saw_INF.pop() elif unknown_sign: return elif not nonpos and not nonneg and pos: return True elif not nonpos and pos: return True elif not pos and not nonneg: return False def _eval_is_extended_nonnegative(self): from sympy.core.exprtools import _monotonic_sign if not self.is_number: c, a = self.as_coeff_Add() if not c.is_zero and a.is_extended_nonnegative: v = _monotonic_sign(a) if v is not None: s = v + c if s != self and s.is_extended_nonnegative: return True if len(self.free_symbols) == 1: v = _monotonic_sign(self) if v is not None and v != self and v.is_extended_nonnegative: return True def _eval_is_extended_nonpositive(self): from sympy.core.exprtools import _monotonic_sign if not self.is_number: c, a = self.as_coeff_Add() if not c.is_zero and a.is_extended_nonpositive: v = _monotonic_sign(a) if v is not None: s = v + c if s != self and s.is_extended_nonpositive: return True if len(self.free_symbols) == 1: v = _monotonic_sign(self) if v is not None and v != self and v.is_extended_nonpositive: return True def _eval_is_extended_negative(self): from sympy.core.exprtools import _monotonic_sign if self.is_number: return super()._eval_is_extended_negative() c, a = self.as_coeff_Add() if not c.is_zero: v = _monotonic_sign(a) if v is not None: s = v + c if s != self and s.is_extended_negative and a.is_extended_nonpositive: return True if len(self.free_symbols) == 1: v = _monotonic_sign(self) if v is not None and v != self and v.is_extended_negative: return True neg = nonpos = nonneg = unknown_sign = False saw_INF = set() args = [a for a in self.args if not a.is_zero] if not args: return False for a in args: isneg = a.is_extended_negative infinite = a.is_infinite if infinite: saw_INF.add(fuzzy_or((isneg, a.is_extended_nonpositive))) if True in saw_INF and False in saw_INF: return if isneg: neg = True continue elif a.is_extended_nonpositive: nonpos = True continue elif a.is_extended_nonnegative: nonneg = True continue if infinite is None: return unknown_sign = True if saw_INF: if len(saw_INF) > 1: return return saw_INF.pop() elif unknown_sign: return elif not nonneg and not nonpos and neg: return True elif not nonneg and neg: return True elif not neg and not nonpos: return False def _eval_subs(self, old, new): if not old.is_Add: if old is S.Infinity and -old in self.args: # foo - oo is foo + (-oo) internally return self.xreplace({-old: -new}) return None coeff_self, terms_self = self.as_coeff_Add() coeff_old, terms_old = old.as_coeff_Add() if coeff_self.is_Rational and coeff_old.is_Rational: if terms_self == terms_old: # (2 + a).subs( 3 + a, y) -> -1 + y return self.func(new, coeff_self, -coeff_old) if terms_self == -terms_old: # (2 + a).subs(-3 - a, y) -> -1 - y return self.func(-new, coeff_self, coeff_old) if coeff_self.is_Rational and coeff_old.is_Rational \ or coeff_self == coeff_old: args_old, args_self = self.func.make_args( terms_old), self.func.make_args(terms_self) if len(args_old) < len(args_self): # (a+b+c).subs(b+c,x) -> a+x self_set = set(args_self) old_set = set(args_old) if old_set < self_set: ret_set = self_set - old_set return self.func(new, coeff_self, -coeff_old, *[s._subs(old, new) for s in ret_set]) args_old = self.func.make_args( -terms_old) # (a+b+c+d).subs(-b-c,x) -> a-x+d old_set = set(args_old) if old_set < self_set: ret_set = self_set - old_set return self.func(-new, coeff_self, coeff_old, *[s._subs(old, new) for s in ret_set]) def removeO(self): args = [a for a in self.args if not a.is_Order] return self._new_rawargs(*args) def getO(self): args = [a for a in self.args if a.is_Order] if args: return self._new_rawargs(*args) @cacheit def extract_leading_order(self, symbols, point=None): """ Returns the leading term and its order. Examples ======== >>> from sympy.abc import x >>> (x + 1 + 1/x**5).extract_leading_order(x) ((x**(-5), O(x**(-5))),) >>> (1 + x).extract_leading_order(x) ((1, O(1)),) >>> (x + x**2).extract_leading_order(x) ((x, O(x)),) """ from sympy import Order lst = [] symbols = list(symbols if is_sequence(symbols) else [symbols]) if not point: point = [0]*len(symbols) seq = [(f, Order(f, *zip(symbols, point))) for f in self.args] for ef, of in seq: for e, o in lst: if o.contains(of) and o != of: of = None break if of is None: continue new_lst = [(ef, of)] for e, o in lst: if of.contains(o) and o != of: continue new_lst.append((e, o)) lst = new_lst return tuple(lst) def as_real_imag(self, deep=True, **hints): """ returns a tuple representing a complex number Examples ======== >>> from sympy import I >>> (7 + 9*I).as_real_imag() (7, 9) >>> ((1 + I)/(1 - I)).as_real_imag() (0, 1) >>> ((1 + 2*I)*(1 + 3*I)).as_real_imag() (-5, 5) """ sargs = self.args re_part, im_part = [], [] for term in sargs: re, im = term.as_real_imag(deep=deep) re_part.append(re) im_part.append(im) return (self.func(*re_part), self.func(*im_part)) def _eval_as_leading_term(self, x, cdir=0): from sympy import expand_mul, Order old = self expr = expand_mul(self) if not expr.is_Add: return expr.as_leading_term(x, cdir=cdir) infinite = [t for t in expr.args if t.is_infinite] leading_terms = [t.as_leading_term(x, cdir=cdir) for t in expr.args] min, new_expr = Order(0), 0 try: for term in leading_terms: order = Order(term, x) if not min or order not in min: min = order new_expr = term elif min in order: new_expr += term except TypeError: return expr new_expr=new_expr.together() if new_expr.is_Add: new_expr = new_expr.simplify() if not new_expr: # simple leading term analysis gave us cancelled terms but we have to send # back a term, so compute the leading term (via series) n0 = min.getn() res = Order(1) incr = S.One while res.is_Order: res = old._eval_nseries(x, n=n0+incr, logx=None, cdir=cdir).cancel().powsimp().trigsimp() incr *= 2 return res.as_leading_term(x, cdir=cdir) elif new_expr is S.NaN: return old.func._from_args(infinite) else: return new_expr def _eval_adjoint(self): return self.func(*[t.adjoint() for t in self.args]) def _eval_conjugate(self): return self.func(*[t.conjugate() for t in self.args]) def _eval_transpose(self): return self.func(*[t.transpose() for t in self.args]) def _sage_(self): s = 0 for x in self.args: s += x._sage_() return s def primitive(self): """ Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```. ``R`` is collected only from the leading coefficient of each term. Examples ======== >>> from sympy.abc import x, y >>> (2*x + 4*y).primitive() (2, x + 2*y) >>> (2*x/3 + 4*y/9).primitive() (2/9, 3*x + 2*y) >>> (2*x/3 + 4.2*y).primitive() (1/3, 2*x + 12.6*y) No subprocessing of term factors is performed: >>> ((2 + 2*x)*x + 2).primitive() (1, x*(2*x + 2) + 2) Recursive processing can be done with the ``as_content_primitive()`` method: >>> ((2 + 2*x)*x + 2).as_content_primitive() (2, x*(x + 1) + 1) See also: primitive() function in polytools.py """ terms = [] inf = False for a in self.args: c, m = a.as_coeff_Mul() if not c.is_Rational: c = S.One m = a inf = inf or m is S.ComplexInfinity terms.append((c.p, c.q, m)) if not inf: ngcd = reduce(igcd, [t[0] for t in terms], 0) dlcm = reduce(ilcm, [t[1] for t in terms], 1) else: ngcd = reduce(igcd, [t[0] for t in terms if t[1]], 0) dlcm = reduce(ilcm, [t[1] for t in terms if t[1]], 1) if ngcd == dlcm == 1: return S.One, self if not inf: for i, (p, q, term) in enumerate(terms): terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term) else: for i, (p, q, term) in enumerate(terms): if q: terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term) else: terms[i] = _keep_coeff(Rational(p, q), term) # we don't need a complete re-flattening since no new terms will join # so we just use the same sort as is used in Add.flatten. When the # coefficient changes, the ordering of terms may change, e.g. # (3*x, 6*y) -> (2*y, x) # # We do need to make sure that term[0] stays in position 0, however. # if terms[0].is_Number or terms[0] is S.ComplexInfinity: c = terms.pop(0) else: c = None _addsort(terms) if c: terms.insert(0, c) return Rational(ngcd, dlcm), self._new_rawargs(*terms) def as_content_primitive(self, radical=False, clear=True): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. If radical is True (default is False) then common radicals will be removed and included as a factor of the primitive expression. Examples ======== >>> from sympy import sqrt >>> (3 + 3*sqrt(2)).as_content_primitive() (3, 1 + sqrt(2)) Radical content can also be factored out of the primitive: >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2)*(1 + 2*sqrt(5))) See docstring of Expr.as_content_primitive for more examples. """ con, prim = self.func(*[_keep_coeff(*a.as_content_primitive( radical=radical, clear=clear)) for a in self.args]).primitive() if not clear and not con.is_Integer and prim.is_Add: con, d = con.as_numer_denom() _p = prim/d if any(a.as_coeff_Mul()[0].is_Integer for a in _p.args): prim = _p else: con /= d if radical and prim.is_Add: # look for common radicals that can be removed args = prim.args rads = [] common_q = None for m in args: term_rads = defaultdict(list) for ai in Mul.make_args(m): if ai.is_Pow: b, e = ai.as_base_exp() if e.is_Rational and b.is_Integer: term_rads[e.q].append(abs(int(b))**e.p) if not term_rads: break if common_q is None: common_q = set(term_rads.keys()) else: common_q = common_q & set(term_rads.keys()) if not common_q: break rads.append(term_rads) else: # process rads # keep only those in common_q for r in rads: for q in list(r.keys()): if q not in common_q: r.pop(q) for q in r: r[q] = prod(r[q]) # find the gcd of bases for each q G = [] for q in common_q: g = reduce(igcd, [r[q] for r in rads], 0) if g != 1: G.append(g**Rational(1, q)) if G: G = Mul(*G) args = [ai/G for ai in args] prim = G*prim.func(*args) return con, prim @property def _sorted_args(self): from sympy.core.compatibility import default_sort_key return tuple(sorted(self.args, key=default_sort_key)) def _eval_difference_delta(self, n, step): from sympy.series.limitseq import difference_delta as dd return self.func(*[dd(a, n, step) for a in self.args]) @property def _mpc_(self): """ Convert self to an mpmath mpc if possible """ from sympy.core.numbers import I, Float re_part, rest = self.as_coeff_Add() im_part, imag_unit = rest.as_coeff_Mul() if not imag_unit == I: # ValueError may seem more reasonable but since it's a @property, # we need to use AttributeError to keep from confusing things like # hasattr. raise AttributeError("Cannot convert Add to mpc. Must be of the form Number + Number*I") return (Float(re_part)._mpf_, Float(im_part)._mpf_) def __neg__(self): if not global_parameters.distribute: return super().__neg__() return Add(*[-i for i in self.args]) add = AssocOpDispatcher('add') from .mul import Mul, _keep_coeff, prod from sympy.core.numbers import Rational
7b5ff63da0e9cc8a438863d5cb9b9826f3f7ca2517d4a15e3bead25da8a839b4
""" Provides functionality for multidimensional usage of scalar-functions. Read the vectorize docstring for more details. """ from sympy.core.decorators import wraps def apply_on_element(f, args, kwargs, n): """ Returns a structure with the same dimension as the specified argument, where each basic element is replaced by the function f applied on it. All other arguments stay the same. """ # Get the specified argument. if isinstance(n, int): structure = args[n] is_arg = True elif isinstance(n, str): structure = kwargs[n] is_arg = False # Define reduced function that is only dependent on the specified argument. def f_reduced(x): if hasattr(x, "__iter__"): return list(map(f_reduced, x)) else: if is_arg: args[n] = x else: kwargs[n] = x return f(*args, **kwargs) # f_reduced will call itself recursively so that in the end f is applied to # all basic elements. return list(map(f_reduced, structure)) def iter_copy(structure): """ Returns a copy of an iterable object (also copying all embedded iterables). """ l = [] for i in structure: if hasattr(i, "__iter__"): l.append(iter_copy(i)) else: l.append(i) return l def structure_copy(structure): """ Returns a copy of the given structure (numpy-array, list, iterable, ..). """ if hasattr(structure, "copy"): return structure.copy() return iter_copy(structure) class vectorize: """ Generalizes a function taking scalars to accept multidimensional arguments. Examples ======== >>> from sympy import diff, sin, symbols, Function >>> from sympy.core.multidimensional import vectorize >>> x, y, z = symbols('x y z') >>> f, g, h = list(map(Function, 'fgh')) >>> @vectorize(0) ... def vsin(x): ... return sin(x) >>> vsin([1, x, y]) [sin(1), sin(x), sin(y)] >>> @vectorize(0, 1) ... def vdiff(f, y): ... return diff(f, y) >>> vdiff([f(x, y, z), g(x, y, z), h(x, y, z)], [x, y, z]) [[Derivative(f(x, y, z), x), Derivative(f(x, y, z), y), Derivative(f(x, y, z), z)], [Derivative(g(x, y, z), x), Derivative(g(x, y, z), y), Derivative(g(x, y, z), z)], [Derivative(h(x, y, z), x), Derivative(h(x, y, z), y), Derivative(h(x, y, z), z)]] """ def __init__(self, *mdargs): """ The given numbers and strings characterize the arguments that will be treated as data structures, where the decorated function will be applied to every single element. If no argument is given, everything is treated multidimensional. """ for a in mdargs: if not isinstance(a, (int, str)): raise TypeError("a is of invalid type") self.mdargs = mdargs def __call__(self, f): """ Returns a wrapper for the one-dimensional function that can handle multidimensional arguments. """ @wraps(f) def wrapper(*args, **kwargs): # Get arguments that should be treated multidimensional if self.mdargs: mdargs = self.mdargs else: mdargs = range(len(args)) + kwargs.keys() arglength = len(args) for n in mdargs: if isinstance(n, int): if n >= arglength: continue entry = args[n] is_arg = True elif isinstance(n, str): try: entry = kwargs[n] except KeyError: continue is_arg = False if hasattr(entry, "__iter__"): # Create now a copy of the given array and manipulate then # the entries directly. if is_arg: args = list(args) args[n] = structure_copy(entry) else: kwargs[n] = structure_copy(entry) result = apply_on_element(wrapper, args, kwargs, n) return result return f(*args, **kwargs) return wrapper
c54ab533b02fef87f8ff03d8215548ac8279de62407c2f973f1cd2b9cf873d20
from typing import Tuple as tTuple from .sympify import sympify, _sympify, SympifyError from .basic import Basic, Atom from .singleton import S from .evalf import EvalfMixin, pure_complex from .decorators import call_highest_priority, sympify_method_args, sympify_return from .cache import cacheit from .compatibility import reduce, as_int, default_sort_key, Iterable from sympy.utilities.misc import func_name from mpmath.libmp import mpf_log, prec_to_dps from collections import defaultdict @sympify_method_args class Expr(Basic, EvalfMixin): """ Base class for algebraic expressions. Explanation =========== Everything that requires arithmetic operations to be defined should subclass this class, instead of Basic (which should be used only for argument storage and expression manipulation, i.e. pattern matching, substitutions, etc). If you want to override the comparisons of expressions: Should use _eval_is_ge for inequality, or _eval_is_eq, with multiple dispatch. _eval_is_ge return true if x >= y, false if x < y, and None if the two types are not comparable or the comparison is indeterminate See Also ======== sympy.core.basic.Basic """ __slots__ = () # type: tTuple[str, ...] is_scalar = True # self derivative is 1 @property def _diff_wrt(self): """Return True if one can differentiate with respect to this object, else False. Explanation =========== Subclasses such as Symbol, Function and Derivative return True to enable derivatives wrt them. The implementation in Derivative separates the Symbol and non-Symbol (_diff_wrt=True) variables and temporarily converts the non-Symbols into Symbols when performing the differentiation. By default, any object deriving from Expr will behave like a scalar with self.diff(self) == 1. If this is not desired then the object must also set `is_scalar = False` or else define an _eval_derivative routine. Note, see the docstring of Derivative for how this should work mathematically. In particular, note that expr.subs(yourclass, Symbol) should be well-defined on a structural level, or this will lead to inconsistent results. Examples ======== >>> from sympy import Expr >>> e = Expr() >>> e._diff_wrt False >>> class MyScalar(Expr): ... _diff_wrt = True ... >>> MyScalar().diff(MyScalar()) 1 >>> class MySymbol(Expr): ... _diff_wrt = True ... is_scalar = False ... >>> MySymbol().diff(MySymbol()) Derivative(MySymbol(), MySymbol()) """ return False @cacheit def sort_key(self, order=None): coeff, expr = self.as_coeff_Mul() if expr.is_Pow: expr, exp = expr.args else: expr, exp = expr, S.One if expr.is_Dummy: args = (expr.sort_key(),) elif expr.is_Atom: args = (str(expr),) else: if expr.is_Add: args = expr.as_ordered_terms(order=order) elif expr.is_Mul: args = expr.as_ordered_factors(order=order) else: args = expr.args args = tuple( [ default_sort_key(arg, order=order) for arg in args ]) args = (len(args), tuple(args)) exp = exp.sort_key(order=order) return expr.class_key(), args, exp, coeff def __hash__(self) -> int: # hash cannot be cached using cache_it because infinite recurrence # occurs as hash is needed for setting cache dictionary keys h = self._mhash if h is None: h = hash((type(self).__name__,) + self._hashable_content()) self._mhash = h return h def _hashable_content(self): """Return a tuple of information about self that can be used to compute the hash. If a class defines additional attributes, like ``name`` in Symbol, then this method should be updated accordingly to return such relevant attributes. Defining more than _hashable_content is necessary if __eq__ has been defined by a class. See note about this in Basic.__eq__.""" return self._args def __eq__(self, other): try: other = _sympify(other) if not isinstance(other, Expr): return False except (SympifyError, SyntaxError): return False # check for pure number expr if not (self.is_Number and other.is_Number) and ( type(self) != type(other)): return False a, b = self._hashable_content(), other._hashable_content() if a != b: return False # check number *in* an expression for a, b in zip(a, b): if not isinstance(a, Expr): continue if a.is_Number and type(a) != type(b): return False return True # *************** # * Arithmetics * # *************** # Expr and its sublcasses use _op_priority to determine which object # passed to a binary special method (__mul__, etc.) will handle the # operation. In general, the 'call_highest_priority' decorator will choose # the object with the highest _op_priority to handle the call. # Custom subclasses that want to define their own binary special methods # should set an _op_priority value that is higher than the default. # # **NOTE**: # This is a temporary fix, and will eventually be replaced with # something better and more powerful. See issue 5510. _op_priority = 10.0 @property def _add_handler(self): return Add @property def _mul_handler(self): return Mul def __pos__(self): return self def __neg__(self): # Mul has its own __neg__ routine, so we just # create a 2-args Mul with the -1 in the canonical # slot 0. c = self.is_commutative return Mul._from_args((S.NegativeOne, self), c) def __abs__(self): from sympy import Abs return Abs(self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__radd__') def __add__(self, other): return Add(self, other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__add__') def __radd__(self, other): return Add(other, self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rsub__') def __sub__(self, other): return Add(self, -other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__sub__') def __rsub__(self, other): return Add(other, -self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rmul__') def __mul__(self, other): return Mul(self, other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__mul__') def __rmul__(self, other): return Mul(other, self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rpow__') def _pow(self, other): return Pow(self, other) def __pow__(self, other, mod=None): if mod is None: return self._pow(other) try: _self, other, mod = as_int(self), as_int(other), as_int(mod) if other >= 0: return pow(_self, other, mod) else: from sympy.core.numbers import mod_inverse return mod_inverse(pow(_self, -other, mod), mod) except ValueError: power = self._pow(other) try: return power%mod except TypeError: return NotImplemented @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__pow__') def __rpow__(self, other): return Pow(other, self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rtruediv__') def __truediv__(self, other): denom = Pow(other, S.NegativeOne) if self is S.One: return denom else: return Mul(self, denom) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__truediv__') def __rtruediv__(self, other): denom = Pow(self, S.NegativeOne) if other is S.One: return denom else: return Mul(other, denom) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rmod__') def __mod__(self, other): return Mod(self, other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__mod__') def __rmod__(self, other): return Mod(other, self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rfloordiv__') def __floordiv__(self, other): from sympy.functions.elementary.integers import floor return floor(self / other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__floordiv__') def __rfloordiv__(self, other): from sympy.functions.elementary.integers import floor return floor(other / self) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__rdivmod__') def __divmod__(self, other): from sympy.functions.elementary.integers import floor return floor(self / other), Mod(self, other) @sympify_return([('other', 'Expr')], NotImplemented) @call_highest_priority('__divmod__') def __rdivmod__(self, other): from sympy.functions.elementary.integers import floor return floor(other / self), Mod(other, self) def __int__(self): # Although we only need to round to the units position, we'll # get one more digit so the extra testing below can be avoided # unless the rounded value rounded to an integer, e.g. if an # expression were equal to 1.9 and we rounded to the unit position # we would get a 2 and would not know if this rounded up or not # without doing a test (as done below). But if we keep an extra # digit we know that 1.9 is not the same as 1 and there is no # need for further testing: our int value is correct. If the value # were 1.99, however, this would round to 2.0 and our int value is # off by one. So...if our round value is the same as the int value # (regardless of how much extra work we do to calculate extra decimal # places) we need to test whether we are off by one. from sympy import Dummy if not self.is_number: raise TypeError("can't convert symbols to int") r = self.round(2) if not r.is_Number: raise TypeError("can't convert complex to int") if r in (S.NaN, S.Infinity, S.NegativeInfinity): raise TypeError("can't convert %s to int" % r) i = int(r) if not i: return 0 # off-by-one check if i == r and not (self - i).equals(0): isign = 1 if i > 0 else -1 x = Dummy() # in the following (self - i).evalf(2) will not always work while # (self - r).evalf(2) and the use of subs does; if the test that # was added when this comment was added passes, it might be safe # to simply use sign to compute this rather than doing this by hand: diff_sign = 1 if (self - x).evalf(2, subs={x: i}) > 0 else -1 if diff_sign != isign: i -= isign return i def __float__(self): # Don't bother testing if it's a number; if it's not this is going # to fail, and if it is we still need to check that it evalf'ed to # a number. result = self.evalf() if result.is_Number: return float(result) if result.is_number and result.as_real_imag()[1]: raise TypeError("can't convert complex to float") raise TypeError("can't convert expression to float") def __complex__(self): result = self.evalf() re, im = result.as_real_imag() return complex(float(re), float(im)) @sympify_return([('other', 'Expr')], NotImplemented) def __ge__(self, other): from .relational import GreaterThan return GreaterThan(self, other) @sympify_return([('other', 'Expr')], NotImplemented) def __le__(self, other): from .relational import LessThan return LessThan(self, other) @sympify_return([('other', 'Expr')], NotImplemented) def __gt__(self, other): from .relational import StrictGreaterThan return StrictGreaterThan(self, other) @sympify_return([('other', 'Expr')], NotImplemented) def __lt__(self, other): from .relational import StrictLessThan return StrictLessThan(self, other) def __trunc__(self): if not self.is_number: raise TypeError("can't truncate symbols and expressions") else: return Integer(self) @staticmethod def _from_mpmath(x, prec): from sympy import Float if hasattr(x, "_mpf_"): return Float._new(x._mpf_, prec) elif hasattr(x, "_mpc_"): re, im = x._mpc_ re = Float._new(re, prec) im = Float._new(im, prec)*S.ImaginaryUnit return re + im else: raise TypeError("expected mpmath number (mpf or mpc)") @property def is_number(self): """Returns True if ``self`` has no free symbols and no undefined functions (AppliedUndef, to be precise). It will be faster than ``if not self.free_symbols``, however, since ``is_number`` will fail as soon as it hits a free symbol or undefined function. Examples ======== >>> from sympy import Integral, cos, sin, pi >>> from sympy.core.function import Function >>> from sympy.abc import x >>> f = Function('f') >>> x.is_number False >>> f(1).is_number False >>> (2*x).is_number False >>> (2 + Integral(2, x)).is_number False >>> (2 + Integral(2, (x, 1, 2))).is_number True Not all numbers are Numbers in the SymPy sense: >>> pi.is_number, pi.is_Number (True, False) If something is a number it should evaluate to a number with real and imaginary parts that are Numbers; the result may not be comparable, however, since the real and/or imaginary part of the result may not have precision. >>> cos(1).is_number and cos(1).is_comparable True >>> z = cos(1)**2 + sin(1)**2 - 1 >>> z.is_number True >>> z.is_comparable False See Also ======== sympy.core.basic.Basic.is_comparable """ return all(obj.is_number for obj in self.args) def _random(self, n=None, re_min=-1, im_min=-1, re_max=1, im_max=1): """Return self evaluated, if possible, replacing free symbols with random complex values, if necessary. Explanation =========== The random complex value for each free symbol is generated by the random_complex_number routine giving real and imaginary parts in the range given by the re_min, re_max, im_min, and im_max values. The returned value is evaluated to a precision of n (if given) else the maximum of 15 and the precision needed to get more than 1 digit of precision. If the expression could not be evaluated to a number, or could not be evaluated to more than 1 digit of precision, then None is returned. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y >>> x._random() # doctest: +SKIP 0.0392918155679172 + 0.916050214307199*I >>> x._random(2) # doctest: +SKIP -0.77 - 0.87*I >>> (x + y/2)._random(2) # doctest: +SKIP -0.57 + 0.16*I >>> sqrt(2)._random(2) 1.4 See Also ======== sympy.testing.randtest.random_complex_number """ free = self.free_symbols prec = 1 if free: from sympy.testing.randtest import random_complex_number a, c, b, d = re_min, re_max, im_min, im_max reps = dict(list(zip(free, [random_complex_number(a, b, c, d, rational=True) for zi in free]))) try: nmag = abs(self.evalf(2, subs=reps)) except (ValueError, TypeError): # if an out of range value resulted in evalf problems # then return None -- XXX is there a way to know how to # select a good random number for a given expression? # e.g. when calculating n! negative values for n should not # be used return None else: reps = {} nmag = abs(self.evalf(2)) if not hasattr(nmag, '_prec'): # e.g. exp_polar(2*I*pi) doesn't evaluate but is_number is True return None if nmag._prec == 1: # increase the precision up to the default maximum # precision to see if we can get any significance from mpmath.libmp.libintmath import giant_steps from sympy.core.evalf import DEFAULT_MAXPREC as target # evaluate for prec in giant_steps(2, target): nmag = abs(self.evalf(prec, subs=reps)) if nmag._prec != 1: break if nmag._prec != 1: if n is None: n = max(prec, 15) return self.evalf(n, subs=reps) # never got any significance return None def is_constant(self, *wrt, **flags): """Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation =========== If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if ``wrt`` is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It won't be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag ``failing_number`` is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples ======== >>> from sympy import cos, sin, Sum, S, pi >>> from sympy.abc import a, n, x, y >>> x.is_constant() False >>> S(2).is_constant() True >>> Sum(x, (x, 1, 10)).is_constant() True >>> Sum(x, (x, 1, n)).is_constant() False >>> Sum(x, (x, 1, n)).is_constant(y) True >>> Sum(x, (x, 1, n)).is_constant(n) False >>> Sum(x, (x, 1, n)).is_constant(x) True >>> eq = a*cos(x)**2 + a*sin(x)**2 - a >>> eq.is_constant() True >>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True >>> (0**x).is_constant() False >>> x.is_constant() False >>> (x**x).is_constant() False >>> one = cos(x)**2 + sin(x)**2 >>> one.is_constant() True >>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True """ def check_denominator_zeros(expression): from sympy.solvers.solvers import denoms retNone = False for den in denoms(expression): z = den.is_zero if z is True: return True if z is None: retNone = True if retNone: return None return False simplify = flags.get('simplify', True) if self.is_number: return True free = self.free_symbols if not free: return True # assume f(1) is some constant # if we are only interested in some symbols and they are not in the # free symbols then this expression is constant wrt those symbols wrt = set(wrt) if wrt and not wrt & free: return True wrt = wrt or free # simplify unless this has already been done expr = self if simplify: expr = expr.simplify() # is_zero should be a quick assumptions check; it can be wrong for # numbers (see test_is_not_constant test), giving False when it # shouldn't, but hopefully it will never give True unless it is sure. if expr.is_zero: return True # try numerical evaluation to see if we get two different values failing_number = None if wrt == free: # try 0 (for a) and 1 (for b) try: a = expr.subs(list(zip(free, [0]*len(free))), simultaneous=True) if a is S.NaN: # evaluation may succeed when substitution fails a = expr._random(None, 0, 0, 0, 0) except ZeroDivisionError: a = None if a is not None and a is not S.NaN: try: b = expr.subs(list(zip(free, [1]*len(free))), simultaneous=True) if b is S.NaN: # evaluation may succeed when substitution fails b = expr._random(None, 1, 0, 1, 0) except ZeroDivisionError: b = None if b is not None and b is not S.NaN and b.equals(a) is False: return False # try random real b = expr._random(None, -1, 0, 1, 0) if b is not None and b is not S.NaN and b.equals(a) is False: return False # try random complex b = expr._random() if b is not None and b is not S.NaN: if b.equals(a) is False: return False failing_number = a if a.is_number else b # now we will test each wrt symbol (or all free symbols) to see if the # expression depends on them or not using differentiation. This is # not sufficient for all expressions, however, so we don't return # False if we get a derivative other than 0 with free symbols. for w in wrt: deriv = expr.diff(w) if simplify: deriv = deriv.simplify() if deriv != 0: if not (pure_complex(deriv, or_real=True)): if flags.get('failing_number', False): return failing_number elif deriv.free_symbols: # dead line provided _random returns None in such cases return None return False cd = check_denominator_zeros(self) if cd is True: return False elif cd is None: return None return True def equals(self, other, failing_expression=False): """Return True if self == other, False if it doesn't, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation =========== If ``self`` is a Number (or complex number) that is not zero, then the result is False. If ``self`` is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. """ from sympy.simplify.simplify import nsimplify, simplify from sympy.solvers.solvers import solve from sympy.polys.polyerrors import NotAlgebraic from sympy.polys.numberfields import minimal_polynomial other = sympify(other) if self == other: return True # they aren't the same so see if we can make the difference 0; # don't worry about doing simplification steps one at a time # because if the expression ever goes to 0 then the subsequent # simplification steps that are done will be very fast. diff = factor_terms(simplify(self - other), radical=True) if not diff: return True if not diff.has(Add, Mod): # if there is no expanding to be done after simplifying # then this can't be a zero return False constant = diff.is_constant(simplify=False, failing_number=True) if constant is False: return False if not diff.is_number: if constant is None: # e.g. unless the right simplification is done, a symbolic # zero is possible (see expression of issue 6829: without # simplification constant will be None). return if constant is True: # this gives a number whether there are free symbols or not ndiff = diff._random() # is_comparable will work whether the result is real # or complex; it could be None, however. if ndiff and ndiff.is_comparable: return False # sometimes we can use a simplified result to give a clue as to # what the expression should be; if the expression is *not* zero # then we should have been able to compute that and so now # we can just consider the cases where the approximation appears # to be zero -- we try to prove it via minimal_polynomial. # # removed # ns = nsimplify(diff) # if diff.is_number and (not ns or ns == diff): # # The thought was that if it nsimplifies to 0 that's a sure sign # to try the following to prove it; or if it changed but wasn't # zero that might be a sign that it's not going to be easy to # prove. But tests seem to be working without that logic. # if diff.is_number: # try to prove via self-consistency surds = [s for s in diff.atoms(Pow) if s.args[0].is_Integer] # it seems to work better to try big ones first surds.sort(key=lambda x: -x.args[0]) for s in surds: try: # simplify is False here -- this expression has already # been identified as being hard to identify as zero; # we will handle the checking ourselves using nsimplify # to see if we are in the right ballpark or not and if so # *then* the simplification will be attempted. sol = solve(diff, s, simplify=False) if sol: if s in sol: # the self-consistent result is present return True if all(si.is_Integer for si in sol): # perfect powers are removed at instantiation # so surd s cannot be an integer return False if all(i.is_algebraic is False for i in sol): # a surd is algebraic return False if any(si in surds for si in sol): # it wasn't equal to s but it is in surds # and different surds are not equal return False if any(nsimplify(s - si) == 0 and simplify(s - si) == 0 for si in sol): return True if s.is_real: if any(nsimplify(si, [s]) == s and simplify(si) == s for si in sol): return True except NotImplementedError: pass # try to prove with minimal_polynomial but know when # *not* to use this or else it can take a long time. e.g. issue 8354 if True: # change True to condition that assures non-hang try: mp = minimal_polynomial(diff) if mp.is_Symbol: return True return False except (NotAlgebraic, NotImplementedError): pass # diff has not simplified to zero; constant is either None, True # or the number with significance (is_comparable) that was randomly # calculated twice as the same value. if constant not in (True, None) and constant != 0: return False if failing_expression: return diff return None def _eval_is_positive(self): finite = self.is_finite if finite is False: return False extended_positive = self.is_extended_positive if finite is True: return extended_positive if extended_positive is False: return False def _eval_is_negative(self): finite = self.is_finite if finite is False: return False extended_negative = self.is_extended_negative if finite is True: return extended_negative if extended_negative is False: return False def _eval_is_extended_positive_negative(self, positive): from sympy.polys.numberfields import minimal_polynomial from sympy.polys.polyerrors import NotAlgebraic if self.is_number: if self.is_extended_real is False: return False # check to see that we can get a value try: n2 = self._eval_evalf(2) # XXX: This shouldn't be caught here # Catches ValueError: hypsum() failed to converge to the requested # 34 bits of accuracy except ValueError: return None if n2 is None: return None if getattr(n2, '_prec', 1) == 1: # no significance return None if n2 is S.NaN: return None r, i = self.evalf(2).as_real_imag() if not i.is_Number or not r.is_Number: return False if r._prec != 1 and i._prec != 1: return bool(not i and ((r > 0) if positive else (r < 0))) elif r._prec == 1 and (not i or i._prec == 1) and \ self.is_algebraic and not self.has(Function): try: if minimal_polynomial(self).is_Symbol: return False except (NotAlgebraic, NotImplementedError): pass def _eval_is_extended_positive(self): return self._eval_is_extended_positive_negative(positive=True) def _eval_is_extended_negative(self): return self._eval_is_extended_positive_negative(positive=False) def _eval_interval(self, x, a, b): """ Returns evaluation over an interval. For most functions this is: self.subs(x, b) - self.subs(x, a), possibly using limit() if NaN is returned from subs, or if singularities are found between a and b. If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x), respectively. """ from sympy.series import limit, Limit from sympy.solvers.solveset import solveset from sympy.sets.sets import Interval from sympy.functions.elementary.exponential import log from sympy.calculus.util import AccumBounds if (a is None and b is None): raise ValueError('Both interval ends cannot be None.') def _eval_endpoint(left): c = a if left else b if c is None: return 0 else: C = self.subs(x, c) if C.has(S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity, AccumBounds): if (a < b) != False: C = limit(self, x, c, "+" if left else "-") else: C = limit(self, x, c, "-" if left else "+") if isinstance(C, Limit): raise NotImplementedError("Could not compute limit") return C if a == b: return 0 A = _eval_endpoint(left=True) if A is S.NaN: return A B = _eval_endpoint(left=False) if (a and b) is None: return B - A value = B - A if a.is_comparable and b.is_comparable: if a < b: domain = Interval(a, b) else: domain = Interval(b, a) # check the singularities of self within the interval # if singularities is a ConditionSet (not iterable), catch the exception and pass singularities = solveset(self.cancel().as_numer_denom()[1], x, domain=domain) for logterm in self.atoms(log): singularities = singularities | solveset(logterm.args[0], x, domain=domain) try: for s in singularities: if value is S.NaN: # no need to keep adding, it will stay NaN break if not s.is_comparable: continue if (a < s) == (s < b) == True: value += -limit(self, x, s, "+") + limit(self, x, s, "-") elif (b < s) == (s < a) == True: value += limit(self, x, s, "+") - limit(self, x, s, "-") except TypeError: pass return value def _eval_power(self, other): # subclass to compute self**other for cases when # other is not NaN, 0, or 1 return None def _eval_conjugate(self): if self.is_extended_real: return self elif self.is_imaginary: return -self def conjugate(self): """Returns the complex conjugate of 'self'.""" from sympy.functions.elementary.complexes import conjugate as c return c(self) def dir(self, x, cdir): from sympy import log minexp = S.Zero if self.is_zero: return S.Zero arg = self while arg: minexp += S.One arg = arg.diff(x) coeff = arg.subs(x, 0) if coeff in (S.NaN, S.ComplexInfinity): try: coeff, _ = arg.leadterm(x) if coeff.has(log(x)): raise ValueError() except ValueError: coeff = arg.limit(x, 0) if coeff != S.Zero: break return coeff*cdir**minexp def _eval_transpose(self): from sympy.functions.elementary.complexes import conjugate if (self.is_complex or self.is_infinite): return self elif self.is_hermitian: return conjugate(self) elif self.is_antihermitian: return -conjugate(self) def transpose(self): from sympy.functions.elementary.complexes import transpose return transpose(self) def _eval_adjoint(self): from sympy.functions.elementary.complexes import conjugate, transpose if self.is_hermitian: return self elif self.is_antihermitian: return -self obj = self._eval_conjugate() if obj is not None: return transpose(obj) obj = self._eval_transpose() if obj is not None: return conjugate(obj) def adjoint(self): from sympy.functions.elementary.complexes import adjoint return adjoint(self) @classmethod def _parse_order(cls, order): """Parse and configure the ordering of terms. """ from sympy.polys.orderings import monomial_key startswith = getattr(order, "startswith", None) if startswith is None: reverse = False else: reverse = startswith('rev-') if reverse: order = order[4:] monom_key = monomial_key(order) def neg(monom): result = [] for m in monom: if isinstance(m, tuple): result.append(neg(m)) else: result.append(-m) return tuple(result) def key(term): _, ((re, im), monom, ncpart) = term monom = neg(monom_key(monom)) ncpart = tuple([e.sort_key(order=order) for e in ncpart]) coeff = ((bool(im), im), (re, im)) return monom, ncpart, coeff return key, reverse def as_ordered_factors(self, order=None): """Return list of ordered factors (if Mul) else [self].""" return [self] def as_poly(self, *gens, **args): """Converts ``self`` to a polynomial or returns ``None``. Explanation =========== >>> from sympy import sin >>> from sympy.abc import x, y >>> print((x**2 + x*y).as_poly()) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + x*y).as_poly(x, y)) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + sin(y)).as_poly(x, y)) None """ from sympy.polys import Poly, PolynomialError try: poly = Poly(self, *gens, **args) if not poly.is_Poly: return None else: return poly except PolynomialError: return None def as_ordered_terms(self, order=None, data=False): """ Transform an expression to an ordered list of terms. Examples ======== >>> from sympy import sin, cos >>> from sympy.abc import x >>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms() [sin(x)**2*cos(x), sin(x)**2, 1] """ from .numbers import Number, NumberSymbol if order is None and self.is_Add: # Spot the special case of Add(Number, Mul(Number, expr)) with the # first number positive and thhe second number nagative key = lambda x:not isinstance(x, (Number, NumberSymbol)) add_args = sorted(Add.make_args(self), key=key) if (len(add_args) == 2 and isinstance(add_args[0], (Number, NumberSymbol)) and isinstance(add_args[1], Mul)): mul_args = sorted(Mul.make_args(add_args[1]), key=key) if (len(mul_args) == 2 and isinstance(mul_args[0], Number) and add_args[0].is_positive and mul_args[0].is_negative): return add_args key, reverse = self._parse_order(order) terms, gens = self.as_terms() if not any(term.is_Order for term, _ in terms): ordered = sorted(terms, key=key, reverse=reverse) else: _terms, _order = [], [] for term, repr in terms: if not term.is_Order: _terms.append((term, repr)) else: _order.append((term, repr)) ordered = sorted(_terms, key=key, reverse=True) \ + sorted(_order, key=key, reverse=True) if data: return ordered, gens else: return [term for term, _ in ordered] def as_terms(self): """Transform an expression to a list of terms. """ from .add import Add from .mul import Mul from .exprtools import decompose_power gens, terms = set(), [] for term in Add.make_args(self): coeff, _term = term.as_coeff_Mul() coeff = complex(coeff) cpart, ncpart = {}, [] if _term is not S.One: for factor in Mul.make_args(_term): if factor.is_number: try: coeff *= complex(factor) except (TypeError, ValueError): pass else: continue if factor.is_commutative: base, exp = decompose_power(factor) cpart[base] = exp gens.add(base) else: ncpart.append(factor) coeff = coeff.real, coeff.imag ncpart = tuple(ncpart) terms.append((term, (coeff, cpart, ncpart))) gens = sorted(gens, key=default_sort_key) k, indices = len(gens), {} for i, g in enumerate(gens): indices[g] = i result = [] for term, (coeff, cpart, ncpart) in terms: monom = [0]*k for base, exp in cpart.items(): monom[indices[base]] = exp result.append((term, (coeff, tuple(monom), ncpart))) return result, gens def removeO(self): """Removes the additive O(..) symbol if there is one""" return self def getO(self): """Returns the additive O(..) symbol if there is one, else None.""" return None def getn(self): """ Returns the order of the expression. Explanation =========== The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples ======== >>> from sympy import O >>> from sympy.abc import x >>> (1 + x + O(x**2)).getn() 2 >>> (1 + x).getn() """ from sympy import Dummy, Symbol o = self.getO() if o is None: return None elif o.is_Order: o = o.expr if o is S.One: return S.Zero if o.is_Symbol: return S.One if o.is_Pow: return o.args[1] if o.is_Mul: # x**n*log(x)**n or x**n/log(x)**n for oi in o.args: if oi.is_Symbol: return S.One if oi.is_Pow: syms = oi.atoms(Symbol) if len(syms) == 1: x = syms.pop() oi = oi.subs(x, Dummy('x', positive=True)) if oi.base.is_Symbol and oi.exp.is_Rational: return abs(oi.exp) raise NotImplementedError('not sure of order of %s' % o) def count_ops(self, visual=None): """wrapper for count_ops that returns the operation count.""" from .function import count_ops return count_ops(self, visual) def args_cnc(self, cset=False, warn=True, split_1=True): """Return [commutative factors, non-commutative factors] of self. Explanation =========== self is treated as a Mul and the ordering of the factors is maintained. If ``cset`` is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting ``warn`` to False. Note: -1 is always separated from a Number unless split_1 is False. Examples ======== >>> from sympy import symbols, oo >>> A, B = symbols('A B', commutative=0) >>> x, y = symbols('x y') >>> (-2*x*y).args_cnc() [[-1, 2, x, y], []] >>> (-2.5*x).args_cnc() [[-1, 2.5, x], []] >>> (-2*x*A*B*y).args_cnc() [[-1, 2, x, y], [A, B]] >>> (-2*x*A*B*y).args_cnc(split_1=False) [[-2, x, y], [A, B]] >>> (-2*x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: >>> (-2 + x + A).args_cnc() [[], [x - 2 + A]] >>> (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] """ if self.is_Mul: args = list(self.args) else: args = [self] for i, mi in enumerate(args): if not mi.is_commutative: c = args[:i] nc = args[i:] break else: c = args nc = [] if c and split_1 and ( c[0].is_Number and c[0].is_extended_negative and c[0] is not S.NegativeOne): c[:1] = [S.NegativeOne, -c[0]] if cset: clen = len(c) c = set(c) if clen and warn and len(c) != clen: raise ValueError('repeated commutative arguments: %s' % [ci for ci in c if list(self.args).count(ci) > 1]) return [c, nc] def coeff(self, x, n=1, right=False): """ Returns the coefficient from the term(s) containing ``x**n``. If ``n`` is zero then all terms independent of ``x`` will be returned. Explanation =========== When ``x`` is noncommutative, the coefficient to the left (default) or right of ``x`` can be returned. The keyword 'right' is ignored when ``x`` is commutative. Examples ======== >>> from sympy import symbols >>> from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: >>> (-x + 2*y).coeff(-1) x >>> (x - 2*y).coeff(-1) 2*y You can select terms with no Rational coefficient: >>> (x + 2*y).coeff(1) x >>> (3 + 2*x + 4*x**2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): >>> (3 + 2*x + 4*x**2).coeff(x, 0) 3 >>> eq = ((x + 1)**3).expand() + 1 >>> eq x**3 + 3*x**2 + 3*x + 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] >>> eq -= 2 >>> [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: >>> (-x - 2*y).coeff(2) -y >>> from sympy import sqrt >>> (x + sqrt(2)*x).coeff(sqrt(2)) x The matching is exact: >>> (3 + 2*x + 4*x**2).coeff(x) 2 >>> (3 + 2*x + 4*x**2).coeff(x**2) 4 >>> (3 + 2*x + 4*x**2).coeff(x**3) 0 >>> (z*(x + y)**2).coeff((x + y)**2) z >>> (z*(x + y)**2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: >>> (x + z*(x + x*y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: >>> from sympy import factor_terms >>> factor_terms(x + z*(x + x*y)).coeff(x) z*(y + 1) + 1 >>> n, m, o = symbols('n m o', commutative=False) >>> n.coeff(n) 1 >>> (3*n).coeff(n) 3 >>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m 1 + m >>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m m If there is more than one possible coefficient 0 is returned: >>> (n*m + m*n).coeff(n) 0 If there is only one possible coefficient, it is returned: >>> (n*m + x*m*n).coeff(m*n) x >>> (n*m + x*m*n).coeff(m*n, right=1) 1 See Also ======== as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used """ x = sympify(x) if not isinstance(x, Basic): return S.Zero n = as_int(n) if not x: return S.Zero if x == self: if n == 1: return S.One return S.Zero if x is S.One: co = [a for a in Add.make_args(self) if a.as_coeff_Mul()[0] is S.One] if not co: return S.Zero return Add(*co) if n == 0: if x.is_Add and self.is_Add: c = self.coeff(x, right=right) if not c: return S.Zero if not right: return self - Add(*[a*x for a in Add.make_args(c)]) return self - Add(*[x*a for a in Add.make_args(c)]) return self.as_independent(x, as_Add=True)[0] # continue with the full method, looking for this power of x: x = x**n def incommon(l1, l2): if not l1 or not l2: return [] n = min(len(l1), len(l2)) for i in range(n): if l1[i] != l2[i]: return l1[:i] return l1[:] def find(l, sub, first=True): """ Find where list sub appears in list l. When ``first`` is True the first occurrence from the left is returned, else the last occurrence is returned. Return None if sub is not in l. Examples ======== >> l = range(5)*2 >> find(l, [2, 3]) 2 >> find(l, [2, 3], first=0) 7 >> find(l, [2, 4]) None """ if not sub or not l or len(sub) > len(l): return None n = len(sub) if not first: l.reverse() sub.reverse() for i in range(0, len(l) - n + 1): if all(l[i + j] == sub[j] for j in range(n)): break else: i = None if not first: l.reverse() sub.reverse() if i is not None and not first: i = len(l) - (i + n) return i co = [] args = Add.make_args(self) self_c = self.is_commutative x_c = x.is_commutative if self_c and not x_c: return S.Zero one_c = self_c or x_c xargs, nx = x.args_cnc(cset=True, warn=bool(not x_c)) # find the parts that pass the commutative terms for a in args: margs, nc = a.args_cnc(cset=True, warn=bool(not self_c)) if nc is None: nc = [] if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): if one_c: co.append(Mul(*(list(resid) + nc))) else: co.append((resid, nc)) if one_c: if co == []: return S.Zero elif co: return Add(*co) else: # both nc # now check the non-comm parts if not co: return S.Zero if all(n == co[0][1] for r, n in co): ii = find(co[0][1], nx, right) if ii is not None: if not right: return Mul(Add(*[Mul(*r) for r, c in co]), Mul(*co[0][1][:ii])) else: return Mul(*co[0][1][ii + len(nx):]) beg = reduce(incommon, (n[1] for n in co)) if beg: ii = find(beg, nx, right) if ii is not None: if not right: gcdc = co[0][0] for i in range(1, len(co)): gcdc = gcdc.intersection(co[i][0]) if not gcdc: break return Mul(*(list(gcdc) + beg[:ii])) else: m = ii + len(nx) return Add(*[Mul(*(list(r) + n[m:])) for r, n in co]) end = list(reversed( reduce(incommon, (list(reversed(n[1])) for n in co)))) if end: ii = find(end, nx, right) if ii is not None: if not right: return Add(*[Mul(*(list(r) + n[:-len(end) + ii])) for r, n in co]) else: return Mul(*end[ii + len(nx):]) # look for single match hit = None for i, (r, n) in enumerate(co): ii = find(n, nx, right) if ii is not None: if not hit: hit = ii, r, n else: break else: if hit: ii, r, n = hit if not right: return Mul(*(list(r) + n[:ii])) else: return Mul(*n[ii + len(nx):]) return S.Zero def as_expr(self, *gens): """ Convert a polynomial to a SymPy expression. Examples ======== >>> from sympy import sin >>> from sympy.abc import x, y >>> f = (x**2 + x*y).as_poly(x, y) >>> f.as_expr() x**2 + x*y >>> sin(x).as_expr() sin(x) """ return self def as_coefficient(self, expr): """ Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples ======== >>> from sympy import E, pi, sin, I, Poly >>> from sympy.abc import x >>> E.as_coefficient(E) 1 >>> (2*E).as_coefficient(E) 2 >>> (2*sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) >>> (2*E + x*E).as_coefficient(E) x + 2 >>> _.args[0] # just want the exact match 2 >>> p = Poly(2*E + x*E); p Poly(x*E + 2*E, x, E, domain='ZZ') >>> p.coeff_monomial(E) 2 >>> p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient ``2*x`` is desired then the ``coeff`` method should be used.) >>> (2*E*x + x).as_coefficient(E) >>> (2*E*x + x).coeff(E) 2*x >>> (E*(x + 1) + x).as_coefficient(E) >>> (2*pi*I).as_coefficient(pi*I) 2 >>> (2*I).as_coefficient(pi*I) See Also ======== coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used """ r = self.extract_multiplicatively(expr) if r and not r.has(expr): return r def as_independent(self, *deps, **hint): """ A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: * separatevars() to change Mul, Add and Pow (including exp) into Mul * .expand(mul=True) to change Add or Mul into Add * .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for `self` of zero regardless of hints. For nonzero `self`, the returned tuple (i, d) has the following interpretation: * i will has no variable that appears in deps * d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) * if self is an Add then self = i + d * if self is a Mul then self = i*d * otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples ======== -- self is an Add >>> from sympy import sin, cos, exp >>> from sympy.abc import x, y, z >>> (x + x*y).as_independent(x) (0, x*y + x) >>> (x + x*y).as_independent(y) (x, x*y) >>> (2*x*sin(x) + y + x + z).as_independent(x) (y + z, 2*x*sin(x) + x) >>> (2*x*sin(x) + y + x + z).as_independent(x, y) (z, 2*x*sin(x) + x + y) -- self is a Mul >>> (x*sin(x)*cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul >>> from sympy import symbols >>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False) >>> (n1 + n1*n2).as_independent(n2) (n1, n1*n2) >>> (n2*n1 + n1*n2).as_independent(n2) (0, n1*n2 + n2*n1) >>> (n1*n2*n3).as_independent(n1) (1, n1*n2*n3) >>> (n1*n2*n3).as_independent(n2) (n1, n2*n3) >>> ((x-n1)*(x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: >>> (sin(x)).as_independent(x) (1, sin(x)) >>> (sin(x)).as_independent(y) (sin(x), 1) >>> exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: >>> (3*x).as_independent(x, as_Add=True) (0, 3*x) -- force self to be treated as a Mul: >>> (3+x).as_independent(x, as_Add=False) (1, x + 3) >>> (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. >>> (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols >>> from sympy import Integral >>> I = Integral(x, (x, 1, 2)) >>> I.has(x) True >>> x in I.free_symbols False >>> I.as_independent(x) == (I, 1) True >>> (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values >>> from sympy import separatevars, log >>> separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) >>> (x + x*y).as_independent(y) (x, x*y) >>> separatevars(x + x*y).as_independent(y) (x, y + 1) >>> (x*(1 + y)).as_independent(y) (x, y + 1) >>> (x*(1 + y)).expand(mul=True).as_independent(y) (x, x*y) >>> a, b=symbols('a b', positive=True) >>> (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also ======== .separatevars(), .expand(log=True), sympy.core.add.Add.as_two_terms(), sympy.core.mul.Mul.as_two_terms(), .as_coeff_add(), .as_coeff_mul() """ from .symbol import Symbol from .add import _unevaluated_Add from .mul import _unevaluated_Mul from sympy.utilities.iterables import sift if self.is_zero: return S.Zero, S.Zero func = self.func if hint.get('as_Add', isinstance(self, Add) ): want = Add else: want = Mul # sift out deps into symbolic and other and ignore # all symbols but those that are in the free symbols sym = set() other = [] for d in deps: if isinstance(d, Symbol): # Symbol.is_Symbol is True sym.add(d) else: other.append(d) def has(e): """return the standard has() if there are no literal symbols, else check to see that symbol-deps are in the free symbols.""" has_other = e.has(*other) if not sym: return has_other return has_other or e.has(*(e.free_symbols & sym)) if (want is not func or func is not Add and func is not Mul): if has(self): return (want.identity, self) else: return (self, want.identity) else: if func is Add: args = list(self.args) else: args, nc = self.args_cnc() d = sift(args, lambda x: has(x)) depend = d[True] indep = d[False] if func is Add: # all terms were treated as commutative return (Add(*indep), _unevaluated_Add(*depend)) else: # handle noncommutative by stopping at first dependent term for i, n in enumerate(nc): if has(n): depend.extend(nc[i:]) break indep.append(n) return Mul(*indep), ( Mul(*depend, evaluate=False) if nc else _unevaluated_Mul(*depend)) def as_real_imag(self, deep=True, **hints): """Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method can't be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. >>> from sympy import symbols, I >>> x, y = symbols('x,y', real=True) >>> (x + y*I).as_real_imag() (x, y) >>> from sympy.abc import z, w >>> (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) """ from sympy import im, re if hints.get('ignore') == self: return None else: return (re(self), im(self)) def as_powers_dict(self): """Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also ======== as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. """ d = defaultdict(int) d.update(dict([self.as_base_exp()])) return d def as_coefficients_dict(self): """Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term. Examples ======== >>> from sympy.abc import a, x >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} """ c, m = self.as_coeff_Mul() if not c.is_Rational: c = S.One m = self d = defaultdict(int) d.update({m: c}) return d def as_base_exp(self): # a -> b ** e return self, S.One def as_coeff_mul(self, *deps, **kwargs): """Return the tuple (c, args) where self is written as a Mul, ``m``. c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. - if you know self is a Mul and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; - if you want to split self into an independent and dependent parts use ``self.as_independent(*deps)`` >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_mul() (3, ()) >>> (3*x*y).as_coeff_mul() (3, (x, y)) >>> (3*x*y).as_coeff_mul(x) (3*y, (x,)) >>> (3*y).as_coeff_mul(x) (3*y, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.One, (self,) def as_coeff_add(self, *deps): """Return the tuple (c, args) where self is written as an Add, ``a``. c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of ``a``; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. - if you know self is an Add and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. - if you want to split self into an independent and dependent parts use ``self.as_independent(*deps)`` >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_add() (3, ()) >>> (3 + x).as_coeff_add() (3, (x,)) >>> (3 + x + y).as_coeff_add(x) (y + 3, (x,)) >>> (3 + y).as_coeff_add(x) (y + 3, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.Zero, (self,) def primitive(self): """Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples ======== >>> from sympy.abc import x >>> (3*(x + 1)**2).primitive() (3, (x + 1)**2) >>> a = (6*x + 2); a.primitive() (2, 3*x + 1) >>> b = (x/2 + 3); b.primitive() (1/2, x + 6) >>> (a*b).primitive() == (1, a*b) True """ if not self: return S.One, S.Zero c, r = self.as_coeff_Mul(rational=True) if c.is_negative: c, r = -c, -r return c, r def as_content_primitive(self, radical=False, clear=True): """This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and ``Mul(*foo.as_content_primitive()) == foo``. The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x, y, z >>> eq = 2 + 2*x + 2*y*(3 + 3*y) The as_content_primitive function is recursive and retains structure: >>> eq.as_content_primitive() (2, x + 3*y*(y + 1) + 1) Integer powers will have Rationals extracted from the base: >>> ((2 + 6*x)**2).as_content_primitive() (4, (3*x + 1)**2) >>> ((2 + 6*x)**(2*y)).as_content_primitive() (1, (2*(3*x + 1))**(2*y)) Terms may end up joining once their as_content_primitives are added: >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (11, x*(y + 1)) >>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() (9, x*(y + 1)) >>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive() (1, 6.0*x*(y + 1) + 3*z*(y + 1)) >>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() (121, x**2*(y + 1)**2) >>> ((x*(1 + y) + 0.4*x*(3 + 3*y))**2).as_content_primitive() (1, 4.84*x**2*(y + 1)**2) Radical content can also be factored out of the primitive: >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2)*(1 + 2*sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. >>> (x/2 + y).as_content_primitive() (1/2, x + 2*y) >>> (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) """ return S.One, self def as_numer_denom(self): """ expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also ======== normal: return a/b instead of a, b """ return self, S.One def normal(self): from .mul import _unevaluated_Mul n, d = self.as_numer_denom() if d is S.One: return n if d.is_Number: return _unevaluated_Mul(n, 1/d) else: return n/d def extract_multiplicatively(self, c): """Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples ======== >>> from sympy import symbols, Rational >>> x, y = symbols('x,y', real=True) >>> ((x*y)**3).extract_multiplicatively(x**2 * y) x*y**2 >>> ((x*y)**3).extract_multiplicatively(x**4 * y) >>> (2*x).extract_multiplicatively(2) x >>> (2*x).extract_multiplicatively(3) >>> (Rational(1, 2)*x).extract_multiplicatively(3) x/6 """ from .add import _unevaluated_Add c = sympify(c) if self is S.NaN: return None if c is S.One: return self elif c == self: return S.One if c.is_Add: cc, pc = c.primitive() if cc is not S.One: c = Mul(cc, pc, evaluate=False) if c.is_Mul: a, b = c.as_two_terms() x = self.extract_multiplicatively(a) if x is not None: return x.extract_multiplicatively(b) else: return x quotient = self / c if self.is_Number: if self is S.Infinity: if c.is_positive: return S.Infinity elif self is S.NegativeInfinity: if c.is_negative: return S.Infinity elif c.is_positive: return S.NegativeInfinity elif self is S.ComplexInfinity: if not c.is_zero: return S.ComplexInfinity elif self.is_Integer: if not quotient.is_Integer: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Rational: if not quotient.is_Rational: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Float: if not quotient.is_Float: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit: if quotient.is_Mul and len(quotient.args) == 2: if quotient.args[0].is_Integer and quotient.args[0].is_positive and quotient.args[1] == self: return quotient elif quotient.is_Integer and c.is_Number: return quotient elif self.is_Add: cs, ps = self.primitive() # assert cs >= 1 if c.is_Number and c is not S.NegativeOne: # assert c != 1 (handled at top) if cs is not S.One: if c.is_negative: xc = -(cs.extract_multiplicatively(-c)) else: xc = cs.extract_multiplicatively(c) if xc is not None: return xc*ps # rely on 2-arg Mul to restore Add return # |c| != 1 can only be extracted from cs if c == ps: return cs # check args of ps newargs = [] for arg in ps.args: newarg = arg.extract_multiplicatively(c) if newarg is None: return # all or nothing newargs.append(newarg) if cs is not S.One: args = [cs*t for t in newargs] # args may be in different order return _unevaluated_Add(*args) else: return Add._from_args(newargs) elif self.is_Mul: args = list(self.args) for i, arg in enumerate(args): newarg = arg.extract_multiplicatively(c) if newarg is not None: args[i] = newarg return Mul(*args) elif self.is_Pow: if c.is_Pow and c.base == self.base: new_exp = self.exp.extract_additively(c.exp) if new_exp is not None: return self.base ** (new_exp) elif c == self.base: new_exp = self.exp.extract_additively(1) if new_exp is not None: return self.base ** (new_exp) def extract_additively(self, c): """Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples ======== >>> from sympy.abc import x, y >>> e = 2*x + 3 >>> e.extract_additively(x + 1) x + 2 >>> e.extract_additively(3*x) >>> e.extract_additively(4) >>> (y*(x + 1)).extract_additively(x + 1) >>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1) (x + 1)*(x + 2*y) + 3 Sometimes auto-expansion will return a less simplified result than desired; gcd_terms might be used in such cases: >>> from sympy import gcd_terms >>> (4*x*(y + 1) + y).extract_additively(x) 4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y >>> gcd_terms(_) x*(4*y + 3) + y See Also ======== extract_multiplicatively coeff as_coefficient """ c = sympify(c) if self is S.NaN: return None if c.is_zero: return self elif c == self: return S.Zero elif self == S.Zero: return None if self.is_Number: if not c.is_Number: return None co = self diff = co - c # XXX should we match types? i.e should 3 - .1 succeed? if (co > 0 and diff > 0 and diff < co or co < 0 and diff < 0 and diff > co): return diff return None if c.is_Number: co, t = self.as_coeff_Add() xa = co.extract_additively(c) if xa is None: return None return xa + t # handle the args[0].is_Number case separately # since we will have trouble looking for the coeff of # a number. if c.is_Add and c.args[0].is_Number: # whole term as a term factor co = self.coeff(c) xa0 = (co.extract_additively(1) or 0)*c if xa0: diff = self - co*c return (xa0 + (diff.extract_additively(c) or diff)) or None # term-wise h, t = c.as_coeff_Add() sh, st = self.as_coeff_Add() xa = sh.extract_additively(h) if xa is None: return None xa2 = st.extract_additively(t) if xa2 is None: return None return xa + xa2 # whole term as a term factor co = self.coeff(c) xa0 = (co.extract_additively(1) or 0)*c if xa0: diff = self - co*c return (xa0 + (diff.extract_additively(c) or diff)) or None # term-wise coeffs = [] for a in Add.make_args(c): ac, at = a.as_coeff_Mul() co = self.coeff(at) if not co: return None coc, cot = co.as_coeff_Add() xa = coc.extract_additively(ac) if xa is None: return None self -= co*at coeffs.append((cot + xa)*at) coeffs.append(self) return Add(*coeffs) @property def expr_free_symbols(self): """ Like ``free_symbols``, but returns the free symbols only if they are contained in an expression node. Examples ======== >>> from sympy.abc import x, y >>> (x + y).expr_free_symbols {x, y} If the expression is contained in a non-expression object, don't return the free symbols. Compare: >>> from sympy import Tuple >>> t = Tuple(x + y) >>> t.expr_free_symbols set() >>> t.free_symbols {x, y} """ return {j for i in self.args for j in i.expr_free_symbols} def could_extract_minus_sign(self): """Return True if self is not in a canonical form with respect to its sign. For most expressions, e, there will be a difference in e and -e. When there is, True will be returned for one and False for the other; False will be returned if there is no difference. Examples ======== >>> from sympy.abc import x, y >>> e = x - y >>> {i.could_extract_minus_sign() for i in (e, -e)} {False, True} """ negative_self = -self if self == negative_self: return False # e.g. zoo*x == -zoo*x self_has_minus = (self.extract_multiplicatively(-1) is not None) negative_self_has_minus = ( (negative_self).extract_multiplicatively(-1) is not None) if self_has_minus != negative_self_has_minus: return self_has_minus else: if self.is_Add: # We choose the one with less arguments with minus signs all_args = len(self.args) negative_args = len([False for arg in self.args if arg.could_extract_minus_sign()]) positive_args = all_args - negative_args if positive_args > negative_args: return False elif positive_args < negative_args: return True elif self.is_Mul: # We choose the one with an odd number of minus signs num, den = self.as_numer_denom() args = Mul.make_args(num) + Mul.make_args(den) arg_signs = [arg.could_extract_minus_sign() for arg in args] negative_args = list(filter(None, arg_signs)) return len(negative_args) % 2 == 1 # As a last resort, we choose the one with greater value of .sort_key() return bool(self.sort_key() < negative_self.sort_key()) def extract_branch_factor(self, allow_half=False): """ Try to write self as ``exp_polar(2*pi*I*n)*z`` in a nice way. Return (z, n). >>> from sympy import exp_polar, I, pi >>> from sympy.abc import x, y >>> exp_polar(I*pi).extract_branch_factor() (exp_polar(I*pi), 0) >>> exp_polar(2*I*pi).extract_branch_factor() (1, 1) >>> exp_polar(-pi*I).extract_branch_factor() (exp_polar(I*pi), -1) >>> exp_polar(3*pi*I + x).extract_branch_factor() (exp_polar(x + I*pi), 1) >>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor() (y*exp_polar(2*pi*x), -1) >>> exp_polar(-I*pi/2).extract_branch_factor() (exp_polar(-I*pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): >>> exp_polar(I*pi).extract_branch_factor(allow_half=True) (1, 1/2) >>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True) (1, 1) >>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True) (1, 3/2) >>> exp_polar(-I*pi).extract_branch_factor(allow_half=True) (1, -1/2) """ from sympy import exp_polar, pi, I, ceiling, Add n = S.Zero res = S.One args = Mul.make_args(self) exps = [] for arg in args: if isinstance(arg, exp_polar): exps += [arg.exp] else: res *= arg piimult = S.Zero extras = [] while exps: exp = exps.pop() if exp.is_Add: exps += exp.args continue if exp.is_Mul: coeff = exp.as_coefficient(pi*I) if coeff is not None: piimult += coeff continue extras += [exp] if piimult.is_number: coeff = piimult tail = () else: coeff, tail = piimult.as_coeff_add(*piimult.free_symbols) # round down to nearest multiple of 2 branchfact = ceiling(coeff/2 - S.Half)*2 n += branchfact/2 c = coeff - branchfact if allow_half: nc = c.extract_additively(1) if nc is not None: n += S.Half c = nc newexp = pi*I*Add(*((c, ) + tail)) + Add(*extras) if newexp != 0: res *= exp_polar(newexp) return res, n def _eval_is_polynomial(self, syms): if self.free_symbols.intersection(syms) == set(): return True return False def is_polynomial(self, *syms): r""" Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are "polynomials" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, \*syms) should work if and only if expr.is_polynomial(\*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples ======== >>> from sympy import Symbol >>> x = Symbol('x') >>> ((x**2 + 1)**4).is_polynomial(x) True >>> ((x**2 + 1)**4).is_polynomial() True >>> (2**x + 1).is_polynomial(x) False >>> n = Symbol('n', nonnegative=True, integer=True) >>> (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. >>> from sympy import sqrt, factor, cancel >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1) >>> a.is_polynomial(y) False >>> factor(a) y + 1 >>> factor(a).is_polynomial(y) True >>> b = (y**2 + 2*y + 1)/(y + 1) >>> b.is_polynomial(y) False >>> cancel(b) y + 1 >>> cancel(b).is_polynomial(y) True See also .is_rational_function() """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set(): # constant polynomial return True else: return self._eval_is_polynomial(syms) def _eval_is_rational_function(self, syms): if self.free_symbols.intersection(syms) == set(): return True return False def is_rational_function(self, *syms): """ Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "rational functions" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples ======== >>> from sympy import Symbol, sin >>> from sympy.abc import x, y >>> (x/y).is_rational_function() True >>> (x**2).is_rational_function() True >>> (x/sin(y)).is_rational_function(y) False >>> n = Symbol('n', integer=True) >>> (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. >>> from sympy import sqrt, factor >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1)/y >>> a.is_rational_function(y) False >>> factor(a) (y + 1)/y >>> factor(a).is_rational_function(y) True See also is_algebraic_expr(). """ if self in [S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity]: return False if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set(): # constant rational function return True else: return self._eval_is_rational_function(syms) def _eval_is_meromorphic(self, x, a): # Default implementation, return True for constants. return None if self.has(x) else True def is_meromorphic(self, x, a): """ This tests whether an expression is meromorphic as a function of the given symbol ``x`` at the point ``a``. This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples ======== >>> from sympy import zoo, log, sin, sqrt >>> from sympy.abc import x >>> f = 1/x**2 + 1 - 2*x**3 >>> f.is_meromorphic(x, 0) True >>> f.is_meromorphic(x, 1) True >>> f.is_meromorphic(x, zoo) True >>> g = x**log(3) >>> g.is_meromorphic(x, 0) False >>> g.is_meromorphic(x, 1) True >>> g.is_meromorphic(x, zoo) False >>> h = sin(1/x)*x**2 >>> h.is_meromorphic(x, 0) False >>> h.is_meromorphic(x, 1) True >>> h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. >>> log(x).is_meromorphic(x, -1) True >>> log(x).is_meromorphic(x, 0) False >>> sqrt(x).is_meromorphic(x, -1) True >>> sqrt(x).is_meromorphic(x, 0) False """ if not x.is_symbol: raise TypeError("{} should be of symbol type".format(x)) a = sympify(a) return self._eval_is_meromorphic(x, a) def _eval_is_algebraic_expr(self, syms): if self.free_symbols.intersection(syms) == set(): return True return False def is_algebraic_expr(self, *syms): """ This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "algebraic expressions" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples ======== >>> from sympy import Symbol, sqrt >>> x = Symbol('x', real=True) >>> sqrt(1 + x).is_rational_function() False >>> sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. >>> from sympy import exp, factor >>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1) >>> a.is_algebraic_expr(x) False >>> factor(a).is_algebraic_expr() True See Also ======== is_rational_function() References ========== - https://en.wikipedia.org/wiki/Algebraic_expression """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set(): # constant algebraic expression return True else: return self._eval_is_algebraic_expr(syms) ################################################################################### ##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ################## ################################################################################### def series(self, x=None, x0=0, n=6, dir="+", logx=None, cdir=0): """ Series expansion of "self" around ``x = x0`` yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of "self" around the point ``x = x0`` with respect to ``x`` up to ``O((x - x0)**n, x, x0)`` (default n is 6). If ``x=None`` and ``self`` is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters ========== expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which ``x`` is calculated. Can be any value from ``-oo`` to ``oo``. n : Value The number of terms upto which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If ``dir="+"``, then (x->x0+). If ``dir="-", then (x->x0-). For infinite ``x0`` (``oo`` or ``-oo``), the ``dir`` argument is determined from the direction of the infinity (i.e., ``dir="-"`` for ``oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples ======== >>> from sympy import cos, exp, tan >>> from sympy.abc import x, y >>> cos(x).series() 1 - x**2/2 + x**4/24 + O(x**6) >>> cos(x).series(n=4) 1 - x**2/2 + O(x**4) >>> cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1)) >>> e = cos(x + exp(y)) >>> e.series(y, n=2) cos(x + 1) - y*sin(x + 1) + O(y**2) >>> e.series(x, n=2) cos(exp(y)) - x*sin(exp(y)) + O(x**2) If ``n=None`` then a generator of the series terms will be returned. >>> term=cos(x).series(n=None) >>> [next(term) for i in range(2)] [1, -x**2/2] For ``dir=+`` (default) the series is calculated from the right and for ``dir=-`` the series from the left. For smooth functions this flag will not alter the results. >>> abs(x).series(dir="+") x >>> abs(x).series(dir="-") -x >>> f = tan(x) >>> f.series(x, 2, 6, "+") tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) + (x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 + 5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 + 2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2)) >>> f.series(x, 2, 3, "-") tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2)) + O((x - 2)**3, (x, 2)) Returns ======= Expr : Expression Series expansion of the expression about x0 Raises ====== TypeError If "n" and "x0" are infinity objects PoleError If "x0" is an infinity object """ from sympy import collect, Dummy, Order, Rational, Symbol, ceiling if x is None: syms = self.free_symbols if not syms: return self elif len(syms) > 1: raise ValueError('x must be given for multivariate functions.') x = syms.pop() if isinstance(x, Symbol): dep = x in self.free_symbols else: d = Dummy() dep = d in self.xreplace({x: d}).free_symbols if not dep: if n is None: return (s for s in [self]) else: return self if len(dir) != 1 or dir not in '+-': raise ValueError("Dir must be '+' or '-'") if x0 in [S.Infinity, S.NegativeInfinity]: sgn = 1 if x0 is S.Infinity else -1 s = self.subs(x, sgn/x).series(x, n=n, dir='+', cdir=cdir) if n is None: return (si.subs(x, sgn/x) for si in s) return s.subs(x, sgn/x) # use rep to shift origin to x0 and change sign (if dir is negative) # and undo the process with rep2 if x0 or dir == '-': if dir == '-': rep = -x + x0 rep2 = -x rep2b = x0 else: rep = x + x0 rep2 = x rep2b = -x0 s = self.subs(x, rep).series(x, x0=0, n=n, dir='+', logx=logx, cdir=cdir) if n is None: # lseries... return (si.subs(x, rep2 + rep2b) for si in s) return s.subs(x, rep2 + rep2b) # from here on it's x0=0 and dir='+' handling if x.is_positive is x.is_negative is None or x.is_Symbol is not True: # replace x with an x that has a positive assumption xpos = Dummy('x', positive=True, finite=True) rv = self.subs(x, xpos).series(xpos, x0, n, dir, logx=logx, cdir=cdir) if n is None: return (s.subs(xpos, x) for s in rv) else: return rv.subs(xpos, x) if n is not None: # nseries handling s1 = self._eval_nseries(x, n=n, logx=logx, cdir=cdir) o = s1.getO() or S.Zero if o: # make sure the requested order is returned ngot = o.getn() if ngot > n: # leave o in its current form (e.g. with x*log(x)) so # it eats terms properly, then replace it below if n != 0: s1 += o.subs(x, x**Rational(n, ngot)) else: s1 += Order(1, x) elif ngot < n: # increase the requested number of terms to get the desired # number keep increasing (up to 9) until the received order # is different than the original order and then predict how # many additional terms are needed for more in range(1, 9): s1 = self._eval_nseries(x, n=n + more, logx=logx, cdir=cdir) newn = s1.getn() if newn != ngot: ndo = n + ceiling((n - ngot)*more/(newn - ngot)) s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir) while s1.getn() < n: s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir) ndo += 1 break else: raise ValueError('Could not calculate %s terms for %s' % (str(n), self)) s1 += Order(x**n, x) o = s1.getO() s1 = s1.removeO() else: o = Order(x**n, x) s1done = s1.doit() if (s1done + o).removeO() == s1done: o = S.Zero try: return collect(s1, x) + o except NotImplementedError: return s1 + o else: # lseries handling def yield_lseries(s): """Return terms of lseries one at a time.""" for si in s: if not si.is_Add: yield si continue # yield terms 1 at a time if possible # by increasing order until all the # terms have been returned yielded = 0 o = Order(si, x)*x ndid = 0 ndo = len(si.args) while 1: do = (si - yielded + o).removeO() o *= x if not do or do.is_Order: continue if do.is_Add: ndid += len(do.args) else: ndid += 1 yield do if ndid == ndo: break yielded += do return yield_lseries(self.removeO()._eval_lseries(x, logx=logx, cdir=cdir)) def aseries(self, x=None, n=6, bound=0, hir=False): """Asymptotic Series expansion of self. This is equivalent to ``self.series(x, oo, n)``. Parameters ========== self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The number of terms upto which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the ``bound`` parameter to give limit on rewriting coefficients in its normalised form. Examples ======== >>> from sympy import sin, exp >>> from sympy.abc import x >>> e = sin(1/x + exp(-x)) - sin(1/x) >>> e.aseries(x) (1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x) >>> e.aseries(x, n=3, hir=True) -exp(-2*x)*sin(1/x)/2 + exp(-x)*cos(1/x) + O(exp(-3*x), (x, oo)) >>> e = exp(exp(x)/(1 - 1/x)) >>> e.aseries(x) exp(exp(x)/(1 - 1/x)) >>> e.aseries(x, bound=3) exp(exp(x)/x**2)*exp(exp(x)/x)*exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x**2)*exp(exp(x)) Returns ======= Expr Asymptotic series expansion of the expression. Notes ===== This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either ``O(x ** (-n))`` or ``O(w ** (-n))`` where ``w`` belongs to the most rapidly varying expression of ``self``. References ========== .. [1] A New Algorithm for Computing Asymptotic Series - Dominik Gruntz .. [2] Gruntz thesis - p90 .. [3] http://en.wikipedia.org/wiki/Asymptotic_expansion See Also ======== Expr.aseries: See the docstring of this function for complete details of this wrapper. """ from sympy import Order, Dummy from sympy.functions import exp, log from sympy.series.gruntz import mrv, rewrite if x.is_positive is x.is_negative is None: xpos = Dummy('x', positive=True) return self.subs(x, xpos).aseries(xpos, n, bound, hir).subs(xpos, x) om, exps = mrv(self, x) # We move one level up by replacing `x` by `exp(x)`, and then # computing the asymptotic series for f(exp(x)). Then asymptotic series # can be obtained by moving one-step back, by replacing x by ln(x). if x in om: s = self.subs(x, exp(x)).aseries(x, n, bound, hir).subs(x, log(x)) if s.getO(): return s + Order(1/x**n, (x, S.Infinity)) return s k = Dummy('k', positive=True) # f is rewritten in terms of omega func, logw = rewrite(exps, om, x, k) if self in om: if bound <= 0: return self s = (self.exp).aseries(x, n, bound=bound) s = s.func(*[t.removeO() for t in s.args]) res = exp(s.subs(x, 1/x).as_leading_term(x).subs(x, 1/x)) func = exp(self.args[0] - res.args[0]) / k logw = log(1/res) s = func.series(k, 0, n) # Hierarchical series if hir: return s.subs(k, exp(logw)) o = s.getO() terms = sorted(Add.make_args(s.removeO()), key=lambda i: int(i.as_coeff_exponent(k)[1])) s = S.Zero has_ord = False # Then we recursively expand these coefficients one by one into # their asymptotic series in terms of their most rapidly varying subexpressions. for t in terms: coeff, expo = t.as_coeff_exponent(k) if coeff.has(x): # Recursive step snew = coeff.aseries(x, n, bound=bound-1) if has_ord and snew.getO(): break elif snew.getO(): has_ord = True s += (snew * k**expo) else: s += t if not o or has_ord: return s.subs(k, exp(logw)) return (s + o).subs(k, exp(logw)) def taylor_term(self, n, x, *previous_terms): """General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the "previous_terms". """ from sympy import Dummy, factorial x = sympify(x) _x = Dummy('x') return self.subs(x, _x).diff(_x, n).subs(_x, x).subs(x, 0) * x**n / factorial(n) def lseries(self, x=None, x0=0, dir='+', logx=None, cdir=0): """ Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don't know how many you should ask for in nseries() using the "n" parameter. See also nseries(). """ return self.series(x, x0, n=None, dir=dir, logx=logx, cdir=cdir) def _eval_lseries(self, x, logx=None, cdir=0): # default implementation of lseries is using nseries(), and adaptively # increasing the "n". As you can see, it is not very efficient, because # we are calculating the series over and over again. Subclasses should # override this method and implement much more efficient yielding of # terms. n = 0 series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir) if not series.is_Order: newseries = series.cancel() if not newseries.is_Order: if series.is_Add: yield series.removeO() else: yield series return else: series = newseries while series.is_Order: n += 1 series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir) e = series.removeO() yield e if e.is_zero: return while 1: while 1: n += 1 series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir).removeO() if e != series: break if (series - self).cancel() is S.Zero: return yield series - e e = series def nseries(self, x=None, x0=0, n=6, dir='+', logx=None, cdir=0): """ Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates "n" terms in the innermost expressions and then builds up the final series just by "cross-multiplying" everything out. The optional ``logx`` parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we don't have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples ======== >>> from sympy import sin, log, Symbol >>> from sympy.abc import x, y >>> sin(x).nseries(x, 0, 6) x - x**3/6 + x**5/120 + O(x**6) >>> log(x+1).nseries(x, 0, 5) x - x**2/2 + x**3/3 - x**4/4 + O(x**5) Handling of the ``logx`` parameter --- in the following example the expansion fails since ``sin`` does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): >>> e = sin(log(x)) >>> e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... >>> logx = Symbol('logx') >>> e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but gives only an Order term unless the ``logx`` parameter is used: >>> e = x**y >>> e.nseries(x, 0, 2) O(log(x)**2) >>> e.nseries(x, 0, 2, logx=logx) exp(logx*y) """ if x and not x in self.free_symbols: return self if x is None or x0 or dir != '+': # {see XPOS above} or (x.is_positive == x.is_negative == None): return self.series(x, x0, n, dir, cdir=cdir) else: return self._eval_nseries(x, n=n, logx=logx, cdir=cdir) def _eval_nseries(self, x, n, logx, cdir): """ Return terms of series for self up to O(x**n) at x=0 from the positive direction. This is a method that should be overridden in subclasses. Users should never call this method directly (use .nseries() instead), so you don't have to write docstrings for _eval_nseries(). """ from sympy.utilities.misc import filldedent raise NotImplementedError(filldedent(""" The _eval_nseries method should be added to %s to give terms up to O(x**n) at x=0 from the positive direction so it is available when nseries calls it.""" % self.func) ) def limit(self, x, xlim, dir='+'): """ Compute limit x->xlim. """ from sympy.series.limits import limit return limit(self, x, xlim, dir) def compute_leading_term(self, x, logx=None): """ as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. """ from sympy import Dummy, log, Piecewise, piecewise_fold from sympy.series.gruntz import calculate_series if self.has(Piecewise): expr = piecewise_fold(self) else: expr = self if self.removeO() == 0: return self if logx is None: d = Dummy('logx') s = calculate_series(expr, x, d).subs(d, log(x)) else: s = calculate_series(expr, x, logx) return s.as_leading_term(x) @cacheit def as_leading_term(self, *symbols, cdir=0): """ Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples ======== >>> from sympy.abc import x >>> (1 + x + x**2).as_leading_term(x) 1 >>> (1/x**2 + x + x**2).as_leading_term(x) x**(-2) """ from sympy import powsimp if len(symbols) > 1: c = self for x in symbols: c = c.as_leading_term(x, cdir=cdir) return c elif not symbols: return self x = sympify(symbols[0]) if not x.is_symbol: raise ValueError('expecting a Symbol but got %s' % x) if x not in self.free_symbols: return self obj = self._eval_as_leading_term(x, cdir=cdir) if obj is not None: return powsimp(obj, deep=True, combine='exp') raise NotImplementedError('as_leading_term(%s, %s)' % (self, x)) def _eval_as_leading_term(self, x, cdir=0): return self def as_coeff_exponent(self, x): """ ``c*x**e -> c,e`` where x can be any symbolic expression. """ from sympy import collect s = collect(self, x) c, p = s.as_coeff_mul(x) if len(p) == 1: b, e = p[0].as_base_exp() if b == x: return c, e return s, S.Zero def leadterm(self, x, cdir=0): """ Returns the leading term a*x**b as a tuple (a, b). Examples ======== >>> from sympy.abc import x >>> (1+x+x**2).leadterm(x) (1, 0) >>> (1/x**2+x+x**2).leadterm(x) (1, -2) """ from sympy import Dummy, log l = self.as_leading_term(x, cdir=cdir) d = Dummy('logx') if l.has(log(x)): l = l.subs(log(x), d) c, e = l.as_coeff_exponent(x) if x in c.free_symbols: from sympy.utilities.misc import filldedent raise ValueError(filldedent(""" cannot compute leadterm(%s, %s). The coefficient should have been free of %s but got %s""" % (self, x, x, c))) c = c.subs(d, log(x)) return c, e def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ return S.One, self def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """ return S.Zero, self def fps(self, x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False): """ Compute formal power power series of self. See the docstring of the :func:`fps` function in sympy.series.formal for more information. """ from sympy.series.formal import fps return fps(self, x, x0, dir, hyper, order, rational, full) def fourier_series(self, limits=None): """Compute fourier sine/cosine series of self. See the docstring of the :func:`fourier_series` in sympy.series.fourier for more information. """ from sympy.series.fourier import fourier_series return fourier_series(self, limits) ################################################################################### ##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS #################### ################################################################################### def diff(self, *symbols, **assumptions): assumptions.setdefault("evaluate", True) return _derivative_dispatch(self, *symbols, **assumptions) ########################################################################### ###################### EXPRESSION EXPANSION METHODS ####################### ########################################################################### # Relevant subclasses should override _eval_expand_hint() methods. See # the docstring of expand() for more info. def _eval_expand_complex(self, **hints): real, imag = self.as_real_imag(**hints) return real + S.ImaginaryUnit*imag @staticmethod def _expand_hint(expr, hint, deep=True, **hints): """ Helper for ``expand()``. Recursively calls ``expr._eval_expand_hint()``. Returns ``(expr, hit)``, where expr is the (possibly) expanded ``expr`` and ``hit`` is ``True`` if ``expr`` was truly expanded and ``False`` otherwise. """ hit = False # XXX: Hack to support non-Basic args # | # V if deep and getattr(expr, 'args', ()) and not expr.is_Atom: sargs = [] for arg in expr.args: arg, arghit = Expr._expand_hint(arg, hint, **hints) hit |= arghit sargs.append(arg) if hit: expr = expr.func(*sargs) if hasattr(expr, hint): newexpr = getattr(expr, hint)(**hints) if newexpr != expr: return (newexpr, True) return (expr, hit) @cacheit def expand(self, deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints): """ Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. """ from sympy.simplify.radsimp import fraction hints.update(power_base=power_base, power_exp=power_exp, mul=mul, log=log, multinomial=multinomial, basic=basic) expr = self if hints.pop('frac', False): n, d = [a.expand(deep=deep, modulus=modulus, **hints) for a in fraction(self)] return n/d elif hints.pop('denom', False): n, d = fraction(self) return n/d.expand(deep=deep, modulus=modulus, **hints) elif hints.pop('numer', False): n, d = fraction(self) return n.expand(deep=deep, modulus=modulus, **hints)/d # Although the hints are sorted here, an earlier hint may get applied # at a given node in the expression tree before another because of how # the hints are applied. e.g. expand(log(x*(y + z))) -> log(x*y + # x*z) because while applying log at the top level, log and mul are # applied at the deeper level in the tree so that when the log at the # upper level gets applied, the mul has already been applied at the # lower level. # Additionally, because hints are only applied once, the expression # may not be expanded all the way. For example, if mul is applied # before multinomial, x*(x + 1)**2 won't be expanded all the way. For # now, we just use a special case to make multinomial run before mul, # so that at least polynomials will be expanded all the way. In the # future, smarter heuristics should be applied. # TODO: Smarter heuristics def _expand_hint_key(hint): """Make multinomial come before mul""" if hint == 'mul': return 'mulz' return hint for hint in sorted(hints.keys(), key=_expand_hint_key): use_hint = hints[hint] if use_hint: hint = '_eval_expand_' + hint expr, hit = Expr._expand_hint(expr, hint, deep=deep, **hints) while True: was = expr if hints.get('multinomial', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_multinomial', deep=deep, **hints) if hints.get('mul', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_mul', deep=deep, **hints) if hints.get('log', False): expr, _ = Expr._expand_hint( expr, '_eval_expand_log', deep=deep, **hints) if expr == was: break if modulus is not None: modulus = sympify(modulus) if not modulus.is_Integer or modulus <= 0: raise ValueError( "modulus must be a positive integer, got %s" % modulus) terms = [] for term in Add.make_args(expr): coeff, tail = term.as_coeff_Mul(rational=True) coeff %= modulus if coeff: terms.append(coeff*tail) expr = Add(*terms) return expr ########################################################################### ################### GLOBAL ACTION VERB WRAPPER METHODS #################### ########################################################################### def integrate(self, *args, **kwargs): """See the integrate function in sympy.integrals""" from sympy.integrals import integrate return integrate(self, *args, **kwargs) def nsimplify(self, constants=[], tolerance=None, full=False): """See the nsimplify function in sympy.simplify""" from sympy.simplify import nsimplify return nsimplify(self, constants, tolerance, full) def separate(self, deep=False, force=False): """See the separate function in sympy.simplify""" from sympy.core.function import expand_power_base return expand_power_base(self, deep=deep, force=force) def collect(self, syms, func=None, evaluate=True, exact=False, distribute_order_term=True): """See the collect function in sympy.simplify""" from sympy.simplify import collect return collect(self, syms, func, evaluate, exact, distribute_order_term) def together(self, *args, **kwargs): """See the together function in sympy.polys""" from sympy.polys import together return together(self, *args, **kwargs) def apart(self, x=None, **args): """See the apart function in sympy.polys""" from sympy.polys import apart return apart(self, x, **args) def ratsimp(self): """See the ratsimp function in sympy.simplify""" from sympy.simplify import ratsimp return ratsimp(self) def trigsimp(self, **args): """See the trigsimp function in sympy.simplify""" from sympy.simplify import trigsimp return trigsimp(self, **args) def radsimp(self, **kwargs): """See the radsimp function in sympy.simplify""" from sympy.simplify import radsimp return radsimp(self, **kwargs) def powsimp(self, *args, **kwargs): """See the powsimp function in sympy.simplify""" from sympy.simplify import powsimp return powsimp(self, *args, **kwargs) def combsimp(self): """See the combsimp function in sympy.simplify""" from sympy.simplify import combsimp return combsimp(self) def gammasimp(self): """See the gammasimp function in sympy.simplify""" from sympy.simplify import gammasimp return gammasimp(self) def factor(self, *gens, **args): """See the factor() function in sympy.polys.polytools""" from sympy.polys import factor return factor(self, *gens, **args) def refine(self, assumption=True): """See the refine function in sympy.assumptions""" from sympy.assumptions import refine return refine(self, assumption) def cancel(self, *gens, **args): """See the cancel function in sympy.polys""" from sympy.polys import cancel return cancel(self, *gens, **args) def invert(self, g, *gens, **args): """Return the multiplicative inverse of ``self`` mod ``g`` where ``self`` (and ``g``) may be symbolic expressions). See Also ======== sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert """ from sympy.polys.polytools import invert from sympy.core.numbers import mod_inverse if self.is_number and getattr(g, 'is_number', True): return mod_inverse(self, g) return invert(self, g, *gens, **args) def round(self, n=None): """Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples ======== >>> from sympy import pi, E, I, S, Number >>> pi.round() 3 >>> pi.round(2) 3.14 >>> (2*pi + E*I).round() 6 + 3*I The round method has a chopping effect: >>> (2*pi + I/10).round() 6 >>> (pi/10 + 2*I).round() 2*I >>> (pi/10 + E*I).round(2) 0.31 + 2.72*I Notes ===== The Python ``round`` function uses the SymPy ``round`` method so it will always return a SymPy number (not a Python float or int): >>> isinstance(round(S(123), -2), Number) True """ from sympy.core.numbers import Float x = self if not x.is_number: raise TypeError("can't round symbolic expression") if not x.is_Atom: if not pure_complex(x.n(2), or_real=True): raise TypeError( 'Expected a number but got %s:' % func_name(x)) elif x in (S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity): return x if not x.is_extended_real: r, i = x.as_real_imag() return r.round(n) + S.ImaginaryUnit*i.round(n) if not x: return S.Zero if n is None else x p = as_int(n or 0) if x.is_Integer: return Integer(round(int(x), p)) digits_to_decimal = _mag(x) # _mag(12) = 2, _mag(.012) = -1 allow = digits_to_decimal + p precs = [f._prec for f in x.atoms(Float)] dps = prec_to_dps(max(precs)) if precs else None if dps is None: # assume everything is exact so use the Python # float default or whatever was requested dps = max(15, allow) else: allow = min(allow, dps) # this will shift all digits to right of decimal # and give us dps to work with as an int shift = -digits_to_decimal + dps extra = 1 # how far we look past known digits # NOTE # mpmath will calculate the binary representation to # an arbitrary number of digits but we must base our # answer on a finite number of those digits, e.g. # .575 2589569785738035/2**52 in binary. # mpmath shows us that the first 18 digits are # >>> Float(.575).n(18) # 0.574999999999999956 # The default precision is 15 digits and if we ask # for 15 we get # >>> Float(.575).n(15) # 0.575000000000000 # mpmath handles rounding at the 15th digit. But we # need to be careful since the user might be asking # for rounding at the last digit and our semantics # are to round toward the even final digit when there # is a tie. So the extra digit will be used to make # that decision. In this case, the value is the same # to 15 digits: # >>> Float(.575).n(16) # 0.5750000000000000 # Now converting this to the 15 known digits gives # 575000000000000.0 # which rounds to integer # 5750000000000000 # And now we can round to the desired digt, e.g. at # the second from the left and we get # 5800000000000000 # and rescaling that gives # 0.58 # as the final result. # If the value is made slightly less than 0.575 we might # still obtain the same value: # >>> Float(.575-1e-16).n(16)*10**15 # 574999999999999.8 # What 15 digits best represents the known digits (which are # to the left of the decimal? 5750000000000000, the same as # before. The only way we will round down (in this case) is # if we declared that we had more than 15 digits of precision. # For example, if we use 16 digits of precision, the integer # we deal with is # >>> Float(.575-1e-16).n(17)*10**16 # 5749999999999998.4 # and this now rounds to 5749999999999998 and (if we round to # the 2nd digit from the left) we get 5700000000000000. # xf = x.n(dps + extra)*Pow(10, shift) xi = Integer(xf) # use the last digit to select the value of xi # nearest to x before rounding at the desired digit sign = 1 if x > 0 else -1 dif2 = sign*(xf - xi).n(extra) if dif2 < 0: raise NotImplementedError( 'not expecting int(x) to round away from 0') if dif2 > .5: xi += sign # round away from 0 elif dif2 == .5: xi += sign if xi%2 else -sign # round toward even # shift p to the new position ip = p - shift # let Python handle the int rounding then rescale xr = round(xi.p, ip) # restore scale rv = Rational(xr, Pow(10, shift)) # return Float or Integer if rv.is_Integer: if n is None: # the single-arg case return rv # use str or else it won't be a float return Float(str(rv), dps) # keep same precision else: if not allow and rv > self: allow += 1 return Float(rv, allow) __round__ = round def _eval_derivative_matrix_lines(self, x): from sympy.matrices.expressions.matexpr import _LeftRightArgs return [_LeftRightArgs([S.One, S.One], higher=self._eval_derivative(x))] class AtomicExpr(Atom, Expr): """ A parent class for object which are both atoms and Exprs. For example: Symbol, Number, Rational, Integer, ... But not: Add, Mul, Pow, ... """ is_number = False is_Atom = True __slots__ = () def _eval_derivative(self, s): if self == s: return S.One return S.Zero def _eval_derivative_n_times(self, s, n): from sympy import Piecewise, Eq from sympy import Tuple, MatrixExpr from sympy.matrices.common import MatrixCommon if isinstance(s, (MatrixCommon, Tuple, Iterable, MatrixExpr)): return super()._eval_derivative_n_times(s, n) if self == s: return Piecewise((self, Eq(n, 0)), (1, Eq(n, 1)), (0, True)) else: return Piecewise((self, Eq(n, 0)), (0, True)) def _eval_is_polynomial(self, syms): return True def _eval_is_rational_function(self, syms): return True def _eval_is_meromorphic(self, x, a): from sympy.calculus.util import AccumBounds return (not self.is_Number or self.is_finite) and not isinstance(self, AccumBounds) def _eval_is_algebraic_expr(self, syms): return True def _eval_nseries(self, x, n, logx, cdir=0): return self @property def expr_free_symbols(self): return {self} def _mag(x): """Return integer ``i`` such that .1 <= x/10**i < 1 Examples ======== >>> from sympy.core.expr import _mag >>> from sympy import Float >>> _mag(Float(.1)) 0 >>> _mag(Float(.01)) -1 >>> _mag(Float(1234)) 4 """ from math import log10, ceil, log from sympy import Float xpos = abs(x.n()) if not xpos: return S.Zero try: mag_first_dig = int(ceil(log10(xpos))) except (ValueError, OverflowError): mag_first_dig = int(ceil(Float(mpf_log(xpos._mpf_, 53))/log(10))) # check that we aren't off by 1 if (xpos/10**mag_first_dig) >= 1: assert 1 <= (xpos/10**mag_first_dig) < 10 mag_first_dig += 1 return mag_first_dig class UnevaluatedExpr(Expr): """ Expression that is not evaluated unless released. Examples ======== >>> from sympy import UnevaluatedExpr >>> from sympy.abc import x >>> x*(1/x) 1 >>> x*UnevaluatedExpr(1/x) x*1/x """ def __new__(cls, arg, **kwargs): arg = _sympify(arg) obj = Expr.__new__(cls, arg, **kwargs) return obj def doit(self, **kwargs): if kwargs.get("deep", True): return self.args[0].doit(**kwargs) else: return self.args[0] def unchanged(func, *args): """Return True if `func` applied to the `args` is unchanged. Can be used instead of `assert foo == foo`. Examples ======== >>> from sympy import Piecewise, cos, pi >>> from sympy.core.expr import unchanged >>> from sympy.abc import x >>> unchanged(cos, 1) # instead of assert cos(1) == cos(1) True >>> unchanged(cos, pi) False Comparison of args uses the builtin capabilities of the object's arguments to test for equality so args can be defined loosely. Here, the ExprCondPair arguments of Piecewise compare as equal to the tuples that can be used to create the Piecewise: >>> unchanged(Piecewise, (x, x > 1), (0, True)) True """ f = func(*args) return f.func == func and f.args == args class ExprBuilder: def __init__(self, op, args=[], validator=None, check=True): if not hasattr(op, "__call__"): raise TypeError("op {} needs to be callable".format(op)) self.op = op self.args = args self.validator = validator if (validator is not None) and check: self.validate() @staticmethod def _build_args(args): return [i.build() if isinstance(i, ExprBuilder) else i for i in args] def validate(self): if self.validator is None: return args = self._build_args(self.args) self.validator(*args) def build(self, check=True): args = self._build_args(self.args) if self.validator and check: self.validator(*args) return self.op(*args) def append_argument(self, arg, check=True): self.args.append(arg) if self.validator and check: self.validate(*self.args) def __getitem__(self, item): if item == 0: return self.op else: return self.args[item-1] def __repr__(self): return str(self.build()) def search_element(self, elem): for i, arg in enumerate(self.args): if isinstance(arg, ExprBuilder): ret = arg.search_index(elem) if ret is not None: return (i,) + ret elif id(arg) == id(elem): return (i,) return None from .mul import Mul from .add import Add from .power import Pow from .function import Function, _derivative_dispatch from .mod import Mod from .exprtools import factor_terms from .numbers import Integer, Rational
6ca916fb6dfd3f859a1ba9ce016e9a43bed469d03a8c97efa4a04d9a787e2c54
from typing import Dict, Union, Type from sympy.utilities.exceptions import SymPyDeprecationWarning from .basic import S, Atom from .compatibility import ordered from .basic import Basic from .evalf import EvalfMixin from .function import AppliedUndef from .sympify import _sympify, SympifyError from .parameters import global_parameters from sympy.core.logic import fuzzy_bool, fuzzy_xor, fuzzy_and, fuzzy_not from sympy.logic.boolalg import Boolean, BooleanAtom __all__ = ( 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan', 'StrictGreaterThan', 'GreaterThan', ) from .expr import Expr from sympy.multipledispatch import dispatch from .containers import Tuple from .symbol import Symbol def _nontrivBool(side): return isinstance(side, Boolean) and \ not isinstance(side, Atom) # Note, see issue 4986. Ideally, we wouldn't want to subclass both Boolean # and Expr. # from .. import Expr def _canonical(cond): # return a condition in which all relationals are canonical reps = {r: r.canonical for r in cond.atoms(Relational)} return cond.xreplace(reps) # XXX: AttributeError was being caught here but it wasn't triggered by any of # the tests so I've removed it... class Relational(Boolean, EvalfMixin): """Base class for all relation types. Explanation =========== Subclasses of Relational should generally be instantiated directly, but Relational can be instantiated with a valid ``rop`` value to dispatch to the appropriate subclass. Parameters ========== rop : str or None Indicates what subclass to instantiate. Valid values can be found in the keys of Relational.ValidRelationOperator. Examples ======== >>> from sympy import Rel >>> from sympy.abc import x, y >>> Rel(y, x + x**2, '==') Eq(y, x**2 + x) """ __slots__ = () ValidRelationOperator = {} # type: Dict[Union[str, None], Type[Relational]] is_Relational = True # ValidRelationOperator - Defined below, because the necessary classes # have not yet been defined def __new__(cls, lhs, rhs, rop=None, **assumptions): # If called by a subclass, do nothing special and pass on to Basic. if cls is not Relational: return Basic.__new__(cls, lhs, rhs, **assumptions) # XXX: Why do this? There should be a separate function to make a # particular subclass of Relational from a string. # # If called directly with an operator, look up the subclass # corresponding to that operator and delegate to it cls = cls.ValidRelationOperator.get(rop, None) if cls is None: raise ValueError("Invalid relational operator symbol: %r" % rop) if not issubclass(cls, (Eq, Ne)): # validate that Booleans are not being used in a relational # other than Eq/Ne; # Note: Symbol is a subclass of Boolean but is considered # acceptable here. if any(map(_nontrivBool, (lhs, rhs))): from sympy.utilities.misc import filldedent raise TypeError(filldedent(''' A Boolean argument can only be used in Eq and Ne; all other relationals expect real expressions. ''')) return cls(lhs, rhs, **assumptions) @property def lhs(self): """The left-hand side of the relation.""" return self._args[0] @property def rhs(self): """The right-hand side of the relation.""" return self._args[1] @property def reversed(self): """Return the relationship with sides reversed. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x >>> Eq(x, 1) Eq(x, 1) >>> _.reversed Eq(1, x) >>> x < 1 x < 1 >>> _.reversed 1 > x """ ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne} a, b = self.args return Relational.__new__(ops.get(self.func, self.func), b, a) @property def reversedsign(self): """Return the relationship with signs reversed. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x >>> Eq(x, 1) Eq(x, 1) >>> _.reversedsign Eq(-x, -1) >>> x < 1 x < 1 >>> _.reversedsign -x > -1 """ a, b = self.args if not (isinstance(a, BooleanAtom) or isinstance(b, BooleanAtom)): ops = {Eq: Eq, Gt: Lt, Ge: Le, Lt: Gt, Le: Ge, Ne: Ne} return Relational.__new__(ops.get(self.func, self.func), -a, -b) else: return self @property def negated(self): """Return the negated relationship. Examples ======== >>> from sympy import Eq >>> from sympy.abc import x >>> Eq(x, 1) Eq(x, 1) >>> _.negated Ne(x, 1) >>> x < 1 x < 1 >>> _.negated x >= 1 Notes ===== This works more or less identical to ``~``/``Not``. The difference is that ``negated`` returns the relationship even if ``evaluate=False``. Hence, this is useful in code when checking for e.g. negated relations to existing ones as it will not be affected by the `evaluate` flag. """ ops = {Eq: Ne, Ge: Lt, Gt: Le, Le: Gt, Lt: Ge, Ne: Eq} # If there ever will be new Relational subclasses, the following line # will work until it is properly sorted out # return ops.get(self.func, lambda a, b, evaluate=False: ~(self.func(a, # b, evaluate=evaluate)))(*self.args, evaluate=False) return Relational.__new__(ops.get(self.func), *self.args) def _eval_evalf(self, prec): return self.func(*[s._evalf(prec) for s in self.args]) @property def canonical(self): """Return a canonical form of the relational by putting a number on the rhs, canonically removing a sign or else ordering the args canonically. No other simplification is attempted. Examples ======== >>> from sympy.abc import x, y >>> x < 2 x < 2 >>> _.reversed.canonical x < 2 >>> (-y < x).canonical x > -y >>> (-y > x).canonical x < -y >>> (-y < -x).canonical x < y """ args = self.args r = self if r.rhs.is_number: if r.rhs.is_Number and r.lhs.is_Number and r.lhs > r.rhs: r = r.reversed elif r.lhs.is_number: r = r.reversed elif tuple(ordered(args)) != args: r = r.reversed LHS_CEMS = getattr(r.lhs, 'could_extract_minus_sign', None) RHS_CEMS = getattr(r.rhs, 'could_extract_minus_sign', None) if isinstance(r.lhs, BooleanAtom) or isinstance(r.rhs, BooleanAtom): return r # Check if first value has negative sign if LHS_CEMS and LHS_CEMS(): return r.reversedsign elif not r.rhs.is_number and RHS_CEMS and RHS_CEMS(): # Right hand side has a minus, but not lhs. # How does the expression with reversed signs behave? # This is so that expressions of the type # Eq(x, -y) and Eq(-x, y) # have the same canonical representation expr1, _ = ordered([r.lhs, -r.rhs]) if expr1 != r.lhs: return r.reversed.reversedsign return r def equals(self, other, failing_expression=False): """Return True if the sides of the relationship are mathematically identical and the type of relationship is the same. If failing_expression is True, return the expression whose truth value was unknown.""" if isinstance(other, Relational): if self == other or self.reversed == other: return True a, b = self, other if a.func in (Eq, Ne) or b.func in (Eq, Ne): if a.func != b.func: return False left, right = [i.equals(j, failing_expression=failing_expression) for i, j in zip(a.args, b.args)] if left is True: return right if right is True: return left lr, rl = [i.equals(j, failing_expression=failing_expression) for i, j in zip(a.args, b.reversed.args)] if lr is True: return rl if rl is True: return lr e = (left, right, lr, rl) if all(i is False for i in e): return False for i in e: if i not in (True, False): return i else: if b.func != a.func: b = b.reversed if a.func != b.func: return False left = a.lhs.equals(b.lhs, failing_expression=failing_expression) if left is False: return False right = a.rhs.equals(b.rhs, failing_expression=failing_expression) if right is False: return False if left is True: return right return left def _eval_simplify(self, **kwargs): from .add import Add r = self r = r.func(*[i.simplify(**kwargs) for i in r.args]) if r.is_Relational: dif = r.lhs - r.rhs # replace dif with a valid Number that will # allow a definitive comparison with 0 v = None if dif.is_comparable: v = dif.n(2) elif dif.equals(0): # XXX this is expensive v = S.Zero if v is not None: r = r.func._eval_relation(v, S.Zero) r = r.canonical # If there is only one symbol in the expression, # try to write it on a simplified form free = list(filter(lambda x: x.is_real is not False, r.free_symbols)) if len(free) == 1: try: from sympy.solvers.solveset import linear_coeffs x = free.pop() dif = r.lhs - r.rhs m, b = linear_coeffs(dif, x) if m.is_zero is False: if m.is_negative: # Dividing with a negative number, so change order of arguments # canonical will put the symbol back on the lhs later r = r.func(-b / m, x) else: r = r.func(x, -b / m) else: r = r.func(b, S.zero) except ValueError: # maybe not a linear function, try polynomial from sympy.polys import Poly, poly, PolynomialError, gcd try: p = poly(dif, x) c = p.all_coeffs() constant = c[-1] c[-1] = 0 scale = gcd(c) c = [ctmp / scale for ctmp in c] r = r.func(Poly.from_list(c, x).as_expr(), -constant / scale) except PolynomialError: pass elif len(free) >= 2: try: from sympy.solvers.solveset import linear_coeffs from sympy.polys import gcd free = list(ordered(free)) dif = r.lhs - r.rhs m = linear_coeffs(dif, *free) constant = m[-1] del m[-1] scale = gcd(m) m = [mtmp / scale for mtmp in m] nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free)))) if scale.is_zero is False: if constant != 0: # lhs: expression, rhs: constant newexpr = Add(*[i * j for i, j in nzm]) r = r.func(newexpr, -constant / scale) else: # keep first term on lhs lhsterm = nzm[0][0] * nzm[0][1] del nzm[0] newexpr = Add(*[i * j for i, j in nzm]) r = r.func(lhsterm, -newexpr) else: r = r.func(constant, S.zero) except ValueError: pass # Did we get a simplified result? r = r.canonical measure = kwargs['measure'] if measure(r) < kwargs['ratio'] * measure(self): return r else: return self def _eval_trigsimp(self, **opts): from sympy.simplify import trigsimp return self.func(trigsimp(self.lhs, **opts), trigsimp(self.rhs, **opts)) def expand(self, **kwargs): args = (arg.expand(**kwargs) for arg in self.args) return self.func(*args) def __bool__(self): raise TypeError("cannot determine truth value of Relational") def _eval_as_set(self): # self is univariate and periodicity(self, x) in (0, None) from sympy.solvers.inequalities import solve_univariate_inequality from sympy.sets.conditionset import ConditionSet syms = self.free_symbols assert len(syms) == 1 x = syms.pop() try: xset = solve_univariate_inequality(self, x, relational=False) except NotImplementedError: # solve_univariate_inequality raises NotImplementedError for # unsolvable equations/inequalities. xset = ConditionSet(x, self, S.Reals) return xset @property def binary_symbols(self): # override where necessary return set() Rel = Relational class Equality(Relational): """An equal relation between two objects. Explanation =========== Represents that two objects are equal. If they can be easily shown to be definitively equal (or unequal), this will reduce to True (or False). Otherwise, the relation is maintained as an unevaluated Equality object. Use the ``simplify`` function on this object for more nontrivial evaluation of the equality relation. As usual, the keyword argument ``evaluate=False`` can be used to prevent any evaluation. Examples ======== >>> from sympy import Eq, simplify, exp, cos >>> from sympy.abc import x, y >>> Eq(y, x + x**2) Eq(y, x**2 + x) >>> Eq(2, 5) False >>> Eq(2, 5, evaluate=False) Eq(2, 5) >>> _.doit() False >>> Eq(exp(x), exp(x).rewrite(cos)) Eq(exp(x), sinh(x) + cosh(x)) >>> simplify(_) True See Also ======== sympy.logic.boolalg.Equivalent : for representing equality between two boolean expressions Notes ===== Python treats 1 and True (and 0 and False) as being equal; SymPy does not. And integer will always compare as unequal to a Boolean: >>> Eq(True, 1), True == 1 (False, True) This class is not the same as the == operator. The == operator tests for exact structural equality between two expressions; this class compares expressions mathematically. If either object defines an `_eval_Eq` method, it can be used in place of the default algorithm. If `lhs._eval_Eq(rhs)` or `rhs._eval_Eq(lhs)` returns anything other than None, that return value will be substituted for the Equality. If None is returned by `_eval_Eq`, an Equality object will be created as usual. Since this object is already an expression, it does not respond to the method `as_expr` if one tries to create `x - y` from Eq(x, y). This can be done with the `rewrite(Add)` method. """ rel_op = '==' __slots__ = () is_Equality = True def __new__(cls, lhs, rhs=None, **options): if rhs is None: SymPyDeprecationWarning( feature="Eq(expr) with rhs default to 0", useinstead="Eq(expr, 0)", issue=16587, deprecated_since_version="1.5" ).warn() rhs = 0 evaluate = options.pop('evaluate', global_parameters.evaluate) lhs = _sympify(lhs) rhs = _sympify(rhs) if evaluate: val = is_eq(lhs, rhs) if val is None: return cls(lhs, rhs, evaluate=False) else: return _sympify(val) return Relational.__new__(cls, lhs, rhs) @classmethod def _eval_relation(cls, lhs, rhs): return _sympify(lhs == rhs) def _eval_rewrite_as_Add(self, *args, **kwargs): """ return Eq(L, R) as L - R. To control the evaluation of the result set pass `evaluate=True` to give L - R; if `evaluate=None` then terms in L and R will not cancel but they will be listed in canonical order; otherwise non-canonical args will be returned. Examples ======== >>> from sympy import Eq, Add >>> from sympy.abc import b, x >>> eq = Eq(x + b, x - b) >>> eq.rewrite(Add) 2*b >>> eq.rewrite(Add, evaluate=None).args (b, b, x, -x) >>> eq.rewrite(Add, evaluate=False).args (b, x, b, -x) """ from .add import _unevaluated_Add, Add L, R = args evaluate = kwargs.get('evaluate', True) if evaluate: # allow cancellation of args return L - R args = Add.make_args(L) + Add.make_args(-R) if evaluate is None: # no cancellation, but canonical return _unevaluated_Add(*args) # no cancellation, not canonical return Add._from_args(args) @property def binary_symbols(self): if S.true in self.args or S.false in self.args: if self.lhs.is_Symbol: return {self.lhs} elif self.rhs.is_Symbol: return {self.rhs} return set() def _eval_simplify(self, **kwargs): from .add import Add from sympy.solvers.solveset import linear_coeffs # standard simplify e = super()._eval_simplify(**kwargs) if not isinstance(e, Equality): return e free = self.free_symbols if len(free) == 1: try: x = free.pop() m, b = linear_coeffs( e.rewrite(Add, evaluate=False), x) if m.is_zero is False: enew = e.func(x, -b / m) else: enew = e.func(m * x, -b) measure = kwargs['measure'] if measure(enew) <= kwargs['ratio'] * measure(e): e = enew except ValueError: pass return e.canonical def integrate(self, *args, **kwargs): """See the integrate function in sympy.integrals""" from sympy.integrals import integrate return integrate(self, *args, **kwargs) def as_poly(self, *gens, **kwargs): '''Returns lhs-rhs as a Poly Examples ======== >>> from sympy import Eq >>> from sympy.abc import x >>> Eq(x**2, 1).as_poly(x) Poly(x**2 - 1, x, domain='ZZ') ''' return (self.lhs - self.rhs).as_poly(*gens, **kwargs) Eq = Equality class Unequality(Relational): """An unequal relation between two objects. Explanation =========== Represents that two objects are not equal. If they can be shown to be definitively equal, this will reduce to False; if definitively unequal, this will reduce to True. Otherwise, the relation is maintained as an Unequality object. Examples ======== >>> from sympy import Ne >>> from sympy.abc import x, y >>> Ne(y, x+x**2) Ne(y, x**2 + x) See Also ======== Equality Notes ===== This class is not the same as the != operator. The != operator tests for exact structural equality between two expressions; this class compares expressions mathematically. This class is effectively the inverse of Equality. As such, it uses the same algorithms, including any available `_eval_Eq` methods. """ rel_op = '!=' __slots__ = () def __new__(cls, lhs, rhs, **options): lhs = _sympify(lhs) rhs = _sympify(rhs) evaluate = options.pop('evaluate', global_parameters.evaluate) if evaluate: val = is_neq(lhs, rhs) if val is None: return cls(lhs, rhs, evaluate=False) else: return _sympify(val) return Relational.__new__(cls, lhs, rhs, **options) @classmethod def _eval_relation(cls, lhs, rhs): return _sympify(lhs != rhs) @property def binary_symbols(self): if S.true in self.args or S.false in self.args: if self.lhs.is_Symbol: return {self.lhs} elif self.rhs.is_Symbol: return {self.rhs} return set() def _eval_simplify(self, **kwargs): # simplify as an equality eq = Equality(*self.args)._eval_simplify(**kwargs) if isinstance(eq, Equality): # send back Ne with the new args return self.func(*eq.args) return eq.negated # result of Ne is the negated Eq Ne = Unequality class _Inequality(Relational): """Internal base class for all *Than types. Each subclass must implement _eval_relation to provide the method for comparing two real numbers. """ __slots__ = () def __new__(cls, lhs, rhs, **options): try: lhs = _sympify(lhs) rhs = _sympify(rhs) except SympifyError: return NotImplemented evaluate = options.pop('evaluate', global_parameters.evaluate) if evaluate: for me in (lhs, rhs): if me.is_extended_real is False: raise TypeError("Invalid comparison of non-real %s" % me) if me is S.NaN: raise TypeError("Invalid NaN comparison") # First we invoke the appropriate inequality method of `lhs` # (e.g., `lhs.__lt__`). That method will try to reduce to # boolean or raise an exception. It may keep calling # superclasses until it reaches `Expr` (e.g., `Expr.__lt__`). # In some cases, `Expr` will just invoke us again (if neither it # nor a subclass was able to reduce to boolean or raise an # exception). In that case, it must call us with # `evaluate=False` to prevent infinite recursion. return cls._eval_relation(lhs, rhs, **options) # make a "non-evaluated" Expr for the inequality return Relational.__new__(cls, lhs, rhs, **options) @classmethod def _eval_relation(cls, lhs, rhs, **options): val = cls._eval_fuzzy_relation(lhs, rhs) if val is None: return cls(lhs, rhs, evaluate=False) else: return _sympify(val) class _Greater(_Inequality): """Not intended for general use _Greater is only used so that GreaterThan and StrictGreaterThan may subclass it for the .gts and .lts properties. """ __slots__ = () @property def gts(self): return self._args[0] @property def lts(self): return self._args[1] class _Less(_Inequality): """Not intended for general use. _Less is only used so that LessThan and StrictLessThan may subclass it for the .gts and .lts properties. """ __slots__ = () @property def gts(self): return self._args[1] @property def lts(self): return self._args[0] class GreaterThan(_Greater): """Class representations of inequalities. Explanation =========== The ``*Than`` classes represent inequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an inequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation: lhs >= rhs In total, there are four ``*Than`` classes, to represent the four inequalities: +-----------------+--------+ |Class Name | Symbol | +=================+========+ |GreaterThan | (>=) | +-----------------+--------+ |LessThan | (<=) | +-----------------+--------+ |StrictGreaterThan| (>) | +-----------------+--------+ |StrictLessThan | (<) | +-----------------+--------+ All classes take two arguments, lhs and rhs. +----------------------------+-----------------+ |Signature Example | Math equivalent | +============================+=================+ |GreaterThan(lhs, rhs) | lhs >= rhs | +----------------------------+-----------------+ |LessThan(lhs, rhs) | lhs <= rhs | +----------------------------+-----------------+ |StrictGreaterThan(lhs, rhs) | lhs > rhs | +----------------------------+-----------------+ |StrictLessThan(lhs, rhs) | lhs < rhs | +----------------------------+-----------------+ In addition to the normal .lhs and .rhs of Relations, ``*Than`` inequality objects also have the .lts and .gts properties, which represent the "less than side" and "greater than side" of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes: >>> from sympy import GreaterThan, StrictGreaterThan >>> from sympy import LessThan, StrictLessThan >>> from sympy import And, Ge, Gt, Le, Lt, Rel, S >>> from sympy.abc import x, y, z >>> from sympy.core.relational import Relational >>> e = GreaterThan(x, 1) >>> e x >= 1 >>> '%s >= %s is the same as %s <= %s' % (e.gts, e.lts, e.lts, e.gts) 'x >= 1 is the same as 1 <= x' Examples ======== One generally does not instantiate these classes directly, but uses various convenience methods: >>> for f in [Ge, Gt, Le, Lt]: # convenience wrappers ... print(f(x, 2)) x >= 2 x > 2 x <= 2 x < 2 Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more "mathematical looking" statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for 'gotcha', below). >>> x >= 2 x >= 2 >>> _ == Ge(x, 2) True However, it is also perfectly valid to instantiate a ``*Than`` class less succinctly and less conveniently: >>> Rel(x, 1, ">") x > 1 >>> Relational(x, 1, ">") x > 1 >>> StrictGreaterThan(x, 1) x > 1 >>> GreaterThan(x, 1) x >= 1 >>> LessThan(x, 1) x <= 1 >>> StrictLessThan(x, 1) x < 1 Notes ===== There are a couple of "gotchas" to be aware of when using Python's operators. The first is that what your write is not always what you get: >>> 1 < x x > 1 Due to the order that Python parses a statement, it may not immediately find two objects comparable. When "1 < x" is evaluated, Python recognizes that the number 1 is a native number and that x is *not*. Because a native Python number does not know how to compare itself with a SymPy object Python will try the reflective operation, "x > 1" and that is the form that gets evaluated, hence returned. If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) "sympify" the literal before comparison >>> S(1) < x 1 < x (2) use one of the wrappers or less succinct methods described above >>> Lt(1, x) 1 < x >>> Relational(1, x, "<") 1 < x The second gotcha involves writing equality tests between relationals when one or both sides of the test involve a literal relational: >>> e = x < 1; e x < 1 >>> e == e # neither side is a literal True >>> e == x < 1 # expecting True, too False >>> e != x < 1 # expecting False x < 1 >>> x < 1 != x < 1 # expecting False or the same thing as before Traceback (most recent call last): ... TypeError: cannot determine truth value of Relational The solution for this case is to wrap literal relationals in parentheses: >>> e == (x < 1) True >>> e != (x < 1) False >>> (x < 1) != (x < 1) False The third gotcha involves chained inequalities not involving '==' or '!='. Occasionally, one may be tempted to write: >>> e = x < y < z Traceback (most recent call last): ... TypeError: symbolic boolean expression has no truth value. Due to an implementation detail or decision of Python [1]_, there is no way for SymPy to create a chained inequality with that syntax so one must use And: >>> e = And(x < y, y < z) >>> type( e ) And >>> e (x < y) & (y < z) Although this can also be done with the '&' operator, it cannot be done with the 'and' operarator: >>> (x < y) & (y < z) (x < y) & (y < z) >>> (x < y) and (y < z) Traceback (most recent call last): ... TypeError: cannot determine truth value of Relational .. [1] This implementation detail is that Python provides no reliable method to determine that a chained inequality is being built. Chained comparison operators are evaluated pairwise, using "and" logic (see http://docs.python.org/2/reference/expressions.html#notin). This is done in an efficient way, so that each object being compared is only evaluated once and the comparison can short-circuit. For example, ``1 > 2 > 3`` is evaluated by Python as ``(1 > 2) and (2 > 3)``. The ``and`` operator coerces each side into a bool, returning the object itself when it short-circuits. The bool of the --Than operators will raise TypeError on purpose, because SymPy cannot determine the mathematical ordering of symbolic expressions. Thus, if we were to compute ``x > y > z``, with ``x``, ``y``, and ``z`` being Symbols, Python converts the statement (roughly) into these steps: (1) x > y > z (2) (x > y) and (y > z) (3) (GreaterThanObject) and (y > z) (4) (GreaterThanObject.__bool__()) and (y > z) (5) TypeError Because of the "and" added at step 2, the statement gets turned into a weak ternary statement, and the first object's __bool__ method will raise TypeError. Thus, creating a chained inequality is not possible. In Python, there is no way to override the ``and`` operator, or to control how it short circuits, so it is impossible to make something like ``x > y > z`` work. There was a PEP to change this, :pep:`335`, but it was officially closed in March, 2012. """ __slots__ = () rel_op = '>=' @classmethod def _eval_fuzzy_relation(cls, lhs, rhs): return is_ge(lhs, rhs) Ge = GreaterThan class LessThan(_Less): __doc__ = GreaterThan.__doc__ __slots__ = () rel_op = '<=' @classmethod def _eval_fuzzy_relation(cls, lhs, rhs): return is_le(lhs, rhs) Le = LessThan class StrictGreaterThan(_Greater): __doc__ = GreaterThan.__doc__ __slots__ = () rel_op = '>' @classmethod def _eval_fuzzy_relation(cls, lhs, rhs): return is_gt(lhs, rhs) Gt = StrictGreaterThan class StrictLessThan(_Less): __doc__ = GreaterThan.__doc__ __slots__ = () rel_op = '<' @classmethod def _eval_fuzzy_relation(cls, lhs, rhs): return is_lt(lhs, rhs) Lt = StrictLessThan # A class-specific (not object-specific) data item used for a minor speedup. # It is defined here, rather than directly in the class, because the classes # that it references have not been defined until now (e.g. StrictLessThan). Relational.ValidRelationOperator = { None: Equality, '==': Equality, 'eq': Equality, '!=': Unequality, '<>': Unequality, 'ne': Unequality, '>=': GreaterThan, 'ge': GreaterThan, '<=': LessThan, 'le': LessThan, '>': StrictGreaterThan, 'gt': StrictGreaterThan, '<': StrictLessThan, 'lt': StrictLessThan, } def _n2(a, b): """Return (a - b).evalf(2) if a and b are comparable, else None. This should only be used when a and b are already sympified. """ # /!\ it is very important (see issue 8245) not to # use a re-evaluated number in the calculation of dif if a.is_comparable and b.is_comparable: dif = (a - b).evalf(2) if dif.is_comparable: return dif @dispatch(Expr, Expr) def _eval_is_ge(lhs, rhs): return None @dispatch(Basic, Basic) def _eval_is_eq(lhs, rhs): return None @dispatch(Tuple, Expr) # type: ignore def _eval_is_eq(lhs, rhs): # noqa:F811 return False @dispatch(Tuple, AppliedUndef) # type: ignore def _eval_is_eq(lhs, rhs): # noqa:F811 return None @dispatch(Tuple, Symbol) # type: ignore def _eval_is_eq(lhs, rhs): # noqa:F811 return None @dispatch(Tuple, Tuple) # type: ignore def _eval_is_eq(lhs, rhs): # noqa:F811 if len(lhs) != len(rhs): return False return fuzzy_and(fuzzy_bool(is_eq(s, o)) for s, o in zip(lhs, rhs)) def is_lt(lhs, rhs): """Fuzzy bool for lhs is strictly less than rhs. See the docstring for is_ge for more """ return fuzzy_not(is_ge(lhs, rhs)) def is_gt(lhs, rhs): """Fuzzy bool for lhs is strictly greater than rhs. See the docstring for is_ge for more """ return fuzzy_not(is_le(lhs, rhs)) def is_le(lhs, rhs): """Fuzzy bool for lhs is less than or equal to rhs. is_gt calls is_lt See the docstring for is_ge for more """ return is_ge(rhs, lhs) def is_ge(lhs, rhs): """ Fuzzy bool for lhs is greater than or equal to rhs. Parameters ========== lhs: Expr The left-hand side of the expression, must be sympified, and an instance of expression. Throws an exception if lhs is not an instance of expression. rhs: Expr The right-hand side of the expression, must be sympified and an instance of expression. Throws an exception if lhs is not an instance of expression. Returns ======= Expr : True if lhs is greater than or equal to rhs, false is lhs is less than rhs, and None if the comparison between lhs and rhs is indeterminate. The four comparison functions ``is_le``, ``is_lt``, ``is_ge``, and ``is_gt`` are each implemented in terms of ``is_ge`` in the following way: is_ge(x, y) := is_ge(x, y) is_le(x, y) := is_ge(y, x) is_lt(x, y) := fuzzy_not(is_ge(x, y)) is_gt(x, y) = fuzzy_not(is_ge(y, x)) To maintain these equivalences in fuzzy logic it is important that in cases where either x or y is non-real all comparisons will give None. InEquality classes, such as Lt, Gt, etc. Use one of is_ge, is_le, etc. To implement comparisons with ``Gt(a, b)`` or ``a > b`` etc for an ``Expr`` subclass it is only necessary to define a dispatcher method for ``_eval_is_ge`` like >>> from sympy.core.relational import is_ge, is_lt, is_gt >>> from sympy.abc import x >>> from sympy import S, Expr, sympify >>> from sympy.multipledispatch import dispatch >>> class MyExpr(Expr): ... def __new__(cls, arg): ... return Expr.__new__(cls, sympify(arg)) ... @property ... def value(self): ... return self.args[0] ... >>> @dispatch(MyExpr, MyExpr) ... def _eval_is_ge(a, b): ... return is_ge(a.value, b.value) ... >>> a = MyExpr(1) >>> b = MyExpr(2) >>> a < b True >>> a <= b True >>> a > b False >>> is_lt(a, b) True Examples ======== >>> is_ge(S(2), S(0)) True >>> is_ge(S(0), S(2)) False >>> is_ge(S(0), x) >>> is_gt(S(2), S(0)) True >>> is_gt(S(0), S(2)) False >>> is_lt(S(0), S(2)) True >>> is_lt(S(2), S(0)) False """ if not (isinstance(lhs, Expr) and isinstance(rhs, Expr)): raise TypeError("Can only compare inequalities with Expr") retval = _eval_is_ge(lhs, rhs) if retval is not None: return retval else: n2 = _n2(lhs, rhs) if n2 is not None: # use float comparison for infinity. # otherwise get stuck in infinite recursion if n2 in (S.Infinity, S.NegativeInfinity): n2 = float(n2) return _sympify(n2 >= 0) if lhs.is_extended_real and rhs.is_extended_real: if (lhs.is_infinite and lhs.is_extended_positive) or (rhs.is_infinite and rhs.is_extended_negative): return True diff = lhs - rhs if diff is not S.NaN: rv = diff.is_extended_nonnegative if rv is not None: return rv def is_neq(lhs, rhs): """Fuzzy bool for lhs does not equal rhs. See the docstring for is_eq for more """ return fuzzy_not(is_eq(lhs, rhs)) def is_eq(lhs, rhs): """ Fuzzy bool representing mathematical equality between lhs and rhs. Parameters ========== lhs: Expr The left-hand side of the expression, must be sympified. rhs: Expr The right-hand side of the expression, must be sympified. Returns ======= True if lhs is equal to rhs, false is lhs is not equal to rhs, and None if the comparison between lhs and rhs is indeterminate. Explanation =========== This function is intended to give a relatively fast determination and deliberately does not attempt slow calculations that might help in obtaining a determination of True or False in more difficult cases. InEquality classes, such as Lt, Gt, etc. Use one of is_ge, is_le, etc. To implement comparisons with ``Gt(a, b)`` or ``a > b`` etc for an ``Expr`` subclass it is only necessary to define a dispatcher method for ``_eval_is_ge`` like >>> from sympy.core.relational import is_eq >>> from sympy.core.relational import is_neq >>> from sympy import S, Basic, Eq, sympify >>> from sympy.abc import x >>> from sympy.multipledispatch import dispatch >>> class MyBasic(Basic): ... def __new__(cls, arg): ... return Basic.__new__(cls, sympify(arg)) ... @property ... def value(self): ... return self.args[0] ... >>> @dispatch(MyBasic, MyBasic) ... def _eval_is_eq(a, b): ... return is_eq(a.value, b.value) ... >>> a = MyBasic(1) >>> b = MyBasic(1) >>> a == b True >>> Eq(a, b) True >>> a != b False >>> is_eq(a, b) True Examples ======== >>> is_eq(S(0), S(0)) True >>> Eq(0, 0) True >>> is_neq(S(0), S(0)) False >>> is_eq(S(0), S(2)) False >>> Eq(0, 2) False >>> is_neq(S(0), S(2)) True >>> is_eq(S(0), x) >>> Eq(S(0), x) Eq(0, x) """ from sympy.core.add import Add from sympy.functions.elementary.complexes import arg from sympy.simplify.simplify import clear_coefficients from sympy.utilities.iterables import sift # here, _eval_Eq is only called for backwards compatibility # new code should use is_eq with multiple dispatch as # outlined in the docstring for side1, side2 in (lhs, rhs), (rhs, lhs): eval_func = getattr(side1, '_eval_Eq', None) if eval_func is not None: retval = eval_func(side2) if retval is not None: return retval retval = _eval_is_eq(lhs, rhs) if retval is not None: return retval if dispatch(type(lhs), type(rhs)) != dispatch(type(rhs), type(lhs)): retval = _eval_is_eq(rhs, lhs) if retval is not None: return retval # retval is still None, so go through the equality logic # If expressions have the same structure, they must be equal. if lhs == rhs: return True # e.g. True == True elif all(isinstance(i, BooleanAtom) for i in (rhs, lhs)): return False # True != False elif not (lhs.is_Symbol or rhs.is_Symbol) and ( isinstance(lhs, Boolean) != isinstance(rhs, Boolean)): return False # only Booleans can equal Booleans if lhs.is_infinite or rhs.is_infinite: if fuzzy_xor([lhs.is_infinite, rhs.is_infinite]): return False if fuzzy_xor([lhs.is_extended_real, rhs.is_extended_real]): return False if fuzzy_and([lhs.is_extended_real, rhs.is_extended_real]): return fuzzy_xor([lhs.is_extended_positive, fuzzy_not(rhs.is_extended_positive)]) # Try to split real/imaginary parts and equate them I = S.ImaginaryUnit def split_real_imag(expr): real_imag = lambda t: ( 'real' if t.is_extended_real else 'imag' if (I * t).is_extended_real else None) return sift(Add.make_args(expr), real_imag) lhs_ri = split_real_imag(lhs) if not lhs_ri[None]: rhs_ri = split_real_imag(rhs) if not rhs_ri[None]: eq_real = Eq(Add(*lhs_ri['real']), Add(*rhs_ri['real'])) eq_imag = Eq(I * Add(*lhs_ri['imag']), I * Add(*rhs_ri['imag'])) return fuzzy_and(map(fuzzy_bool, [eq_real, eq_imag])) # Compare e.g. zoo with 1+I*oo by comparing args arglhs = arg(lhs) argrhs = arg(rhs) # Guard against Eq(nan, nan) -> Falsesymp if not (arglhs == S.NaN and argrhs == S.NaN): return fuzzy_bool(Eq(arglhs, argrhs)) if all(isinstance(i, Expr) for i in (lhs, rhs)): # see if the difference evaluates dif = lhs - rhs z = dif.is_zero if z is not None: if z is False and dif.is_commutative: # issue 10728 return False if z: return True n2 = _n2(lhs, rhs) if n2 is not None: return _sympify(n2 == 0) # see if the ratio evaluates n, d = dif.as_numer_denom() rv = None if n.is_zero: rv = d.is_nonzero elif n.is_finite: if d.is_infinite: rv = True elif n.is_zero is False: rv = d.is_infinite if rv is None: # if the condition that makes the denominator # infinite does not make the original expression # True then False can be returned l, r = clear_coefficients(d, S.Infinity) args = [_.subs(l, r) for _ in (lhs, rhs)] if args != [lhs, rhs]: rv = fuzzy_bool(Eq(*args)) if rv is True: rv = None elif any(a.is_infinite for a in Add.make_args(n)): # (inf or nan)/x != 0 rv = False if rv is not None: return rv
660de38c82f84706f0ded11714b86fe4e9ce71822153ee5c441e0d9afa01c097
import numbers import decimal import fractions import math import re as regex from .containers import Tuple from .sympify import (SympifyError, converter, sympify, _convert_numpy_types, _sympify, _is_numpy_instance) from .singleton import S, Singleton from .expr import Expr, AtomicExpr from .evalf import pure_complex from .decorators import _sympifyit from .cache import cacheit, clear_cache from .logic import fuzzy_not from sympy.core.compatibility import (as_int, HAS_GMPY, SYMPY_INTS, int_info, gmpy) from sympy.core.cache import lru_cache from sympy.multipledispatch import dispatch import mpmath import mpmath.libmp as mlib from mpmath.libmp import bitcount from mpmath.libmp.backend import MPZ from mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed from mpmath.ctx_mp import mpnumeric from mpmath.libmp.libmpf import ( finf as _mpf_inf, fninf as _mpf_ninf, fnan as _mpf_nan, fzero, _normalize as mpf_normalize, prec_to_dps) from sympy.utilities.misc import debug, filldedent from .parameters import global_parameters from sympy.utilities.exceptions import SymPyDeprecationWarning rnd = mlib.round_nearest _LOG2 = math.log(2) def comp(z1, z2, tol=None): """Return a bool indicating whether the error between z1 and z2 is <= tol. Examples ======== If ``tol`` is None then True will be returned if ``abs(z1 - z2)*10**p <= 5`` where ``p`` is minimum value of the decimal precision of each value. >>> from sympy.core.numbers import comp, pi >>> pi4 = pi.n(4); pi4 3.142 >>> comp(_, 3.142) True >>> comp(pi4, 3.141) False >>> comp(pi4, 3.143) False A comparison of strings will be made if ``z1`` is a Number and ``z2`` is a string or ``tol`` is ''. >>> comp(pi4, 3.1415) True >>> comp(pi4, 3.1415, '') False When ``tol`` is provided and ``z2`` is non-zero and ``|z1| > 1`` the error is normalized by ``|z1|``: >>> abs(pi4 - 3.14)/pi4 0.000509791731426756 >>> comp(pi4, 3.14, .001) # difference less than 0.1% True >>> comp(pi4, 3.14, .0005) # difference less than 0.1% False When ``|z1| <= 1`` the absolute error is used: >>> 1/pi4 0.3183 >>> abs(1/pi4 - 0.3183)/(1/pi4) 3.07371499106316e-5 >>> abs(1/pi4 - 0.3183) 9.78393554684764e-6 >>> comp(1/pi4, 0.3183, 1e-5) True To see if the absolute error between ``z1`` and ``z2`` is less than or equal to ``tol``, call this as ``comp(z1 - z2, 0, tol)`` or ``comp(z1 - z2, tol=tol)``: >>> abs(pi4 - 3.14) 0.00160156249999988 >>> comp(pi4 - 3.14, 0, .002) True >>> comp(pi4 - 3.14, 0, .001) False """ if type(z2) is str: if not pure_complex(z1, or_real=True): raise ValueError('when z2 is a str z1 must be a Number') return str(z1) == z2 if not z1: z1, z2 = z2, z1 if not z1: return True if not tol: a, b = z1, z2 if tol == '': return str(a) == str(b) if tol is None: a, b = sympify(a), sympify(b) if not all(i.is_number for i in (a, b)): raise ValueError('expecting 2 numbers') fa = a.atoms(Float) fb = b.atoms(Float) if not fa and not fb: # no floats -- compare exactly return a == b # get a to be pure_complex for do in range(2): ca = pure_complex(a, or_real=True) if not ca: if fa: a = a.n(prec_to_dps(min([i._prec for i in fa]))) ca = pure_complex(a, or_real=True) break else: fa, fb = fb, fa a, b = b, a cb = pure_complex(b) if not cb and fb: b = b.n(prec_to_dps(min([i._prec for i in fb]))) cb = pure_complex(b, or_real=True) if ca and cb and (ca[1] or cb[1]): return all(comp(i, j) for i, j in zip(ca, cb)) tol = 10**prec_to_dps(min(a._prec, getattr(b, '_prec', a._prec))) return int(abs(a - b)*tol) <= 5 diff = abs(z1 - z2) az1 = abs(z1) if z2 and az1 > 1: return diff/az1 <= tol else: return diff <= tol def mpf_norm(mpf, prec): """Return the mpf tuple normalized appropriately for the indicated precision after doing a check to see if zero should be returned or not when the mantissa is 0. ``mpf_normlize`` always assumes that this is zero, but it may not be since the mantissa for mpf's values "+inf", "-inf" and "nan" have a mantissa of zero, too. Note: this is not intended to validate a given mpf tuple, so sending mpf tuples that were not created by mpmath may produce bad results. This is only a wrapper to ``mpf_normalize`` which provides the check for non- zero mpfs that have a 0 for the mantissa. """ sign, man, expt, bc = mpf if not man: # hack for mpf_normalize which does not do this; # it assumes that if man is zero the result is 0 # (see issue 6639) if not bc: return fzero else: # don't change anything; this should already # be a well formed mpf tuple return mpf # Necessary if mpmath is using the gmpy backend from mpmath.libmp.backend import MPZ rv = mpf_normalize(sign, MPZ(man), expt, bc, prec, rnd) return rv # TODO: we should use the warnings module _errdict = {"divide": False} def seterr(divide=False): """ Should sympy raise an exception on 0/0 or return a nan? divide == True .... raise an exception divide == False ... return nan """ if _errdict["divide"] != divide: clear_cache() _errdict["divide"] = divide def _as_integer_ratio(p): neg_pow, man, expt, bc = getattr(p, '_mpf_', mpmath.mpf(p)._mpf_) p = [1, -1][neg_pow % 2]*man if expt < 0: q = 2**-expt else: q = 1 p *= 2**expt return int(p), int(q) def _decimal_to_Rational_prec(dec): """Convert an ordinary decimal instance to a Rational.""" if not dec.is_finite(): raise TypeError("dec must be finite, got %s." % dec) s, d, e = dec.as_tuple() prec = len(d) if e >= 0: # it's an integer rv = Integer(int(dec)) else: s = (-1)**s d = sum([di*10**i for i, di in enumerate(reversed(d))]) rv = Rational(s*d, 10**-e) return rv, prec _floatpat = regex.compile(r"[-+]?((\d*\.\d+)|(\d+\.?))") def _literal_float(f): """Return True if n starts like a floating point number.""" return bool(_floatpat.match(f)) # (a,b) -> gcd(a,b) # TODO caching with decorator, but not to degrade performance @lru_cache(1024) def igcd(*args): """Computes nonnegative integer greatest common divisor. Explanation =========== The algorithm is based on the well known Euclid's algorithm. To improve speed, igcd() has its own caching mechanism implemented. Examples ======== >>> from sympy.core.numbers import igcd >>> igcd(2, 4) 2 >>> igcd(5, 10, 15) 5 """ if len(args) < 2: raise TypeError( 'igcd() takes at least 2 arguments (%s given)' % len(args)) args_temp = [abs(as_int(i)) for i in args] if 1 in args_temp: return 1 a = args_temp.pop() if HAS_GMPY: # Using gmpy if present to speed up. for b in args_temp: a = gmpy.gcd(a, b) if b else a return as_int(a) for b in args_temp: a = igcd2(a, b) if b else a return a def _igcd2_python(a, b): """Compute gcd of two Python integers a and b.""" if (a.bit_length() > BIGBITS and b.bit_length() > BIGBITS): return igcd_lehmer(a, b) a, b = abs(a), abs(b) while b: a, b = b, a % b return a try: from math import gcd as igcd2 except ImportError: igcd2 = _igcd2_python # Use Lehmer's algorithm only for very large numbers. BIGBITS = 5000 def igcd_lehmer(a, b): """Computes greatest common divisor of two integers. Explanation =========== Euclid's algorithm for the computation of the greatest common divisor gcd(a, b) of two (positive) integers a and b is based on the division identity a = q*b + r, where the quotient q and the remainder r are integers and 0 <= r < b. Then each common divisor of a and b divides r, and it follows that gcd(a, b) == gcd(b, r). The algorithm works by constructing the sequence r0, r1, r2, ..., where r0 = a, r1 = b, and each rn is the remainder from the division of the two preceding elements. In Python, q = a // b and r = a % b are obtained by the floor division and the remainder operations, respectively. These are the most expensive arithmetic operations, especially for large a and b. Lehmer's algorithm is based on the observation that the quotients qn = r(n-1) // rn are in general small integers even when a and b are very large. Hence the quotients can be usually determined from a relatively small number of most significant bits. The efficiency of the algorithm is further enhanced by not computing each long remainder in Euclid's sequence. The remainders are linear combinations of a and b with integer coefficients derived from the quotients. The coefficients can be computed as far as the quotients can be determined from the chosen most significant parts of a and b. Only then a new pair of consecutive remainders is computed and the algorithm starts anew with this pair. References ========== .. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm """ a, b = abs(as_int(a)), abs(as_int(b)) if a < b: a, b = b, a # The algorithm works by using one or two digit division # whenever possible. The outer loop will replace the # pair (a, b) with a pair of shorter consecutive elements # of the Euclidean gcd sequence until a and b # fit into two Python (long) int digits. nbits = 2*int_info.bits_per_digit while a.bit_length() > nbits and b != 0: # Quotients are mostly small integers that can # be determined from most significant bits. n = a.bit_length() - nbits x, y = int(a >> n), int(b >> n) # most significant bits # Elements of the Euclidean gcd sequence are linear # combinations of a and b with integer coefficients. # Compute the coefficients of consecutive pairs # a' = A*a + B*b, b' = C*a + D*b # using small integer arithmetic as far as possible. A, B, C, D = 1, 0, 0, 1 # initial values while True: # The coefficients alternate in sign while looping. # The inner loop combines two steps to keep track # of the signs. # At this point we have # A > 0, B <= 0, C <= 0, D > 0, # x' = x + B <= x < x" = x + A, # y' = y + C <= y < y" = y + D, # and # x'*N <= a' < x"*N, y'*N <= b' < y"*N, # where N = 2**n. # Now, if y' > 0, and x"//y' and x'//y" agree, # then their common value is equal to q = a'//b'. # In addition, # x'%y" = x' - q*y" < x" - q*y' = x"%y', # and # (x'%y")*N < a'%b' < (x"%y')*N. # On the other hand, we also have x//y == q, # and therefore # x'%y" = x + B - q*(y + D) = x%y + B', # x"%y' = x + A - q*(y + C) = x%y + A', # where # B' = B - q*D < 0, A' = A - q*C > 0. if y + C <= 0: break q = (x + A) // (y + C) # Now x'//y" <= q, and equality holds if # x' - q*y" = (x - q*y) + (B - q*D) >= 0. # This is a minor optimization to avoid division. x_qy, B_qD = x - q*y, B - q*D if x_qy + B_qD < 0: break # Next step in the Euclidean sequence. x, y = y, x_qy A, B, C, D = C, D, A - q*C, B_qD # At this point the signs of the coefficients # change and their roles are interchanged. # A <= 0, B > 0, C > 0, D < 0, # x' = x + A <= x < x" = x + B, # y' = y + D < y < y" = y + C. if y + D <= 0: break q = (x + B) // (y + D) x_qy, A_qC = x - q*y, A - q*C if x_qy + A_qC < 0: break x, y = y, x_qy A, B, C, D = C, D, A_qC, B - q*D # Now the conditions on top of the loop # are again satisfied. # A > 0, B < 0, C < 0, D > 0. if B == 0: # This can only happen when y == 0 in the beginning # and the inner loop does nothing. # Long division is forced. a, b = b, a % b continue # Compute new long arguments using the coefficients. a, b = A*a + B*b, C*a + D*b # Small divisors. Finish with the standard algorithm. while b: a, b = b, a % b return a def ilcm(*args): """Computes integer least common multiple. Examples ======== >>> from sympy.core.numbers import ilcm >>> ilcm(5, 10) 10 >>> ilcm(7, 3) 21 >>> ilcm(5, 10, 15) 30 """ if len(args) < 2: raise TypeError( 'ilcm() takes at least 2 arguments (%s given)' % len(args)) if 0 in args: return 0 a = args[0] for b in args[1:]: a = a // igcd(a, b) * b # since gcd(a,b) | a return a def igcdex(a, b): """Returns x, y, g such that g = x*a + y*b = gcd(a, b). Examples ======== >>> from sympy.core.numbers import igcdex >>> igcdex(2, 3) (-1, 1, 1) >>> igcdex(10, 12) (-1, 1, 2) >>> x, y, g = igcdex(100, 2004) >>> x, y, g (-20, 1, 4) >>> x*100 + y*2004 4 """ if (not a) and (not b): return (0, 1, 0) if not a: return (0, b//abs(b), abs(b)) if not b: return (a//abs(a), 0, abs(a)) if a < 0: a, x_sign = -a, -1 else: x_sign = 1 if b < 0: b, y_sign = -b, -1 else: y_sign = 1 x, y, r, s = 1, 0, 0, 1 while b: (c, q) = (a % b, a // b) (a, b, r, s, x, y) = (b, c, x - q*r, y - q*s, r, s) return (x*x_sign, y*y_sign, a) def mod_inverse(a, m): """ Return the number c such that, (a * c) = 1 (mod m) where c has the same sign as m. If no such value exists, a ValueError is raised. Examples ======== >>> from sympy import S >>> from sympy.core.numbers import mod_inverse Suppose we wish to find multiplicative inverse x of 3 modulo 11. This is the same as finding x such that 3 * x = 1 (mod 11). One value of x that satisfies this congruence is 4. Because 3 * 4 = 12 and 12 = 1 (mod 11). This is the value returned by mod_inverse: >>> mod_inverse(3, 11) 4 >>> mod_inverse(-3, 11) 7 When there is a common factor between the numerators of ``a`` and ``m`` the inverse does not exist: >>> mod_inverse(2, 4) Traceback (most recent call last): ... ValueError: inverse of 2 mod 4 does not exist >>> mod_inverse(S(2)/7, S(5)/2) 7/2 References ========== .. [1] https://en.wikipedia.org/wiki/Modular_multiplicative_inverse .. [2] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm """ c = None try: a, m = as_int(a), as_int(m) if m != 1 and m != -1: x, y, g = igcdex(a, m) if g == 1: c = x % m except ValueError: a, m = sympify(a), sympify(m) if not (a.is_number and m.is_number): raise TypeError(filldedent(''' Expected numbers for arguments; symbolic `mod_inverse` is not implemented but symbolic expressions can be handled with the similar function, sympy.polys.polytools.invert''')) big = (m > 1) if not (big is S.true or big is S.false): raise ValueError('m > 1 did not evaluate; try to simplify %s' % m) elif big: c = 1/a if c is None: raise ValueError('inverse of %s (mod %s) does not exist' % (a, m)) return c class Number(AtomicExpr): """Represents atomic numbers in SymPy. Explanation =========== Floating point numbers are represented by the Float class. Rational numbers (of any size) are represented by the Rational class. Integer numbers (of any size) are represented by the Integer class. Float and Rational are subclasses of Number; Integer is a subclass of Rational. For example, ``2/3`` is represented as ``Rational(2, 3)`` which is a different object from the floating point number obtained with Python division ``2/3``. Even for numbers that are exactly represented in binary, there is a difference between how two forms, such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy. The rational form is to be preferred in symbolic computations. Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or complex numbers ``3 + 4*I``, are not instances of Number class as they are not atomic. See Also ======== Float, Integer, Rational """ is_commutative = True is_number = True is_Number = True __slots__ = () # Used to make max(x._prec, y._prec) return x._prec when only x is a float _prec = -1 def __new__(cls, *obj): if len(obj) == 1: obj = obj[0] if isinstance(obj, Number): return obj if isinstance(obj, SYMPY_INTS): return Integer(obj) if isinstance(obj, tuple) and len(obj) == 2: return Rational(*obj) if isinstance(obj, (float, mpmath.mpf, decimal.Decimal)): return Float(obj) if isinstance(obj, str): _obj = obj.lower() # float('INF') == float('inf') if _obj == 'nan': return S.NaN elif _obj == 'inf': return S.Infinity elif _obj == '+inf': return S.Infinity elif _obj == '-inf': return S.NegativeInfinity val = sympify(obj) if isinstance(val, Number): return val else: raise ValueError('String "%s" does not denote a Number' % obj) msg = "expected str|int|long|float|Decimal|Number object but got %r" raise TypeError(msg % type(obj).__name__) def invert(self, other, *gens, **args): from sympy.polys.polytools import invert if getattr(other, 'is_number', True): return mod_inverse(self, other) return invert(self, other, *gens, **args) def __divmod__(self, other): from .containers import Tuple from sympy.functions.elementary.complexes import sign try: other = Number(other) if self.is_infinite or S.NaN in (self, other): return (S.NaN, S.NaN) except TypeError: return NotImplemented if not other: raise ZeroDivisionError('modulo by zero') if self.is_Integer and other.is_Integer: return Tuple(*divmod(self.p, other.p)) elif isinstance(other, Float): rat = self/Rational(other) else: rat = self/other if other.is_finite: w = int(rat) if rat >= 0 else int(rat) - 1 r = self - other*w else: w = 0 if not self or (sign(self) == sign(other)) else -1 r = other if w else self return Tuple(w, r) def __rdivmod__(self, other): try: other = Number(other) except TypeError: return NotImplemented return divmod(other, self) def _as_mpf_val(self, prec): """Evaluation of mpf tuple accurate to at least prec bits.""" raise NotImplementedError('%s needs ._as_mpf_val() method' % (self.__class__.__name__)) def _eval_evalf(self, prec): return Float._new(self._as_mpf_val(prec), prec) def _as_mpf_op(self, prec): prec = max(prec, self._prec) return self._as_mpf_val(prec), prec def __float__(self): return mlib.to_float(self._as_mpf_val(53)) def floor(self): raise NotImplementedError('%s needs .floor() method' % (self.__class__.__name__)) def ceiling(self): raise NotImplementedError('%s needs .ceiling() method' % (self.__class__.__name__)) def __floor__(self): return self.floor() def __ceil__(self): return self.ceiling() def _eval_conjugate(self): return self def _eval_order(self, *symbols): from sympy import Order # Order(5, x, y) -> Order(1,x,y) return Order(S.One, *symbols) def _eval_subs(self, old, new): if old == -self: return -new return self # there is no other possibility def _eval_is_finite(self): return True @classmethod def class_key(cls): return 1, 0, 'Number' @cacheit def sort_key(self, order=None): return self.class_key(), (0, ()), (), self @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NaN: return S.NaN elif other is S.Infinity: return S.Infinity elif other is S.NegativeInfinity: return S.NegativeInfinity return AtomicExpr.__add__(self, other) @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NaN: return S.NaN elif other is S.Infinity: return S.NegativeInfinity elif other is S.NegativeInfinity: return S.Infinity return AtomicExpr.__sub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NaN: return S.NaN elif other is S.Infinity: if self.is_zero: return S.NaN elif self.is_positive: return S.Infinity else: return S.NegativeInfinity elif other is S.NegativeInfinity: if self.is_zero: return S.NaN elif self.is_positive: return S.NegativeInfinity else: return S.Infinity elif isinstance(other, Tuple): return NotImplemented return AtomicExpr.__mul__(self, other) @_sympifyit('other', NotImplemented) def __truediv__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NaN: return S.NaN elif other is S.Infinity or other is S.NegativeInfinity: return S.Zero return AtomicExpr.__truediv__(self, other) def __eq__(self, other): raise NotImplementedError('%s needs .__eq__() method' % (self.__class__.__name__)) def __ne__(self, other): raise NotImplementedError('%s needs .__ne__() method' % (self.__class__.__name__)) def __lt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s < %s" % (self, other)) raise NotImplementedError('%s needs .__lt__() method' % (self.__class__.__name__)) def __le__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s <= %s" % (self, other)) raise NotImplementedError('%s needs .__le__() method' % (self.__class__.__name__)) def __gt__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s > %s" % (self, other)) return _sympify(other).__lt__(self) def __ge__(self, other): try: other = _sympify(other) except SympifyError: raise TypeError("Invalid comparison %s >= %s" % (self, other)) return _sympify(other).__le__(self) def __hash__(self): return super().__hash__() def is_constant(self, *wrt, **flags): return True def as_coeff_mul(self, *deps, rational=True, **kwargs): # a -> c*t if self.is_Rational or not rational: return self, tuple() elif self.is_negative: return S.NegativeOne, (-self,) return S.One, (self,) def as_coeff_add(self, *deps): # a -> c + t if self.is_Rational: return self, tuple() return S.Zero, (self,) def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ if rational and not self.is_Rational: return S.One, self return (self, S.One) if self else (S.One, self) def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """ if not rational: return self, S.Zero return S.Zero, self def gcd(self, other): """Compute GCD of `self` and `other`. """ from sympy.polys import gcd return gcd(self, other) def lcm(self, other): """Compute LCM of `self` and `other`. """ from sympy.polys import lcm return lcm(self, other) def cofactors(self, other): """Compute GCD and cofactors of `self` and `other`. """ from sympy.polys import cofactors return cofactors(self, other) class Float(Number): """Represent a floating-point number of arbitrary precision. Examples ======== >>> from sympy import Float >>> Float(3.5) 3.50000000000000 >>> Float(3) 3.00000000000000 Creating Floats from strings (and Python ``int`` and ``long`` types) will give a minimum precision of 15 digits, but the precision will automatically increase to capture all digits entered. >>> Float(1) 1.00000000000000 >>> Float(10**20) 100000000000000000000. >>> Float('1e20') 100000000000000000000. However, *floating-point* numbers (Python ``float`` types) retain only 15 digits of precision: >>> Float(1e20) 1.00000000000000e+20 >>> Float(1.23456789123456789) 1.23456789123457 It may be preferable to enter high-precision decimal numbers as strings: >>> Float('1.23456789123456789') 1.23456789123456789 The desired number of digits can also be specified: >>> Float('1e-3', 3) 0.00100 >>> Float(100, 4) 100.0 Float can automatically count significant figures if a null string is sent for the precision; spaces or underscores are also allowed. (Auto- counting is only allowed for strings, ints and longs). >>> Float('123 456 789.123_456', '') 123456789.123456 >>> Float('12e-3', '') 0.012 >>> Float(3, '') 3. If a number is written in scientific notation, only the digits before the exponent are considered significant if a decimal appears, otherwise the "e" signifies only how to move the decimal: >>> Float('60.e2', '') # 2 digits significant 6.0e+3 >>> Float('60e2', '') # 4 digits significant 6000. >>> Float('600e-2', '') # 3 digits significant 6.00 Notes ===== Floats are inexact by their nature unless their value is a binary-exact value. >>> approx, exact = Float(.1, 1), Float(.125, 1) For calculation purposes, evalf needs to be able to change the precision but this will not increase the accuracy of the inexact value. The following is the most accurate 5-digit approximation of a value of 0.1 that had only 1 digit of precision: >>> approx.evalf(5) 0.099609 By contrast, 0.125 is exact in binary (as it is in base 10) and so it can be passed to Float or evalf to obtain an arbitrary precision with matching accuracy: >>> Float(exact, 5) 0.12500 >>> exact.evalf(20) 0.12500000000000000000 Trying to make a high-precision Float from a float is not disallowed, but one must keep in mind that the *underlying float* (not the apparent decimal value) is being obtained with high precision. For example, 0.3 does not have a finite binary representation. The closest rational is the fraction 5404319552844595/2**54. So if you try to obtain a Float of 0.3 to 20 digits of precision you will not see the same thing as 0.3 followed by 19 zeros: >>> Float(0.3, 20) 0.29999999999999998890 If you want a 20-digit value of the decimal 0.3 (not the floating point approximation of 0.3) you should send the 0.3 as a string. The underlying representation is still binary but a higher precision than Python's float is used: >>> Float('0.3', 20) 0.30000000000000000000 Although you can increase the precision of an existing Float using Float it will not increase the accuracy -- the underlying value is not changed: >>> def show(f): # binary rep of Float ... from sympy import Mul, Pow ... s, m, e, b = f._mpf_ ... v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False) ... print('%s at prec=%s' % (v, f._prec)) ... >>> t = Float('0.3', 3) >>> show(t) 4915/2**14 at prec=13 >>> show(Float(t, 20)) # higher prec, not higher accuracy 4915/2**14 at prec=70 >>> show(Float(t, 2)) # lower prec 307/2**10 at prec=10 The same thing happens when evalf is used on a Float: >>> show(t.evalf(20)) 4915/2**14 at prec=70 >>> show(t.evalf(2)) 307/2**10 at prec=10 Finally, Floats can be instantiated with an mpf tuple (n, c, p) to produce the number (-1)**n*c*2**p: >>> n, c, p = 1, 5, 0 >>> (-1)**n*c*2**p -5 >>> Float((1, 5, 0)) -5.00000000000000 An actual mpf tuple also contains the number of bits in c as the last element of the tuple: >>> _._mpf_ (1, 5, 0, 3) This is not needed for instantiation and is not the same thing as the precision. The mpf tuple and the precision are two separate quantities that Float tracks. In SymPy, a Float is a number that can be computed with arbitrary precision. Although floating point 'inf' and 'nan' are not such numbers, Float can create these numbers: >>> Float('-inf') -oo >>> _.is_Float False """ __slots__ = ('_mpf_', '_prec') # A Float represents many real numbers, # both rational and irrational. is_rational = None is_irrational = None is_number = True is_real = True is_extended_real = True is_Float = True def __new__(cls, num, dps=None, prec=None, precision=None): if prec is not None: SymPyDeprecationWarning( feature="Using 'prec=XX' to denote decimal precision", useinstead="'dps=XX' for decimal precision and 'precision=XX' "\ "for binary precision", issue=12820, deprecated_since_version="1.1").warn() dps = prec del prec # avoid using this deprecated kwarg if dps is not None and precision is not None: raise ValueError('Both decimal and binary precision supplied. ' 'Supply only one. ') if isinstance(num, str): # Float accepts spaces as digit separators num = num.replace(' ', '').lower() # in Py 3.6 # underscores are allowed. In anticipation of that, we ignore # legally placed underscores if '_' in num: parts = num.split('_') if not (all(parts) and all(parts[i][-1].isdigit() for i in range(0, len(parts), 2)) and all(parts[i][0].isdigit() for i in range(1, len(parts), 2))): # copy Py 3.6 error raise ValueError("could not convert string to float: '%s'" % num) num = ''.join(parts) if num.startswith('.') and len(num) > 1: num = '0' + num elif num.startswith('-.') and len(num) > 2: num = '-0.' + num[2:] elif num in ('inf', '+inf'): return S.Infinity elif num == '-inf': return S.NegativeInfinity elif isinstance(num, float) and num == 0: num = '0' elif isinstance(num, float) and num == float('inf'): return S.Infinity elif isinstance(num, float) and num == float('-inf'): return S.NegativeInfinity elif isinstance(num, float) and num == float('nan'): return S.NaN elif isinstance(num, (SYMPY_INTS, Integer)): num = str(num) elif num is S.Infinity: return num elif num is S.NegativeInfinity: return num elif num is S.NaN: return num elif _is_numpy_instance(num): # support for numpy datatypes num = _convert_numpy_types(num) elif isinstance(num, mpmath.mpf): if precision is None: if dps is None: precision = num.context.prec num = num._mpf_ if dps is None and precision is None: dps = 15 if isinstance(num, Float): return num if isinstance(num, str) and _literal_float(num): try: Num = decimal.Decimal(num) except decimal.InvalidOperation: pass else: isint = '.' not in num num, dps = _decimal_to_Rational_prec(Num) if num.is_Integer and isint: dps = max(dps, len(str(num).lstrip('-'))) dps = max(15, dps) precision = mlib.libmpf.dps_to_prec(dps) elif precision == '' and dps is None or precision is None and dps == '': if not isinstance(num, str): raise ValueError('The null string can only be used when ' 'the number to Float is passed as a string or an integer.') ok = None if _literal_float(num): try: Num = decimal.Decimal(num) except decimal.InvalidOperation: pass else: isint = '.' not in num num, dps = _decimal_to_Rational_prec(Num) if num.is_Integer and isint: dps = max(dps, len(str(num).lstrip('-'))) precision = mlib.libmpf.dps_to_prec(dps) ok = True if ok is None: raise ValueError('string-float not recognized: %s' % num) # decimal precision(dps) is set and maybe binary precision(precision) # as well.From here on binary precision is used to compute the Float. # Hence, if supplied use binary precision else translate from decimal # precision. if precision is None or precision == '': precision = mlib.libmpf.dps_to_prec(dps) precision = int(precision) if isinstance(num, float): _mpf_ = mlib.from_float(num, precision, rnd) elif isinstance(num, str): _mpf_ = mlib.from_str(num, precision, rnd) elif isinstance(num, decimal.Decimal): if num.is_finite(): _mpf_ = mlib.from_str(str(num), precision, rnd) elif num.is_nan(): return S.NaN elif num.is_infinite(): if num > 0: return S.Infinity return S.NegativeInfinity else: raise ValueError("unexpected decimal value %s" % str(num)) elif isinstance(num, tuple) and len(num) in (3, 4): if type(num[1]) is str: # it's a hexadecimal (coming from a pickled object) # assume that it is in standard form num = list(num) # If we're loading an object pickled in Python 2 into # Python 3, we may need to strip a tailing 'L' because # of a shim for int on Python 3, see issue #13470. if num[1].endswith('L'): num[1] = num[1][:-1] num[1] = MPZ(num[1], 16) _mpf_ = tuple(num) else: if len(num) == 4: # handle normalization hack return Float._new(num, precision) else: if not all(( num[0] in (0, 1), num[1] >= 0, all(type(i) in (int, int) for i in num) )): raise ValueError('malformed mpf: %s' % (num,)) # don't compute number or else it may # over/underflow return Float._new( (num[0], num[1], num[2], bitcount(num[1])), precision) else: try: _mpf_ = num._as_mpf_val(precision) except (NotImplementedError, AttributeError): _mpf_ = mpmath.mpf(num, prec=precision)._mpf_ return cls._new(_mpf_, precision, zero=False) @classmethod def _new(cls, _mpf_, _prec, zero=True): # special cases if zero and _mpf_ == fzero: return S.Zero # Float(0) -> 0.0; Float._new((0,0,0,0)) -> 0 elif _mpf_ == _mpf_nan: return S.NaN elif _mpf_ == _mpf_inf: return S.Infinity elif _mpf_ == _mpf_ninf: return S.NegativeInfinity obj = Expr.__new__(cls) obj._mpf_ = mpf_norm(_mpf_, _prec) obj._prec = _prec return obj # mpz can't be pickled def __getnewargs__(self): return (mlib.to_pickable(self._mpf_),) def __getstate__(self): return {'_prec': self._prec} def _hashable_content(self): return (self._mpf_, self._prec) def floor(self): return Integer(int(mlib.to_int( mlib.mpf_floor(self._mpf_, self._prec)))) def ceiling(self): return Integer(int(mlib.to_int( mlib.mpf_ceil(self._mpf_, self._prec)))) def __floor__(self): return self.floor() def __ceil__(self): return self.ceiling() @property def num(self): return mpmath.mpf(self._mpf_) def _as_mpf_val(self, prec): rv = mpf_norm(self._mpf_, prec) if rv != self._mpf_ and self._prec == prec: debug(self._mpf_, rv) return rv def _as_mpf_op(self, prec): return self._mpf_, max(prec, self._prec) def _eval_is_finite(self): if self._mpf_ in (_mpf_inf, _mpf_ninf): return False return True def _eval_is_infinite(self): if self._mpf_ in (_mpf_inf, _mpf_ninf): return True return False def _eval_is_integer(self): return self._mpf_ == fzero def _eval_is_negative(self): if self._mpf_ == _mpf_ninf or self._mpf_ == _mpf_inf: return False return self.num < 0 def _eval_is_positive(self): if self._mpf_ == _mpf_ninf or self._mpf_ == _mpf_inf: return False return self.num > 0 def _eval_is_extended_negative(self): if self._mpf_ == _mpf_ninf: return True if self._mpf_ == _mpf_inf: return False return self.num < 0 def _eval_is_extended_positive(self): if self._mpf_ == _mpf_inf: return True if self._mpf_ == _mpf_ninf: return False return self.num > 0 def _eval_is_zero(self): return self._mpf_ == fzero def __bool__(self): return self._mpf_ != fzero def __neg__(self): return Float._new(mlib.mpf_neg(self._mpf_), self._prec) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number) and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec) return Number.__add__(self, other) @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number) and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_sub(self._mpf_, rhs, prec, rnd), prec) return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number) and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec) return Number.__mul__(self, other) @_sympifyit('other', NotImplemented) def __truediv__(self, other): if isinstance(other, Number) and other != 0 and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec) return Number.__truediv__(self, other) @_sympifyit('other', NotImplemented) def __mod__(self, other): if isinstance(other, Rational) and other.q != 1 and global_parameters.evaluate: # calculate mod with Rationals, *then* round the result return Float(Rational.__mod__(Rational(self), other), precision=self._prec) if isinstance(other, Float) and global_parameters.evaluate: r = self/other if r == int(r): return Float(0, precision=max(self._prec, other._prec)) if isinstance(other, Number) and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mod(self._mpf_, rhs, prec, rnd), prec) return Number.__mod__(self, other) @_sympifyit('other', NotImplemented) def __rmod__(self, other): if isinstance(other, Float) and global_parameters.evaluate: return other.__mod__(self) if isinstance(other, Number) and global_parameters.evaluate: rhs, prec = other._as_mpf_op(self._prec) return Float._new(mlib.mpf_mod(rhs, self._mpf_, prec, rnd), prec) return Number.__rmod__(self, other) def _eval_power(self, expt): """ expt is symbolic object but not equal to 0, 1 (-p)**r -> exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) -> -> p**r*(sin(Pi*r) + cos(Pi*r)*I) """ if self == 0: if expt.is_positive: return S.Zero if expt.is_negative: return S.Infinity if isinstance(expt, Number): if isinstance(expt, Integer): prec = self._prec return Float._new( mlib.mpf_pow_int(self._mpf_, expt.p, prec, rnd), prec) elif isinstance(expt, Rational) and \ expt.p == 1 and expt.q % 2 and self.is_negative: return Pow(S.NegativeOne, expt, evaluate=False)*( -self)._eval_power(expt) expt, prec = expt._as_mpf_op(self._prec) mpfself = self._mpf_ try: y = mpf_pow(mpfself, expt, prec, rnd) return Float._new(y, prec) except mlib.ComplexResult: re, im = mlib.mpc_pow( (mpfself, fzero), (expt, fzero), prec, rnd) return Float._new(re, prec) + \ Float._new(im, prec)*S.ImaginaryUnit def __abs__(self): return Float._new(mlib.mpf_abs(self._mpf_), self._prec) def __int__(self): if self._mpf_ == fzero: return 0 return int(mlib.to_int(self._mpf_)) # uses round_fast = round_down def __eq__(self, other): from sympy.logic.boolalg import Boolean try: other = _sympify(other) except SympifyError: return NotImplemented if not self: return not other if isinstance(other, Boolean): return False if other.is_NumberSymbol: if other.is_irrational: return False return other.__eq__(self) if other.is_Float: # comparison is exact # so Float(.1, 3) != Float(.1, 33) return self._mpf_ == other._mpf_ if other.is_Rational: return other.__eq__(self) if other.is_Number: # numbers should compare at the same precision; # all _as_mpf_val routines should be sure to abide # by the request to change the prec if necessary; if # they don't, the equality test will fail since it compares # the mpf tuples ompf = other._as_mpf_val(self._prec) return bool(mlib.mpf_eq(self._mpf_, ompf)) return False # Float != non-Number def __ne__(self, other): return not self == other def _Frel(self, other, op): from sympy.core.numbers import prec_to_dps try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Rational: # test self*other.q <?> other.p without losing precision ''' >>> f = Float(.1,2) >>> i = 1234567890 >>> (f*i)._mpf_ (0, 471, 18, 9) >>> mlib.mpf_mul(f._mpf_, mlib.from_int(i)) (0, 505555550955, -12, 39) ''' smpf = mlib.mpf_mul(self._mpf_, mlib.from_int(other.q)) ompf = mlib.from_int(other.p) return _sympify(bool(op(smpf, ompf))) elif other.is_Float: return _sympify(bool( op(self._mpf_, other._mpf_))) elif other.is_comparable and other not in ( S.Infinity, S.NegativeInfinity): other = other.evalf(prec_to_dps(self._prec)) if other._prec > 1: if other.is_Number: return _sympify(bool( op(self._mpf_, other._as_mpf_val(self._prec)))) def __gt__(self, other): if isinstance(other, NumberSymbol): return other.__lt__(self) rv = self._Frel(other, mlib.mpf_gt) if rv is None: return Expr.__gt__(self, other) return rv def __ge__(self, other): if isinstance(other, NumberSymbol): return other.__le__(self) rv = self._Frel(other, mlib.mpf_ge) if rv is None: return Expr.__ge__(self, other) return rv def __lt__(self, other): if isinstance(other, NumberSymbol): return other.__gt__(self) rv = self._Frel(other, mlib.mpf_lt) if rv is None: return Expr.__lt__(self, other) return rv def __le__(self, other): if isinstance(other, NumberSymbol): return other.__ge__(self) rv = self._Frel(other, mlib.mpf_le) if rv is None: return Expr.__le__(self, other) return rv def __hash__(self): return super().__hash__() def epsilon_eq(self, other, epsilon="1e-15"): return abs(self - other) < Float(epsilon) def _sage_(self): import sage.all as sage return sage.RealNumber(str(self)) def __format__(self, format_spec): return format(decimal.Decimal(str(self)), format_spec) # Add sympify converters converter[float] = converter[decimal.Decimal] = Float # this is here to work nicely in Sage RealNumber = Float class Rational(Number): """Represents rational numbers (p/q) of any size. Examples ======== >>> from sympy import Rational, nsimplify, S, pi >>> Rational(1, 2) 1/2 Rational is unprejudiced in accepting input. If a float is passed, the underlying value of the binary representation will be returned: >>> Rational(.5) 1/2 >>> Rational(.2) 3602879701896397/18014398509481984 If the simpler representation of the float is desired then consider limiting the denominator to the desired value or convert the float to a string (which is roughly equivalent to limiting the denominator to 10**12): >>> Rational(str(.2)) 1/5 >>> Rational(.2).limit_denominator(10**12) 1/5 An arbitrarily precise Rational is obtained when a string literal is passed: >>> Rational("1.23") 123/100 >>> Rational('1e-2') 1/100 >>> Rational(".1") 1/10 >>> Rational('1e-2/3.2') 1/320 The conversion of other types of strings can be handled by the sympify() function, and conversion of floats to expressions or simple fractions can be handled with nsimplify: >>> S('.[3]') # repeating digits in brackets 1/3 >>> S('3**2/10') # general expressions 9/10 >>> nsimplify(.3) # numbers that have a simple form 3/10 But if the input does not reduce to a literal Rational, an error will be raised: >>> Rational(pi) Traceback (most recent call last): ... TypeError: invalid input: pi Low-level --------- Access numerator and denominator as .p and .q: >>> r = Rational(3, 4) >>> r 3/4 >>> r.p 3 >>> r.q 4 Note that p and q return integers (not SymPy Integers) so some care is needed when using them in expressions: >>> r.p/r.q 0.75 See Also ======== sympy.core.sympify.sympify, sympy.simplify.simplify.nsimplify """ is_real = True is_integer = False is_rational = True is_number = True __slots__ = ('p', 'q') is_Rational = True @cacheit def __new__(cls, p, q=None, gcd=None): if q is None: if isinstance(p, Rational): return p if isinstance(p, SYMPY_INTS): pass else: if isinstance(p, (float, Float)): return Rational(*_as_integer_ratio(p)) if not isinstance(p, str): try: p = sympify(p) except (SympifyError, SyntaxError): pass # error will raise below else: if p.count('/') > 1: raise TypeError('invalid input: %s' % p) p = p.replace(' ', '') pq = p.rsplit('/', 1) if len(pq) == 2: p, q = pq fp = fractions.Fraction(p) fq = fractions.Fraction(q) p = fp/fq try: p = fractions.Fraction(p) except ValueError: pass # error will raise below else: return Rational(p.numerator, p.denominator, 1) if not isinstance(p, Rational): raise TypeError('invalid input: %s' % p) q = 1 gcd = 1 else: p = Rational(p) q = Rational(q) if isinstance(q, Rational): p *= q.q q = q.p if isinstance(p, Rational): q *= p.q p = p.p # p and q are now integers if q == 0: if p == 0: if _errdict["divide"]: raise ValueError("Indeterminate 0/0") else: return S.NaN return S.ComplexInfinity if q < 0: q = -q p = -p if not gcd: gcd = igcd(abs(p), q) if gcd > 1: p //= gcd q //= gcd if q == 1: return Integer(p) if p == 1 and q == 2: return S.Half obj = Expr.__new__(cls) obj.p = p obj.q = q return obj def limit_denominator(self, max_denominator=1000000): """Closest Rational to self with denominator at most max_denominator. Examples ======== >>> from sympy import Rational >>> Rational('3.141592653589793').limit_denominator(10) 22/7 >>> Rational('3.141592653589793').limit_denominator(100) 311/99 """ f = fractions.Fraction(self.p, self.q) return Rational(f.limit_denominator(fractions.Fraction(int(max_denominator)))) def __getnewargs__(self): return (self.p, self.q) def _hashable_content(self): return (self.p, self.q) def _eval_is_positive(self): return self.p > 0 def _eval_is_zero(self): return self.p == 0 def __neg__(self): return Rational(-self.p, self.q) @_sympifyit('other', NotImplemented) def __add__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): return Rational(self.p + self.q*other.p, self.q, 1) elif isinstance(other, Rational): #TODO: this can probably be optimized more return Rational(self.p*other.q + self.q*other.p, self.q*other.q) elif isinstance(other, Float): return other + self else: return Number.__add__(self, other) return Number.__add__(self, other) __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): return Rational(self.p - self.q*other.p, self.q, 1) elif isinstance(other, Rational): return Rational(self.p*other.q - self.q*other.p, self.q*other.q) elif isinstance(other, Float): return -other + self else: return Number.__sub__(self, other) return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __rsub__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): return Rational(self.q*other.p - self.p, self.q, 1) elif isinstance(other, Rational): return Rational(self.q*other.p - self.p*other.q, self.q*other.q) elif isinstance(other, Float): return -self + other else: return Number.__rsub__(self, other) return Number.__rsub__(self, other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): return Rational(self.p*other.p, self.q, igcd(other.p, self.q)) elif isinstance(other, Rational): return Rational(self.p*other.p, self.q*other.q, igcd(self.p, other.q)*igcd(self.q, other.p)) elif isinstance(other, Float): return other*self else: return Number.__mul__(self, other) return Number.__mul__(self, other) __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __truediv__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): if self.p and other.p == S.Zero: return S.ComplexInfinity else: return Rational(self.p, self.q*other.p, igcd(self.p, other.p)) elif isinstance(other, Rational): return Rational(self.p*other.q, self.q*other.p, igcd(self.p, other.p)*igcd(self.q, other.q)) elif isinstance(other, Float): return self*(1/other) else: return Number.__truediv__(self, other) return Number.__truediv__(self, other) @_sympifyit('other', NotImplemented) def __rtruediv__(self, other): if global_parameters.evaluate: if isinstance(other, Integer): return Rational(other.p*self.q, self.p, igcd(self.p, other.p)) elif isinstance(other, Rational): return Rational(other.p*self.q, other.q*self.p, igcd(self.p, other.p)*igcd(self.q, other.q)) elif isinstance(other, Float): return other*(1/self) else: return Number.__rtruediv__(self, other) return Number.__rtruediv__(self, other) @_sympifyit('other', NotImplemented) def __mod__(self, other): if global_parameters.evaluate: if isinstance(other, Rational): n = (self.p*other.q) // (other.p*self.q) return Rational(self.p*other.q - n*other.p*self.q, self.q*other.q) if isinstance(other, Float): # calculate mod with Rationals, *then* round the answer return Float(self.__mod__(Rational(other)), precision=other._prec) return Number.__mod__(self, other) return Number.__mod__(self, other) @_sympifyit('other', NotImplemented) def __rmod__(self, other): if isinstance(other, Rational): return Rational.__mod__(other, self) return Number.__rmod__(self, other) def _eval_power(self, expt): if isinstance(expt, Number): if isinstance(expt, Float): return self._eval_evalf(expt._prec)**expt if expt.is_extended_negative: # (3/4)**-2 -> (4/3)**2 ne = -expt if (ne is S.One): return Rational(self.q, self.p) if self.is_negative: return S.NegativeOne**expt*Rational(self.q, -self.p)**ne else: return Rational(self.q, self.p)**ne if expt is S.Infinity: # -oo already caught by test for negative if self.p > self.q: # (3/2)**oo -> oo return S.Infinity if self.p < -self.q: # (-3/2)**oo -> oo + I*oo return S.Infinity + S.Infinity*S.ImaginaryUnit return S.Zero if isinstance(expt, Integer): # (4/3)**2 -> 4**2 / 3**2 return Rational(self.p**expt.p, self.q**expt.p, 1) if isinstance(expt, Rational): if self.p != 1: # (4/3)**(5/6) -> 4**(5/6)*3**(-5/6) return Integer(self.p)**expt*Integer(self.q)**(-expt) # as the above caught negative self.p, now self is positive return Integer(self.q)**Rational( expt.p*(expt.q - 1), expt.q) / \ Integer(self.q)**Integer(expt.p) if self.is_extended_negative and expt.is_even: return (-self)**expt return def _as_mpf_val(self, prec): return mlib.from_rational(self.p, self.q, prec, rnd) def _mpmath_(self, prec, rnd): return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd)) def __abs__(self): return Rational(abs(self.p), self.q) def __int__(self): p, q = self.p, self.q if p < 0: return -int(-p//q) return int(p//q) def floor(self): return Integer(self.p // self.q) def ceiling(self): return -Integer(-self.p // self.q) def __floor__(self): return self.floor() def __ceil__(self): return self.ceiling() def __eq__(self, other): from sympy.core.power import integer_log try: other = _sympify(other) except SympifyError: return NotImplemented if not isinstance(other, Number): # S(0) == S.false is False # S(0) == False is True return False if not self: return not other if other.is_NumberSymbol: if other.is_irrational: return False return other.__eq__(self) if other.is_Rational: # a Rational is always in reduced form so will never be 2/4 # so we can just check equivalence of args return self.p == other.p and self.q == other.q if other.is_Float: # all Floats have a denominator that is a power of 2 # so if self doesn't, it can't be equal to other if self.q & (self.q - 1): return False s, m, t = other._mpf_[:3] if s: m = -m if not t: # other is an odd integer if not self.is_Integer or self.is_even: return False return m == self.p if t > 0: # other is an even integer if not self.is_Integer: return False # does m*2**t == self.p return self.p and not self.p % m and \ integer_log(self.p//m, 2) == (t, True) # does non-integer s*m/2**-t = p/q? if self.is_Integer: return False return m == self.p and integer_log(self.q, 2) == (-t, True) return False def __ne__(self, other): return not self == other def _Rrel(self, other, attr): # if you want self < other, pass self, other, __gt__ try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Number: op = None s, o = self, other if other.is_NumberSymbol: op = getattr(o, attr) elif other.is_Float: op = getattr(o, attr) elif other.is_Rational: s, o = Integer(s.p*o.q), Integer(s.q*o.p) op = getattr(o, attr) if op: return op(s) if o.is_number and o.is_extended_real: return Integer(s.p), s.q*o def __gt__(self, other): rv = self._Rrel(other, '__lt__') if rv is None: rv = self, other elif not type(rv) is tuple: return rv return Expr.__gt__(*rv) def __ge__(self, other): rv = self._Rrel(other, '__le__') if rv is None: rv = self, other elif not type(rv) is tuple: return rv return Expr.__ge__(*rv) def __lt__(self, other): rv = self._Rrel(other, '__gt__') if rv is None: rv = self, other elif not type(rv) is tuple: return rv return Expr.__lt__(*rv) def __le__(self, other): rv = self._Rrel(other, '__ge__') if rv is None: rv = self, other elif not type(rv) is tuple: return rv return Expr.__le__(*rv) def __hash__(self): return super().__hash__() def factors(self, limit=None, use_trial=True, use_rho=False, use_pm1=False, verbose=False, visual=False): """A wrapper to factorint which return factors of self that are smaller than limit (or cheap to compute). Special methods of factoring are disabled by default so that only trial division is used. """ from sympy.ntheory import factorrat return factorrat(self, limit=limit, use_trial=use_trial, use_rho=use_rho, use_pm1=use_pm1, verbose=verbose).copy() def numerator(self): return self.p def denominator(self): return self.q @_sympifyit('other', NotImplemented) def gcd(self, other): if isinstance(other, Rational): if other == S.Zero: return other return Rational( Integer(igcd(self.p, other.p)), Integer(ilcm(self.q, other.q))) return Number.gcd(self, other) @_sympifyit('other', NotImplemented) def lcm(self, other): if isinstance(other, Rational): return Rational( self.p // igcd(self.p, other.p) * other.p, igcd(self.q, other.q)) return Number.lcm(self, other) def as_numer_denom(self): return Integer(self.p), Integer(self.q) def _sage_(self): import sage.all as sage return sage.Integer(self.p)/sage.Integer(self.q) def as_content_primitive(self, radical=False, clear=True): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import S >>> (S(-3)/2).as_content_primitive() (3/2, -1) See docstring of Expr.as_content_primitive for more examples. """ if self: if self.is_positive: return self, S.One return -self, S.NegativeOne return S.One, self def as_coeff_Mul(self, rational=False): """Efficiently extract the coefficient of a product. """ return self, S.One def as_coeff_Add(self, rational=False): """Efficiently extract the coefficient of a summation. """ return self, S.Zero class Integer(Rational): """Represents integer numbers of any size. Examples ======== >>> from sympy import Integer >>> Integer(3) 3 If a float or a rational is passed to Integer, the fractional part will be discarded; the effect is of rounding toward zero. >>> Integer(3.8) 3 >>> Integer(-3.8) -3 A string is acceptable input if it can be parsed as an integer: >>> Integer("9" * 20) 99999999999999999999 It is rarely needed to explicitly instantiate an Integer, because Python integers are automatically converted to Integer when they are used in SymPy expressions. """ q = 1 is_integer = True is_number = True is_Integer = True __slots__ = ('p',) def _as_mpf_val(self, prec): return mlib.from_int(self.p, prec, rnd) def _mpmath_(self, prec, rnd): return mpmath.make_mpf(self._as_mpf_val(prec)) @cacheit def __new__(cls, i): if isinstance(i, str): i = i.replace(' ', '') # whereas we cannot, in general, make a Rational from an # arbitrary expression, we can make an Integer unambiguously # (except when a non-integer expression happens to round to # an integer). So we proceed by taking int() of the input and # let the int routines determine whether the expression can # be made into an int or whether an error should be raised. try: ival = int(i) except TypeError: raise TypeError( "Argument of Integer should be of numeric type, got %s." % i) # We only work with well-behaved integer types. This converts, for # example, numpy.int32 instances. if ival == 1: return S.One if ival == -1: return S.NegativeOne if ival == 0: return S.Zero obj = Expr.__new__(cls) obj.p = ival return obj def __getnewargs__(self): return (self.p,) # Arithmetic operations are here for efficiency def __int__(self): return self.p def floor(self): return Integer(self.p) def ceiling(self): return Integer(self.p) def __floor__(self): return self.floor() def __ceil__(self): return self.ceiling() def __neg__(self): return Integer(-self.p) def __abs__(self): if self.p >= 0: return self else: return Integer(-self.p) def __divmod__(self, other): from .containers import Tuple if isinstance(other, Integer) and global_parameters.evaluate: return Tuple(*(divmod(self.p, other.p))) else: return Number.__divmod__(self, other) def __rdivmod__(self, other): from .containers import Tuple if isinstance(other, int) and global_parameters.evaluate: return Tuple(*(divmod(other, self.p))) else: try: other = Number(other) except TypeError: msg = "unsupported operand type(s) for divmod(): '%s' and '%s'" oname = type(other).__name__ sname = type(self).__name__ raise TypeError(msg % (oname, sname)) return Number.__divmod__(other, self) # TODO make it decorator + bytecodehacks? def __add__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(self.p + other) elif isinstance(other, Integer): return Integer(self.p + other.p) elif isinstance(other, Rational): return Rational(self.p*other.q + other.p, other.q, 1) return Rational.__add__(self, other) else: return Add(self, other) def __radd__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(other + self.p) elif isinstance(other, Rational): return Rational(other.p + self.p*other.q, other.q, 1) return Rational.__radd__(self, other) return Rational.__radd__(self, other) def __sub__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(self.p - other) elif isinstance(other, Integer): return Integer(self.p - other.p) elif isinstance(other, Rational): return Rational(self.p*other.q - other.p, other.q, 1) return Rational.__sub__(self, other) return Rational.__sub__(self, other) def __rsub__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(other - self.p) elif isinstance(other, Rational): return Rational(other.p - self.p*other.q, other.q, 1) return Rational.__rsub__(self, other) return Rational.__rsub__(self, other) def __mul__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(self.p*other) elif isinstance(other, Integer): return Integer(self.p*other.p) elif isinstance(other, Rational): return Rational(self.p*other.p, other.q, igcd(self.p, other.q)) return Rational.__mul__(self, other) return Rational.__mul__(self, other) def __rmul__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(other*self.p) elif isinstance(other, Rational): return Rational(other.p*self.p, other.q, igcd(self.p, other.q)) return Rational.__rmul__(self, other) return Rational.__rmul__(self, other) def __mod__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(self.p % other) elif isinstance(other, Integer): return Integer(self.p % other.p) return Rational.__mod__(self, other) return Rational.__mod__(self, other) def __rmod__(self, other): if global_parameters.evaluate: if isinstance(other, int): return Integer(other % self.p) elif isinstance(other, Integer): return Integer(other.p % self.p) return Rational.__rmod__(self, other) return Rational.__rmod__(self, other) def __eq__(self, other): if isinstance(other, int): return (self.p == other) elif isinstance(other, Integer): return (self.p == other.p) return Rational.__eq__(self, other) def __ne__(self, other): return not self == other def __gt__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Integer: return _sympify(self.p > other.p) return Rational.__gt__(self, other) def __lt__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Integer: return _sympify(self.p < other.p) return Rational.__lt__(self, other) def __ge__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Integer: return _sympify(self.p >= other.p) return Rational.__ge__(self, other) def __le__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if other.is_Integer: return _sympify(self.p <= other.p) return Rational.__le__(self, other) def __hash__(self): return hash(self.p) def __index__(self): return self.p ######################################## def _eval_is_odd(self): return bool(self.p % 2) def _eval_power(self, expt): """ Tries to do some simplifications on self**expt Returns None if no further simplifications can be done. Explanation =========== When exponent is a fraction (so we have for example a square root), we try to find a simpler representation by factoring the argument up to factors of 2**15, e.g. - sqrt(4) becomes 2 - sqrt(-4) becomes 2*I - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7) Further simplification would require a special call to factorint on the argument which is not done here for sake of speed. """ from sympy.ntheory.factor_ import perfect_power if expt is S.Infinity: if self.p > S.One: return S.Infinity # cases -1, 0, 1 are done in their respective classes return S.Infinity + S.ImaginaryUnit*S.Infinity if expt is S.NegativeInfinity: return Rational(1, self)**S.Infinity if not isinstance(expt, Number): # simplify when expt is even # (-2)**k --> 2**k if self.is_negative and expt.is_even: return (-self)**expt if isinstance(expt, Float): # Rational knows how to exponentiate by a Float return super()._eval_power(expt) if not isinstance(expt, Rational): return if expt is S.Half and self.is_negative: # we extract I for this special case since everyone is doing so return S.ImaginaryUnit*Pow(-self, expt) if expt.is_negative: # invert base and change sign on exponent ne = -expt if self.is_negative: return S.NegativeOne**expt*Rational(1, -self)**ne else: return Rational(1, self.p)**ne # see if base is a perfect root, sqrt(4) --> 2 x, xexact = integer_nthroot(abs(self.p), expt.q) if xexact: # if it's a perfect root we've finished result = Integer(x**abs(expt.p)) if self.is_negative: result *= S.NegativeOne**expt return result # The following is an algorithm where we collect perfect roots # from the factors of base. # if it's not an nth root, it still might be a perfect power b_pos = int(abs(self.p)) p = perfect_power(b_pos) if p is not False: dict = {p[0]: p[1]} else: dict = Integer(b_pos).factors(limit=2**15) # now process the dict of factors out_int = 1 # integer part out_rad = 1 # extracted radicals sqr_int = 1 sqr_gcd = 0 sqr_dict = {} for prime, exponent in dict.items(): exponent *= expt.p # remove multiples of expt.q: (2**12)**(1/10) -> 2*(2**2)**(1/10) div_e, div_m = divmod(exponent, expt.q) if div_e > 0: out_int *= prime**div_e if div_m > 0: # see if the reduced exponent shares a gcd with e.q # (2**2)**(1/10) -> 2**(1/5) g = igcd(div_m, expt.q) if g != 1: out_rad *= Pow(prime, Rational(div_m//g, expt.q//g)) else: sqr_dict[prime] = div_m # identify gcd of remaining powers for p, ex in sqr_dict.items(): if sqr_gcd == 0: sqr_gcd = ex else: sqr_gcd = igcd(sqr_gcd, ex) if sqr_gcd == 1: break for k, v in sqr_dict.items(): sqr_int *= k**(v//sqr_gcd) if sqr_int == b_pos and out_int == 1 and out_rad == 1: result = None else: result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q)) if self.is_negative: result *= Pow(S.NegativeOne, expt) return result def _eval_is_prime(self): from sympy.ntheory import isprime return isprime(self) def _eval_is_composite(self): if self > 1: return fuzzy_not(self.is_prime) else: return False def as_numer_denom(self): return self, S.One @_sympifyit('other', NotImplemented) def __floordiv__(self, other): if not isinstance(other, Expr): return NotImplemented if isinstance(other, Integer): return Integer(self.p // other) return Integer(divmod(self, other)[0]) def __rfloordiv__(self, other): return Integer(Integer(other).p // self.p) # Add sympify converters converter[int] = Integer class AlgebraicNumber(Expr): """Class for representing algebraic numbers in SymPy. """ __slots__ = ('rep', 'root', 'alias', 'minpoly') is_AlgebraicNumber = True is_algebraic = True is_number = True def __new__(cls, expr, coeffs=None, alias=None, **args): """Construct a new algebraic number. """ from sympy import Poly from sympy.polys.polyclasses import ANP, DMP from sympy.polys.numberfields import minimal_polynomial from sympy.core.symbol import Symbol expr = sympify(expr) if isinstance(expr, (tuple, Tuple)): minpoly, root = expr if not minpoly.is_Poly: minpoly = Poly(minpoly) elif expr.is_AlgebraicNumber: minpoly, root = expr.minpoly, expr.root else: minpoly, root = minimal_polynomial( expr, args.get('gen'), polys=True), expr dom = minpoly.get_domain() if coeffs is not None: if not isinstance(coeffs, ANP): rep = DMP.from_sympy_list(sympify(coeffs), 0, dom) scoeffs = Tuple(*coeffs) else: rep = DMP.from_list(coeffs.to_list(), 0, dom) scoeffs = Tuple(*coeffs.to_list()) if rep.degree() >= minpoly.degree(): rep = rep.rem(minpoly.rep) else: rep = DMP.from_list([1, 0], 0, dom) scoeffs = Tuple(1, 0) sargs = (root, scoeffs) if alias is not None: if not isinstance(alias, Symbol): alias = Symbol(alias) sargs = sargs + (alias,) obj = Expr.__new__(cls, *sargs) obj.rep = rep obj.root = root obj.alias = alias obj.minpoly = minpoly return obj def __hash__(self): return super().__hash__() def _eval_evalf(self, prec): return self.as_expr()._evalf(prec) @property def is_aliased(self): """Returns ``True`` if ``alias`` was set. """ return self.alias is not None def as_poly(self, x=None): """Create a Poly instance from ``self``. """ from sympy import Dummy, Poly, PurePoly if x is not None: return Poly.new(self.rep, x) else: if self.alias is not None: return Poly.new(self.rep, self.alias) else: return PurePoly.new(self.rep, Dummy('x')) def as_expr(self, x=None): """Create a Basic expression from ``self``. """ return self.as_poly(x or self.root).as_expr().expand() def coeffs(self): """Returns all SymPy coefficients of an algebraic number. """ return [ self.rep.dom.to_sympy(c) for c in self.rep.all_coeffs() ] def native_coeffs(self): """Returns all native coefficients of an algebraic number. """ return self.rep.all_coeffs() def to_algebraic_integer(self): """Convert ``self`` to an algebraic integer. """ from sympy import Poly f = self.minpoly if f.LC() == 1: return self coeff = f.LC()**(f.degree() - 1) poly = f.compose(Poly(f.gen/f.LC())) minpoly = poly*coeff root = f.LC()*self.root return AlgebraicNumber((minpoly, root), self.coeffs()) def _eval_simplify(self, **kwargs): from sympy.polys import CRootOf, minpoly measure, ratio = kwargs['measure'], kwargs['ratio'] for r in [r for r in self.minpoly.all_roots() if r.func != CRootOf]: if minpoly(self.root - r).is_Symbol: # use the matching root if it's simpler if measure(r) < ratio*measure(self.root): return AlgebraicNumber(r) return self class RationalConstant(Rational): """ Abstract base class for rationals with specific behaviors Derived classes must define class attributes p and q and should probably all be singletons. """ __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) class IntegerConstant(Integer): __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) class Zero(IntegerConstant, metaclass=Singleton): """The number zero. Zero is a singleton, and can be accessed by ``S.Zero`` Examples ======== >>> from sympy import S, Integer >>> Integer(0) is S.Zero True >>> 1/S.Zero zoo References ========== .. [1] https://en.wikipedia.org/wiki/Zero """ p = 0 q = 1 is_positive = False is_negative = False is_zero = True is_number = True is_comparable = True __slots__ = () def __getnewargs__(self): return () @staticmethod def __abs__(): return S.Zero @staticmethod def __neg__(): return S.Zero def _eval_power(self, expt): if expt.is_positive: return self if expt.is_negative: return S.ComplexInfinity if expt.is_extended_real is False: return S.NaN # infinities are already handled with pos and neg # tests above; now throw away leading numbers on Mul # exponent coeff, terms = expt.as_coeff_Mul() if coeff.is_negative: return S.ComplexInfinity**terms if coeff is not S.One: # there is a Number to discard return self**terms def _eval_order(self, *symbols): # Order(0,x) -> 0 return self def __bool__(self): return False def as_coeff_Mul(self, rational=False): # XXX this routine should be deleted """Efficiently extract the coefficient of a summation. """ return S.One, self class One(IntegerConstant, metaclass=Singleton): """The number one. One is a singleton, and can be accessed by ``S.One``. Examples ======== >>> from sympy import S, Integer >>> Integer(1) is S.One True References ========== .. [1] https://en.wikipedia.org/wiki/1_%28number%29 """ is_number = True p = 1 q = 1 __slots__ = () def __getnewargs__(self): return () @staticmethod def __abs__(): return S.One @staticmethod def __neg__(): return S.NegativeOne def _eval_power(self, expt): return self def _eval_order(self, *symbols): return @staticmethod def factors(limit=None, use_trial=True, use_rho=False, use_pm1=False, verbose=False, visual=False): if visual: return S.One else: return {} class NegativeOne(IntegerConstant, metaclass=Singleton): """The number negative one. NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``. Examples ======== >>> from sympy import S, Integer >>> Integer(-1) is S.NegativeOne True See Also ======== One References ========== .. [1] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29 """ is_number = True p = -1 q = 1 __slots__ = () def __getnewargs__(self): return () @staticmethod def __abs__(): return S.One @staticmethod def __neg__(): return S.One def _eval_power(self, expt): if expt.is_odd: return S.NegativeOne if expt.is_even: return S.One if isinstance(expt, Number): if isinstance(expt, Float): return Float(-1.0)**expt if expt is S.NaN: return S.NaN if expt is S.Infinity or expt is S.NegativeInfinity: return S.NaN if expt is S.Half: return S.ImaginaryUnit if isinstance(expt, Rational): if expt.q == 2: return S.ImaginaryUnit**Integer(expt.p) i, r = divmod(expt.p, expt.q) if i: return self**i*self**Rational(r, expt.q) return class Half(RationalConstant, metaclass=Singleton): """The rational number 1/2. Half is a singleton, and can be accessed by ``S.Half``. Examples ======== >>> from sympy import S, Rational >>> Rational(1, 2) is S.Half True References ========== .. [1] https://en.wikipedia.org/wiki/One_half """ is_number = True p = 1 q = 2 __slots__ = () def __getnewargs__(self): return () @staticmethod def __abs__(): return S.Half class Infinity(Number, metaclass=Singleton): r"""Positive infinite quantity. Explanation =========== In real analysis the symbol `\infty` denotes an unbounded limit: `x\to\infty` means that `x` grows without bound. Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled `+\infty` and `-\infty` can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. Infinity is a singleton, and can be accessed by ``S.Infinity``, or can be imported as ``oo``. Examples ======== >>> from sympy import oo, exp, limit, Symbol >>> 1 + oo oo >>> 42/oo 0 >>> x = Symbol('x') >>> limit(exp(x), x, oo) oo See Also ======== NegativeInfinity, NaN References ========== .. [1] https://en.wikipedia.org/wiki/Infinity """ is_commutative = True is_number = True is_complex = False is_extended_real = True is_infinite = True is_comparable = True is_extended_positive = True is_prime = False __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\infty" def _eval_subs(self, old, new): if self == old: return new def _eval_evalf(self, prec=None): return Float('inf') def evalf(self, prec=None, **options): return self._eval_evalf(prec) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NegativeInfinity or other is S.NaN: return S.NaN return self return Number.__add__(self, other) __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.Infinity or other is S.NaN: return S.NaN return self return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __rsub__(self, other): return (-self).__add__(other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other.is_zero or other is S.NaN: return S.NaN if other.is_extended_positive: return self return S.NegativeInfinity return Number.__mul__(self, other) __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __truediv__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.Infinity or \ other is S.NegativeInfinity or \ other is S.NaN: return S.NaN if other.is_extended_nonnegative: return self return S.NegativeInfinity return Number.__truediv__(self, other) def __abs__(self): return S.Infinity def __neg__(self): return S.NegativeInfinity def _eval_power(self, expt): """ ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``oo ** nan`` ``nan`` ``oo ** -p`` ``0`` ``p`` is number, ``oo`` ================ ======= ============================== See Also ======== Pow NaN NegativeInfinity """ from sympy.functions import re if expt.is_extended_positive: return S.Infinity if expt.is_extended_negative: return S.Zero if expt is S.NaN: return S.NaN if expt is S.ComplexInfinity: return S.NaN if expt.is_extended_real is False and expt.is_number: expt_real = re(expt) if expt_real.is_positive: return S.ComplexInfinity if expt_real.is_negative: return S.Zero if expt_real.is_zero: return S.NaN return self**expt.evalf() def _as_mpf_val(self, prec): return mlib.finf def _sage_(self): import sage.all as sage return sage.oo def __hash__(self): return super().__hash__() def __eq__(self, other): return other is S.Infinity or other == float('inf') def __ne__(self, other): return other is not S.Infinity and other != float('inf') __gt__ = Expr.__gt__ __ge__ = Expr.__ge__ __lt__ = Expr.__lt__ __le__ = Expr.__le__ @_sympifyit('other', NotImplemented) def __mod__(self, other): if not isinstance(other, Expr): return NotImplemented return S.NaN __rmod__ = __mod__ def floor(self): return self def ceiling(self): return self oo = S.Infinity class NegativeInfinity(Number, metaclass=Singleton): """Negative infinite quantity. NegativeInfinity is a singleton, and can be accessed by ``S.NegativeInfinity``. See Also ======== Infinity """ is_extended_real = True is_complex = False is_commutative = True is_infinite = True is_comparable = True is_extended_negative = True is_number = True is_prime = False __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"-\infty" def _eval_subs(self, old, new): if self == old: return new def _eval_evalf(self, prec=None): return Float('-inf') def evalf(self, prec=None, **options): return self._eval_evalf(prec) @_sympifyit('other', NotImplemented) def __add__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.Infinity or other is S.NaN: return S.NaN return self return Number.__add__(self, other) __radd__ = __add__ @_sympifyit('other', NotImplemented) def __sub__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.NegativeInfinity or other is S.NaN: return S.NaN return self return Number.__sub__(self, other) @_sympifyit('other', NotImplemented) def __rsub__(self, other): return (-self).__add__(other) @_sympifyit('other', NotImplemented) def __mul__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other.is_zero or other is S.NaN: return S.NaN if other.is_extended_positive: return self return S.Infinity return Number.__mul__(self, other) __rmul__ = __mul__ @_sympifyit('other', NotImplemented) def __truediv__(self, other): if isinstance(other, Number) and global_parameters.evaluate: if other is S.Infinity or \ other is S.NegativeInfinity or \ other is S.NaN: return S.NaN if other.is_extended_nonnegative: return self return S.Infinity return Number.__truediv__(self, other) def __abs__(self): return S.Infinity def __neg__(self): return S.Infinity def _eval_power(self, expt): """ ``expt`` is symbolic object but not equal to 0 or 1. ================ ======= ============================== Expression Result Notes ================ ======= ============================== ``(-oo) ** nan`` ``nan`` ``(-oo) ** oo`` ``nan`` ``(-oo) ** -oo`` ``nan`` ``(-oo) ** e`` ``oo`` ``e`` is positive even integer ``(-oo) ** o`` ``-oo`` ``o`` is positive odd integer ================ ======= ============================== See Also ======== Infinity Pow NaN """ if expt.is_number: if expt is S.NaN or \ expt is S.Infinity or \ expt is S.NegativeInfinity: return S.NaN if isinstance(expt, Integer) and expt.is_extended_positive: if expt.is_odd: return S.NegativeInfinity else: return S.Infinity return S.NegativeOne**expt*S.Infinity**expt def _as_mpf_val(self, prec): return mlib.fninf def _sage_(self): import sage.all as sage return -(sage.oo) def __hash__(self): return super().__hash__() def __eq__(self, other): return other is S.NegativeInfinity or other == float('-inf') def __ne__(self, other): return other is not S.NegativeInfinity and other != float('-inf') __gt__ = Expr.__gt__ __ge__ = Expr.__ge__ __lt__ = Expr.__lt__ __le__ = Expr.__le__ @_sympifyit('other', NotImplemented) def __mod__(self, other): if not isinstance(other, Expr): return NotImplemented return S.NaN __rmod__ = __mod__ def floor(self): return self def ceiling(self): return self def as_powers_dict(self): return {S.NegativeOne: 1, S.Infinity: 1} class NaN(Number, metaclass=Singleton): """ Not a Number. Explanation =========== This serves as a place holder for numeric values that are indeterminate. Most operations on NaN, produce another NaN. Most indeterminate forms, such as ``0/0`` or ``oo - oo` produce NaN. Two exceptions are ``0**0`` and ``oo**0``, which all produce ``1`` (this is consistent with Python's float). NaN is loosely related to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python ``float('nan')``. Differences are noted below. NaN is mathematically not equal to anything else, even NaN itself. This explains the initially counter-intuitive results with ``Eq`` and ``==`` in the examples below. NaN is not comparable so inequalities raise a TypeError. This is in contrast with floating point nan where all inequalities are false. NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported as ``nan``. Examples ======== >>> from sympy import nan, S, oo, Eq >>> nan is S.NaN True >>> oo - oo nan >>> nan + 1 nan >>> Eq(nan, nan) # mathematical equality False >>> nan == nan # structural equality True References ========== .. [1] https://en.wikipedia.org/wiki/NaN """ is_commutative = True is_extended_real = None is_real = None is_rational = None is_algebraic = None is_transcendental = None is_integer = None is_comparable = False is_finite = None is_zero = None is_prime = None is_positive = None is_negative = None is_number = True __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\text{NaN}" def __neg__(self): return self @_sympifyit('other', NotImplemented) def __add__(self, other): return self @_sympifyit('other', NotImplemented) def __sub__(self, other): return self @_sympifyit('other', NotImplemented) def __mul__(self, other): return self @_sympifyit('other', NotImplemented) def __truediv__(self, other): return self def floor(self): return self def ceiling(self): return self def _as_mpf_val(self, prec): return _mpf_nan def _sage_(self): import sage.all as sage return sage.NaN def __hash__(self): return super().__hash__() def __eq__(self, other): # NaN is structurally equal to another NaN return other is S.NaN def __ne__(self, other): return other is not S.NaN # Expr will _sympify and raise TypeError __gt__ = Expr.__gt__ __ge__ = Expr.__ge__ __lt__ = Expr.__lt__ __le__ = Expr.__le__ nan = S.NaN @dispatch(NaN, Expr) # type:ignore def _eval_is_eq(a, b): # noqa:F811 return False class ComplexInfinity(AtomicExpr, metaclass=Singleton): r"""Complex infinity. Explanation =========== In complex analysis the symbol `\tilde\infty`, called "complex infinity", represents a quantity with infinite magnitude, but undetermined complex phase. ComplexInfinity is a singleton, and can be accessed by ``S.ComplexInfinity``, or can be imported as ``zoo``. Examples ======== >>> from sympy import zoo >>> zoo + 42 zoo >>> 42/zoo 0 >>> zoo + zoo nan >>> zoo*zoo zoo See Also ======== Infinity """ is_commutative = True is_infinite = True is_number = True is_prime = False is_complex = False is_extended_real = False __slots__ = () def __new__(cls): return AtomicExpr.__new__(cls) def _latex(self, printer): return r"\tilde{\infty}" @staticmethod def __abs__(): return S.Infinity def floor(self): return self def ceiling(self): return self @staticmethod def __neg__(): return S.ComplexInfinity def _eval_power(self, expt): if expt is S.ComplexInfinity: return S.NaN if isinstance(expt, Number): if expt.is_zero: return S.NaN else: if expt.is_positive: return S.ComplexInfinity else: return S.Zero def _sage_(self): import sage.all as sage return sage.UnsignedInfinityRing.gen() zoo = S.ComplexInfinity class NumberSymbol(AtomicExpr): is_commutative = True is_finite = True is_number = True __slots__ = () is_NumberSymbol = True def __new__(cls): return AtomicExpr.__new__(cls) def approximation(self, number_cls): """ Return an interval with number_cls endpoints that contains the value of NumberSymbol. If not implemented, then return None. """ def _eval_evalf(self, prec): return Float._new(self._as_mpf_val(prec), prec) def __eq__(self, other): try: other = _sympify(other) except SympifyError: return NotImplemented if self is other: return True if other.is_Number and self.is_irrational: return False return False # NumberSymbol != non-(Number|self) def __ne__(self, other): return not self == other def __le__(self, other): if self is other: return S.true return Expr.__le__(self, other) def __ge__(self, other): if self is other: return S.true return Expr.__ge__(self, other) def __int__(self): # subclass with appropriate return value raise NotImplementedError def __hash__(self): return super().__hash__() class Exp1(NumberSymbol, metaclass=Singleton): r"""The `e` constant. Explanation =========== The transcendental number `e = 2.718281828\ldots` is the base of the natural logarithm and of the exponential function, `e = \exp(1)`. Sometimes called Euler's number or Napier's constant. Exp1 is a singleton, and can be accessed by ``S.Exp1``, or can be imported as ``E``. Examples ======== >>> from sympy import exp, log, E >>> E is exp(1) True >>> log(E) 1 References ========== .. [1] https://en.wikipedia.org/wiki/E_%28mathematical_constant%29 """ is_real = True is_positive = True is_negative = False # XXX Forces is_negative/is_nonnegative is_irrational = True is_number = True is_algebraic = False is_transcendental = True __slots__ = () def _latex(self, printer): return r"e" @staticmethod def __abs__(): return S.Exp1 def __int__(self): return 2 def _as_mpf_val(self, prec): return mpf_e(prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (Integer(2), Integer(3)) elif issubclass(number_cls, Rational): pass def _eval_power(self, expt): from sympy import exp return exp(expt) def _eval_rewrite_as_sin(self, **kwargs): from sympy import sin I = S.ImaginaryUnit return sin(I + S.Pi/2) - I*sin(I) def _eval_rewrite_as_cos(self, **kwargs): from sympy import cos I = S.ImaginaryUnit return cos(I) + I*cos(I + S.Pi/2) def _sage_(self): import sage.all as sage return sage.e E = S.Exp1 class Pi(NumberSymbol, metaclass=Singleton): r"""The `\pi` constant. Explanation =========== The transcendental number `\pi = 3.141592654\ldots` represents the ratio of a circle's circumference to its diameter, the area of the unit circle, the half-period of trigonometric functions, and many other things in mathematics. Pi is a singleton, and can be accessed by ``S.Pi``, or can be imported as ``pi``. Examples ======== >>> from sympy import S, pi, oo, sin, exp, integrate, Symbol >>> S.Pi pi >>> pi > 3 True >>> pi.is_irrational True >>> x = Symbol('x') >>> sin(x + 2*pi) sin(x) >>> integrate(exp(-x**2), (x, -oo, oo)) sqrt(pi) References ========== .. [1] https://en.wikipedia.org/wiki/Pi """ is_real = True is_positive = True is_negative = False is_irrational = True is_number = True is_algebraic = False is_transcendental = True __slots__ = () def _latex(self, printer): return r"\pi" @staticmethod def __abs__(): return S.Pi def __int__(self): return 3 def _as_mpf_val(self, prec): return mpf_pi(prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (Integer(3), Integer(4)) elif issubclass(number_cls, Rational): return (Rational(223, 71), Rational(22, 7)) def _sage_(self): import sage.all as sage return sage.pi pi = S.Pi class GoldenRatio(NumberSymbol, metaclass=Singleton): r"""The golden ratio, `\phi`. Explanation =========== `\phi = \frac{1 + \sqrt{5}}{2}` is algebraic number. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities, i.e. their maximum. GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``. Examples ======== >>> from sympy import S >>> S.GoldenRatio > 1 True >>> S.GoldenRatio.expand(func=True) 1/2 + sqrt(5)/2 >>> S.GoldenRatio.is_irrational True References ========== .. [1] https://en.wikipedia.org/wiki/Golden_ratio """ is_real = True is_positive = True is_negative = False is_irrational = True is_number = True is_algebraic = True is_transcendental = False __slots__ = () def _latex(self, printer): return r"\phi" def __int__(self): return 1 def _as_mpf_val(self, prec): # XXX track down why this has to be increased rv = mlib.from_man_exp(phi_fixed(prec + 10), -prec - 10) return mpf_norm(rv, prec) def _eval_expand_func(self, **hints): from sympy import sqrt return S.Half + S.Half*sqrt(5) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.One, Rational(2)) elif issubclass(number_cls, Rational): pass def _sage_(self): import sage.all as sage return sage.golden_ratio _eval_rewrite_as_sqrt = _eval_expand_func class TribonacciConstant(NumberSymbol, metaclass=Singleton): r"""The tribonacci constant. Explanation =========== The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`, and also satisfies the equation `x + x^{-3} = 2`. TribonacciConstant is a singleton, and can be accessed by ``S.TribonacciConstant``. Examples ======== >>> from sympy import S >>> S.TribonacciConstant > 1 True >>> S.TribonacciConstant.expand(func=True) 1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3 >>> S.TribonacciConstant.is_irrational True >>> S.TribonacciConstant.n(20) 1.8392867552141611326 References ========== .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers """ is_real = True is_positive = True is_negative = False is_irrational = True is_number = True is_algebraic = True is_transcendental = False __slots__ = () def _latex(self, printer): return r"\text{TribonacciConstant}" def __int__(self): return 2 def _eval_evalf(self, prec): rv = self._eval_expand_func(function=True)._eval_evalf(prec + 4) return Float(rv, precision=prec) def _eval_expand_func(self, **hints): from sympy import sqrt, cbrt return (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3 def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.One, Rational(2)) elif issubclass(number_cls, Rational): pass _eval_rewrite_as_sqrt = _eval_expand_func class EulerGamma(NumberSymbol, metaclass=Singleton): r"""The Euler-Mascheroni constant. Explanation =========== `\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical constant recurring in analysis and number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm: .. math:: \gamma = \lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right) EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``. Examples ======== >>> from sympy import S >>> S.EulerGamma.is_irrational >>> S.EulerGamma > 0 True >>> S.EulerGamma > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant """ is_real = True is_positive = True is_negative = False is_irrational = None is_number = True __slots__ = () def _latex(self, printer): return r"\gamma" def __int__(self): return 0 def _as_mpf_val(self, prec): # XXX track down why this has to be increased v = mlib.libhyper.euler_fixed(prec + 10) rv = mlib.from_man_exp(v, -prec - 10) return mpf_norm(rv, prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.Zero, S.One) elif issubclass(number_cls, Rational): return (S.Half, Rational(3, 5)) def _sage_(self): import sage.all as sage return sage.euler_gamma class Catalan(NumberSymbol, metaclass=Singleton): r"""Catalan's constant. Explanation =========== `K = 0.91596559\ldots` is given by the infinite series .. math:: K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2} Catalan is a singleton, and can be accessed by ``S.Catalan``. Examples ======== >>> from sympy import S >>> S.Catalan.is_irrational >>> S.Catalan > 0 True >>> S.Catalan > 1 False References ========== .. [1] https://en.wikipedia.org/wiki/Catalan%27s_constant """ is_real = True is_positive = True is_negative = False is_irrational = None is_number = True __slots__ = () def __int__(self): return 0 def _as_mpf_val(self, prec): # XXX track down why this has to be increased v = mlib.catalan_fixed(prec + 10) rv = mlib.from_man_exp(v, -prec - 10) return mpf_norm(rv, prec) def approximation_interval(self, number_cls): if issubclass(number_cls, Integer): return (S.Zero, S.One) elif issubclass(number_cls, Rational): return (Rational(9, 10), S.One) def _eval_rewrite_as_Sum(self, k_sym=None, symbols=None): from sympy import Sum, Dummy if (k_sym is not None) or (symbols is not None): return self k = Dummy('k', integer=True, nonnegative=True) return Sum((-1)**k / (2*k+1)**2, (k, 0, S.Infinity)) def _sage_(self): import sage.all as sage return sage.catalan class ImaginaryUnit(AtomicExpr, metaclass=Singleton): r"""The imaginary unit, `i = \sqrt{-1}`. I is a singleton, and can be accessed by ``S.I``, or can be imported as ``I``. Examples ======== >>> from sympy import I, sqrt >>> sqrt(-1) I >>> I*I -1 >>> 1/I -I References ========== .. [1] https://en.wikipedia.org/wiki/Imaginary_unit """ is_commutative = True is_imaginary = True is_finite = True is_number = True is_algebraic = True is_transcendental = False __slots__ = () def _latex(self, printer): return printer._settings['imaginary_unit_latex'] @staticmethod def __abs__(): return S.One def _eval_evalf(self, prec): return self def _eval_conjugate(self): return -S.ImaginaryUnit def _eval_power(self, expt): """ b is I = sqrt(-1) e is symbolic object but not equal to 0, 1 I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal I**0 mod 4 -> 1 I**1 mod 4 -> I I**2 mod 4 -> -1 I**3 mod 4 -> -I """ if isinstance(expt, Number): if isinstance(expt, Integer): expt = expt.p % 4 if expt == 0: return S.One if expt == 1: return S.ImaginaryUnit if expt == 2: return -S.One return -S.ImaginaryUnit return def as_base_exp(self): return S.NegativeOne, S.Half def _sage_(self): import sage.all as sage return sage.I @property def _mpc_(self): return (Float(0)._mpf_, Float(1)._mpf_) I = S.ImaginaryUnit @dispatch(Tuple, Number) # type:ignore def _eval_is_eq(self, other): # noqa: F811 return False def sympify_fractions(f): return Rational(f.numerator, f.denominator, 1) converter[fractions.Fraction] = sympify_fractions if HAS_GMPY: def sympify_mpz(x): return Integer(int(x)) # XXX: The sympify_mpq function here was never used because it is # overridden by the other sympify_mpq function below. Maybe it should just # be removed or maybe it should be used for something... def sympify_mpq(x): return Rational(int(x.numerator), int(x.denominator)) converter[type(gmpy.mpz(1))] = sympify_mpz converter[type(gmpy.mpq(1, 2))] = sympify_mpq def sympify_mpmath_mpq(x): p, q = x._mpq_ return Rational(p, q, 1) converter[type(mpmath.rational.mpq(1, 2))] = sympify_mpmath_mpq def sympify_mpmath(x): return Expr._from_mpmath(x, x.context.prec) converter[mpnumeric] = sympify_mpmath def sympify_complex(a): real, imag = list(map(sympify, (a.real, a.imag))) return real + S.ImaginaryUnit*imag converter[complex] = sympify_complex from .power import Pow, integer_nthroot from .mul import Mul Mul.identity = One() from .add import Add Add.identity = Zero() def _register_classes(): numbers.Number.register(Number) numbers.Real.register(Float) numbers.Rational.register(Rational) numbers.Rational.register(Integer) _register_classes()
11358109ea8e14556ccb640cb53591f7bc4e05414471e255d18dc10bd698dff4
from operator import attrgetter from typing import Tuple, Type from collections import defaultdict from sympy.utilities.exceptions import SymPyDeprecationWarning from sympy.core.sympify import _sympify as _sympify_, sympify from sympy.core.basic import Basic from sympy.core.cache import cacheit from sympy.core.compatibility import ordered from sympy.core.logic import fuzzy_and from sympy.core.parameters import global_parameters from sympy.utilities.iterables import sift from sympy.multipledispatch.dispatcher import (Dispatcher, ambiguity_register_error_ignore_dup, str_signature, RaiseNotImplementedError) class AssocOp(Basic): """ Associative operations, can separate noncommutative and commutative parts. (a op b) op c == a op (b op c) == a op b op c. Base class for Add and Mul. This is an abstract base class, concrete derived classes must define the attribute `identity`. Parameters ========== *args : Arguments which are operated evaluate : bool, optional Evaluate the operation. If not passed, refer to ``global_parameters.evaluate``. """ # for performance reason, we don't let is_commutative go to assumptions, # and keep it right here __slots__ = ('is_commutative',) # type: Tuple[str, ...] _args_type = None # type: Type[Basic] @cacheit def __new__(cls, *args, evaluate=None, _sympify=True): from sympy import Order # Allow faster processing by passing ``_sympify=False``, if all arguments # are already sympified. if _sympify: args = list(map(_sympify_, args)) # Disallow non-Expr args in Add/Mul typ = cls._args_type if typ is not None: from sympy.core.relational import Relational if any(isinstance(arg, Relational) for arg in args): raise TypeError("Relational can not be used in %s" % cls.__name__) # This should raise TypeError once deprecation period is over: if not all(isinstance(arg, typ) for arg in args): SymPyDeprecationWarning( feature="Add/Mul with non-Expr args", useinstead="Expr args", issue=19445, deprecated_since_version="1.7" ).warn() if evaluate is None: evaluate = global_parameters.evaluate if not evaluate: obj = cls._from_args(args) obj = cls._exec_constructor_postprocessors(obj) return obj args = [a for a in args if a is not cls.identity] if len(args) == 0: return cls.identity if len(args) == 1: return args[0] c_part, nc_part, order_symbols = cls.flatten(args) is_commutative = not nc_part obj = cls._from_args(c_part + nc_part, is_commutative) obj = cls._exec_constructor_postprocessors(obj) if order_symbols is not None: return Order(obj, *order_symbols) return obj @classmethod def _from_args(cls, args, is_commutative=None): """Create new instance with already-processed args. If the args are not in canonical order, then a non-canonical result will be returned, so use with caution. The order of args may change if the sign of the args is changed.""" if len(args) == 0: return cls.identity elif len(args) == 1: return args[0] obj = super().__new__(cls, *args) if is_commutative is None: is_commutative = fuzzy_and(a.is_commutative for a in args) obj.is_commutative = is_commutative return obj def _new_rawargs(self, *args, reeval=True, **kwargs): """Create new instance of own class with args exactly as provided by caller but returning the self class identity if args is empty. Examples ======== This is handy when we want to optimize things, e.g. >>> from sympy import Mul, S >>> from sympy.abc import x, y >>> e = Mul(3, x, y) >>> e.args (3, x, y) >>> Mul(*e.args[1:]) x*y >>> e._new_rawargs(*e.args[1:]) # the same as above, but faster x*y Note: use this with caution. There is no checking of arguments at all. This is best used when you are rebuilding an Add or Mul after simply removing one or more args. If, for example, modifications, result in extra 1s being inserted they will show up in the result: >>> m = (x*y)._new_rawargs(S.One, x); m 1*x >>> m == x False >>> m.is_Mul True Another issue to be aware of is that the commutativity of the result is based on the commutativity of self. If you are rebuilding the terms that came from a commutative object then there will be no problem, but if self was non-commutative then what you are rebuilding may now be commutative. Although this routine tries to do as little as possible with the input, getting the commutativity right is important, so this level of safety is enforced: commutativity will always be recomputed if self is non-commutative and kwarg `reeval=False` has not been passed. """ if reeval and self.is_commutative is False: is_commutative = None else: is_commutative = self.is_commutative return self._from_args(args, is_commutative) @classmethod def flatten(cls, seq): """Return seq so that none of the elements are of type `cls`. This is the vanilla routine that will be used if a class derived from AssocOp does not define its own flatten routine.""" # apply associativity, no commutativity property is used new_seq = [] while seq: o = seq.pop() if o.__class__ is cls: # classes must match exactly seq.extend(o.args) else: new_seq.append(o) new_seq.reverse() # c_part, nc_part, order_symbols return [], new_seq, None def _matches_commutative(self, expr, repl_dict={}, old=False): """ Matches Add/Mul "pattern" to an expression "expr". repl_dict ... a dictionary of (wild: expression) pairs, that get returned with the results This function is the main workhorse for Add/Mul. Examples ======== >>> from sympy import symbols, Wild, sin >>> a = Wild("a") >>> b = Wild("b") >>> c = Wild("c") >>> x, y, z = symbols("x y z") >>> (a+sin(b)*c)._matches_commutative(x+sin(y)*z) {a_: x, b_: y, c_: z} In the example above, "a+sin(b)*c" is the pattern, and "x+sin(y)*z" is the expression. The repl_dict contains parts that were already matched. For example here: >>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z, repl_dict={a: x}) {a_: x, b_: y, c_: z} the only function of the repl_dict is to return it in the result, e.g. if you omit it: >>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z) {b_: y, c_: z} the "a: x" is not returned in the result, but otherwise it is equivalent. """ # make sure expr is Expr if pattern is Expr from .expr import Add, Expr from sympy import Mul repl_dict = repl_dict.copy() if isinstance(self, Expr) and not isinstance(expr, Expr): return None # handle simple patterns if self == expr: return repl_dict d = self._matches_simple(expr, repl_dict) if d is not None: return d # eliminate exact part from pattern: (2+a+w1+w2).matches(expr) -> (w1+w2).matches(expr-a-2) from .function import WildFunction from .symbol import Wild wild_part, exact_part = sift(self.args, lambda p: p.has(Wild, WildFunction) and not expr.has(p), binary=True) if not exact_part: wild_part = list(ordered(wild_part)) if self.is_Add: # in addition to normal ordered keys, impose # sorting on Muls with leading Number to put # them in order wild_part = sorted(wild_part, key=lambda x: x.args[0] if x.is_Mul and x.args[0].is_Number else 0) else: exact = self._new_rawargs(*exact_part) free = expr.free_symbols if free and (exact.free_symbols - free): # there are symbols in the exact part that are not # in the expr; but if there are no free symbols, let # the matching continue return None newexpr = self._combine_inverse(expr, exact) if not old and (expr.is_Add or expr.is_Mul): if newexpr.count_ops() > expr.count_ops(): return None newpattern = self._new_rawargs(*wild_part) return newpattern.matches(newexpr, repl_dict) # now to real work ;) i = 0 saw = set() while expr not in saw: saw.add(expr) args = tuple(ordered(self.make_args(expr))) if self.is_Add and expr.is_Add: # in addition to normal ordered keys, impose # sorting on Muls with leading Number to put # them in order args = tuple(sorted(args, key=lambda x: x.args[0] if x.is_Mul and x.args[0].is_Number else 0)) expr_list = (self.identity,) + args for last_op in reversed(expr_list): for w in reversed(wild_part): d1 = w.matches(last_op, repl_dict) if d1 is not None: d2 = self.xreplace(d1).matches(expr, d1) if d2 is not None: return d2 if i == 0: if self.is_Mul: # make e**i look like Mul if expr.is_Pow and expr.exp.is_Integer: if expr.exp > 0: expr = Mul(*[expr.base, expr.base**(expr.exp - 1)], evaluate=False) else: expr = Mul(*[1/expr.base, expr.base**(expr.exp + 1)], evaluate=False) i += 1 continue elif self.is_Add: # make i*e look like Add c, e = expr.as_coeff_Mul() if abs(c) > 1: if c > 0: expr = Add(*[e, (c - 1)*e], evaluate=False) else: expr = Add(*[-e, (c + 1)*e], evaluate=False) i += 1 continue # try collection on non-Wild symbols from sympy.simplify.radsimp import collect was = expr did = set() for w in reversed(wild_part): c, w = w.as_coeff_mul(Wild) free = c.free_symbols - did if free: did.update(free) expr = collect(expr, free) if expr != was: i += 0 continue break # if we didn't continue, there is nothing more to do return def _has_matcher(self): """Helper for .has()""" def _ncsplit(expr): # this is not the same as args_cnc because here # we don't assume expr is a Mul -- hence deal with args -- # and always return a set. cpart, ncpart = sift(expr.args, lambda arg: arg.is_commutative is True, binary=True) return set(cpart), ncpart c, nc = _ncsplit(self) cls = self.__class__ def is_in(expr): if expr == self: return True elif not isinstance(expr, Basic): return False elif isinstance(expr, cls): _c, _nc = _ncsplit(expr) if (c & _c) == c: if not nc: return True elif len(nc) <= len(_nc): for i in range(len(_nc) - len(nc) + 1): if _nc[i:i + len(nc)] == nc: return True return False return is_in def _eval_evalf(self, prec): """ Evaluate the parts of self that are numbers; if the whole thing was a number with no functions it would have been evaluated, but it wasn't so we must judiciously extract the numbers and reconstruct the object. This is *not* simply replacing numbers with evaluated numbers. Numbers should be handled in the largest pure-number expression as possible. So the code below separates ``self`` into number and non-number parts and evaluates the number parts and walks the args of the non-number part recursively (doing the same thing). """ from .add import Add from .mul import Mul from .symbol import Symbol from .function import AppliedUndef if isinstance(self, (Mul, Add)): x, tail = self.as_independent(Symbol, AppliedUndef) # if x is an AssocOp Function then the _evalf below will # call _eval_evalf (here) so we must break the recursion if not (tail is self.identity or isinstance(x, AssocOp) and x.is_Function or x is self.identity and isinstance(tail, AssocOp)): # here, we have a number so we just call to _evalf with prec; # prec is not the same as n, it is the binary precision so # that's why we don't call to evalf. x = x._evalf(prec) if x is not self.identity else self.identity args = [] tail_args = tuple(self.func.make_args(tail)) for a in tail_args: # here we call to _eval_evalf since we don't know what we # are dealing with and all other _eval_evalf routines should # be doing the same thing (i.e. taking binary prec and # finding the evalf-able args) newa = a._eval_evalf(prec) if newa is None: args.append(a) else: args.append(newa) return self.func(x, *args) # this is the same as above, but there were no pure-number args to # deal with args = [] for a in self.args: newa = a._eval_evalf(prec) if newa is None: args.append(a) else: args.append(newa) return self.func(*args) @classmethod def make_args(cls, expr): """ Return a sequence of elements `args` such that cls(*args) == expr Examples ======== >>> from sympy import Symbol, Mul, Add >>> x, y = map(Symbol, 'xy') >>> Mul.make_args(x*y) (x, y) >>> Add.make_args(x*y) (x*y,) >>> set(Add.make_args(x*y + y)) == set([y, x*y]) True """ if isinstance(expr, cls): return expr.args else: return (sympify(expr),) def doit(self, **hints): if hints.get('deep', True): terms = [term.doit(**hints) for term in self.args] else: terms = self.args return self.func(*terms, evaluate=True) class ShortCircuit(Exception): pass class LatticeOp(AssocOp): """ Join/meet operations of an algebraic lattice[1]. Explanation =========== These binary operations are associative (op(op(a, b), c) = op(a, op(b, c))), commutative (op(a, b) = op(b, a)) and idempotent (op(a, a) = op(a) = a). Common examples are AND, OR, Union, Intersection, max or min. They have an identity element (op(identity, a) = a) and an absorbing element conventionally called zero (op(zero, a) = zero). This is an abstract base class, concrete derived classes must declare attributes zero and identity. All defining properties are then respected. Examples ======== >>> from sympy import Integer >>> from sympy.core.operations import LatticeOp >>> class my_join(LatticeOp): ... zero = Integer(0) ... identity = Integer(1) >>> my_join(2, 3) == my_join(3, 2) True >>> my_join(2, my_join(3, 4)) == my_join(2, 3, 4) True >>> my_join(0, 1, 4, 2, 3, 4) 0 >>> my_join(1, 2) 2 References: .. [1] https://en.wikipedia.org/wiki/Lattice_%28order%29 """ is_commutative = True def __new__(cls, *args, **options): args = (_sympify_(arg) for arg in args) try: # /!\ args is a generator and _new_args_filter # must be careful to handle as such; this # is done so short-circuiting can be done # without having to sympify all values _args = frozenset(cls._new_args_filter(args)) except ShortCircuit: return sympify(cls.zero) if not _args: return sympify(cls.identity) elif len(_args) == 1: return set(_args).pop() else: # XXX in almost every other case for __new__, *_args is # passed along, but the expectation here is for _args obj = super(AssocOp, cls).__new__(cls, _args) obj._argset = _args return obj @classmethod def _new_args_filter(cls, arg_sequence, call_cls=None): """Generator filtering args""" ncls = call_cls or cls for arg in arg_sequence: if arg == ncls.zero: raise ShortCircuit(arg) elif arg == ncls.identity: continue elif arg.func == ncls: yield from arg.args else: yield arg @classmethod def make_args(cls, expr): """ Return a set of args such that cls(*arg_set) == expr. """ if isinstance(expr, cls): return expr._argset else: return frozenset([sympify(expr)]) # XXX: This should be cached on the object rather than using cacheit # Maybe _argset can just be sorted in the constructor? @property # type: ignore @cacheit def args(self): return tuple(ordered(self._argset)) @staticmethod def _compare_pretty(a, b): return (str(a) > str(b)) - (str(a) < str(b)) class AssocOpDispatcher: """ Handler dispatcher for associative operators .. notes:: This approach is experimental, and can be replaced or deleted in the future. See https://github.com/sympy/sympy/pull/19463. Explanation =========== If arguments of different types are passed, the classes which handle the operation for each type are collected. Then, a class which performs the operation is selected by recursive binary dispatching. Dispatching relation can be registered by ``register_handlerclass`` method. Priority registration is unordered. You cannot make ``A*B`` and ``B*A`` refer to different handler classes. All logic dealing with the order of arguments must be implemented in the handler class. Examples ======== >>> from sympy import Add, Expr, Symbol >>> from sympy.core.add import add >>> class NewExpr(Expr): ... @property ... def _add_handler(self): ... return NewAdd >>> class NewAdd(NewExpr, Add): ... pass >>> add.register_handlerclass((Add, NewAdd), NewAdd) >>> a, b = Symbol('a'), NewExpr() >>> add(a, b) == NewAdd(a, b) True """ def __init__(self, name, doc=None): self.name = name self.doc = doc self.handlerattr = "_%s_handler" % name self._handlergetter = attrgetter(self.handlerattr) self._dispatcher = Dispatcher(name) def __repr__(self): return "<dispatched %s>" % self.name def register_handlerclass(self, classes, typ, on_ambiguity=ambiguity_register_error_ignore_dup): """ Register the handler class for two classes, in both straight and reversed order. Paramteters =========== classes : tuple of two types Classes who are compared with each other. typ: Class which is registered to represent *cls1* and *cls2*. Handler method of *self* must be implemented in this class. """ if not len(classes) == 2: raise RuntimeError( "Only binary dispatch is supported, but got %s types: <%s>." % ( len(classes), str_signature(classes) )) if len(set(classes)) == 1: raise RuntimeError( "Duplicate types <%s> cannot be dispatched." % str_signature(classes) ) self._dispatcher.add(tuple(classes), typ, on_ambiguity=on_ambiguity) self._dispatcher.add(tuple(reversed(classes)), typ, on_ambiguity=on_ambiguity) @cacheit def __call__(self, *args, _sympify=True, **kwargs): """ Parameters ========== *args : Arguments which are operated """ if _sympify: args = tuple(map(_sympify_, args)) handlers = frozenset(map(self._handlergetter, args)) # no need to sympify again return self.dispatch(handlers)(*args, _sympify=False, **kwargs) @cacheit def dispatch(self, handlers): """ Select the handler class, and return its handler method. """ # Quick exit for the case where all handlers are same if len(handlers) == 1: h, = handlers if not isinstance(h, type): raise RuntimeError("Handler {!r} is not a type.".format(h)) return h # Recursively select with registered binary priority for i, typ in enumerate(handlers): if not isinstance(typ, type): raise RuntimeError("Handler {!r} is not a type.".format(typ)) if i == 0: handler = typ else: prev_handler = handler handler = self._dispatcher.dispatch(prev_handler, typ) if not isinstance(handler, type): raise RuntimeError( "Dispatcher for {!r} and {!r} must return a type, but got {!r}".format( prev_handler, typ, handler )) # return handler class return handler @property def __doc__(self): docs = [ "Multiply dispatched associative operator: %s" % self.name, "Note that support for this is experimental, see the docs for :class:`AssocOpDispatcher` for details" ] if self.doc: docs.append(self.doc) s = "Registered handler classes\n" s += '=' * len(s) docs.append(s) amb_sigs = [] typ_sigs = defaultdict(list) for sigs in self._dispatcher.ordering[::-1]: key = self._dispatcher.funcs[sigs] typ_sigs[key].append(sigs) for typ, sigs in typ_sigs.items(): sigs_str = ', '.join('<%s>' % str_signature(sig) for sig in sigs) if isinstance(typ, RaiseNotImplementedError): amb_sigs.append(sigs_str) continue s = 'Inputs: %s\n' % sigs_str s += '-' * len(s) + '\n' s += typ.__name__ docs.append(s) if amb_sigs: s = "Ambiguous handler classes\n" s += '=' * len(s) docs.append(s) s = '\n'.join(amb_sigs) docs.append(s) return '\n\n'.join(docs)
3e6e15503b7bf6decad6b20dfeaa361fd749c221a6e2d0633a14bdf1ebe6338e
from sympy.core.numbers import nan from .function import Function class Mod(Function): """Represents a modulo operation on symbolic expressions. Parameters ========== p : Expr Dividend. q : Expr Divisor. Notes ===== The convention used is the same as Python's: the remainder always has the same sign as the divisor. Examples ======== >>> from sympy.abc import x, y >>> x**2 % y Mod(x**2, y) >>> _.subs({x: 5, y: 6}) 1 """ @classmethod def eval(cls, p, q): from sympy.core.add import Add from sympy.core.mul import Mul from sympy.core.singleton import S from sympy.core.exprtools import gcd_terms from sympy.polys.polytools import gcd def doit(p, q): """Try to return p % q if both are numbers or +/-p is known to be less than or equal q. """ if q.is_zero: raise ZeroDivisionError("Modulo by zero") if p.is_finite is False or q.is_finite is False or p is nan or q is nan: return nan if p is S.Zero or p == q or p == -q or (p.is_integer and q == 1): return S.Zero if q.is_Number: if p.is_Number: return p%q if q == 2: if p.is_even: return S.Zero elif p.is_odd: return S.One if hasattr(p, '_eval_Mod'): rv = getattr(p, '_eval_Mod')(q) if rv is not None: return rv # by ratio r = p/q if r.is_integer: return S.Zero try: d = int(r) except TypeError: pass else: if isinstance(d, int): rv = p - d*q if (rv*q < 0) == True: rv += q return rv # by difference # -2|q| < p < 2|q| d = abs(p) for _ in range(2): d -= abs(q) if d.is_negative: if q.is_positive: if p.is_positive: return d + q elif p.is_negative: return -d elif q.is_negative: if p.is_positive: return d elif p.is_negative: return -d + q break rv = doit(p, q) if rv is not None: return rv # denest if isinstance(p, cls): qinner = p.args[1] if qinner % q == 0: return cls(p.args[0], q) elif (qinner*(q - qinner)).is_nonnegative: # |qinner| < |q| and have same sign return p elif isinstance(-p, cls): qinner = (-p).args[1] if qinner % q == 0: return cls(-(-p).args[0], q) elif (qinner*(q + qinner)).is_nonpositive: # |qinner| < |q| and have different sign return p elif isinstance(p, Add): # separating into modulus and non modulus both_l = non_mod_l, mod_l = [], [] for arg in p.args: both_l[isinstance(arg, cls)].append(arg) # if q same for all if mod_l and all(inner.args[1] == q for inner in mod_l): net = Add(*non_mod_l) + Add(*[i.args[0] for i in mod_l]) return cls(net, q) elif isinstance(p, Mul): # separating into modulus and non modulus both_l = non_mod_l, mod_l = [], [] for arg in p.args: both_l[isinstance(arg, cls)].append(arg) if mod_l and all(inner.args[1] == q for inner in mod_l): # finding distributive term non_mod_l = [cls(x, q) for x in non_mod_l] mod = [] non_mod = [] for j in non_mod_l: if isinstance(j, cls): mod.append(j.args[0]) else: non_mod.append(j) prod_mod = Mul(*mod) prod_non_mod = Mul(*non_mod) prod_mod1 = Mul(*[i.args[0] for i in mod_l]) net = prod_mod1*prod_mod return prod_non_mod*cls(net, q) if q.is_Integer and q is not S.One: _ = [] for i in non_mod_l: if i.is_Integer and (i % q is not S.Zero): _.append(i%q) else: _.append(i) non_mod_l = _ p = Mul(*(non_mod_l + mod_l)) # XXX other possibilities? # extract gcd; any further simplification should be done by the user G = gcd(p, q) if G != 1: p, q = [ gcd_terms(i/G, clear=False, fraction=False) for i in (p, q)] pwas, qwas = p, q # simplify terms # (x + y + 2) % x -> Mod(y + 2, x) if p.is_Add: args = [] for i in p.args: a = cls(i, q) if a.count(cls) > i.count(cls): args.append(i) else: args.append(a) if args != list(p.args): p = Add(*args) else: # handle coefficients if they are not Rational # since those are not handled by factor_terms # e.g. Mod(.6*x, .3*y) -> 0.3*Mod(2*x, y) cp, p = p.as_coeff_Mul() cq, q = q.as_coeff_Mul() ok = False if not cp.is_Rational or not cq.is_Rational: r = cp % cq if r == 0: G *= cq p *= int(cp/cq) ok = True if not ok: p = cp*p q = cq*q # simple -1 extraction if p.could_extract_minus_sign() and q.could_extract_minus_sign(): G, p, q = [-i for i in (G, p, q)] # check again to see if p and q can now be handled as numbers rv = doit(p, q) if rv is not None: return rv*G # put 1.0 from G on inside if G.is_Float and G == 1: p *= G return cls(p, q, evaluate=False) elif G.is_Mul and G.args[0].is_Float and G.args[0] == 1: p = G.args[0]*p G = Mul._from_args(G.args[1:]) return G*cls(p, q, evaluate=(p, q) != (pwas, qwas)) def _eval_is_integer(self): from sympy.core.logic import fuzzy_and, fuzzy_not p, q = self.args if fuzzy_and([p.is_integer, q.is_integer, fuzzy_not(q.is_zero)]): return True def _eval_is_nonnegative(self): if self.args[1].is_positive: return True def _eval_is_nonpositive(self): if self.args[1].is_negative: return True def _eval_rewrite_as_floor(self, a, b, **kwargs): from sympy.functions.elementary.integers import floor return a - b*floor(a/b)
da20ccf1235a3a9c490484ae34b02bafbbaad7c79b00287331e270e3244ec35e
from sympy.core.assumptions import StdFactKB, _assume_defined from sympy.core.compatibility import is_sequence, ordered from .basic import Basic, Atom from .sympify import sympify from .singleton import S from .expr import Expr, AtomicExpr from .cache import cacheit from .function import FunctionClass from sympy.core.logic import fuzzy_bool from sympy.logic.boolalg import Boolean from sympy.utilities.iterables import cartes, sift from sympy.core.containers import Tuple import string import re as _re import random class Str(Atom): """ Represents string in SymPy. Explanation =========== Previously, ``Symbol`` was used where string is needed in ``args`` of SymPy objects, e.g. denoting the name of the instance. However, since ``Symbol`` represents mathematical scalar, this class should be used instead. """ __slots__ = ('name',) def __new__(cls, name, **kwargs): if not isinstance(name, str): raise TypeError("name should be a string, not %s" % repr(type(name))) obj = Expr.__new__(cls, **kwargs) obj.name = name return obj def __getnewargs__(self): return (self.name,) def _hashable_content(self): return (self.name,) def _filter_assumptions(kwargs): """Split the given dict into assumptions and non-assumptions. Keys are taken as assumptions if they correspond to an entry in ``_assume_defined``. """ assumptions, nonassumptions = map(dict, sift(kwargs.items(), lambda i: i[0] in _assume_defined, binary=True)) Symbol._sanitize(assumptions) return assumptions, nonassumptions def _symbol(s, matching_symbol=None, **assumptions): """Return s if s is a Symbol, else if s is a string, return either the matching_symbol if the names are the same or else a new symbol with the same assumptions as the matching symbol (or the assumptions as provided). Examples ======== >>> from sympy import Symbol >>> from sympy.core.symbol import _symbol >>> _symbol('y') y >>> _.is_real is None True >>> _symbol('y', real=True).is_real True >>> x = Symbol('x') >>> _symbol(x, real=True) x >>> _.is_real is None # ignore attribute if s is a Symbol True Below, the variable sym has the name 'foo': >>> sym = Symbol('foo', real=True) Since 'x' is not the same as sym's name, a new symbol is created: >>> _symbol('x', sym).name 'x' It will acquire any assumptions give: >>> _symbol('x', sym, real=False).is_real False Since 'foo' is the same as sym's name, sym is returned >>> _symbol('foo', sym) foo Any assumptions given are ignored: >>> _symbol('foo', sym, real=False).is_real True NB: the symbol here may not be the same as a symbol with the same name defined elsewhere as a result of different assumptions. See Also ======== sympy.core.symbol.Symbol """ if isinstance(s, str): if matching_symbol and matching_symbol.name == s: return matching_symbol return Symbol(s, **assumptions) elif isinstance(s, Symbol): return s else: raise ValueError('symbol must be string for symbol name or Symbol') def uniquely_named_symbol(xname, exprs=(), compare=str, modify=None, **assumptions): """Return a symbol which, when printed, will have a name unique from any other already in the expressions given. The name is made unique by appending numbers (default) but this can be customized with the keyword 'modify'. Parameters ========== xname : a string or a Symbol (when symbol xname <- str(xname)) compare : a single arg function that takes a symbol and returns a string to be compared with xname (the default is the str function which indicates how the name will look when it is printed, e.g. this includes underscores that appear on Dummy symbols) modify : a single arg function that changes its string argument in some way (the default is to append numbers) Examples ======== >>> from sympy.core.symbol import uniquely_named_symbol >>> from sympy.abc import x >>> uniquely_named_symbol('x', x) x0 """ from sympy.core.function import AppliedUndef def numbered_string_incr(s, start=0): if not s: return str(start) i = len(s) - 1 while i != -1: if not s[i].isdigit(): break i -= 1 n = str(int(s[i + 1:] or start - 1) + 1) return s[:i + 1] + n default = None if is_sequence(xname): xname, default = xname x = str(xname) if not exprs: return _symbol(x, default, **assumptions) if not is_sequence(exprs): exprs = [exprs] names = set().union( [i.name for e in exprs for i in e.atoms(Symbol)] + [i.func.name for e in exprs for i in e.atoms(AppliedUndef)]) if modify is None: modify = numbered_string_incr while any(x == compare(s) for s in names): x = modify(x) return _symbol(x, default, **assumptions) _uniquely_named_symbol = uniquely_named_symbol class Symbol(AtomicExpr, Boolean): """ Assumptions: commutative = True You can override the default assumptions in the constructor. Examples ======== >>> from sympy import symbols >>> A,B = symbols('A,B', commutative = False) >>> bool(A*B != B*A) True >>> bool(A*B*2 == 2*A*B) == True # multiplication by scalars is commutative True """ is_comparable = False __slots__ = ('name',) is_Symbol = True is_symbol = True @property def _diff_wrt(self): """Allow derivatives wrt Symbols. Examples ======== >>> from sympy import Symbol >>> x = Symbol('x') >>> x._diff_wrt True """ return True @staticmethod def _sanitize(assumptions, obj=None): """Remove None, covert values to bool, check commutativity *in place*. """ # be strict about commutativity: cannot be None is_commutative = fuzzy_bool(assumptions.get('commutative', True)) if is_commutative is None: whose = '%s ' % obj.__name__ if obj else '' raise ValueError( '%scommutativity must be True or False.' % whose) # sanitize other assumptions so 1 -> True and 0 -> False for key in list(assumptions.keys()): v = assumptions[key] if v is None: assumptions.pop(key) continue assumptions[key] = bool(v) def _merge(self, assumptions): base = self.assumptions0 for k in set(assumptions) & set(base): if assumptions[k] != base[k]: from sympy.utilities.misc import filldedent raise ValueError(filldedent(''' non-matching assumptions for %s: existing value is %s and new value is %s''' % ( k, base[k], assumptions[k]))) base.update(assumptions) return base def __new__(cls, name, **assumptions): """Symbols are identified by name and assumptions:: >>> from sympy import Symbol >>> Symbol("x") == Symbol("x") True >>> Symbol("x", real=True) == Symbol("x", real=False) False """ cls._sanitize(assumptions, cls) return Symbol.__xnew_cached_(cls, name, **assumptions) def __new_stage2__(cls, name, **assumptions): if not isinstance(name, str): raise TypeError("name should be a string, not %s" % repr(type(name))) obj = Expr.__new__(cls) obj.name = name # TODO: Issue #8873: Forcing the commutative assumption here means # later code such as ``srepr()`` cannot tell whether the user # specified ``commutative=True`` or omitted it. To workaround this, # we keep a copy of the assumptions dict, then create the StdFactKB, # and finally overwrite its ``._generator`` with the dict copy. This # is a bit of a hack because we assume StdFactKB merely copies the # given dict as ``._generator``, but future modification might, e.g., # compute a minimal equivalent assumption set. tmp_asm_copy = assumptions.copy() # be strict about commutativity is_commutative = fuzzy_bool(assumptions.get('commutative', True)) assumptions['commutative'] = is_commutative obj._assumptions = StdFactKB(assumptions) obj._assumptions._generator = tmp_asm_copy # Issue #8873 return obj __xnew__ = staticmethod( __new_stage2__) # never cached (e.g. dummy) __xnew_cached_ = staticmethod( cacheit(__new_stage2__)) # symbols are always cached def __getnewargs__(self): return (self.name,) def __getstate__(self): return {'_assumptions': self._assumptions} def _hashable_content(self): # Note: user-specified assumptions not hashed, just derived ones return (self.name,) + tuple(sorted(self.assumptions0.items())) def _eval_subs(self, old, new): from sympy.core.power import Pow if old.is_Pow: return Pow(self, S.One, evaluate=False)._eval_subs(old, new) @property def assumptions0(self): return {key: value for key, value in self._assumptions.items() if value is not None} @cacheit def sort_key(self, order=None): return self.class_key(), (1, (self.name,)), S.One.sort_key(), S.One def as_dummy(self): # only put commutativity in explicitly if it is False return Dummy(self.name) if self.is_commutative is not False \ else Dummy(self.name, commutative=self.is_commutative) def as_real_imag(self, deep=True, **hints): from sympy import im, re if hints.get('ignore') == self: return None else: return (re(self), im(self)) def _sage_(self): import sage.all as sage return sage.var(self.name) def is_constant(self, *wrt, **flags): if not wrt: return False return not self in wrt @property def free_symbols(self): return {self} binary_symbols = free_symbols # in this case, not always def as_set(self): return S.UniversalSet class Dummy(Symbol): """Dummy symbols are each unique, even if they have the same name: Examples ======== >>> from sympy import Dummy >>> Dummy("x") == Dummy("x") False If a name is not supplied then a string value of an internal count will be used. This is useful when a temporary variable is needed and the name of the variable used in the expression is not important. >>> Dummy() #doctest: +SKIP _Dummy_10 """ # In the rare event that a Dummy object needs to be recreated, both the # `name` and `dummy_index` should be passed. This is used by `srepr` for # example: # >>> d1 = Dummy() # >>> d2 = eval(srepr(d1)) # >>> d2 == d1 # True # # If a new session is started between `srepr` and `eval`, there is a very # small chance that `d2` will be equal to a previously-created Dummy. _count = 0 _prng = random.Random() _base_dummy_index = _prng.randint(10**6, 9*10**6) __slots__ = ('dummy_index',) is_Dummy = True def __new__(cls, name=None, dummy_index=None, **assumptions): if dummy_index is not None: assert name is not None, "If you specify a dummy_index, you must also provide a name" if name is None: name = "Dummy_" + str(Dummy._count) if dummy_index is None: dummy_index = Dummy._base_dummy_index + Dummy._count Dummy._count += 1 cls._sanitize(assumptions, cls) obj = Symbol.__xnew__(cls, name, **assumptions) obj.dummy_index = dummy_index return obj def __getstate__(self): return {'_assumptions': self._assumptions, 'dummy_index': self.dummy_index} @cacheit def sort_key(self, order=None): return self.class_key(), ( 2, (self.name, self.dummy_index)), S.One.sort_key(), S.One def _hashable_content(self): return Symbol._hashable_content(self) + (self.dummy_index,) class Wild(Symbol): """ A Wild symbol matches anything, or anything without whatever is explicitly excluded. Parameters ========== name : str Name of the Wild instance. exclude : iterable, optional Instances in ``exclude`` will not be matched. properties : iterable of functions, optional Functions, each taking an expressions as input and returns a ``bool``. All functions in ``properties`` need to return ``True`` in order for the Wild instance to match the expression. Examples ======== >>> from sympy import Wild, WildFunction, cos, pi >>> from sympy.abc import x, y, z >>> a = Wild('a') >>> x.match(a) {a_: x} >>> pi.match(a) {a_: pi} >>> (3*x**2).match(a*x) {a_: 3*x} >>> cos(x).match(a) {a_: cos(x)} >>> b = Wild('b', exclude=[x]) >>> (3*x**2).match(b*x) >>> b.match(a) {a_: b_} >>> A = WildFunction('A') >>> A.match(a) {a_: A_} Tips ==== When using Wild, be sure to use the exclude keyword to make the pattern more precise. Without the exclude pattern, you may get matches that are technically correct, but not what you wanted. For example, using the above without exclude: >>> from sympy import symbols >>> a, b = symbols('a b', cls=Wild) >>> (2 + 3*y).match(a*x + b*y) {a_: 2/x, b_: 3} This is technically correct, because (2/x)*x + 3*y == 2 + 3*y, but you probably wanted it to not match at all. The issue is that you really didn't want a and b to include x and y, and the exclude parameter lets you specify exactly this. With the exclude parameter, the pattern will not match. >>> a = Wild('a', exclude=[x, y]) >>> b = Wild('b', exclude=[x, y]) >>> (2 + 3*y).match(a*x + b*y) Exclude also helps remove ambiguity from matches. >>> E = 2*x**3*y*z >>> a, b = symbols('a b', cls=Wild) >>> E.match(a*b) {a_: 2*y*z, b_: x**3} >>> a = Wild('a', exclude=[x, y]) >>> E.match(a*b) {a_: z, b_: 2*x**3*y} >>> a = Wild('a', exclude=[x, y, z]) >>> E.match(a*b) {a_: 2, b_: x**3*y*z} Wild also accepts a ``properties`` parameter: >>> a = Wild('a', properties=[lambda k: k.is_Integer]) >>> E.match(a*b) {a_: 2, b_: x**3*y*z} """ is_Wild = True __slots__ = ('exclude', 'properties') def __new__(cls, name, exclude=(), properties=(), **assumptions): exclude = tuple([sympify(x) for x in exclude]) properties = tuple(properties) cls._sanitize(assumptions, cls) return Wild.__xnew__(cls, name, exclude, properties, **assumptions) def __getnewargs__(self): return (self.name, self.exclude, self.properties) @staticmethod @cacheit def __xnew__(cls, name, exclude, properties, **assumptions): obj = Symbol.__xnew__(cls, name, **assumptions) obj.exclude = exclude obj.properties = properties return obj def _hashable_content(self): return super()._hashable_content() + (self.exclude, self.properties) # TODO add check against another Wild def matches(self, expr, repl_dict={}, old=False): if any(expr.has(x) for x in self.exclude): return None if any(not f(expr) for f in self.properties): return None repl_dict = repl_dict.copy() repl_dict[self] = expr return repl_dict _range = _re.compile('([0-9]*:[0-9]+|[a-zA-Z]?:[a-zA-Z])') def symbols(names, *, cls=Symbol, **args): r""" Transform strings into instances of :class:`Symbol` class. :func:`symbols` function returns a sequence of symbols with names taken from ``names`` argument, which can be a comma or whitespace delimited string, or a sequence of strings:: >>> from sympy import symbols, Function >>> x, y, z = symbols('x,y,z') >>> a, b, c = symbols('a b c') The type of output is dependent on the properties of input arguments:: >>> symbols('x') x >>> symbols('x,') (x,) >>> symbols('x,y') (x, y) >>> symbols(('a', 'b', 'c')) (a, b, c) >>> symbols(['a', 'b', 'c']) [a, b, c] >>> symbols({'a', 'b', 'c'}) {a, b, c} If an iterable container is needed for a single symbol, set the ``seq`` argument to ``True`` or terminate the symbol name with a comma:: >>> symbols('x', seq=True) (x,) To reduce typing, range syntax is supported to create indexed symbols. Ranges are indicated by a colon and the type of range is determined by the character to the right of the colon. If the character is a digit then all contiguous digits to the left are taken as the nonnegative starting value (or 0 if there is no digit left of the colon) and all contiguous digits to the right are taken as 1 greater than the ending value:: >>> symbols('x:10') (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9) >>> symbols('x5:10') (x5, x6, x7, x8, x9) >>> symbols('x5(:2)') (x50, x51) >>> symbols('x5:10,y:5') (x5, x6, x7, x8, x9, y0, y1, y2, y3, y4) >>> symbols(('x5:10', 'y:5')) ((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4)) If the character to the right of the colon is a letter, then the single letter to the left (or 'a' if there is none) is taken as the start and all characters in the lexicographic range *through* the letter to the right are used as the range:: >>> symbols('x:z') (x, y, z) >>> symbols('x:c') # null range () >>> symbols('x(:c)') (xa, xb, xc) >>> symbols(':c') (a, b, c) >>> symbols('a:d, x:z') (a, b, c, d, x, y, z) >>> symbols(('a:d', 'x:z')) ((a, b, c, d), (x, y, z)) Multiple ranges are supported; contiguous numerical ranges should be separated by parentheses to disambiguate the ending number of one range from the starting number of the next:: >>> symbols('x:2(1:3)') (x01, x02, x11, x12) >>> symbols(':3:2') # parsing is from left to right (00, 01, 10, 11, 20, 21) Only one pair of parentheses surrounding ranges are removed, so to include parentheses around ranges, double them. And to include spaces, commas, or colons, escape them with a backslash:: >>> symbols('x((a:b))') (x(a), x(b)) >>> symbols(r'x(:1\,:2)') # or r'x((:1)\,(:2))' (x(0,0), x(0,1)) All newly created symbols have assumptions set according to ``args``:: >>> a = symbols('a', integer=True) >>> a.is_integer True >>> x, y, z = symbols('x,y,z', real=True) >>> x.is_real and y.is_real and z.is_real True Despite its name, :func:`symbols` can create symbol-like objects like instances of Function or Wild classes. To achieve this, set ``cls`` keyword argument to the desired type:: >>> symbols('f,g,h', cls=Function) (f, g, h) >>> type(_[0]) <class 'sympy.core.function.UndefinedFunction'> """ result = [] if isinstance(names, str): marker = 0 literals = [r'\,', r'\:', r'\ '] for i in range(len(literals)): lit = literals.pop(0) if lit in names: while chr(marker) in names: marker += 1 lit_char = chr(marker) marker += 1 names = names.replace(lit, lit_char) literals.append((lit_char, lit[1:])) def literal(s): if literals: for c, l in literals: s = s.replace(c, l) return s names = names.strip() as_seq = names.endswith(',') if as_seq: names = names[:-1].rstrip() if not names: raise ValueError('no symbols given') # split on commas names = [n.strip() for n in names.split(',')] if not all(n for n in names): raise ValueError('missing symbol between commas') # split on spaces for i in range(len(names) - 1, -1, -1): names[i: i + 1] = names[i].split() seq = args.pop('seq', as_seq) for name in names: if not name: raise ValueError('missing symbol') if ':' not in name: symbol = cls(literal(name), **args) result.append(symbol) continue split = _range.split(name) # remove 1 layer of bounding parentheses around ranges for i in range(len(split) - 1): if i and ':' in split[i] and split[i] != ':' and \ split[i - 1].endswith('(') and \ split[i + 1].startswith(')'): split[i - 1] = split[i - 1][:-1] split[i + 1] = split[i + 1][1:] for i, s in enumerate(split): if ':' in s: if s[-1].endswith(':'): raise ValueError('missing end range') a, b = s.split(':') if b[-1] in string.digits: a = 0 if not a else int(a) b = int(b) split[i] = [str(c) for c in range(a, b)] else: a = a or 'a' split[i] = [string.ascii_letters[c] for c in range( string.ascii_letters.index(a), string.ascii_letters.index(b) + 1)] # inclusive if not split[i]: break else: split[i] = [s] else: seq = True if len(split) == 1: names = split[0] else: names = [''.join(s) for s in cartes(*split)] if literals: result.extend([cls(literal(s), **args) for s in names]) else: result.extend([cls(s, **args) for s in names]) if not seq and len(result) <= 1: if not result: return () return result[0] return tuple(result) else: for name in names: result.append(symbols(name, **args)) return type(names)(result) def var(names, **args): """ Create symbols and inject them into the global namespace. Explanation =========== This calls :func:`symbols` with the same arguments and puts the results into the *global* namespace. It's recommended not to use :func:`var` in library code, where :func:`symbols` has to be used:: Examples ======== >>> from sympy import var >>> var('x') x >>> x # noqa: F821 x >>> var('a,ab,abc') (a, ab, abc) >>> abc # noqa: F821 abc >>> var('x,y', real=True) (x, y) >>> x.is_real and y.is_real # noqa: F821 True See :func:`symbols` documentation for more details on what kinds of arguments can be passed to :func:`var`. """ def traverse(symbols, frame): """Recursively inject symbols to the global namespace. """ for symbol in symbols: if isinstance(symbol, Basic): frame.f_globals[symbol.name] = symbol elif isinstance(symbol, FunctionClass): frame.f_globals[symbol.__name__] = symbol else: traverse(symbol, frame) from inspect import currentframe frame = currentframe().f_back try: syms = symbols(names, **args) if syms is not None: if isinstance(syms, Basic): frame.f_globals[syms.name] = syms elif isinstance(syms, FunctionClass): frame.f_globals[syms.__name__] = syms else: traverse(syms, frame) finally: del frame # break cyclic dependencies as stated in inspect docs return syms def disambiguate(*iter): """ Return a Tuple containing the passed expressions with symbols that appear the same when printed replaced with numerically subscripted symbols, and all Dummy symbols replaced with Symbols. Parameters ========== iter: list of symbols or expressions. Examples ======== >>> from sympy.core.symbol import disambiguate >>> from sympy import Dummy, Symbol, Tuple >>> from sympy.abc import y >>> tup = Symbol('_x'), Dummy('x'), Dummy('x') >>> disambiguate(*tup) (x_2, x, x_1) >>> eqs = Tuple(Symbol('x')/y, Dummy('x')/y) >>> disambiguate(*eqs) (x_1/y, x/y) >>> ix = Symbol('x', integer=True) >>> vx = Symbol('x') >>> disambiguate(vx + ix) (x + x_1,) To make your own mapping of symbols to use, pass only the free symbols of the expressions and create a dictionary: >>> free = eqs.free_symbols >>> mapping = dict(zip(free, disambiguate(*free))) >>> eqs.xreplace(mapping) (x_1/y, x/y) """ new_iter = Tuple(*iter) key = lambda x:tuple(sorted(x.assumptions0.items())) syms = ordered(new_iter.free_symbols, keys=key) mapping = {} for s in syms: mapping.setdefault(str(s).lstrip('_'), []).append(s) reps = {} for k in mapping: # the first or only symbol doesn't get subscripted but make # sure that it's a Symbol, not a Dummy mapk0 = Symbol("%s" % (k), **mapping[k][0].assumptions0) if mapping[k][0] != mapk0: reps[mapping[k][0]] = mapk0 # the others get subscripts (and are made into Symbols) skip = 0 for i in range(1, len(mapping[k])): while True: name = "%s_%i" % (k, i + skip) if name not in mapping: break skip += 1 ki = mapping[k][i] reps[ki] = Symbol(name, **ki.assumptions0) return new_iter.xreplace(reps)
b7b0409ccb826a29ac08c1d4d72e7c900af0c93d13273586bcd312cde4ab4b5e
"""sympify -- convert objects SymPy internal format""" import typing if typing.TYPE_CHECKING: from typing import Any, Callable, Dict, Type from inspect import getmro from .compatibility import iterable from .parameters import global_parameters class SympifyError(ValueError): def __init__(self, expr, base_exc=None): self.expr = expr self.base_exc = base_exc def __str__(self): if self.base_exc is None: return "SympifyError: %r" % (self.expr,) return ("Sympify of expression '%s' failed, because of exception being " "raised:\n%s: %s" % (self.expr, self.base_exc.__class__.__name__, str(self.base_exc))) # See sympify docstring. converter = {} # type: Dict[Type[Any], Callable[[Any], Basic]] class CantSympify: """ Mix in this trait to a class to disallow sympification of its instances. Examples ======== >>> from sympy.core.sympify import sympify, CantSympify >>> class Something(dict): ... pass ... >>> sympify(Something()) {} >>> class Something(dict, CantSympify): ... pass ... >>> sympify(Something()) Traceback (most recent call last): ... SympifyError: SympifyError: {} """ pass def _is_numpy_instance(a): """ Checks if an object is an instance of a type from the numpy module. """ # This check avoids unnecessarily importing NumPy. We check the whole # __mro__ in case any base type is a numpy type. return any(type_.__module__ == 'numpy' for type_ in type(a).__mro__) def _convert_numpy_types(a, **sympify_args): """ Converts a numpy datatype input to an appropriate SymPy type. """ import numpy as np if not isinstance(a, np.floating): if np.iscomplex(a): return converter[complex](a.item()) else: return sympify(a.item(), **sympify_args) else: try: from sympy.core.numbers import Float prec = np.finfo(a).nmant + 1 # E.g. double precision means prec=53 but nmant=52 # Leading bit of mantissa is always 1, so is not stored a = str(list(np.reshape(np.asarray(a), (1, np.size(a)))[0]))[1:-1] return Float(a, precision=prec) except NotImplementedError: raise SympifyError('Translation for numpy float : %s ' 'is not implemented' % a) def sympify(a, locals=None, convert_xor=True, strict=False, rational=False, evaluate=None): """ Converts an arbitrary expression to a type that can be used inside SymPy. Explanation =========== It will convert Python ints into instances of sympy.Integer, floats into instances of sympy.Float, etc. It is also able to coerce symbolic expressions which inherit from Basic. This can be useful in cooperation with SAGE. .. warning:: Note that this function uses ``eval``, and thus shouldn't be used on unsanitized input. If the argument is already a type that SymPy understands, it will do nothing but return that value. This can be used at the beginning of a function to ensure you are working with the correct type. Examples ======== >>> from sympy import sympify >>> sympify(2).is_integer True >>> sympify(2).is_real True >>> sympify(2.0).is_real True >>> sympify("2.0").is_real True >>> sympify("2e-45").is_real True If the expression could not be converted, a SympifyError is raised. >>> sympify("x***2") Traceback (most recent call last): ... SympifyError: SympifyError: "could not parse 'x***2'" Locals ------ The sympification happens with access to everything that is loaded by ``from sympy import *``; anything used in a string that is not defined by that import will be converted to a symbol. In the following, the ``bitcount`` function is treated as a symbol and the ``O`` is interpreted as the Order object (used with series) and it raises an error when used improperly: >>> s = 'bitcount(42)' >>> sympify(s) bitcount(42) >>> sympify("O(x)") O(x) >>> sympify("O + 1") Traceback (most recent call last): ... TypeError: unbound method... In order to have ``bitcount`` be recognized it can be imported into a namespace dictionary and passed as locals: >>> from sympy.core.compatibility import exec_ >>> ns = {} >>> exec_('from sympy.core.evalf import bitcount', ns) >>> sympify(s, locals=ns) 6 In order to have the ``O`` interpreted as a Symbol, identify it as such in the namespace dictionary. This can be done in a variety of ways; all three of the following are possibilities: >>> from sympy import Symbol >>> ns["O"] = Symbol("O") # method 1 >>> exec_('from sympy.abc import O', ns) # method 2 >>> ns.update(dict(O=Symbol("O"))) # method 3 >>> sympify("O + 1", locals=ns) O + 1 If you want *all* single-letter and Greek-letter variables to be symbols then you can use the clashing-symbols dictionaries that have been defined there as private variables: _clash1 (single-letter variables), _clash2 (the multi-letter Greek names) or _clash (both single and multi-letter names that are defined in abc). >>> from sympy.abc import _clash1 >>> _clash1 {'C': C, 'E': E, 'I': I, 'N': N, 'O': O, 'Q': Q, 'S': S} >>> sympify('I & Q', _clash1) I & Q Strict ------ If the option ``strict`` is set to ``True``, only the types for which an explicit conversion has been defined are converted. In the other cases, a SympifyError is raised. >>> print(sympify(None)) None >>> sympify(None, strict=True) Traceback (most recent call last): ... SympifyError: SympifyError: None Evaluation ---------- If the option ``evaluate`` is set to ``False``, then arithmetic and operators will be converted into their SymPy equivalents and the ``evaluate=False`` option will be added. Nested ``Add`` or ``Mul`` will be denested first. This is done via an AST transformation that replaces operators with their SymPy equivalents, so if an operand redefines any of those operations, the redefined operators will not be used. If argument a is not a string, the mathematical expression is evaluated before being passed to sympify, so adding evaluate=False will still return the evaluated result of expression. >>> sympify('2**2 / 3 + 5') 19/3 >>> sympify('2**2 / 3 + 5', evaluate=False) 2**2/3 + 5 >>> sympify('4/2+7', evaluate=True) 9 >>> sympify('4/2+7', evaluate=False) 4/2 + 7 >>> sympify(4/2+7, evaluate=False) 9.00000000000000 Extending --------- To extend ``sympify`` to convert custom objects (not derived from ``Basic``), just define a ``_sympy_`` method to your class. You can do that even to classes that you do not own by subclassing or adding the method at runtime. >>> from sympy import Matrix >>> class MyList1(object): ... def __iter__(self): ... yield 1 ... yield 2 ... return ... def __getitem__(self, i): return list(self)[i] ... def _sympy_(self): return Matrix(self) >>> sympify(MyList1()) Matrix([ [1], [2]]) If you do not have control over the class definition you could also use the ``converter`` global dictionary. The key is the class and the value is a function that takes a single argument and returns the desired SymPy object, e.g. ``converter[MyList] = lambda x: Matrix(x)``. >>> class MyList2(object): # XXX Do not do this if you control the class! ... def __iter__(self): # Use _sympy_! ... yield 1 ... yield 2 ... return ... def __getitem__(self, i): return list(self)[i] >>> from sympy.core.sympify import converter >>> converter[MyList2] = lambda x: Matrix(x) >>> sympify(MyList2()) Matrix([ [1], [2]]) Notes ===== The keywords ``rational`` and ``convert_xor`` are only used when the input is a string. convert_xor ----------- >>> sympify('x^y',convert_xor=True) x**y >>> sympify('x^y',convert_xor=False) x ^ y rational -------- >>> sympify('0.1',rational=False) 0.1 >>> sympify('0.1',rational=True) 1/10 Sometimes autosimplification during sympification results in expressions that are very different in structure than what was entered. Until such autosimplification is no longer done, the ``kernS`` function might be of some use. In the example below you can see how an expression reduces to -1 by autosimplification, but does not do so when ``kernS`` is used. >>> from sympy.core.sympify import kernS >>> from sympy.abc import x >>> -2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1 -1 >>> s = '-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1' >>> sympify(s) -1 >>> kernS(s) -2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1 Parameters ========== a : - any object defined in SymPy - standard numeric python types: int, long, float, Decimal - strings (like "0.09", "2e-19" or 'sin(x)') - booleans, including ``None`` (will leave ``None`` unchanged) - dict, lists, sets or tuples containing any of the above convert_xor : boolean, optional If true, treats XOR as exponentiation. If False, treats XOR as XOR itself. Used only when input is a string. locals : any object defined in SymPy, optional In order to have strings be recognized it can be imported into a namespace dictionary and passed as locals. strict : boolean, optional If the option strict is set to True, only the types for which an explicit conversion has been defined are converted. In the other cases, a SympifyError is raised. rational : boolean, optional If true, converts floats into Rational. If false, it lets floats remain as it is. Used only when input is a string. evaluate : boolean, optional If False, then arithmetic and operators will be converted into their SymPy equivalents. If True the expression will be evaluated and the result will be returned. """ # XXX: If a is a Basic subclass rather than instance (e.g. sin rather than # sin(x)) then a.__sympy__ will be the property. Only on the instance will # a.__sympy__ give the *value* of the property (True). Since sympify(sin) # was used for a long time we allow it to pass. However if strict=True as # is the case in internal calls to _sympify then we only allow # is_sympy=True. # # https://github.com/sympy/sympy/issues/20124 is_sympy = getattr(a, '__sympy__', None) if is_sympy is True: return a elif is_sympy is not None: if not strict: return a else: raise SympifyError(a) if isinstance(a, CantSympify): raise SympifyError(a) cls = getattr(a, "__class__", None) if cls is None: cls = type(a) # Probably an old-style class conv = converter.get(cls, None) if conv is not None: return conv(a) for superclass in getmro(cls): try: return converter[superclass](a) except KeyError: continue if cls is type(None): if strict: raise SympifyError(a) else: return a if evaluate is None: evaluate = global_parameters.evaluate # Support for basic numpy datatypes if _is_numpy_instance(a): import numpy as np if np.isscalar(a): return _convert_numpy_types(a, locals=locals, convert_xor=convert_xor, strict=strict, rational=rational, evaluate=evaluate) _sympy_ = getattr(a, "_sympy_", None) if _sympy_ is not None: try: return a._sympy_() # XXX: Catches AttributeError: 'SympyConverter' object has no # attribute 'tuple' # This is probably a bug somewhere but for now we catch it here. except AttributeError: pass if not strict: # Put numpy array conversion _before_ float/int, see # <https://github.com/sympy/sympy/issues/13924>. flat = getattr(a, "flat", None) if flat is not None: shape = getattr(a, "shape", None) if shape is not None: from ..tensor.array import Array return Array(a.flat, a.shape) # works with e.g. NumPy arrays if not isinstance(a, str): if _is_numpy_instance(a): import numpy as np assert not isinstance(a, np.number) if isinstance(a, np.ndarray): # Scalar arrays (those with zero dimensions) have sympify # called on the scalar element. if a.ndim == 0: try: return sympify(a.item(), locals=locals, convert_xor=convert_xor, strict=strict, rational=rational, evaluate=evaluate) except SympifyError: pass else: # float and int can coerce size-one numpy arrays to their lone # element. See issue https://github.com/numpy/numpy/issues/10404. for coerce in (float, int): try: return sympify(coerce(a)) except (TypeError, ValueError, AttributeError, SympifyError): continue if strict: raise SympifyError(a) if iterable(a): try: return type(a)([sympify(x, locals=locals, convert_xor=convert_xor, rational=rational) for x in a]) except TypeError: # Not all iterables are rebuildable with their type. pass if isinstance(a, dict): try: return type(a)([sympify(x, locals=locals, convert_xor=convert_xor, rational=rational) for x in a.items()]) except TypeError: # Not all iterables are rebuildable with their type. pass if not isinstance(a, str): try: a = str(a) except Exception as exc: raise SympifyError(a, exc) from sympy.utilities.exceptions import SymPyDeprecationWarning SymPyDeprecationWarning( feature="String fallback in sympify", useinstead= \ 'sympify(str(obj)) or ' + \ 'sympy.core.sympify.converter or obj._sympy_', issue=18066, deprecated_since_version='1.6' ).warn() from sympy.parsing.sympy_parser import (parse_expr, TokenError, standard_transformations) from sympy.parsing.sympy_parser import convert_xor as t_convert_xor from sympy.parsing.sympy_parser import rationalize as t_rationalize transformations = standard_transformations if rational: transformations += (t_rationalize,) if convert_xor: transformations += (t_convert_xor,) try: a = a.replace('\n', '') expr = parse_expr(a, local_dict=locals, transformations=transformations, evaluate=evaluate) except (TokenError, SyntaxError) as exc: raise SympifyError('could not parse %r' % a, exc) return expr def _sympify(a): """ Short version of sympify for internal usage for __add__ and __eq__ methods where it is ok to allow some things (like Python integers and floats) in the expression. This excludes things (like strings) that are unwise to allow into such an expression. >>> from sympy import Integer >>> Integer(1) == 1 True >>> Integer(1) == '1' False >>> from sympy.abc import x >>> x + 1 x + 1 >>> x + '1' Traceback (most recent call last): ... TypeError: unsupported operand type(s) for +: 'Symbol' and 'str' see: sympify """ return sympify(a, strict=True) def kernS(s): """Use a hack to try keep autosimplification from distributing a a number into an Add; this modification doesn't prevent the 2-arg Mul from becoming an Add, however. Examples ======== >>> from sympy.core.sympify import kernS >>> from sympy.abc import x, y The 2-arg Mul distributes a number (or minus sign) across the terms of an expression, but kernS will prevent that: >>> 2*(x + y), -(x + 1) (2*x + 2*y, -x - 1) >>> kernS('2*(x + y)') 2*(x + y) >>> kernS('-(x + 1)') -(x + 1) If use of the hack fails, the un-hacked string will be passed to sympify... and you get what you get. XXX This hack should not be necessary once issue 4596 has been resolved. """ import string from random import choice from sympy.core.symbol import Symbol hit = False quoted = '"' in s or "'" in s if '(' in s and not quoted: if s.count('(') != s.count(")"): raise SympifyError('unmatched left parenthesis') # strip all space from s s = ''.join(s.split()) olds = s # now use space to represent a symbol that # will # step 1. turn potential 2-arg Muls into 3-arg versions # 1a. *( -> * *( s = s.replace('*(', '* *(') # 1b. close up exponentials s = s.replace('** *', '**') # 2. handle the implied multiplication of a negated # parenthesized expression in two steps # 2a: -(...) --> -( *(...) target = '-( *(' s = s.replace('-(', target) # 2b: double the matching closing parenthesis # -( *(...) --> -( *(...)) i = nest = 0 assert target.endswith('(') # assumption below while True: j = s.find(target, i) if j == -1: break j += len(target) - 1 for j in range(j, len(s)): if s[j] == "(": nest += 1 elif s[j] == ")": nest -= 1 if nest == 0: break s = s[:j] + ")" + s[j:] i = j + 2 # the first char after 2nd ) if ' ' in s: # get a unique kern kern = '_' while kern in s: kern += choice(string.ascii_letters + string.digits) s = s.replace(' ', kern) hit = kern in s else: hit = False for i in range(2): try: expr = sympify(s) break except TypeError: # the kern might cause unknown errors... if hit: s = olds # maybe it didn't like the kern; use un-kerned s hit = False continue expr = sympify(s) # let original error raise if not hit: return expr rep = {Symbol(kern): 1} def _clear(expr): if isinstance(expr, (list, tuple, set)): return type(expr)([_clear(e) for e in expr]) if hasattr(expr, 'subs'): return expr.subs(rep, hack2=True) return expr expr = _clear(expr) # hope that kern is not there anymore return expr # Avoid circular import from .basic import Basic
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from sympy import Expr, Add, Mul, Pow, sympify, Matrix, Tuple from sympy.utilities import default_sort_key def _is_scalar(e): """ Helper method used in Tr""" # sympify to set proper attributes e = sympify(e) if isinstance(e, Expr): if (e.is_Integer or e.is_Float or e.is_Rational or e.is_Number or (e.is_Symbol and e.is_commutative) ): return True return False def _cycle_permute(l): """ Cyclic permutations based on canonical ordering Explanation =========== This method does the sort based ascii values while a better approach would be to used lexicographic sort. TODO: Handle condition such as symbols have subscripts/superscripts in case of lexicographic sort """ if len(l) == 1: return l min_item = min(l, key=default_sort_key) indices = [i for i, x in enumerate(l) if x == min_item] le = list(l) le.extend(l) # duplicate and extend string for easy processing # adding the first min_item index back for easier looping indices.append(len(l) + indices[0]) # create sublist of items with first item as min_item and last_item # in each of the sublist is item just before the next occurrence of # minitem in the cycle formed. sublist = [[le[indices[i]:indices[i + 1]]] for i in range(len(indices) - 1)] # we do comparison of strings by comparing elements # in each sublist idx = sublist.index(min(sublist)) ordered_l = le[indices[idx]:indices[idx] + len(l)] return ordered_l def _rearrange_args(l): """ this just moves the last arg to first position to enable expansion of args A,B,A ==> A**2,B """ if len(l) == 1: return l x = list(l[-1:]) x.extend(l[0:-1]) return Mul(*x).args class Tr(Expr): """ Generic Trace operation than can trace over: a) sympy matrix b) operators c) outer products Parameters ========== o : operator, matrix, expr i : tuple/list indices (optional) Examples ======== # TODO: Need to handle printing a) Trace(A+B) = Tr(A) + Tr(B) b) Trace(scalar*Operator) = scalar*Trace(Operator) >>> from sympy.core.trace import Tr >>> from sympy import symbols, Matrix >>> a, b = symbols('a b', commutative=True) >>> A, B = symbols('A B', commutative=False) >>> Tr(a*A,[2]) a*Tr(A) >>> m = Matrix([[1,2],[1,1]]) >>> Tr(m) 2 """ def __new__(cls, *args): """ Construct a Trace object. Parameters ========== args = sympy expression indices = tuple/list if indices, optional """ # expect no indices,int or a tuple/list/Tuple if (len(args) == 2): if not isinstance(args[1], (list, Tuple, tuple)): indices = Tuple(args[1]) else: indices = Tuple(*args[1]) expr = args[0] elif (len(args) == 1): indices = Tuple() expr = args[0] else: raise ValueError("Arguments to Tr should be of form " "(expr[, [indices]])") if isinstance(expr, Matrix): return expr.trace() elif hasattr(expr, 'trace') and callable(expr.trace): #for any objects that have trace() defined e.g numpy return expr.trace() elif isinstance(expr, Add): return Add(*[Tr(arg, indices) for arg in expr.args]) elif isinstance(expr, Mul): c_part, nc_part = expr.args_cnc() if len(nc_part) == 0: return Mul(*c_part) else: obj = Expr.__new__(cls, Mul(*nc_part), indices ) #this check is needed to prevent cached instances #being returned even if len(c_part)==0 return Mul(*c_part)*obj if len(c_part) > 0 else obj elif isinstance(expr, Pow): if (_is_scalar(expr.args[0]) and _is_scalar(expr.args[1])): return expr else: return Expr.__new__(cls, expr, indices) else: if (_is_scalar(expr)): return expr return Expr.__new__(cls, expr, indices) def doit(self, **kwargs): """ Perform the trace operation. #TODO: Current version ignores the indices set for partial trace. >>> from sympy.core.trace import Tr >>> from sympy.physics.quantum.operator import OuterProduct >>> from sympy.physics.quantum.spin import JzKet, JzBra >>> t = Tr(OuterProduct(JzKet(1,1), JzBra(1,1))) >>> t.doit() 1 """ if hasattr(self.args[0], '_eval_trace'): return self.args[0]._eval_trace(indices=self.args[1]) return self @property def is_number(self): # TODO : improve this implementation return True #TODO: Review if the permute method is needed # and if it needs to return a new instance def permute(self, pos): """ Permute the arguments cyclically. Parameters ========== pos : integer, if positive, shift-right, else shift-left Examples ======== >>> from sympy.core.trace import Tr >>> from sympy import symbols >>> A, B, C, D = symbols('A B C D', commutative=False) >>> t = Tr(A*B*C*D) >>> t.permute(2) Tr(C*D*A*B) >>> t.permute(-2) Tr(C*D*A*B) """ if pos > 0: pos = pos % len(self.args[0].args) else: pos = -(abs(pos) % len(self.args[0].args)) args = list(self.args[0].args[-pos:] + self.args[0].args[0:-pos]) return Tr(Mul(*(args))) def _hashable_content(self): if isinstance(self.args[0], Mul): args = _cycle_permute(_rearrange_args(self.args[0].args)) else: args = [self.args[0]] return tuple(args) + (self.args[1], )
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""" Adaptive numerical evaluation of SymPy expressions, using mpmath for mathematical functions. """ from typing import Tuple import math import mpmath.libmp as libmp from mpmath import ( make_mpc, make_mpf, mp, mpc, mpf, nsum, quadts, quadosc, workprec) from mpmath import inf as mpmath_inf from mpmath.libmp import (from_int, from_man_exp, from_rational, fhalf, fnan, fnone, fone, fzero, mpf_abs, mpf_add, mpf_atan, mpf_atan2, mpf_cmp, mpf_cos, mpf_e, mpf_exp, mpf_log, mpf_lt, mpf_mul, mpf_neg, mpf_pi, mpf_pow, mpf_pow_int, mpf_shift, mpf_sin, mpf_sqrt, normalize, round_nearest, to_int, to_str) from mpmath.libmp import bitcount as mpmath_bitcount from mpmath.libmp.backend import MPZ from mpmath.libmp.libmpc import _infs_nan from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps from mpmath.libmp.gammazeta import mpf_bernoulli from .compatibility import SYMPY_INTS from .sympify import sympify from .singleton import S from sympy.utilities.iterables import is_sequence LG10 = math.log(10, 2) rnd = round_nearest def bitcount(n): """Return smallest integer, b, such that |n|/2**b < 1. """ return mpmath_bitcount(abs(int(n))) # Used in a few places as placeholder values to denote exponents and # precision levels, e.g. of exact numbers. Must be careful to avoid # passing these to mpmath functions or returning them in final results. INF = float(mpmath_inf) MINUS_INF = float(-mpmath_inf) # ~= 100 digits. Real men set this to INF. DEFAULT_MAXPREC = 333 class PrecisionExhausted(ArithmeticError): pass #----------------------------------------------------------------------------# # # # Helper functions for arithmetic and complex parts # # # #----------------------------------------------------------------------------# """ An mpf value tuple is a tuple of integers (sign, man, exp, bc) representing a floating-point number: [1, -1][sign]*man*2**exp where sign is 0 or 1 and bc should correspond to the number of bits used to represent the mantissa (man) in binary notation, e.g. Explanation =========== >>> from sympy.core.evalf import bitcount >>> sign, man, exp, bc = 0, 5, 1, 3 >>> n = [1, -1][sign]*man*2**exp >>> n, bitcount(man) (10, 3) A temporary result is a tuple (re, im, re_acc, im_acc) where re and im are nonzero mpf value tuples representing approximate numbers, or None to denote exact zeros. re_acc, im_acc are integers denoting log2(e) where e is the estimated relative accuracy of the respective complex part, but may be anything if the corresponding complex part is None. """ def fastlog(x): """Fast approximation of log2(x) for an mpf value tuple x. Explanation =========== Calculated as exponent + width of mantissa. This is an approximation for two reasons: 1) it gives the ceil(log2(abs(x))) value and 2) it is too high by 1 in the case that x is an exact power of 2. Although this is easy to remedy by testing to see if the odd mpf mantissa is 1 (indicating that one was dealing with an exact power of 2) that would decrease the speed and is not necessary as this is only being used as an approximation for the number of bits in x. The correct return value could be written as "x[2] + (x[3] if x[1] != 1 else 0)". Since mpf tuples always have an odd mantissa, no check is done to see if the mantissa is a multiple of 2 (in which case the result would be too large by 1). Examples ======== >>> from sympy import log >>> from sympy.core.evalf import fastlog, bitcount >>> s, m, e = 0, 5, 1 >>> bc = bitcount(m) >>> n = [1, -1][s]*m*2**e >>> n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc)) (10, 3.3, 4) """ if not x or x == fzero: return MINUS_INF return x[2] + x[3] def pure_complex(v, or_real=False): """Return a and b if v matches a + I*b where b is not zero and a and b are Numbers, else None. If `or_real` is True then 0 will be returned for `b` if `v` is a real number. Examples ======== >>> from sympy.core.evalf import pure_complex >>> from sympy import sqrt, I, S >>> a, b, surd = S(2), S(3), sqrt(2) >>> pure_complex(a) >>> pure_complex(a, or_real=True) (2, 0) >>> pure_complex(surd) >>> pure_complex(a + b*I) (2, 3) >>> pure_complex(I) (0, 1) """ h, t = v.as_coeff_Add() if not t: if or_real: return h, t return c, i = t.as_coeff_Mul() if i is S.ImaginaryUnit: return h, c def scaled_zero(mag, sign=1): """Return an mpf representing a power of two with magnitude ``mag`` and -1 for precision. Or, if ``mag`` is a scaled_zero tuple, then just remove the sign from within the list that it was initially wrapped in. Examples ======== >>> from sympy.core.evalf import scaled_zero >>> from sympy import Float >>> z, p = scaled_zero(100) >>> z, p (([0], 1, 100, 1), -1) >>> ok = scaled_zero(z) >>> ok (0, 1, 100, 1) >>> Float(ok) 1.26765060022823e+30 >>> Float(ok, p) 0.e+30 >>> ok, p = scaled_zero(100, -1) >>> Float(scaled_zero(ok), p) -0.e+30 """ if type(mag) is tuple and len(mag) == 4 and iszero(mag, scaled=True): return (mag[0][0],) + mag[1:] elif isinstance(mag, SYMPY_INTS): if sign not in [-1, 1]: raise ValueError('sign must be +/-1') rv, p = mpf_shift(fone, mag), -1 s = 0 if sign == 1 else 1 rv = ([s],) + rv[1:] return rv, p else: raise ValueError('scaled zero expects int or scaled_zero tuple.') def iszero(mpf, scaled=False): if not scaled: return not mpf or not mpf[1] and not mpf[-1] return mpf and type(mpf[0]) is list and mpf[1] == mpf[-1] == 1 def complex_accuracy(result): """ Returns relative accuracy of a complex number with given accuracies for the real and imaginary parts. The relative accuracy is defined in the complex norm sense as ||z|+|error|| / |z| where error is equal to (real absolute error) + (imag absolute error)*i. The full expression for the (logarithmic) error can be approximated easily by using the max norm to approximate the complex norm. In the worst case (re and im equal), this is wrong by a factor sqrt(2), or by log2(sqrt(2)) = 0.5 bit. """ re, im, re_acc, im_acc = result if not im: if not re: return INF return re_acc if not re: return im_acc re_size = fastlog(re) im_size = fastlog(im) absolute_error = max(re_size - re_acc, im_size - im_acc) relative_error = absolute_error - max(re_size, im_size) return -relative_error def get_abs(expr, prec, options): re, im, re_acc, im_acc = evalf(expr, prec + 2, options) if not re: re, re_acc, im, im_acc = im, im_acc, re, re_acc if im: if expr.is_number: abs_expr, _, acc, _ = evalf(abs(N(expr, prec + 2)), prec + 2, options) return abs_expr, None, acc, None else: if 'subs' in options: return libmp.mpc_abs((re, im), prec), None, re_acc, None return abs(expr), None, prec, None elif re: return mpf_abs(re), None, re_acc, None else: return None, None, None, None def get_complex_part(expr, no, prec, options): """no = 0 for real part, no = 1 for imaginary part""" workprec = prec i = 0 while 1: res = evalf(expr, workprec, options) value, accuracy = res[no::2] # XXX is the last one correct? Consider re((1+I)**2).n() if (not value) or accuracy >= prec or -value[2] > prec: return value, None, accuracy, None workprec += max(30, 2**i) i += 1 def evalf_abs(expr, prec, options): return get_abs(expr.args[0], prec, options) def evalf_re(expr, prec, options): return get_complex_part(expr.args[0], 0, prec, options) def evalf_im(expr, prec, options): return get_complex_part(expr.args[0], 1, prec, options) def finalize_complex(re, im, prec): if re == fzero and im == fzero: raise ValueError("got complex zero with unknown accuracy") elif re == fzero: return None, im, None, prec elif im == fzero: return re, None, prec, None size_re = fastlog(re) size_im = fastlog(im) if size_re > size_im: re_acc = prec im_acc = prec + min(-(size_re - size_im), 0) else: im_acc = prec re_acc = prec + min(-(size_im - size_re), 0) return re, im, re_acc, im_acc def chop_parts(value, prec): """ Chop off tiny real or complex parts. """ re, im, re_acc, im_acc = value # Method 1: chop based on absolute value if re and re not in _infs_nan and (fastlog(re) < -prec + 4): re, re_acc = None, None if im and im not in _infs_nan and (fastlog(im) < -prec + 4): im, im_acc = None, None # Method 2: chop if inaccurate and relatively small if re and im: delta = fastlog(re) - fastlog(im) if re_acc < 2 and (delta - re_acc <= -prec + 4): re, re_acc = None, None if im_acc < 2 and (delta - im_acc >= prec - 4): im, im_acc = None, None return re, im, re_acc, im_acc def check_target(expr, result, prec): a = complex_accuracy(result) if a < prec: raise PrecisionExhausted("Failed to distinguish the expression: \n\n%s\n\n" "from zero. Try simplifying the input, using chop=True, or providing " "a higher maxn for evalf" % (expr)) def get_integer_part(expr, no, options, return_ints=False): """ With no = 1, computes ceiling(expr) With no = -1, computes floor(expr) Note: this function either gives the exact result or signals failure. """ from sympy.functions.elementary.complexes import re, im # The expression is likely less than 2^30 or so assumed_size = 30 ire, iim, ire_acc, iim_acc = evalf(expr, assumed_size, options) # We now know the size, so we can calculate how much extra precision # (if any) is needed to get within the nearest integer if ire and iim: gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc) elif ire: gap = fastlog(ire) - ire_acc elif iim: gap = fastlog(iim) - iim_acc else: # ... or maybe the expression was exactly zero if return_ints: return 0, 0 else: return None, None, None, None margin = 10 if gap >= -margin: prec = margin + assumed_size + gap ire, iim, ire_acc, iim_acc = evalf( expr, prec, options) else: prec = assumed_size # We can now easily find the nearest integer, but to find floor/ceil, we # must also calculate whether the difference to the nearest integer is # positive or negative (which may fail if very close). def calc_part(re_im, nexpr): from sympy.core.add import Add n, c, p, b = nexpr is_int = (p == 0) nint = int(to_int(nexpr, rnd)) if is_int: # make sure that we had enough precision to distinguish # between nint and the re or im part (re_im) of expr that # was passed to calc_part ire, iim, ire_acc, iim_acc = evalf( re_im - nint, 10, options) # don't need much precision assert not iim size = -fastlog(ire) + 2 # -ve b/c ire is less than 1 if size > prec: ire, iim, ire_acc, iim_acc = evalf( re_im, size, options) assert not iim nexpr = ire n, c, p, b = nexpr is_int = (p == 0) nint = int(to_int(nexpr, rnd)) if not is_int: # if there are subs and they all contain integer re/im parts # then we can (hopefully) safely substitute them into the # expression s = options.get('subs', False) if s: doit = True from sympy.core.compatibility import as_int # use strict=False with as_int because we take # 2.0 == 2 for v in s.values(): try: as_int(v, strict=False) except ValueError: try: [as_int(i, strict=False) for i in v.as_real_imag()] continue except (ValueError, AttributeError): doit = False break if doit: re_im = re_im.subs(s) re_im = Add(re_im, -nint, evaluate=False) x, _, x_acc, _ = evalf(re_im, 10, options) try: check_target(re_im, (x, None, x_acc, None), 3) except PrecisionExhausted: if not re_im.equals(0): raise PrecisionExhausted x = fzero nint += int(no*(mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, INF re_, im_, re_acc, im_acc = None, None, None, None if ire: re_, re_acc = calc_part(re(expr, evaluate=False), ire) if iim: im_, im_acc = calc_part(im(expr, evaluate=False), iim) if return_ints: return int(to_int(re_ or fzero)), int(to_int(im_ or fzero)) return re_, im_, re_acc, im_acc def evalf_ceiling(expr, prec, options): return get_integer_part(expr.args[0], 1, options) def evalf_floor(expr, prec, options): return get_integer_part(expr.args[0], -1, options) #----------------------------------------------------------------------------# # # # Arithmetic operations # # # #----------------------------------------------------------------------------# def add_terms(terms, prec, target_prec): """ Helper for evalf_add. Adds a list of (mpfval, accuracy) terms. Returns ======= - None, None if there are no non-zero terms; - terms[0] if there is only 1 term; - scaled_zero if the sum of the terms produces a zero by cancellation e.g. mpfs representing 1 and -1 would produce a scaled zero which need special handling since they are not actually zero and they are purposely malformed to ensure that they can't be used in anything but accuracy calculations; - a tuple that is scaled to target_prec that corresponds to the sum of the terms. The returned mpf tuple will be normalized to target_prec; the input prec is used to define the working precision. XXX explain why this is needed and why one can't just loop using mpf_add """ terms = [t for t in terms if not iszero(t[0])] if not terms: return None, None elif len(terms) == 1: return terms[0] # see if any argument is NaN or oo and thus warrants a special return special = [] from sympy.core.numbers import Float for t in terms: arg = Float._new(t[0], 1) if arg is S.NaN or arg.is_infinite: special.append(arg) if special: from sympy.core.add import Add rv = evalf(Add(*special), prec + 4, {}) return rv[0], rv[2] working_prec = 2*prec sum_man, sum_exp, absolute_error = 0, 0, MINUS_INF for x, accuracy in terms: sign, man, exp, bc = x if sign: man = -man absolute_error = max(absolute_error, bc + exp - accuracy) delta = exp - sum_exp if exp >= sum_exp: # x much larger than existing sum? # first: quick test if ((delta > working_prec) and ((not sum_man) or delta - bitcount(abs(sum_man)) > working_prec)): sum_man = man sum_exp = exp else: sum_man += (man << delta) else: delta = -delta # x much smaller than existing sum? if delta - bc > working_prec: if not sum_man: sum_man, sum_exp = man, exp else: sum_man = (sum_man << delta) + man sum_exp = exp if not sum_man: return scaled_zero(absolute_error) if sum_man < 0: sum_sign = 1 sum_man = -sum_man else: sum_sign = 0 sum_bc = bitcount(sum_man) sum_accuracy = sum_exp + sum_bc - absolute_error r = normalize(sum_sign, sum_man, sum_exp, sum_bc, target_prec, rnd), sum_accuracy return r def evalf_add(v, prec, options): res = pure_complex(v) if res: h, c = res re, _, re_acc, _ = evalf(h, prec, options) im, _, im_acc, _ = evalf(c, prec, options) return re, im, re_acc, im_acc oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC) i = 0 target_prec = prec while 1: options['maxprec'] = min(oldmaxprec, 2*prec) terms = [evalf(arg, prec + 10, options) for arg in v.args] re, re_acc = add_terms( [a[0::2] for a in terms if a[0]], prec, target_prec) im, im_acc = add_terms( [a[1::2] for a in terms if a[1]], prec, target_prec) acc = complex_accuracy((re, im, re_acc, im_acc)) if acc >= target_prec: if options.get('verbose'): print("ADD: wanted", target_prec, "accurate bits, got", re_acc, im_acc) break else: if (prec - target_prec) > options['maxprec']: break prec = prec + max(10 + 2**i, target_prec - acc) i += 1 if options.get('verbose'): print("ADD: restarting with prec", prec) options['maxprec'] = oldmaxprec if iszero(re, scaled=True): re = scaled_zero(re) if iszero(im, scaled=True): im = scaled_zero(im) return re, im, re_acc, im_acc def evalf_mul(v, prec, options): res = pure_complex(v) if res: # the only pure complex that is a mul is h*I _, h = res im, _, im_acc, _ = evalf(h, prec, options) return None, im, None, im_acc args = list(v.args) # see if any argument is NaN or oo and thus warrants a special return special = [] from sympy.core.numbers import Float for arg in args: arg = evalf(arg, prec, options) if arg[0] is None: continue arg = Float._new(arg[0], 1) if arg is S.NaN or arg.is_infinite: special.append(arg) if special: from sympy.core.mul import Mul special = Mul(*special) return evalf(special, prec + 4, {}) # With guard digits, multiplication in the real case does not destroy # accuracy. This is also true in the complex case when considering the # total accuracy; however accuracy for the real or imaginary parts # separately may be lower. acc = prec # XXX: big overestimate working_prec = prec + len(args) + 5 # Empty product is 1 start = man, exp, bc = MPZ(1), 0, 1 # First, we multiply all pure real or pure imaginary numbers. # direction tells us that the result should be multiplied by # I**direction; all other numbers get put into complex_factors # to be multiplied out after the first phase. last = len(args) direction = 0 args.append(S.One) complex_factors = [] for i, arg in enumerate(args): if i != last and pure_complex(arg): args[-1] = (args[-1]*arg).expand() continue elif i == last and arg is S.One: continue re, im, re_acc, im_acc = evalf(arg, working_prec, options) if re and im: complex_factors.append((re, im, re_acc, im_acc)) continue elif re: (s, m, e, b), w_acc = re, re_acc elif im: (s, m, e, b), w_acc = im, im_acc direction += 1 else: return None, None, None, None direction += 2*s man *= m exp += e bc += b if bc > 3*working_prec: man >>= working_prec exp += working_prec acc = min(acc, w_acc) sign = (direction & 2) >> 1 if not complex_factors: v = normalize(sign, man, exp, bitcount(man), prec, rnd) # multiply by i if direction & 1: return None, v, None, acc else: return v, None, acc, None else: # initialize with the first term if (man, exp, bc) != start: # there was a real part; give it an imaginary part re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0) i0 = 0 else: # there is no real part to start (other than the starting 1) wre, wim, wre_acc, wim_acc = complex_factors[0] acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc))) re = wre im = wim i0 = 1 for wre, wim, wre_acc, wim_acc in complex_factors[i0:]: # acc is the overall accuracy of the product; we aren't # computing exact accuracies of the product. acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc))) use_prec = working_prec A = mpf_mul(re, wre, use_prec) B = mpf_mul(mpf_neg(im), wim, use_prec) C = mpf_mul(re, wim, use_prec) D = mpf_mul(im, wre, use_prec) re = mpf_add(A, B, use_prec) im = mpf_add(C, D, use_prec) if options.get('verbose'): print("MUL: wanted", prec, "accurate bits, got", acc) # multiply by I if direction & 1: re, im = mpf_neg(im), re return re, im, acc, acc def evalf_pow(v, prec, options): target_prec = prec base, exp = v.args # We handle x**n separately. This has two purposes: 1) it is much # faster, because we avoid calling evalf on the exponent, and 2) it # allows better handling of real/imaginary parts that are exactly zero if exp.is_Integer: p = exp.p # Exact if not p: return fone, None, prec, None # Exponentiation by p magnifies relative error by |p|, so the # base must be evaluated with increased precision if p is large prec += int(math.log(abs(p), 2)) re, im, re_acc, im_acc = evalf(base, prec + 5, options) # Real to integer power if re and not im: return mpf_pow_int(re, p, target_prec), None, target_prec, None # (x*I)**n = I**n * x**n if im and not re: z = mpf_pow_int(im, p, target_prec) case = p % 4 if case == 0: return z, None, target_prec, None if case == 1: return None, z, None, target_prec if case == 2: return mpf_neg(z), None, target_prec, None if case == 3: return None, mpf_neg(z), None, target_prec # Zero raised to an integer power if not re: return None, None, None, None # General complex number to arbitrary integer power re, im = libmp.mpc_pow_int((re, im), p, prec) # Assumes full accuracy in input return finalize_complex(re, im, target_prec) # Pure square root if exp is S.Half: xre, xim, _, _ = evalf(base, prec + 5, options) # General complex square root if xim: re, im = libmp.mpc_sqrt((xre or fzero, xim), prec) return finalize_complex(re, im, prec) if not xre: return None, None, None, None # Square root of a negative real number if mpf_lt(xre, fzero): return None, mpf_sqrt(mpf_neg(xre), prec), None, prec # Positive square root return mpf_sqrt(xre, prec), None, prec, None # We first evaluate the exponent to find its magnitude # This determines the working precision that must be used prec += 10 yre, yim, _, _ = evalf(exp, prec, options) # Special cases: x**0 if not (yre or yim): return fone, None, prec, None ysize = fastlog(yre) # Restart if too big # XXX: prec + ysize might exceed maxprec if ysize > 5: prec += ysize yre, yim, _, _ = evalf(exp, prec, options) # Pure exponential function; no need to evalf the base if base is S.Exp1: if yim: re, im = libmp.mpc_exp((yre or fzero, yim), prec) return finalize_complex(re, im, target_prec) return mpf_exp(yre, target_prec), None, target_prec, None xre, xim, _, _ = evalf(base, prec + 5, options) # 0**y if not (xre or xim): return None, None, None, None # (real ** complex) or (complex ** complex) if yim: re, im = libmp.mpc_pow( (xre or fzero, xim or fzero), (yre or fzero, yim), target_prec) return finalize_complex(re, im, target_prec) # complex ** real if xim: re, im = libmp.mpc_pow_mpf((xre or fzero, xim), yre, target_prec) return finalize_complex(re, im, target_prec) # negative ** real elif mpf_lt(xre, fzero): re, im = libmp.mpc_pow_mpf((xre, fzero), yre, target_prec) return finalize_complex(re, im, target_prec) # positive ** real else: return mpf_pow(xre, yre, target_prec), None, target_prec, None #----------------------------------------------------------------------------# # # # Special functions # # # #----------------------------------------------------------------------------# def evalf_trig(v, prec, options): """ This function handles sin and cos of complex arguments. TODO: should also handle tan of complex arguments. """ from sympy import cos, sin if isinstance(v, cos): func = mpf_cos elif isinstance(v, sin): func = mpf_sin else: raise NotImplementedError arg = v.args[0] # 20 extra bits is possibly overkill. It does make the need # to restart very unlikely xprec = prec + 20 re, im, re_acc, im_acc = evalf(arg, xprec, options) if im: if 'subs' in options: v = v.subs(options['subs']) return evalf(v._eval_evalf(prec), prec, options) if not re: if isinstance(v, cos): return fone, None, prec, None elif isinstance(v, sin): return None, None, None, None else: raise NotImplementedError # For trigonometric functions, we are interested in the # fixed-point (absolute) accuracy of the argument. xsize = fastlog(re) # Magnitude <= 1.0. OK to compute directly, because there is no # danger of hitting the first root of cos (with sin, magnitude # <= 2.0 would actually be ok) if xsize < 1: return func(re, prec, rnd), None, prec, None # Very large if xsize >= 10: xprec = prec + xsize re, im, re_acc, im_acc = evalf(arg, xprec, options) # Need to repeat in case the argument is very close to a # multiple of pi (or pi/2), hitting close to a root while 1: y = func(re, prec, rnd) ysize = fastlog(y) gap = -ysize accuracy = (xprec - xsize) - gap if accuracy < prec: if options.get('verbose'): print("SIN/COS", accuracy, "wanted", prec, "gap", gap) print(to_str(y, 10)) if xprec > options.get('maxprec', DEFAULT_MAXPREC): return y, None, accuracy, None xprec += gap re, im, re_acc, im_acc = evalf(arg, xprec, options) continue else: return y, None, prec, None def evalf_log(expr, prec, options): from sympy import Abs, Add, log if len(expr.args)>1: expr = expr.doit() return evalf(expr, prec, options) arg = expr.args[0] workprec = prec + 10 xre, xim, xacc, _ = evalf(arg, workprec, options) if xim: # XXX: use get_abs etc instead re = evalf_log( log(Abs(arg, evaluate=False), evaluate=False), prec, options) im = mpf_atan2(xim, xre or fzero, prec) return re[0], im, re[2], prec imaginary_term = (mpf_cmp(xre, fzero) < 0) re = mpf_log(mpf_abs(xre), prec, rnd) size = fastlog(re) if prec - size > workprec and re != fzero: # We actually need to compute 1+x accurately, not x arg = Add(S.NegativeOne, arg, evaluate=False) xre, xim, _, _ = evalf_add(arg, prec, options) prec2 = workprec - fastlog(xre) # xre is now x - 1 so we add 1 back here to calculate x re = mpf_log(mpf_abs(mpf_add(xre, fone, prec2)), prec, rnd) re_acc = prec if imaginary_term: return re, mpf_pi(prec), re_acc, prec else: return re, None, re_acc, None def evalf_atan(v, prec, options): arg = v.args[0] xre, xim, reacc, imacc = evalf(arg, prec + 5, options) if xre is xim is None: return (None,)*4 if xim: raise NotImplementedError return mpf_atan(xre, prec, rnd), None, prec, None def evalf_subs(prec, subs): """ Change all Float entries in `subs` to have precision prec. """ newsubs = {} for a, b in subs.items(): b = S(b) if b.is_Float: b = b._eval_evalf(prec) newsubs[a] = b return newsubs def evalf_piecewise(expr, prec, options): from sympy import Float, Integer if 'subs' in options: expr = expr.subs(evalf_subs(prec, options['subs'])) newopts = options.copy() del newopts['subs'] if hasattr(expr, 'func'): return evalf(expr, prec, newopts) if type(expr) == float: return evalf(Float(expr), prec, newopts) if type(expr) == int: return evalf(Integer(expr), prec, newopts) # We still have undefined symbols raise NotImplementedError def evalf_bernoulli(expr, prec, options): arg = expr.args[0] if not arg.is_Integer: raise ValueError("Bernoulli number index must be an integer") n = int(arg) b = mpf_bernoulli(n, prec, rnd) if b == fzero: return None, None, None, None return b, None, prec, None #----------------------------------------------------------------------------# # # # High-level operations # # # #----------------------------------------------------------------------------# def as_mpmath(x, prec, options): from sympy.core.numbers import Infinity, NegativeInfinity, Zero x = sympify(x) if isinstance(x, Zero) or x == 0: return mpf(0) if isinstance(x, Infinity): return mpf('inf') if isinstance(x, NegativeInfinity): return mpf('-inf') # XXX re, im, _, _ = evalf(x, prec, options) if im: return mpc(re or fzero, im) return mpf(re) def do_integral(expr, prec, options): func = expr.args[0] x, xlow, xhigh = expr.args[1] if xlow == xhigh: xlow = xhigh = 0 elif x not in func.free_symbols: # only the difference in limits matters in this case # so if there is a symbol in common that will cancel # out when taking the difference, then use that # difference if xhigh.free_symbols & xlow.free_symbols: diff = xhigh - xlow if diff.is_number: xlow, xhigh = 0, diff oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC) options['maxprec'] = min(oldmaxprec, 2*prec) with workprec(prec + 5): xlow = as_mpmath(xlow, prec + 15, options) xhigh = as_mpmath(xhigh, prec + 15, options) # Integration is like summation, and we can phone home from # the integrand function to update accuracy summation style # Note that this accuracy is inaccurate, since it fails # to account for the variable quadrature weights, # but it is better than nothing from sympy import cos, sin, Wild have_part = [False, False] max_real_term = [MINUS_INF] max_imag_term = [MINUS_INF] def f(t): re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}}) have_part[0] = re or have_part[0] have_part[1] = im or have_part[1] max_real_term[0] = max(max_real_term[0], fastlog(re)) max_imag_term[0] = max(max_imag_term[0], fastlog(im)) if im: return mpc(re or fzero, im) return mpf(re or fzero) if options.get('quad') == 'osc': A = Wild('A', exclude=[x]) B = Wild('B', exclude=[x]) D = Wild('D') m = func.match(cos(A*x + B)*D) if not m: m = func.match(sin(A*x + B)*D) if not m: raise ValueError("An integrand of the form sin(A*x+B)*f(x) " "or cos(A*x+B)*f(x) is required for oscillatory quadrature") period = as_mpmath(2*S.Pi/m[A], prec + 15, options) result = quadosc(f, [xlow, xhigh], period=period) # XXX: quadosc does not do error detection yet quadrature_error = MINUS_INF else: result, quadrature_error = quadts(f, [xlow, xhigh], error=1) quadrature_error = fastlog(quadrature_error._mpf_) options['maxprec'] = oldmaxprec if have_part[0]: re = result.real._mpf_ if re == fzero: re, re_acc = scaled_zero( min(-prec, -max_real_term[0], -quadrature_error)) re = scaled_zero(re) # handled ok in evalf_integral else: re_acc = -max(max_real_term[0] - fastlog(re) - prec, quadrature_error) else: re, re_acc = None, None if have_part[1]: im = result.imag._mpf_ if im == fzero: im, im_acc = scaled_zero( min(-prec, -max_imag_term[0], -quadrature_error)) im = scaled_zero(im) # handled ok in evalf_integral else: im_acc = -max(max_imag_term[0] - fastlog(im) - prec, quadrature_error) else: im, im_acc = None, None result = re, im, re_acc, im_acc return result def evalf_integral(expr, prec, options): limits = expr.limits if len(limits) != 1 or len(limits[0]) != 3: raise NotImplementedError workprec = prec i = 0 maxprec = options.get('maxprec', INF) while 1: result = do_integral(expr, workprec, options) accuracy = complex_accuracy(result) if accuracy >= prec: # achieved desired precision break if workprec >= maxprec: # can't increase accuracy any more break if accuracy == -1: # maybe the answer really is zero and maybe we just haven't increased # the precision enough. So increase by doubling to not take too long # to get to maxprec. workprec *= 2 else: workprec += max(prec, 2**i) workprec = min(workprec, maxprec) i += 1 return result def check_convergence(numer, denom, n): """ Returns ======= (h, g, p) where -- h is: > 0 for convergence of rate 1/factorial(n)**h < 0 for divergence of rate factorial(n)**(-h) = 0 for geometric or polynomial convergence or divergence -- abs(g) is: > 1 for geometric convergence of rate 1/h**n < 1 for geometric divergence of rate h**n = 1 for polynomial convergence or divergence (g < 0 indicates an alternating series) -- p is: > 1 for polynomial convergence of rate 1/n**h <= 1 for polynomial divergence of rate n**(-h) """ from sympy import Poly npol = Poly(numer, n) dpol = Poly(denom, n) p = npol.degree() q = dpol.degree() rate = q - p if rate: return rate, None, None constant = dpol.LC() / npol.LC() if abs(constant) != 1: return rate, constant, None if npol.degree() == dpol.degree() == 0: return rate, constant, 0 pc = npol.all_coeffs()[1] qc = dpol.all_coeffs()[1] return rate, constant, (qc - pc)/dpol.LC() def hypsum(expr, n, start, prec): """ Sum a rapidly convergent infinite hypergeometric series with given general term, e.g. e = hypsum(1/factorial(n), n). The quotient between successive terms must be a quotient of integer polynomials. """ from sympy import Float, hypersimp, lambdify if prec == float('inf'): raise NotImplementedError('does not support inf prec') if start: expr = expr.subs(n, n + start) hs = hypersimp(expr, n) if hs is None: raise NotImplementedError("a hypergeometric series is required") num, den = hs.as_numer_denom() func1 = lambdify(n, num) func2 = lambdify(n, den) h, g, p = check_convergence(num, den, n) if h < 0: raise ValueError("Sum diverges like (n!)^%i" % (-h)) term = expr.subs(n, 0) if not term.is_Rational: raise NotImplementedError("Non rational term functionality is not implemented.") # Direct summation if geometric or faster if h > 0 or (h == 0 and abs(g) > 1): term = (MPZ(term.p) << prec) // term.q s = term k = 1 while abs(term) > 5: term *= MPZ(func1(k - 1)) term //= MPZ(func2(k - 1)) s += term k += 1 return from_man_exp(s, -prec) else: alt = g < 0 if abs(g) < 1: raise ValueError("Sum diverges like (%i)^n" % abs(1/g)) if p < 1 or (p == 1 and not alt): raise ValueError("Sum diverges like n^%i" % (-p)) # We have polynomial convergence: use Richardson extrapolation vold = None ndig = prec_to_dps(prec) while True: # Need to use at least quad precision because a lot of cancellation # might occur in the extrapolation process; we check the answer to # make sure that the desired precision has been reached, too. prec2 = 4*prec term0 = (MPZ(term.p) << prec2) // term.q def summand(k, _term=[term0]): if k: k = int(k) _term[0] *= MPZ(func1(k - 1)) _term[0] //= MPZ(func2(k - 1)) return make_mpf(from_man_exp(_term[0], -prec2)) with workprec(prec): v = nsum(summand, [0, mpmath_inf], method='richardson') vf = Float(v, ndig) if vold is not None and vold == vf: break prec += prec # double precision each time vold = vf return v._mpf_ def evalf_prod(expr, prec, options): from sympy import Sum if all((l[1] - l[2]).is_Integer for l in expr.limits): re, im, re_acc, im_acc = evalf(expr.doit(), prec=prec, options=options) else: re, im, re_acc, im_acc = evalf(expr.rewrite(Sum), prec=prec, options=options) return re, im, re_acc, im_acc def evalf_sum(expr, prec, options): from sympy import Float if 'subs' in options: expr = expr.subs(options['subs']) func = expr.function limits = expr.limits if len(limits) != 1 or len(limits[0]) != 3: raise NotImplementedError if func.is_zero: return None, None, prec, None prec2 = prec + 10 try: n, a, b = limits[0] if b != S.Infinity or a != int(a): raise NotImplementedError # Use fast hypergeometric summation if possible v = hypsum(func, n, int(a), prec2) delta = prec - fastlog(v) if fastlog(v) < -10: v = hypsum(func, n, int(a), delta) return v, None, min(prec, delta), None except NotImplementedError: # Euler-Maclaurin summation for general series eps = Float(2.0)**(-prec) for i in range(1, 5): m = n = 2**i * prec s, err = expr.euler_maclaurin(m=m, n=n, eps=eps, eval_integral=False) err = err.evalf() if err <= eps: break err = fastlog(evalf(abs(err), 20, options)[0]) re, im, re_acc, im_acc = evalf(s, prec2, options) if re_acc is None: re_acc = -err if im_acc is None: im_acc = -err return re, im, re_acc, im_acc #----------------------------------------------------------------------------# # # # Symbolic interface # # # #----------------------------------------------------------------------------# def evalf_symbol(x, prec, options): val = options['subs'][x] if isinstance(val, mpf): if not val: return None, None, None, None return val._mpf_, None, prec, None else: if not '_cache' in options: options['_cache'] = {} cache = options['_cache'] cached, cached_prec = cache.get(x, (None, MINUS_INF)) if cached_prec >= prec: return cached v = evalf(sympify(val), prec, options) cache[x] = (v, prec) return v evalf_table = None def _create_evalf_table(): global evalf_table from sympy.functions.combinatorial.numbers import bernoulli from sympy.concrete.products import Product from sympy.concrete.summations import Sum from sympy.core.add import Add from sympy.core.mul import Mul from sympy.core.numbers import Exp1, Float, Half, ImaginaryUnit, Integer, NaN, NegativeOne, One, Pi, Rational, Zero from sympy.core.power import Pow from sympy.core.symbol import Dummy, Symbol from sympy.functions.elementary.complexes import Abs, im, re from sympy.functions.elementary.exponential import exp, log from sympy.functions.elementary.integers import ceiling, floor from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import atan, cos, sin from sympy.integrals.integrals import Integral evalf_table = { Symbol: evalf_symbol, Dummy: evalf_symbol, Float: lambda x, prec, options: (x._mpf_, None, prec, None), Rational: lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None), Integer: lambda x, prec, options: (from_int(x.p, prec), None, prec, None), Zero: lambda x, prec, options: (None, None, prec, None), One: lambda x, prec, options: (fone, None, prec, None), Half: lambda x, prec, options: (fhalf, None, prec, None), Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None), Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None), ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec), NegativeOne: lambda x, prec, options: (fnone, None, prec, None), NaN: lambda x, prec, options: (fnan, None, prec, None), exp: lambda x, prec, options: evalf_pow( Pow(S.Exp1, x.args[0], evaluate=False), prec, options), cos: evalf_trig, sin: evalf_trig, Add: evalf_add, Mul: evalf_mul, Pow: evalf_pow, log: evalf_log, atan: evalf_atan, Abs: evalf_abs, re: evalf_re, im: evalf_im, floor: evalf_floor, ceiling: evalf_ceiling, Integral: evalf_integral, Sum: evalf_sum, Product: evalf_prod, Piecewise: evalf_piecewise, bernoulli: evalf_bernoulli, } def evalf(x, prec, options): from sympy import re as re_, im as im_ try: rf = evalf_table[x.func] r = rf(x, prec, options) except KeyError: # Fall back to ordinary evalf if possible if 'subs' in options: x = x.subs(evalf_subs(prec, options['subs'])) xe = x._eval_evalf(prec) if xe is None: raise NotImplementedError as_real_imag = getattr(xe, "as_real_imag", None) if as_real_imag is None: raise NotImplementedError # e.g. FiniteSet(-1.0, 1.0).evalf() re, im = as_real_imag() if re.has(re_) or im.has(im_): raise NotImplementedError if re == 0: re = None reprec = None elif re.is_number: re = re._to_mpmath(prec, allow_ints=False)._mpf_ reprec = prec else: raise NotImplementedError if im == 0: im = None imprec = None elif im.is_number: im = im._to_mpmath(prec, allow_ints=False)._mpf_ imprec = prec else: raise NotImplementedError r = re, im, reprec, imprec if options.get("verbose"): print("### input", x) print("### output", to_str(r[0] or fzero, 50)) print("### raw", r) # r[0], r[2] print() chop = options.get('chop', False) if chop: if chop is True: chop_prec = prec else: # convert (approximately) from given tolerance; # the formula here will will make 1e-i rounds to 0 for # i in the range +/-27 while 2e-i will not be chopped chop_prec = int(round(-3.321*math.log10(chop) + 2.5)) if chop_prec == 3: chop_prec -= 1 r = chop_parts(r, chop_prec) if options.get("strict"): check_target(x, r, prec) return r class EvalfMixin: """Mixin class adding evalf capabililty.""" __slots__ = () # type: Tuple[str, ...] def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False): """ Evaluate the given formula to an accuracy of *n* digits. Parameters ========== subs : dict, optional Substitute numerical values for symbols, e.g. ``subs={x:3, y:1+pi}``. The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of "small" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise ``PrecisionExhausted`` if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try ``quad='osc'``. verbose : bool, optional Print debug information. Notes ===== When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: >>> from sympy.abc import x, y, z >>> values = {x: 1e16, y: 1, z: 1e16} >>> (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: >>> (x + y - z).evalf(subs=values) 1.00000000000000 """ from sympy import Float, Number n = n if n is not None else 15 if subs and is_sequence(subs): raise TypeError('subs must be given as a dictionary') # for sake of sage that doesn't like evalf(1) if n == 1 and isinstance(self, Number): from sympy.core.expr import _mag rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose) m = _mag(rv) rv = rv.round(1 - m) return rv if not evalf_table: _create_evalf_table() prec = dps_to_prec(n) options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop, 'strict': strict, 'verbose': verbose} if subs is not None: options['subs'] = subs if quad is not None: options['quad'] = quad try: result = evalf(self, prec + 4, options) except NotImplementedError: # Fall back to the ordinary evalf v = self._eval_evalf(prec) if v is None: return self elif not v.is_number: return v try: # If the result is numerical, normalize it result = evalf(v, prec, options) except NotImplementedError: # Probably contains symbols or unknown functions return v re, im, re_acc, im_acc = result if re: p = max(min(prec, re_acc), 1) re = Float._new(re, p) else: re = S.Zero if im: p = max(min(prec, im_acc), 1) im = Float._new(im, p) return re + im*S.ImaginaryUnit else: return re n = evalf def _evalf(self, prec): """Helper for evalf. Does the same thing but takes binary precision""" r = self._eval_evalf(prec) if r is None: r = self return r def _eval_evalf(self, prec): return def _to_mpmath(self, prec, allow_ints=True): # mpmath functions accept ints as input errmsg = "cannot convert to mpmath number" if allow_ints and self.is_Integer: return self.p if hasattr(self, '_as_mpf_val'): return make_mpf(self._as_mpf_val(prec)) try: re, im, _, _ = evalf(self, prec, {}) if im: if not re: re = fzero return make_mpc((re, im)) elif re: return make_mpf(re) else: return make_mpf(fzero) except NotImplementedError: v = self._eval_evalf(prec) if v is None: raise ValueError(errmsg) if v.is_Float: return make_mpf(v._mpf_) # Number + Number*I is also fine re, im = v.as_real_imag() if allow_ints and re.is_Integer: re = from_int(re.p) elif re.is_Float: re = re._mpf_ else: raise ValueError(errmsg) if allow_ints and im.is_Integer: im = from_int(im.p) elif im.is_Float: im = im._mpf_ else: raise ValueError(errmsg) return make_mpc((re, im)) def N(x, n=15, **options): r""" Calls x.evalf(n, \*\*options). Explanations ============ Both .n() and N() are equivalent to .evalf(); use the one that you like better. See also the docstring of .evalf() for information on the options. Examples ======== >>> from sympy import Sum, oo, N >>> from sympy.abc import k >>> Sum(1/k**k, (k, 1, oo)) Sum(k**(-k), (k, 1, oo)) >>> N(_, 4) 1.291 """ # by using rational=True, any evaluation of a string # will be done using exact values for the Floats return sympify(x, rational=True).evalf(n, **options)
a1a144fb7022f19405cf304bd4eca82dbda3d043872f75bf07529ed7bb10ca8d
"""Module for SymPy containers (SymPy objects that store other SymPy objects) The containers implemented in this module are subclassed to Basic. They are supposed to work seamlessly within the SymPy framework. """ from collections import OrderedDict from sympy.core.basic import Basic from sympy.core.compatibility import as_int, MutableSet from sympy.core.sympify import _sympify, sympify, converter, SympifyError from sympy.utilities.iterables import iterable class Tuple(Basic): """ Wrapper around the builtin tuple object. Explanation =========== The Tuple is a subclass of Basic, so that it works well in the SymPy framework. The wrapped tuple is available as self.args, but you can also access elements or slices with [:] syntax. Parameters ========== sympify : bool If ``False``, ``sympify`` is not called on ``args``. This can be used for speedups for very large tuples where the elements are known to already be sympy objects. Examples ======== >>> from sympy import symbols >>> from sympy.core.containers import Tuple >>> a, b, c, d = symbols('a b c d') >>> Tuple(a, b, c)[1:] (b, c) >>> Tuple(a, b, c).subs(a, d) (d, b, c) """ def __new__(cls, *args, **kwargs): if kwargs.get('sympify', True): args = (sympify(arg) for arg in args) obj = Basic.__new__(cls, *args) return obj def __getitem__(self, i): if isinstance(i, slice): indices = i.indices(len(self)) return Tuple(*(self.args[j] for j in range(*indices))) return self.args[i] def __len__(self): return len(self.args) def __contains__(self, item): return item in self.args def __iter__(self): return iter(self.args) def __add__(self, other): if isinstance(other, Tuple): return Tuple(*(self.args + other.args)) elif isinstance(other, tuple): return Tuple(*(self.args + other)) else: return NotImplemented def __radd__(self, other): if isinstance(other, Tuple): return Tuple(*(other.args + self.args)) elif isinstance(other, tuple): return Tuple(*(other + self.args)) else: return NotImplemented def __mul__(self, other): try: n = as_int(other) except ValueError: raise TypeError("Can't multiply sequence by non-integer of type '%s'" % type(other)) return self.func(*(self.args*n)) __rmul__ = __mul__ def __eq__(self, other): if isinstance(other, Basic): return super().__eq__(other) return self.args == other def __ne__(self, other): if isinstance(other, Basic): return super().__ne__(other) return self.args != other def __hash__(self): return hash(self.args) def _to_mpmath(self, prec): return tuple(a._to_mpmath(prec) for a in self.args) def __lt__(self, other): return _sympify(self.args < other.args) def __le__(self, other): return _sympify(self.args <= other.args) # XXX: Basic defines count() as something different, so we can't # redefine it here. Originally this lead to cse() test failure. def tuple_count(self, value): """T.count(value) -> integer -- return number of occurrences of value""" return self.args.count(value) def index(self, value, start=None, stop=None): """Searches and returns the first index of the value.""" # XXX: One would expect: # # return self.args.index(value, start, stop) # # here. Any trouble with that? Yes: # # >>> (1,).index(1, None, None) # Traceback (most recent call last): # File "<stdin>", line 1, in <module> # TypeError: slice indices must be integers or None or have an __index__ method # # See: http://bugs.python.org/issue13340 if start is None and stop is None: return self.args.index(value) elif stop is None: return self.args.index(value, start) else: return self.args.index(value, start, stop) converter[tuple] = lambda tup: Tuple(*tup) def tuple_wrapper(method): """ Decorator that converts any tuple in the function arguments into a Tuple. Explanation =========== The motivation for this is to provide simple user interfaces. The user can call a function with regular tuples in the argument, and the wrapper will convert them to Tuples before handing them to the function. Explanation =========== >>> from sympy.core.containers import tuple_wrapper >>> def f(*args): ... return args >>> g = tuple_wrapper(f) The decorated function g sees only the Tuple argument: >>> g(0, (1, 2), 3) (0, (1, 2), 3) """ def wrap_tuples(*args, **kw_args): newargs = [] for arg in args: if type(arg) is tuple: newargs.append(Tuple(*arg)) else: newargs.append(arg) return method(*newargs, **kw_args) return wrap_tuples class Dict(Basic): """ Wrapper around the builtin dict object Explanation =========== The Dict is a subclass of Basic, so that it works well in the SymPy framework. Because it is immutable, it may be included in sets, but its values must all be given at instantiation and cannot be changed afterwards. Otherwise it behaves identically to the Python dict. Examples ======== >>> from sympy import Symbol >>> from sympy.core.containers import Dict >>> D = Dict({1: 'one', 2: 'two'}) >>> for key in D: ... if key == 1: ... print('%s %s' % (key, D[key])) 1 one The args are sympified so the 1 and 2 are Integers and the values are Symbols. Queries automatically sympify args so the following work: >>> 1 in D True >>> D.has(Symbol('one')) # searches keys and values True >>> 'one' in D # not in the keys False >>> D[1] one """ def __new__(cls, *args): if len(args) == 1 and isinstance(args[0], (dict, Dict)): items = [Tuple(k, v) for k, v in args[0].items()] elif iterable(args) and all(len(arg) == 2 for arg in args): items = [Tuple(k, v) for k, v in args] else: raise TypeError('Pass Dict args as Dict((k1, v1), ...) or Dict({k1: v1, ...})') elements = frozenset(items) obj = Basic.__new__(cls, elements) obj.elements = elements obj._dict = dict(items) # In case Tuple decides it wants to sympify return obj def __getitem__(self, key): """x.__getitem__(y) <==> x[y]""" try: key = _sympify(key) except SympifyError: raise KeyError(key) return self._dict[key] def __setitem__(self, key, value): raise NotImplementedError("SymPy Dicts are Immutable") @property def args(self): """Returns a tuple of arguments of 'self'. See Also ======== sympy.core.basic.Basic.args """ return tuple(self.elements) def items(self): '''Returns a set-like object providing a view on dict's items. ''' return self._dict.items() def keys(self): '''Returns the list of the dict's keys.''' return self._dict.keys() def values(self): '''Returns the list of the dict's values.''' return self._dict.values() def __iter__(self): '''x.__iter__() <==> iter(x)''' return iter(self._dict) def __len__(self): '''x.__len__() <==> len(x)''' return self._dict.__len__() def get(self, key, default=None): '''Returns the value for key if the key is in the dictionary.''' try: key = _sympify(key) except SympifyError: return default return self._dict.get(key, default) def __contains__(self, key): '''D.__contains__(k) -> True if D has a key k, else False''' try: key = _sympify(key) except SympifyError: return False return key in self._dict def __lt__(self, other): return _sympify(self.args < other.args) @property def _sorted_args(self): from sympy.utilities import default_sort_key return tuple(sorted(self.args, key=default_sort_key)) # this handles dict, defaultdict, OrderedDict converter[dict] = lambda d: Dict(*d.items()) class OrderedSet(MutableSet): def __init__(self, iterable=None): if iterable: self.map = OrderedDict((item, None) for item in iterable) else: self.map = OrderedDict() def __len__(self): return len(self.map) def __contains__(self, key): return key in self.map def add(self, key): self.map[key] = None def discard(self, key): self.map.pop(key) def pop(self, last=True): return self.map.popitem(last=last)[0] def __iter__(self): yield from self.map.keys() def __repr__(self): if not self.map: return '%s()' % (self.__class__.__name__,) return '%s(%r)' % (self.__class__.__name__, list(self.map.keys())) def intersection(self, other): result = [] for val in self: if val in other: result.append(val) return self.__class__(result) def difference(self, other): result = [] for val in self: if val not in other: result.append(val) return self.__class__(result) def update(self, iterable): for val in iterable: self.add(val)
76ee7589112d5d9ff9e8bbdb9aada1e88d85cfdbb3c3f3c912fd5ebf5dee65b8
from collections import defaultdict from functools import cmp_to_key import operator from .sympify import sympify from .basic import Basic from .singleton import S from .operations import AssocOp, AssocOpDispatcher from .cache import cacheit from .logic import fuzzy_not, _fuzzy_group, fuzzy_and from .compatibility import reduce from .expr import Expr from .parameters import global_parameters # internal marker to indicate: # "there are still non-commutative objects -- don't forget to process them" class NC_Marker: is_Order = False is_Mul = False is_Number = False is_Poly = False is_commutative = False # Key for sorting commutative args in canonical order _args_sortkey = cmp_to_key(Basic.compare) def _mulsort(args): # in-place sorting of args args.sort(key=_args_sortkey) def _unevaluated_Mul(*args): """Return a well-formed unevaluated Mul: Numbers are collected and put in slot 0, any arguments that are Muls will be flattened, and args are sorted. Use this when args have changed but you still want to return an unevaluated Mul. Examples ======== >>> from sympy.core.mul import _unevaluated_Mul as uMul >>> from sympy import S, sqrt, Mul >>> from sympy.abc import x >>> a = uMul(*[S(3.0), x, S(2)]) >>> a.args[0] 6.00000000000000 >>> a.args[1] x Two unevaluated Muls with the same arguments will always compare as equal during testing: >>> m = uMul(sqrt(2), sqrt(3)) >>> m == uMul(sqrt(3), sqrt(2)) True >>> u = Mul(sqrt(3), sqrt(2), evaluate=False) >>> m == uMul(u) True >>> m == Mul(*m.args) False """ args = list(args) newargs = [] ncargs = [] co = S.One while args: a = args.pop() if a.is_Mul: c, nc = a.args_cnc() args.extend(c) if nc: ncargs.append(Mul._from_args(nc)) elif a.is_Number: co *= a else: newargs.append(a) _mulsort(newargs) if co is not S.One: newargs.insert(0, co) if ncargs: newargs.append(Mul._from_args(ncargs)) return Mul._from_args(newargs) class Mul(Expr, AssocOp): __slots__ = () is_Mul = True _args_type = Expr def __neg__(self): c, args = self.as_coeff_mul() c = -c if c is not S.One: if args[0].is_Number: args = list(args) if c is S.NegativeOne: args[0] = -args[0] else: args[0] *= c else: args = (c,) + args return self._from_args(args, self.is_commutative) @classmethod def flatten(cls, seq): """Return commutative, noncommutative and order arguments by combining related terms. Notes ===== * In an expression like ``a*b*c``, python process this through sympy as ``Mul(Mul(a, b), c)``. This can have undesirable consequences. - Sometimes terms are not combined as one would like: {c.f. https://github.com/sympy/sympy/issues/4596} >>> from sympy import Mul, sqrt >>> from sympy.abc import x, y, z >>> 2*(x + 1) # this is the 2-arg Mul behavior 2*x + 2 >>> y*(x + 1)*2 2*y*(x + 1) >>> 2*(x + 1)*y # 2-arg result will be obtained first y*(2*x + 2) >>> Mul(2, x + 1, y) # all 3 args simultaneously processed 2*y*(x + 1) >>> 2*((x + 1)*y) # parentheses can control this behavior 2*y*(x + 1) Powers with compound bases may not find a single base to combine with unless all arguments are processed at once. Post-processing may be necessary in such cases. {c.f. https://github.com/sympy/sympy/issues/5728} >>> a = sqrt(x*sqrt(y)) >>> a**3 (x*sqrt(y))**(3/2) >>> Mul(a,a,a) (x*sqrt(y))**(3/2) >>> a*a*a x*sqrt(y)*sqrt(x*sqrt(y)) >>> _.subs(a.base, z).subs(z, a.base) (x*sqrt(y))**(3/2) - If more than two terms are being multiplied then all the previous terms will be re-processed for each new argument. So if each of ``a``, ``b`` and ``c`` were :class:`Mul` expression, then ``a*b*c`` (or building up the product with ``*=``) will process all the arguments of ``a`` and ``b`` twice: once when ``a*b`` is computed and again when ``c`` is multiplied. Using ``Mul(a, b, c)`` will process all arguments once. * The results of Mul are cached according to arguments, so flatten will only be called once for ``Mul(a, b, c)``. If you can structure a calculation so the arguments are most likely to be repeats then this can save time in computing the answer. For example, say you had a Mul, M, that you wished to divide by ``d[i]`` and multiply by ``n[i]`` and you suspect there are many repeats in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the product, ``M*n[i]`` will be returned without flattening -- the cached value will be returned. If you divide by the ``d[i]`` first (and those are more unique than the ``n[i]``) then that will create a new Mul, ``M/d[i]`` the args of which will be traversed again when it is multiplied by ``n[i]``. {c.f. https://github.com/sympy/sympy/issues/5706} This consideration is moot if the cache is turned off. NB -- The validity of the above notes depends on the implementation details of Mul and flatten which may change at any time. Therefore, you should only consider them when your code is highly performance sensitive. Removal of 1 from the sequence is already handled by AssocOp.__new__. """ from sympy.calculus.util import AccumBounds from sympy.matrices.expressions import MatrixExpr rv = None if len(seq) == 2: a, b = seq if b.is_Rational: a, b = b, a seq = [a, b] assert not a is S.One if not a.is_zero and a.is_Rational: r, b = b.as_coeff_Mul() if b.is_Add: if r is not S.One: # 2-arg hack # leave the Mul as a Mul? ar = a*r if ar is S.One: arb = b else: arb = cls(a*r, b, evaluate=False) rv = [arb], [], None elif global_parameters.distribute and b.is_commutative: r, b = b.as_coeff_Add() bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)] _addsort(bargs) ar = a*r if ar: bargs.insert(0, ar) bargs = [Add._from_args(bargs)] rv = bargs, [], None if rv: return rv # apply associativity, separate commutative part of seq c_part = [] # out: commutative factors nc_part = [] # out: non-commutative factors nc_seq = [] coeff = S.One # standalone term # e.g. 3 * ... c_powers = [] # (base,exp) n # e.g. (x,n) for x num_exp = [] # (num-base, exp) y # e.g. (3, y) for ... * 3 * ... neg1e = S.Zero # exponent on -1 extracted from Number-based Pow and I pnum_rat = {} # (num-base, Rat-exp) 1/2 # e.g. (3, 1/2) for ... * 3 * ... order_symbols = None # --- PART 1 --- # # "collect powers and coeff": # # o coeff # o c_powers # o num_exp # o neg1e # o pnum_rat # # NOTE: this is optimized for all-objects-are-commutative case for o in seq: # O(x) if o.is_Order: o, order_symbols = o.as_expr_variables(order_symbols) # Mul([...]) if o.is_Mul: if o.is_commutative: seq.extend(o.args) # XXX zerocopy? else: # NCMul can have commutative parts as well for q in o.args: if q.is_commutative: seq.append(q) else: nc_seq.append(q) # append non-commutative marker, so we don't forget to # process scheduled non-commutative objects seq.append(NC_Marker) continue # 3 elif o.is_Number: if o is S.NaN or coeff is S.ComplexInfinity and o.is_zero: # we know for sure the result will be nan return [S.NaN], [], None elif coeff.is_Number or isinstance(coeff, AccumBounds): # it could be zoo coeff *= o if coeff is S.NaN: # we know for sure the result will be nan return [S.NaN], [], None continue elif isinstance(o, AccumBounds): coeff = o.__mul__(coeff) continue elif o is S.ComplexInfinity: if not coeff: # 0 * zoo = NaN return [S.NaN], [], None coeff = S.ComplexInfinity continue elif o is S.ImaginaryUnit: neg1e += S.Half continue elif o.is_commutative: # e # o = b b, e = o.as_base_exp() # y # 3 if o.is_Pow: if b.is_Number: # get all the factors with numeric base so they can be # combined below, but don't combine negatives unless # the exponent is an integer if e.is_Rational: if e.is_Integer: coeff *= Pow(b, e) # it is an unevaluated power continue elif e.is_negative: # also a sign of an unevaluated power seq.append(Pow(b, e)) continue elif b.is_negative: neg1e += e b = -b if b is not S.One: pnum_rat.setdefault(b, []).append(e) continue elif b.is_positive or e.is_integer: num_exp.append((b, e)) continue c_powers.append((b, e)) # NON-COMMUTATIVE # TODO: Make non-commutative exponents not combine automatically else: if o is not NC_Marker: nc_seq.append(o) # process nc_seq (if any) while nc_seq: o = nc_seq.pop(0) if not nc_part: nc_part.append(o) continue # b c b+c # try to combine last terms: a * a -> a o1 = nc_part.pop() b1, e1 = o1.as_base_exp() b2, e2 = o.as_base_exp() new_exp = e1 + e2 # Only allow powers to combine if the new exponent is # not an Add. This allow things like a**2*b**3 == a**5 # if a.is_commutative == False, but prohibits # a**x*a**y and x**a*x**b from combining (x,y commute). if b1 == b2 and (not new_exp.is_Add): o12 = b1 ** new_exp # now o12 could be a commutative object if o12.is_commutative: seq.append(o12) continue else: nc_seq.insert(0, o12) else: nc_part.append(o1) nc_part.append(o) # We do want a combined exponent if it would not be an Add, such as # y 2y 3y # x * x -> x # We determine if two exponents have the same term by using # as_coeff_Mul. # # Unfortunately, this isn't smart enough to consider combining into # exponents that might already be adds, so things like: # z - y y # x * x will be left alone. This is because checking every possible # combination can slow things down. # gather exponents of common bases... def _gather(c_powers): common_b = {} # b:e for b, e in c_powers: co = e.as_coeff_Mul() common_b.setdefault(b, {}).setdefault( co[1], []).append(co[0]) for b, d in common_b.items(): for di, li in d.items(): d[di] = Add(*li) new_c_powers = [] for b, e in common_b.items(): new_c_powers.extend([(b, c*t) for t, c in e.items()]) return new_c_powers # in c_powers c_powers = _gather(c_powers) # and in num_exp num_exp = _gather(num_exp) # --- PART 2 --- # # o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow) # o combine collected powers (2**x * 3**x -> 6**x) # with numeric base # ................................ # now we have: # - coeff: # - c_powers: (b, e) # - num_exp: (2, e) # - pnum_rat: {(1/3, [1/3, 2/3, 1/4])} # 0 1 # x -> 1 x -> x # this should only need to run twice; if it fails because # it needs to be run more times, perhaps this should be # changed to a "while True" loop -- the only reason it # isn't such now is to allow a less-than-perfect result to # be obtained rather than raising an error or entering an # infinite loop for i in range(2): new_c_powers = [] changed = False for b, e in c_powers: if e.is_zero: # canceling out infinities yields NaN if (b.is_Add or b.is_Mul) and any(infty in b.args for infty in (S.ComplexInfinity, S.Infinity, S.NegativeInfinity)): return [S.NaN], [], None continue if e is S.One: if b.is_Number: coeff *= b continue p = b if e is not S.One: p = Pow(b, e) # check to make sure that the base doesn't change # after exponentiation; to allow for unevaluated # Pow, we only do so if b is not already a Pow if p.is_Pow and not b.is_Pow: bi = b b, e = p.as_base_exp() if b != bi: changed = True c_part.append(p) new_c_powers.append((b, e)) # there might have been a change, but unless the base # matches some other base, there is nothing to do if changed and len({ b for b, e in new_c_powers}) != len(new_c_powers): # start over again c_part = [] c_powers = _gather(new_c_powers) else: break # x x x # 2 * 3 -> 6 inv_exp_dict = {} # exp:Mul(num-bases) x x # e.g. x:6 for ... * 2 * 3 * ... for b, e in num_exp: inv_exp_dict.setdefault(e, []).append(b) for e, b in inv_exp_dict.items(): inv_exp_dict[e] = cls(*b) c_part.extend([Pow(b, e) for e, b in inv_exp_dict.items() if e]) # b, e -> e' = sum(e), b # {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])} comb_e = {} for b, e in pnum_rat.items(): comb_e.setdefault(Add(*e), []).append(b) del pnum_rat # process them, reducing exponents to values less than 1 # and updating coeff if necessary else adding them to # num_rat for further processing num_rat = [] for e, b in comb_e.items(): b = cls(*b) if e.q == 1: coeff *= Pow(b, e) continue if e.p > e.q: e_i, ep = divmod(e.p, e.q) coeff *= Pow(b, e_i) e = Rational(ep, e.q) num_rat.append((b, e)) del comb_e # extract gcd of bases in num_rat # 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4) pnew = defaultdict(list) i = 0 # steps through num_rat which may grow while i < len(num_rat): bi, ei = num_rat[i] grow = [] for j in range(i + 1, len(num_rat)): bj, ej = num_rat[j] g = bi.gcd(bj) if g is not S.One: # 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2 # this might have a gcd with something else e = ei + ej if e.q == 1: coeff *= Pow(g, e) else: if e.p > e.q: e_i, ep = divmod(e.p, e.q) # change e in place coeff *= Pow(g, e_i) e = Rational(ep, e.q) grow.append((g, e)) # update the jth item num_rat[j] = (bj/g, ej) # update bi that we are checking with bi = bi/g if bi is S.One: break if bi is not S.One: obj = Pow(bi, ei) if obj.is_Number: coeff *= obj else: # changes like sqrt(12) -> 2*sqrt(3) for obj in Mul.make_args(obj): if obj.is_Number: coeff *= obj else: assert obj.is_Pow bi, ei = obj.args pnew[ei].append(bi) num_rat.extend(grow) i += 1 # combine bases of the new powers for e, b in pnew.items(): pnew[e] = cls(*b) # handle -1 and I if neg1e: # treat I as (-1)**(1/2) and compute -1's total exponent p, q = neg1e.as_numer_denom() # if the integer part is odd, extract -1 n, p = divmod(p, q) if n % 2: coeff = -coeff # if it's a multiple of 1/2 extract I if q == 2: c_part.append(S.ImaginaryUnit) elif p: # see if there is any positive base this power of # -1 can join neg1e = Rational(p, q) for e, b in pnew.items(): if e == neg1e and b.is_positive: pnew[e] = -b break else: # keep it separate; we've already evaluated it as # much as possible so evaluate=False c_part.append(Pow(S.NegativeOne, neg1e, evaluate=False)) # add all the pnew powers c_part.extend([Pow(b, e) for e, b in pnew.items()]) # oo, -oo if (coeff is S.Infinity) or (coeff is S.NegativeInfinity): def _handle_for_oo(c_part, coeff_sign): new_c_part = [] for t in c_part: if t.is_extended_positive: continue if t.is_extended_negative: coeff_sign *= -1 continue new_c_part.append(t) return new_c_part, coeff_sign c_part, coeff_sign = _handle_for_oo(c_part, 1) nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign) coeff *= coeff_sign # zoo if coeff is S.ComplexInfinity: # zoo might be # infinite_real + bounded_im # bounded_real + infinite_im # infinite_real + infinite_im # and non-zero real or imaginary will not change that status. c_part = [c for c in c_part if not (fuzzy_not(c.is_zero) and c.is_extended_real is not None)] nc_part = [c for c in nc_part if not (fuzzy_not(c.is_zero) and c.is_extended_real is not None)] # 0 elif coeff.is_zero: # we know for sure the result will be 0 except the multiplicand # is infinity or a matrix if any(isinstance(c, MatrixExpr) for c in nc_part): return [coeff], nc_part, order_symbols if any(c.is_finite == False for c in c_part): return [S.NaN], [], order_symbols return [coeff], [], order_symbols # check for straggling Numbers that were produced _new = [] for i in c_part: if i.is_Number: coeff *= i else: _new.append(i) c_part = _new # order commutative part canonically _mulsort(c_part) # current code expects coeff to be always in slot-0 if coeff is not S.One: c_part.insert(0, coeff) # we are done if (global_parameters.distribute and not nc_part and len(c_part) == 2 and c_part[0].is_Number and c_part[0].is_finite and c_part[1].is_Add): # 2*(1+a) -> 2 + 2 * a coeff = c_part[0] c_part = [Add(*[coeff*f for f in c_part[1].args])] return c_part, nc_part, order_symbols def _eval_power(self, e): # don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B cargs, nc = self.args_cnc(split_1=False) if e.is_Integer: return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * \ Pow(Mul._from_args(nc), e, evaluate=False) if e.is_Rational and e.q == 2: from sympy.core.power import integer_nthroot from sympy.functions.elementary.complexes import sign if self.is_imaginary: a = self.as_real_imag()[1] if a.is_Rational: n, d = abs(a/2).as_numer_denom() n, t = integer_nthroot(n, 2) if t: d, t = integer_nthroot(d, 2) if t: r = sympify(n)/d return _unevaluated_Mul(r**e.p, (1 + sign(a)*S.ImaginaryUnit)**e.p) p = Pow(self, e, evaluate=False) if e.is_Rational or e.is_Float: return p._eval_expand_power_base() return p @classmethod def class_key(cls): return 3, 0, cls.__name__ def _eval_evalf(self, prec): c, m = self.as_coeff_Mul() if c is S.NegativeOne: if m.is_Mul: rv = -AssocOp._eval_evalf(m, prec) else: mnew = m._eval_evalf(prec) if mnew is not None: m = mnew rv = -m else: rv = AssocOp._eval_evalf(self, prec) if rv.is_number: return rv.expand() return rv @property def _mpc_(self): """ Convert self to an mpmath mpc if possible """ from sympy.core.numbers import I, Float im_part, imag_unit = self.as_coeff_Mul() if not imag_unit == I: # ValueError may seem more reasonable but since it's a @property, # we need to use AttributeError to keep from confusing things like # hasattr. raise AttributeError("Cannot convert Mul to mpc. Must be of the form Number*I") return (Float(0)._mpf_, Float(im_part)._mpf_) @cacheit def as_two_terms(self): """Return head and tail of self. This is the most efficient way to get the head and tail of an expression. - if you want only the head, use self.args[0]; - if you want to process the arguments of the tail then use self.as_coef_mul() which gives the head and a tuple containing the arguments of the tail when treated as a Mul. - if you want the coefficient when self is treated as an Add then use self.as_coeff_add()[0] Examples ======== >>> from sympy.abc import x, y >>> (3*x*y).as_two_terms() (3, x*y) """ args = self.args if len(args) == 1: return S.One, self elif len(args) == 2: return args else: return args[0], self._new_rawargs(*args[1:]) @cacheit def as_coefficients_dict(self): """Return a dictionary mapping terms to their coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. The dictionary is considered to have a single term. Examples ======== >>> from sympy.abc import a, x >>> (3*a*x).as_coefficients_dict() {a*x: 3} >>> _[a] 0 """ d = defaultdict(int) args = self.args if len(args) == 1 or not args[0].is_Number: d[self] = S.One else: d[self._new_rawargs(*args[1:])] = args[0] return d @cacheit def as_coeff_mul(self, *deps, rational=True, **kwargs): if deps: from sympy.utilities.iterables import sift l1, l2 = sift(self.args, lambda x: x.has(*deps), binary=True) return self._new_rawargs(*l2), tuple(l1) args = self.args if args[0].is_Number: if not rational or args[0].is_Rational: return args[0], args[1:] elif args[0].is_extended_negative: return S.NegativeOne, (-args[0],) + args[1:] return S.One, args def as_coeff_Mul(self, rational=False): """ Efficiently extract the coefficient of a product. """ coeff, args = self.args[0], self.args[1:] if coeff.is_Number: if not rational or coeff.is_Rational: if len(args) == 1: return coeff, args[0] else: return coeff, self._new_rawargs(*args) elif coeff.is_extended_negative: return S.NegativeOne, self._new_rawargs(*((-coeff,) + args)) return S.One, self def as_real_imag(self, deep=True, **hints): from sympy import Abs, expand_mul, im, re other = [] coeffr = [] coeffi = [] addterms = S.One for a in self.args: r, i = a.as_real_imag() if i.is_zero: coeffr.append(r) elif r.is_zero: coeffi.append(i*S.ImaginaryUnit) elif a.is_commutative: # search for complex conjugate pairs: for i, x in enumerate(other): if x == a.conjugate(): coeffr.append(Abs(x)**2) del other[i] break else: if a.is_Add: addterms *= a else: other.append(a) else: other.append(a) m = self.func(*other) if hints.get('ignore') == m: return if len(coeffi) % 2: imco = im(coeffi.pop(0)) # all other pairs make a real factor; they will be # put into reco below else: imco = S.Zero reco = self.func(*(coeffr + coeffi)) r, i = (reco*re(m), reco*im(m)) if addterms == 1: if m == 1: if imco.is_zero: return (reco, S.Zero) else: return (S.Zero, reco*imco) if imco is S.Zero: return (r, i) return (-imco*i, imco*r) addre, addim = expand_mul(addterms, deep=False).as_real_imag() if imco is S.Zero: return (r*addre - i*addim, i*addre + r*addim) else: r, i = -imco*i, imco*r return (r*addre - i*addim, r*addim + i*addre) @staticmethod def _expandsums(sums): """ Helper function for _eval_expand_mul. sums must be a list of instances of Basic. """ L = len(sums) if L == 1: return sums[0].args terms = [] left = Mul._expandsums(sums[:L//2]) right = Mul._expandsums(sums[L//2:]) terms = [Mul(a, b) for a in left for b in right] added = Add(*terms) return Add.make_args(added) # it may have collapsed down to one term def _eval_expand_mul(self, **hints): from sympy import fraction # Handle things like 1/(x*(x + 1)), which are automatically converted # to 1/x*1/(x + 1) expr = self n, d = fraction(expr) if d.is_Mul: n, d = [i._eval_expand_mul(**hints) if i.is_Mul else i for i in (n, d)] expr = n/d if not expr.is_Mul: return expr plain, sums, rewrite = [], [], False for factor in expr.args: if factor.is_Add: sums.append(factor) rewrite = True else: if factor.is_commutative: plain.append(factor) else: sums.append(Basic(factor)) # Wrapper if not rewrite: return expr else: plain = self.func(*plain) if sums: deep = hints.get("deep", False) terms = self.func._expandsums(sums) args = [] for term in terms: t = self.func(plain, term) if t.is_Mul and any(a.is_Add for a in t.args) and deep: t = t._eval_expand_mul() args.append(t) return Add(*args) else: return plain @cacheit def _eval_derivative(self, s): args = list(self.args) terms = [] for i in range(len(args)): d = args[i].diff(s) if d: # Note: reduce is used in step of Mul as Mul is unable to # handle subtypes and operation priority: terms.append(reduce(lambda x, y: x*y, (args[:i] + [d] + args[i + 1:]), S.One)) return Add.fromiter(terms) @cacheit def _eval_derivative_n_times(self, s, n): from sympy import Integer, factorial, prod, Sum, Max from sympy.ntheory.multinomial import multinomial_coefficients_iterator from .function import AppliedUndef from .symbol import Symbol, symbols, Dummy if not isinstance(s, AppliedUndef) and not isinstance(s, Symbol): # other types of s may not be well behaved, e.g. # (cos(x)*sin(y)).diff([[x, y, z]]) return super()._eval_derivative_n_times(s, n) args = self.args m = len(args) if isinstance(n, (int, Integer)): # https://en.wikipedia.org/wiki/General_Leibniz_rule#More_than_two_factors terms = [] for kvals, c in multinomial_coefficients_iterator(m, n): p = prod([arg.diff((s, k)) for k, arg in zip(kvals, args)]) terms.append(c * p) return Add(*terms) kvals = symbols("k1:%i" % m, cls=Dummy) klast = n - sum(kvals) nfact = factorial(n) e, l = (# better to use the multinomial? nfact/prod(map(factorial, kvals))/factorial(klast)*\ prod([args[t].diff((s, kvals[t])) for t in range(m-1)])*\ args[-1].diff((s, Max(0, klast))), [(k, 0, n) for k in kvals]) return Sum(e, *l) def _eval_difference_delta(self, n, step): from sympy.series.limitseq import difference_delta as dd arg0 = self.args[0] rest = Mul(*self.args[1:]) return (arg0.subs(n, n + step) * dd(rest, n, step) + dd(arg0, n, step) * rest) def _matches_simple(self, expr, repl_dict): # handle (w*3).matches('x*5') -> {w: x*5/3} coeff, terms = self.as_coeff_Mul() terms = Mul.make_args(terms) if len(terms) == 1: newexpr = self.__class__._combine_inverse(expr, coeff) return terms[0].matches(newexpr, repl_dict) return def matches(self, expr, repl_dict={}, old=False): expr = sympify(expr) repl_dict = repl_dict.copy() if self.is_commutative and expr.is_commutative: return self._matches_commutative(expr, repl_dict, old) elif self.is_commutative is not expr.is_commutative: return None # Proceed only if both both expressions are non-commutative c1, nc1 = self.args_cnc() c2, nc2 = expr.args_cnc() c1, c2 = [c or [1] for c in [c1, c2]] # TODO: Should these be self.func? comm_mul_self = Mul(*c1) comm_mul_expr = Mul(*c2) repl_dict = comm_mul_self.matches(comm_mul_expr, repl_dict, old) # If the commutative arguments didn't match and aren't equal, then # then the expression as a whole doesn't match if repl_dict is None and c1 != c2: return None # Now match the non-commutative arguments, expanding powers to # multiplications nc1 = Mul._matches_expand_pows(nc1) nc2 = Mul._matches_expand_pows(nc2) repl_dict = Mul._matches_noncomm(nc1, nc2, repl_dict) return repl_dict or None @staticmethod def _matches_expand_pows(arg_list): new_args = [] for arg in arg_list: if arg.is_Pow and arg.exp > 0: new_args.extend([arg.base] * arg.exp) else: new_args.append(arg) return new_args @staticmethod def _matches_noncomm(nodes, targets, repl_dict={}): """Non-commutative multiplication matcher. `nodes` is a list of symbols within the matcher multiplication expression, while `targets` is a list of arguments in the multiplication expression being matched against. """ repl_dict = repl_dict.copy() # List of possible future states to be considered agenda = [] # The current matching state, storing index in nodes and targets state = (0, 0) node_ind, target_ind = state # Mapping between wildcard indices and the index ranges they match wildcard_dict = {} repl_dict = repl_dict.copy() while target_ind < len(targets) and node_ind < len(nodes): node = nodes[node_ind] if node.is_Wild: Mul._matches_add_wildcard(wildcard_dict, state) states_matches = Mul._matches_new_states(wildcard_dict, state, nodes, targets) if states_matches: new_states, new_matches = states_matches agenda.extend(new_states) if new_matches: for match in new_matches: repl_dict[match] = new_matches[match] if not agenda: return None else: state = agenda.pop() node_ind, target_ind = state return repl_dict @staticmethod def _matches_add_wildcard(dictionary, state): node_ind, target_ind = state if node_ind in dictionary: begin, end = dictionary[node_ind] dictionary[node_ind] = (begin, target_ind) else: dictionary[node_ind] = (target_ind, target_ind) @staticmethod def _matches_new_states(dictionary, state, nodes, targets): node_ind, target_ind = state node = nodes[node_ind] target = targets[target_ind] # Don't advance at all if we've exhausted the targets but not the nodes if target_ind >= len(targets) - 1 and node_ind < len(nodes) - 1: return None if node.is_Wild: match_attempt = Mul._matches_match_wilds(dictionary, node_ind, nodes, targets) if match_attempt: # If the same node has been matched before, don't return # anything if the current match is diverging from the previous # match other_node_inds = Mul._matches_get_other_nodes(dictionary, nodes, node_ind) for ind in other_node_inds: other_begin, other_end = dictionary[ind] curr_begin, curr_end = dictionary[node_ind] other_targets = targets[other_begin:other_end + 1] current_targets = targets[curr_begin:curr_end + 1] for curr, other in zip(current_targets, other_targets): if curr != other: return None # A wildcard node can match more than one target, so only the # target index is advanced new_state = [(node_ind, target_ind + 1)] # Only move on to the next node if there is one if node_ind < len(nodes) - 1: new_state.append((node_ind + 1, target_ind + 1)) return new_state, match_attempt else: # If we're not at a wildcard, then make sure we haven't exhausted # nodes but not targets, since in this case one node can only match # one target if node_ind >= len(nodes) - 1 and target_ind < len(targets) - 1: return None match_attempt = node.matches(target) if match_attempt: return [(node_ind + 1, target_ind + 1)], match_attempt elif node == target: return [(node_ind + 1, target_ind + 1)], None else: return None @staticmethod def _matches_match_wilds(dictionary, wildcard_ind, nodes, targets): """Determine matches of a wildcard with sub-expression in `target`.""" wildcard = nodes[wildcard_ind] begin, end = dictionary[wildcard_ind] terms = targets[begin:end + 1] # TODO: Should this be self.func? mul = Mul(*terms) if len(terms) > 1 else terms[0] return wildcard.matches(mul) @staticmethod def _matches_get_other_nodes(dictionary, nodes, node_ind): """Find other wildcards that may have already been matched.""" other_node_inds = [] for ind in dictionary: if nodes[ind] == nodes[node_ind]: other_node_inds.append(ind) return other_node_inds @staticmethod def _combine_inverse(lhs, rhs): """ Returns lhs/rhs, but treats arguments like symbols, so things like oo/oo return 1 (instead of a nan) and ``I`` behaves like a symbol instead of sqrt(-1). """ from .symbol import Dummy if lhs == rhs: return S.One def check(l, r): if l.is_Float and r.is_comparable: # if both objects are added to 0 they will share the same "normalization" # and are more likely to compare the same. Since Add(foo, 0) will not allow # the 0 to pass, we use __add__ directly. return l.__add__(0) == r.evalf().__add__(0) return False if check(lhs, rhs) or check(rhs, lhs): return S.One if any(i.is_Pow or i.is_Mul for i in (lhs, rhs)): # gruntz and limit wants a literal I to not combine # with a power of -1 d = Dummy('I') _i = {S.ImaginaryUnit: d} i_ = {d: S.ImaginaryUnit} a = lhs.xreplace(_i).as_powers_dict() b = rhs.xreplace(_i).as_powers_dict() blen = len(b) for bi in tuple(b.keys()): if bi in a: a[bi] -= b.pop(bi) if not a[bi]: a.pop(bi) if len(b) != blen: lhs = Mul(*[k**v for k, v in a.items()]).xreplace(i_) rhs = Mul(*[k**v for k, v in b.items()]).xreplace(i_) return lhs/rhs def as_powers_dict(self): d = defaultdict(int) for term in self.args: for b, e in term.as_powers_dict().items(): d[b] += e return d def as_numer_denom(self): # don't use _from_args to rebuild the numerators and denominators # as the order is not guaranteed to be the same once they have # been separated from each other numers, denoms = list(zip(*[f.as_numer_denom() for f in self.args])) return self.func(*numers), self.func(*denoms) def as_base_exp(self): e1 = None bases = [] nc = 0 for m in self.args: b, e = m.as_base_exp() if not b.is_commutative: nc += 1 if e1 is None: e1 = e elif e != e1 or nc > 1: return self, S.One bases.append(b) return self.func(*bases), e1 def _eval_is_polynomial(self, syms): return all(term._eval_is_polynomial(syms) for term in self.args) def _eval_is_rational_function(self, syms): return all(term._eval_is_rational_function(syms) for term in self.args) def _eval_is_meromorphic(self, x, a): return _fuzzy_group((arg.is_meromorphic(x, a) for arg in self.args), quick_exit=True) def _eval_is_algebraic_expr(self, syms): return all(term._eval_is_algebraic_expr(syms) for term in self.args) _eval_is_commutative = lambda self: _fuzzy_group( a.is_commutative for a in self.args) def _eval_is_complex(self): comp = _fuzzy_group(a.is_complex for a in self.args) if comp is False: if any(a.is_infinite for a in self.args): if any(a.is_zero is not False for a in self.args): return None return False return comp def _eval_is_finite(self): if all(a.is_finite for a in self.args): return True if any(a.is_infinite for a in self.args): if all(a.is_zero is False for a in self.args): return False def _eval_is_infinite(self): if any(a.is_infinite for a in self.args): if any(a.is_zero for a in self.args): return S.NaN.is_infinite if any(a.is_zero is None for a in self.args): return None return True def _eval_is_rational(self): r = _fuzzy_group((a.is_rational for a in self.args), quick_exit=True) if r: return r elif r is False: return self.is_zero def _eval_is_algebraic(self): r = _fuzzy_group((a.is_algebraic for a in self.args), quick_exit=True) if r: return r elif r is False: return self.is_zero def _eval_is_zero(self): zero = infinite = False for a in self.args: z = a.is_zero if z: if infinite: return # 0*oo is nan and nan.is_zero is None zero = True else: if not a.is_finite: if zero: return # 0*oo is nan and nan.is_zero is None infinite = True if zero is False and z is None: # trap None zero = None return zero def _eval_is_integer(self): from sympy import fraction from sympy.core.numbers import Float is_rational = self._eval_is_rational() if is_rational is False: return False # use exact=True to avoid recomputing num or den n, d = fraction(self, exact=True) if is_rational: if d is S.One: return True if d.is_even: if d.is_prime: # literal or symbolic 2 return n.is_even if n.is_odd: return False # true even if d = 0 if n == d: return fuzzy_and([not bool(self.atoms(Float)), fuzzy_not(d.is_zero)]) def _eval_is_polar(self): has_polar = any(arg.is_polar for arg in self.args) return has_polar and \ all(arg.is_polar or arg.is_positive for arg in self.args) def _eval_is_extended_real(self): return self._eval_real_imag(True) def _eval_real_imag(self, real): zero = False t_not_re_im = None for t in self.args: if (t.is_complex or t.is_infinite) is False and t.is_extended_real is False: return False elif t.is_imaginary: # I real = not real elif t.is_extended_real: # 2 if not zero: z = t.is_zero if not z and zero is False: zero = z elif z: if all(a.is_finite for a in self.args): return True return elif t.is_extended_real is False: # symbolic or literal like `2 + I` or symbolic imaginary if t_not_re_im: return # complex terms might cancel t_not_re_im = t elif t.is_imaginary is False: # symbolic like `2` or `2 + I` if t_not_re_im: return # complex terms might cancel t_not_re_im = t else: return if t_not_re_im: if t_not_re_im.is_extended_real is False: if real: # like 3 return zero # 3*(smthng like 2 + I or i) is not real if t_not_re_im.is_imaginary is False: # symbolic 2 or 2 + I if not real: # like I return zero # I*(smthng like 2 or 2 + I) is not real elif zero is False: return real # can't be trumped by 0 elif real: return real # doesn't matter what zero is def _eval_is_imaginary(self): z = self.is_zero if z: return False if self.is_finite is False: return False elif z is False and self.is_finite is True: return self._eval_real_imag(False) def _eval_is_hermitian(self): return self._eval_herm_antiherm(True) def _eval_herm_antiherm(self, real): one_nc = zero = one_neither = False for t in self.args: if not t.is_commutative: if one_nc: return one_nc = True if t.is_antihermitian: real = not real elif t.is_hermitian: if not zero: z = t.is_zero if not z and zero is False: zero = z elif z: if all(a.is_finite for a in self.args): return True return elif t.is_hermitian is False: if one_neither: return one_neither = True else: return if one_neither: if real: return zero elif zero is False or real: return real def _eval_is_antihermitian(self): z = self.is_zero if z: return False elif z is False: return self._eval_herm_antiherm(False) def _eval_is_irrational(self): for t in self.args: a = t.is_irrational if a: others = list(self.args) others.remove(t) if all((x.is_rational and fuzzy_not(x.is_zero)) is True for x in others): return True return if a is None: return if all(x.is_real for x in self.args): return False def _eval_is_extended_positive(self): """Return True if self is positive, False if not, and None if it cannot be determined. Explanation =========== This algorithm is non-recursive and works by keeping track of the sign which changes when a negative or nonpositive is encountered. Whether a nonpositive or nonnegative is seen is also tracked since the presence of these makes it impossible to return True, but possible to return False if the end result is nonpositive. e.g. pos * neg * nonpositive -> pos or zero -> None is returned pos * neg * nonnegative -> neg or zero -> False is returned """ return self._eval_pos_neg(1) def _eval_pos_neg(self, sign): saw_NON = saw_NOT = False for t in self.args: if t.is_extended_positive: continue elif t.is_extended_negative: sign = -sign elif t.is_zero: if all(a.is_finite for a in self.args): return False return elif t.is_extended_nonpositive: sign = -sign saw_NON = True elif t.is_extended_nonnegative: saw_NON = True # FIXME: is_positive/is_negative is False doesn't take account of # Symbol('x', infinite=True, extended_real=True) which has # e.g. is_positive is False but has uncertain sign. elif t.is_positive is False: sign = -sign if saw_NOT: return saw_NOT = True elif t.is_negative is False: if saw_NOT: return saw_NOT = True else: return if sign == 1 and saw_NON is False and saw_NOT is False: return True if sign < 0: return False def _eval_is_extended_negative(self): return self._eval_pos_neg(-1) def _eval_is_odd(self): is_integer = self.is_integer if is_integer: r, acc = True, 1 for t in self.args: if not t.is_integer: return None elif t.is_even: r = False elif t.is_integer: if r is False: pass elif acc != 1 and (acc + t).is_odd: r = False elif t.is_odd is None: r = None acc = t return r # !integer -> !odd elif is_integer is False: return False def _eval_is_even(self): is_integer = self.is_integer if is_integer: return fuzzy_not(self.is_odd) elif is_integer is False: return False def _eval_is_composite(self): """ Here we count the number of arguments that have a minimum value greater than two. If there are more than one of such a symbol then the result is composite. Else, the result cannot be determined. """ number_of_args = 0 # count of symbols with minimum value greater than one for arg in self.args: if not (arg.is_integer and arg.is_positive): return None if (arg-1).is_positive: number_of_args += 1 if number_of_args > 1: return True def _eval_subs(self, old, new): from sympy.functions.elementary.complexes import sign from sympy.ntheory.factor_ import multiplicity from sympy.simplify.powsimp import powdenest from sympy.simplify.radsimp import fraction if not old.is_Mul: return None # try keep replacement literal so -2*x doesn't replace 4*x if old.args[0].is_Number and old.args[0] < 0: if self.args[0].is_Number: if self.args[0] < 0: return self._subs(-old, -new) return None def base_exp(a): # if I and -1 are in a Mul, they get both end up with # a -1 base (see issue 6421); all we want here are the # true Pow or exp separated into base and exponent from sympy import exp if a.is_Pow or isinstance(a, exp): return a.as_base_exp() return a, S.One def breakup(eq): """break up powers of eq when treated as a Mul: b**(Rational*e) -> b**e, Rational commutatives come back as a dictionary {b**e: Rational} noncommutatives come back as a list [(b**e, Rational)] """ (c, nc) = (defaultdict(int), list()) for a in Mul.make_args(eq): a = powdenest(a) (b, e) = base_exp(a) if e is not S.One: (co, _) = e.as_coeff_mul() b = Pow(b, e/co) e = co if a.is_commutative: c[b] += e else: nc.append([b, e]) return (c, nc) def rejoin(b, co): """ Put rational back with exponent; in general this is not ok, but since we took it from the exponent for analysis, it's ok to put it back. """ (b, e) = base_exp(b) return Pow(b, e*co) def ndiv(a, b): """if b divides a in an extractive way (like 1/4 divides 1/2 but not vice versa, and 2/5 does not divide 1/3) then return the integer number of times it divides, else return 0. """ if not b.q % a.q or not a.q % b.q: return int(a/b) return 0 # give Muls in the denominator a chance to be changed (see issue 5651) # rv will be the default return value rv = None n, d = fraction(self) self2 = self if d is not S.One: self2 = n._subs(old, new)/d._subs(old, new) if not self2.is_Mul: return self2._subs(old, new) if self2 != self: rv = self2 # Now continue with regular substitution. # handle the leading coefficient and use it to decide if anything # should even be started; we always know where to find the Rational # so it's a quick test co_self = self2.args[0] co_old = old.args[0] co_xmul = None if co_old.is_Rational and co_self.is_Rational: # if coeffs are the same there will be no updating to do # below after breakup() step; so skip (and keep co_xmul=None) if co_old != co_self: co_xmul = co_self.extract_multiplicatively(co_old) elif co_old.is_Rational: return rv # break self and old into factors (c, nc) = breakup(self2) (old_c, old_nc) = breakup(old) # update the coefficients if we had an extraction # e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5 # then co_self in c is replaced by (3/5)**2 and co_residual # is 2*(1/7)**2 if co_xmul and co_xmul.is_Rational and abs(co_old) != 1: mult = S(multiplicity(abs(co_old), co_self)) c.pop(co_self) if co_old in c: c[co_old] += mult else: c[co_old] = mult co_residual = co_self/co_old**mult else: co_residual = 1 # do quick tests to see if we can't succeed ok = True if len(old_nc) > len(nc): # more non-commutative terms ok = False elif len(old_c) > len(c): # more commutative terms ok = False elif {i[0] for i in old_nc}.difference({i[0] for i in nc}): # unmatched non-commutative bases ok = False elif set(old_c).difference(set(c)): # unmatched commutative terms ok = False elif any(sign(c[b]) != sign(old_c[b]) for b in old_c): # differences in sign ok = False if not ok: return rv if not old_c: cdid = None else: rat = [] for (b, old_e) in old_c.items(): c_e = c[b] rat.append(ndiv(c_e, old_e)) if not rat[-1]: return rv cdid = min(rat) if not old_nc: ncdid = None for i in range(len(nc)): nc[i] = rejoin(*nc[i]) else: ncdid = 0 # number of nc replacements we did take = len(old_nc) # how much to look at each time limit = cdid or S.Infinity # max number that we can take failed = [] # failed terms will need subs if other terms pass i = 0 while limit and i + take <= len(nc): hit = False # the bases must be equivalent in succession, and # the powers must be extractively compatible on the # first and last factor but equal in between. rat = [] for j in range(take): if nc[i + j][0] != old_nc[j][0]: break elif j == 0: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif j == take - 1: rat.append(ndiv(nc[i + j][1], old_nc[j][1])) elif nc[i + j][1] != old_nc[j][1]: break else: rat.append(1) j += 1 else: ndo = min(rat) if ndo: if take == 1: if cdid: ndo = min(cdid, ndo) nc[i] = Pow(new, ndo)*rejoin(nc[i][0], nc[i][1] - ndo*old_nc[0][1]) else: ndo = 1 # the left residual l = rejoin(nc[i][0], nc[i][1] - ndo* old_nc[0][1]) # eliminate all middle terms mid = new # the right residual (which may be the same as the middle if take == 2) ir = i + take - 1 r = (nc[ir][0], nc[ir][1] - ndo* old_nc[-1][1]) if r[1]: if i + take < len(nc): nc[i:i + take] = [l*mid, r] else: r = rejoin(*r) nc[i:i + take] = [l*mid*r] else: # there was nothing left on the right nc[i:i + take] = [l*mid] limit -= ndo ncdid += ndo hit = True if not hit: # do the subs on this failing factor failed.append(i) i += 1 else: if not ncdid: return rv # although we didn't fail, certain nc terms may have # failed so we rebuild them after attempting a partial # subs on them failed.extend(range(i, len(nc))) for i in failed: nc[i] = rejoin(*nc[i]).subs(old, new) # rebuild the expression if cdid is None: do = ncdid elif ncdid is None: do = cdid else: do = min(ncdid, cdid) margs = [] for b in c: if b in old_c: # calculate the new exponent e = c[b] - old_c[b]*do margs.append(rejoin(b, e)) else: margs.append(rejoin(b.subs(old, new), c[b])) if cdid and not ncdid: # in case we are replacing commutative with non-commutative, # we want the new term to come at the front just like the # rest of this routine margs = [Pow(new, cdid)] + margs return co_residual*self2.func(*margs)*self2.func(*nc) def _eval_nseries(self, x, n, logx, cdir=0): from sympy import degree, Mul, Order, ceiling, powsimp, PolynomialError from itertools import product def coeff_exp(term, x): coeff, exp = S.One, S.Zero for factor in Mul.make_args(term): if factor.has(x): base, exp = factor.as_base_exp() if base != x: try: return term.leadterm(x) except ValueError: return term, S.Zero else: coeff *= factor return coeff, exp ords = [] try: for t in self.args: coeff, exp = t.leadterm(x) if not coeff.has(x): ords.append((t, exp)) else: raise ValueError n0 = sum(t[1] for t in ords) facs = [] for t, m in ords: n1 = ceiling(n - n0 + m) s = t.nseries(x, n=n1, logx=logx, cdir=cdir) ns = s.getn() if ns is not None: if ns < n1: # less than expected n -= n1 - ns # reduce n facs.append(s.removeO()) except (ValueError, NotImplementedError, TypeError, AttributeError): facs = [t.nseries(x, n=n, logx=logx, cdir=cdir) for t in self.args] res = powsimp(self.func(*facs).expand(), combine='exp', deep=True) if res.has(Order): res += Order(x**n, x) return res res = 0 ords2 = [Add.make_args(factor) for factor in facs] for fac in product(*ords2): ords3 = [coeff_exp(term, x) for term in fac] coeffs, powers = zip(*ords3) power = sum(powers) if power < n: res += Mul(*coeffs)*(x**power) if self.is_polynomial(x): try: if degree(self, x) != degree(res, x): res += Order(x**n, x) except PolynomialError: pass else: return res for i in (1, 2, 3): if (res - self).subs(x, i) is not S.Zero: res += Order(x**n, x) break return res def _eval_as_leading_term(self, x, cdir=0): return self.func(*[t.as_leading_term(x, cdir=cdir) for t in self.args]) def _eval_conjugate(self): return self.func(*[t.conjugate() for t in self.args]) def _eval_transpose(self): return self.func(*[t.transpose() for t in self.args[::-1]]) def _eval_adjoint(self): return self.func(*[t.adjoint() for t in self.args[::-1]]) def _sage_(self): s = 1 for x in self.args: s *= x._sage_() return s def as_content_primitive(self, radical=False, clear=True): """Return the tuple (R, self/R) where R is the positive Rational extracted from self. Examples ======== >>> from sympy import sqrt >>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive() (6, -sqrt(2)*(1 - sqrt(2))) See docstring of Expr.as_content_primitive for more examples. """ coef = S.One args = [] for i, a in enumerate(self.args): c, p = a.as_content_primitive(radical=radical, clear=clear) coef *= c if p is not S.One: args.append(p) # don't use self._from_args here to reconstruct args # since there may be identical args now that should be combined # e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x)) return coef, self.func(*args) def as_ordered_factors(self, order=None): """Transform an expression into an ordered list of factors. Examples ======== >>> from sympy import sin, cos >>> from sympy.abc import x, y >>> (2*x*y*sin(x)*cos(x)).as_ordered_factors() [2, x, y, sin(x), cos(x)] """ cpart, ncpart = self.args_cnc() cpart.sort(key=lambda expr: expr.sort_key(order=order)) return cpart + ncpart @property def _sorted_args(self): return tuple(self.as_ordered_factors()) mul = AssocOpDispatcher('mul') def prod(a, start=1): """Return product of elements of a. Start with int 1 so if only ints are included then an int result is returned. Examples ======== >>> from sympy import prod, S >>> prod(range(3)) 0 >>> type(_) is int True >>> prod([S(2), 3]) 6 >>> _.is_Integer True You can start the product at something other than 1: >>> prod([1, 2], 3) 6 """ return reduce(operator.mul, a, start) def _keep_coeff(coeff, factors, clear=True, sign=False): """Return ``coeff*factors`` unevaluated if necessary. If ``clear`` is False, do not keep the coefficient as a factor if it can be distributed on a single factor such that one or more terms will still have integer coefficients. If ``sign`` is True, allow a coefficient of -1 to remain factored out. Examples ======== >>> from sympy.core.mul import _keep_coeff >>> from sympy.abc import x, y >>> from sympy import S >>> _keep_coeff(S.Half, x + 2) (x + 2)/2 >>> _keep_coeff(S.Half, x + 2, clear=False) x/2 + 1 >>> _keep_coeff(S.Half, (x + 2)*y, clear=False) y*(x + 2)/2 >>> _keep_coeff(S(-1), x + y) -x - y >>> _keep_coeff(S(-1), x + y, sign=True) -(x + y) """ if not coeff.is_Number: if factors.is_Number: factors, coeff = coeff, factors else: return coeff*factors if coeff is S.One: return factors elif coeff is S.NegativeOne and not sign: return -factors elif factors.is_Add: if not clear and coeff.is_Rational and coeff.q != 1: q = S(coeff.q) for i in factors.args: c, t = i.as_coeff_Mul() r = c/q if r == int(r): return coeff*factors return Mul(coeff, factors, evaluate=False) elif factors.is_Mul: margs = list(factors.args) if margs[0].is_Number: margs[0] *= coeff if margs[0] == 1: margs.pop(0) else: margs.insert(0, coeff) return Mul._from_args(margs) else: return coeff*factors def expand_2arg(e): from sympy.simplify.simplify import bottom_up def do(e): if e.is_Mul: c, r = e.as_coeff_Mul() if c.is_Number and r.is_Add: return _unevaluated_Add(*[c*ri for ri in r.args]) return e return bottom_up(e, do) from .numbers import Rational from .power import Pow from .add import Add, _addsort, _unevaluated_Add
9a07c5a69f52535e7db40fcf336a34e16672ccca167ceb5666e8117079cfe367
"""Configuration utilities for polynomial manipulation algorithms. """ from contextlib import contextmanager _default_config = { 'USE_COLLINS_RESULTANT': False, 'USE_SIMPLIFY_GCD': True, 'USE_HEU_GCD': True, 'USE_IRREDUCIBLE_IN_FACTOR': False, 'USE_CYCLOTOMIC_FACTOR': True, 'EEZ_RESTART_IF_NEEDED': True, 'EEZ_NUMBER_OF_CONFIGS': 3, 'EEZ_NUMBER_OF_TRIES': 5, 'EEZ_MODULUS_STEP': 2, 'GF_IRRED_METHOD': 'rabin', 'GF_FACTOR_METHOD': 'zassenhaus', 'GROEBNER': 'buchberger', } _current_config = {} @contextmanager def using(**kwargs): for k, v in kwargs.items(): setup(k, v) yield for k in kwargs.keys(): setup(k) def setup(key, value=None): """Assign a value to (or reset) a configuration item. """ key = key.upper() if value is not None: _current_config[key] = value else: _current_config[key] = _default_config[key] def query(key): """Ask for a value of the given configuration item. """ return _current_config.get(key.upper(), None) def configure(): """Initialized configuration of polys module. """ from os import getenv for key, default in _default_config.items(): value = getenv('SYMPY_' + key) if value is not None: try: _current_config[key] = eval(value) except NameError: _current_config[key] = value else: _current_config[key] = default configure()
3f39dfc54487672cc5fe769ae1780ab0d6702c9ce433bfb62a899a845a61a326
from sympy.matrices.dense import MutableDenseMatrix from sympy.polys.polytools import Poly from sympy.polys.domains import EX class MutablePolyDenseMatrix(MutableDenseMatrix): """ A mutable matrix of objects from poly module or to operate with them. Examples ======== >>> from sympy.polys.polymatrix import PolyMatrix >>> from sympy import Symbol, Poly, ZZ >>> x = Symbol('x') >>> pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(x**3, x), Poly(-1 + x, x)]]) >>> v1 = PolyMatrix([[1, 0], [-1, 0]]) >>> pm1*v1 Matrix([ [ Poly(x**2 + x, x, domain='ZZ'), Poly(0, x, domain='ZZ')], [Poly(x**3 - x + 1, x, domain='ZZ'), Poly(0, x, domain='ZZ')]]) >>> pm1.ring ZZ[x] >>> v1*pm1 Matrix([ [ Poly(x**2, x, domain='ZZ'), Poly(-x, x, domain='ZZ')], [Poly(-x**2, x, domain='ZZ'), Poly(x, x, domain='ZZ')]]) >>> pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(1, x, domain='QQ'), \ Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]]) >>> v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring=ZZ) >>> v2.ring ZZ >>> pm2*v2 Matrix([[Poly(x**2, x, domain='QQ')]]) """ _class_priority = 10 # we don't want to sympify the elements of PolyMatrix _sympify = staticmethod(lambda x: x) def __init__(self, *args, ring=EX, **kwargs): # kwargs are consumed by __new__, so need to be allowed here. # if any non-Poly element is given as input then # 'ring' defaults 'EX' if all(isinstance(p, Poly) for p in self._mat) and self._mat: domain = tuple([p.domain[p.gens] for p in self._mat]) ring = domain[0] for i in range(1, len(domain)): ring = ring.unify(domain[i]) self.ring = ring def _eval_matrix_mul(self, other): self_cols = self.cols other_rows, other_cols = other.rows, other.cols other_len = other_rows*other_cols new_mat_rows = self.rows new_mat_cols = other.cols new_mat = [0]*new_mat_rows*new_mat_cols if self.cols != 0 and other.rows != 0: mat = self._mat other_mat = other._mat for i in range(len(new_mat)): row, col = i // new_mat_cols, i % new_mat_cols row_indices = range(self_cols*row, self_cols*(row+1)) col_indices = range(col, other_len, other_cols) vec = (mat[a]*other_mat[b] for a,b in zip(row_indices, col_indices)) # 'Add' shouldn't be used here new_mat[i] = sum(vec) return self.__class__(new_mat_rows, new_mat_cols, new_mat, copy=False) def _eval_scalar_mul(self, other): mat = [Poly(a.as_expr()*other, *a.gens) if isinstance(a, Poly) else a*other for a in self._mat] return self.__class__(self.rows, self.cols, mat, copy=False) def _eval_scalar_rmul(self, other): mat = [Poly(other*a.as_expr(), *a.gens) if isinstance(a, Poly) else other*a for a in self._mat] return self.__class__(self.rows, self.cols, mat, copy=False) MutablePolyMatrix = PolyMatrix = MutablePolyDenseMatrix
a947c5b77e6aebb0ab797e54f18d0f6fbfd07d931d356a934bbfa20651d74291
"""Definitions of monomial orderings. """ from typing import Optional __all__ = ["lex", "grlex", "grevlex", "ilex", "igrlex", "igrevlex"] from sympy.core import Symbol from sympy.core.compatibility import iterable class MonomialOrder: """Base class for monomial orderings. """ alias = None # type: Optional[str] is_global = None # type: Optional[bool] is_default = False def __repr__(self): return self.__class__.__name__ + "()" def __str__(self): return self.alias def __call__(self, monomial): raise NotImplementedError def __eq__(self, other): return self.__class__ == other.__class__ def __hash__(self): return hash(self.__class__) def __ne__(self, other): return not (self == other) class LexOrder(MonomialOrder): """Lexicographic order of monomials. """ alias = 'lex' is_global = True is_default = True def __call__(self, monomial): return monomial class GradedLexOrder(MonomialOrder): """Graded lexicographic order of monomials. """ alias = 'grlex' is_global = True def __call__(self, monomial): return (sum(monomial), monomial) class ReversedGradedLexOrder(MonomialOrder): """Reversed graded lexicographic order of monomials. """ alias = 'grevlex' is_global = True def __call__(self, monomial): return (sum(monomial), tuple(reversed([-m for m in monomial]))) class ProductOrder(MonomialOrder): """ A product order built from other monomial orders. Given (not necessarily total) orders O1, O2, ..., On, their product order P is defined as M1 > M2 iff there exists i such that O1(M1) = O2(M2), ..., Oi(M1) = Oi(M2), O{i+1}(M1) > O{i+1}(M2). Product orders are typically built from monomial orders on different sets of variables. ProductOrder is constructed by passing a list of pairs [(O1, L1), (O2, L2), ...] where Oi are MonomialOrders and Li are callables. Upon comparison, the Li are passed the total monomial, and should filter out the part of the monomial to pass to Oi. Examples ======== We can use a lexicographic order on x_1, x_2 and also on y_1, y_2, y_3, and their product on {x_i, y_i} as follows: >>> from sympy.polys.orderings import lex, grlex, ProductOrder >>> P = ProductOrder( ... (lex, lambda m: m[:2]), # lex order on x_1 and x_2 of monomial ... (grlex, lambda m: m[2:]) # grlex on y_1, y_2, y_3 ... ) >>> P((2, 1, 1, 0, 0)) > P((1, 10, 0, 2, 0)) True Here the exponent `2` of `x_1` in the first monomial (`x_1^2 x_2 y_1`) is bigger than the exponent `1` of `x_1` in the second monomial (`x_1 x_2^10 y_2^2`), so the first monomial is greater in the product ordering. >>> P((2, 1, 1, 0, 0)) < P((2, 1, 0, 2, 0)) True Here the exponents of `x_1` and `x_2` agree, so the grlex order on `y_1, y_2, y_3` is used to decide the ordering. In this case the monomial `y_2^2` is ordered larger than `y_1`, since for the grlex order the degree of the monomial is most important. """ def __init__(self, *args): self.args = args def __call__(self, monomial): return tuple(O(lamda(monomial)) for (O, lamda) in self.args) def __repr__(self): contents = [repr(x[0]) for x in self.args] return self.__class__.__name__ + '(' + ", ".join(contents) + ')' def __str__(self): contents = [str(x[0]) for x in self.args] return self.__class__.__name__ + '(' + ", ".join(contents) + ')' def __eq__(self, other): if not isinstance(other, ProductOrder): return False return self.args == other.args def __hash__(self): return hash((self.__class__, self.args)) @property def is_global(self): if all(o.is_global is True for o, _ in self.args): return True if all(o.is_global is False for o, _ in self.args): return False return None class InverseOrder(MonomialOrder): """ The "inverse" of another monomial order. If O is any monomial order, we can construct another monomial order iO such that `A >_{iO} B` if and only if `B >_O A`. This is useful for constructing local orders. Note that many algorithms only work with *global* orders. For example, in the inverse lexicographic order on a single variable `x`, high powers of `x` count as small: >>> from sympy.polys.orderings import lex, InverseOrder >>> ilex = InverseOrder(lex) >>> ilex((5,)) < ilex((0,)) True """ def __init__(self, O): self.O = O def __str__(self): return "i" + str(self.O) def __call__(self, monomial): def inv(l): if iterable(l): return tuple(inv(x) for x in l) return -l return inv(self.O(monomial)) @property def is_global(self): if self.O.is_global is True: return False if self.O.is_global is False: return True return None def __eq__(self, other): return isinstance(other, InverseOrder) and other.O == self.O def __hash__(self): return hash((self.__class__, self.O)) lex = LexOrder() grlex = GradedLexOrder() grevlex = ReversedGradedLexOrder() ilex = InverseOrder(lex) igrlex = InverseOrder(grlex) igrevlex = InverseOrder(grevlex) _monomial_key = { 'lex': lex, 'grlex': grlex, 'grevlex': grevlex, 'ilex': ilex, 'igrlex': igrlex, 'igrevlex': igrevlex } def monomial_key(order=None, gens=None): """ Return a function defining admissible order on monomials. The result of a call to :func:`monomial_key` is a function which should be used as a key to :func:`sorted` built-in function, to provide order in a set of monomials of the same length. Currently supported monomial orderings are: 1. lex - lexicographic order (default) 2. grlex - graded lexicographic order 3. grevlex - reversed graded lexicographic order 4. ilex, igrlex, igrevlex - the corresponding inverse orders If the ``order`` input argument is not a string but has ``__call__`` attribute, then it will pass through with an assumption that the callable object defines an admissible order on monomials. If the ``gens`` input argument contains a list of generators, the resulting key function can be used to sort SymPy ``Expr`` objects. """ if order is None: order = lex if isinstance(order, Symbol): order = str(order) if isinstance(order, str): try: order = _monomial_key[order] except KeyError: raise ValueError("supported monomial orderings are 'lex', 'grlex' and 'grevlex', got %r" % order) if hasattr(order, '__call__'): if gens is not None: def _order(expr): return order(expr.as_poly(*gens).degree_list()) return _order return order else: raise ValueError("monomial ordering specification must be a string or a callable, got %s" % order) class _ItemGetter: """Helper class to return a subsequence of values.""" def __init__(self, seq): self.seq = tuple(seq) def __call__(self, m): return tuple(m[idx] for idx in self.seq) def __eq__(self, other): if not isinstance(other, _ItemGetter): return False return self.seq == other.seq def build_product_order(arg, gens): """ Build a monomial order on ``gens``. ``arg`` should be a tuple of iterables. The first element of each iterable should be a string or monomial order (will be passed to monomial_key), the others should be subsets of the generators. This function will build the corresponding product order. For example, build a product of two grlex orders: >>> from sympy.polys.orderings import build_product_order >>> from sympy.abc import x, y, z, t >>> O = build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) >>> O((1, 2, 3, 4)) ((3, (1, 2)), (7, (3, 4))) """ gens2idx = {} for i, g in enumerate(gens): gens2idx[g] = i order = [] for expr in arg: name = expr[0] var = expr[1:] def makelambda(var): return _ItemGetter(gens2idx[g] for g in var) order.append((monomial_key(name), makelambda(var))) return ProductOrder(*order)
ee1cb206a22e6f650bbfe0482b59c0766d1807679cc074509bf09dd6cb3dea4f
"""OO layer for several polynomial representations. """ from sympy import oo from sympy.core.sympify import CantSympify from sympy.polys.polyerrors import CoercionFailed, NotReversible, NotInvertible from sympy.polys.polyutils import PicklableWithSlots class GenericPoly(PicklableWithSlots): """Base class for low-level polynomial representations. """ def ground_to_ring(f): """Make the ground domain a ring. """ return f.set_domain(f.dom.get_ring()) def ground_to_field(f): """Make the ground domain a field. """ return f.set_domain(f.dom.get_field()) def ground_to_exact(f): """Make the ground domain exact. """ return f.set_domain(f.dom.get_exact()) @classmethod def _perify_factors(per, result, include): if include: coeff, factors = result else: coeff = result factors = [ (per(g), k) for g, k in factors ] if include: return coeff, factors else: return factors from sympy.polys.densebasic import ( dmp_validate, dup_normal, dmp_normal, dup_convert, dmp_convert, dmp_from_sympy, dup_strip, dup_degree, dmp_degree_in, dmp_degree_list, dmp_negative_p, dup_LC, dmp_ground_LC, dup_TC, dmp_ground_TC, dmp_ground_nth, dmp_one, dmp_ground, dmp_zero_p, dmp_one_p, dmp_ground_p, dup_from_dict, dmp_from_dict, dmp_to_dict, dmp_deflate, dmp_inject, dmp_eject, dmp_terms_gcd, dmp_list_terms, dmp_exclude, dmp_slice_in, dmp_permute, dmp_to_tuple,) from sympy.polys.densearith import ( dmp_add_ground, dmp_sub_ground, dmp_mul_ground, dmp_quo_ground, dmp_exquo_ground, dmp_abs, dup_neg, dmp_neg, dup_add, dmp_add, dup_sub, dmp_sub, dup_mul, dmp_mul, dmp_sqr, dup_pow, dmp_pow, dmp_pdiv, dmp_prem, dmp_pquo, dmp_pexquo, dmp_div, dup_rem, dmp_rem, dmp_quo, dmp_exquo, dmp_add_mul, dmp_sub_mul, dmp_max_norm, dmp_l1_norm) from sympy.polys.densetools import ( dmp_clear_denoms, dmp_integrate_in, dmp_diff_in, dmp_eval_in, dup_revert, dmp_ground_trunc, dmp_ground_content, dmp_ground_primitive, dmp_ground_monic, dmp_compose, dup_decompose, dup_shift, dup_transform, dmp_lift) from sympy.polys.euclidtools import ( dup_half_gcdex, dup_gcdex, dup_invert, dmp_subresultants, dmp_resultant, dmp_discriminant, dmp_inner_gcd, dmp_gcd, dmp_lcm, dmp_cancel) from sympy.polys.sqfreetools import ( dup_gff_list, dmp_norm, dmp_sqf_p, dmp_sqf_norm, dmp_sqf_part, dmp_sqf_list, dmp_sqf_list_include) from sympy.polys.factortools import ( dup_cyclotomic_p, dmp_irreducible_p, dmp_factor_list, dmp_factor_list_include) from sympy.polys.rootisolation import ( dup_isolate_real_roots_sqf, dup_isolate_real_roots, dup_isolate_all_roots_sqf, dup_isolate_all_roots, dup_refine_real_root, dup_count_real_roots, dup_count_complex_roots, dup_sturm) from sympy.polys.polyerrors import ( UnificationFailed, PolynomialError) def init_normal_DMP(rep, lev, dom): return DMP(dmp_normal(rep, lev, dom), dom, lev) class DMP(PicklableWithSlots, CantSympify): """Dense Multivariate Polynomials over `K`. """ __slots__ = ('rep', 'lev', 'dom', 'ring') def __init__(self, rep, dom, lev=None, ring=None): if lev is not None: if type(rep) is dict: rep = dmp_from_dict(rep, lev, dom) elif type(rep) is not list: rep = dmp_ground(dom.convert(rep), lev) else: rep, lev = dmp_validate(rep) self.rep = rep self.lev = lev self.dom = dom self.ring = ring def __repr__(f): return "%s(%s, %s, %s)" % (f.__class__.__name__, f.rep, f.dom, f.ring) def __hash__(f): return hash((f.__class__.__name__, f.to_tuple(), f.lev, f.dom, f.ring)) def unify(f, g): """Unify representations of two multivariate polynomials. """ if not isinstance(g, DMP) or f.lev != g.lev: raise UnificationFailed("can't unify %s with %s" % (f, g)) if f.dom == g.dom and f.ring == g.ring: return f.lev, f.dom, f.per, f.rep, g.rep else: lev, dom = f.lev, f.dom.unify(g.dom) ring = f.ring if g.ring is not None: if ring is not None: ring = ring.unify(g.ring) else: ring = g.ring F = dmp_convert(f.rep, lev, f.dom, dom) G = dmp_convert(g.rep, lev, g.dom, dom) def per(rep, dom=dom, lev=lev, kill=False): if kill: if not lev: return rep else: lev -= 1 return DMP(rep, dom, lev, ring) return lev, dom, per, F, G def per(f, rep, dom=None, kill=False, ring=None): """Create a DMP out of the given representation. """ lev = f.lev if kill: if not lev: return rep else: lev -= 1 if dom is None: dom = f.dom if ring is None: ring = f.ring return DMP(rep, dom, lev, ring) @classmethod def zero(cls, lev, dom, ring=None): return DMP(0, dom, lev, ring) @classmethod def one(cls, lev, dom, ring=None): return DMP(1, dom, lev, ring) @classmethod def from_list(cls, rep, lev, dom): """Create an instance of ``cls`` given a list of native coefficients. """ return cls(dmp_convert(rep, lev, None, dom), dom, lev) @classmethod def from_sympy_list(cls, rep, lev, dom): """Create an instance of ``cls`` given a list of SymPy coefficients. """ return cls(dmp_from_sympy(rep, lev, dom), dom, lev) def to_dict(f, zero=False): """Convert ``f`` to a dict representation with native coefficients. """ return dmp_to_dict(f.rep, f.lev, f.dom, zero=zero) def to_sympy_dict(f, zero=False): """Convert ``f`` to a dict representation with SymPy coefficients. """ rep = dmp_to_dict(f.rep, f.lev, f.dom, zero=zero) for k, v in rep.items(): rep[k] = f.dom.to_sympy(v) return rep def to_list(f): """Convert ``f`` to a list representation with native coefficients. """ return f.rep def to_sympy_list(f): """Convert ``f`` to a list representation with SymPy coefficients. """ def sympify_nested_list(rep): out = [] for val in rep: if isinstance(val, list): out.append(sympify_nested_list(val)) else: out.append(f.dom.to_sympy(val)) return out return sympify_nested_list(f.rep) def to_tuple(f): """ Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. """ return dmp_to_tuple(f.rep, f.lev) @classmethod def from_dict(cls, rep, lev, dom): """Construct and instance of ``cls`` from a ``dict`` representation. """ return cls(dmp_from_dict(rep, lev, dom), dom, lev) @classmethod def from_monoms_coeffs(cls, monoms, coeffs, lev, dom, ring=None): return DMP(dict(list(zip(monoms, coeffs))), dom, lev, ring) def to_ring(f): """Make the ground domain a ring. """ return f.convert(f.dom.get_ring()) def to_field(f): """Make the ground domain a field. """ return f.convert(f.dom.get_field()) def to_exact(f): """Make the ground domain exact. """ return f.convert(f.dom.get_exact()) def convert(f, dom): """Convert the ground domain of ``f``. """ if f.dom == dom: return f else: return DMP(dmp_convert(f.rep, f.lev, f.dom, dom), dom, f.lev) def slice(f, m, n, j=0): """Take a continuous subsequence of terms of ``f``. """ return f.per(dmp_slice_in(f.rep, m, n, j, f.lev, f.dom)) def coeffs(f, order=None): """Returns all non-zero coefficients from ``f`` in lex order. """ return [ c for _, c in dmp_list_terms(f.rep, f.lev, f.dom, order=order) ] def monoms(f, order=None): """Returns all non-zero monomials from ``f`` in lex order. """ return [ m for m, _ in dmp_list_terms(f.rep, f.lev, f.dom, order=order) ] def terms(f, order=None): """Returns all non-zero terms from ``f`` in lex order. """ return dmp_list_terms(f.rep, f.lev, f.dom, order=order) def all_coeffs(f): """Returns all coefficients from ``f``. """ if not f.lev: if not f: return [f.dom.zero] else: return [ c for c in f.rep ] else: raise PolynomialError('multivariate polynomials not supported') def all_monoms(f): """Returns all monomials from ``f``. """ if not f.lev: n = dup_degree(f.rep) if n < 0: return [(0,)] else: return [ (n - i,) for i, c in enumerate(f.rep) ] else: raise PolynomialError('multivariate polynomials not supported') def all_terms(f): """Returns all terms from a ``f``. """ if not f.lev: n = dup_degree(f.rep) if n < 0: return [((0,), f.dom.zero)] else: return [ ((n - i,), c) for i, c in enumerate(f.rep) ] else: raise PolynomialError('multivariate polynomials not supported') def lift(f): """Convert algebraic coefficients to rationals. """ return f.per(dmp_lift(f.rep, f.lev, f.dom), dom=f.dom.dom) def deflate(f): """Reduce degree of `f` by mapping `x_i^m` to `y_i`. """ J, F = dmp_deflate(f.rep, f.lev, f.dom) return J, f.per(F) def inject(f, front=False): """Inject ground domain generators into ``f``. """ F, lev = dmp_inject(f.rep, f.lev, f.dom, front=front) return f.__class__(F, f.dom.dom, lev) def eject(f, dom, front=False): """Eject selected generators into the ground domain. """ F = dmp_eject(f.rep, f.lev, dom, front=front) return f.__class__(F, dom, f.lev - len(dom.symbols)) def exclude(f): r""" Remove useless generators from ``f``. Returns the removed generators and the new excluded ``f``. Examples ======== >>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude() ([2], DMP([[1], [1, 2]], ZZ, None)) """ J, F, u = dmp_exclude(f.rep, f.lev, f.dom) return J, f.__class__(F, f.dom, u) def permute(f, P): r""" Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. Examples ======== >>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2]) DMP([[[2], []], [[1, 0], []]], ZZ, None) >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0]) DMP([[[1], []], [[2, 0], []]], ZZ, None) """ return f.per(dmp_permute(f.rep, P, f.lev, f.dom)) def terms_gcd(f): """Remove GCD of terms from the polynomial ``f``. """ J, F = dmp_terms_gcd(f.rep, f.lev, f.dom) return J, f.per(F) def add_ground(f, c): """Add an element of the ground domain to ``f``. """ return f.per(dmp_add_ground(f.rep, f.dom.convert(c), f.lev, f.dom)) def sub_ground(f, c): """Subtract an element of the ground domain from ``f``. """ return f.per(dmp_sub_ground(f.rep, f.dom.convert(c), f.lev, f.dom)) def mul_ground(f, c): """Multiply ``f`` by a an element of the ground domain. """ return f.per(dmp_mul_ground(f.rep, f.dom.convert(c), f.lev, f.dom)) def quo_ground(f, c): """Quotient of ``f`` by a an element of the ground domain. """ return f.per(dmp_quo_ground(f.rep, f.dom.convert(c), f.lev, f.dom)) def exquo_ground(f, c): """Exact quotient of ``f`` by a an element of the ground domain. """ return f.per(dmp_exquo_ground(f.rep, f.dom.convert(c), f.lev, f.dom)) def abs(f): """Make all coefficients in ``f`` positive. """ return f.per(dmp_abs(f.rep, f.lev, f.dom)) def neg(f): """Negate all coefficients in ``f``. """ return f.per(dmp_neg(f.rep, f.lev, f.dom)) def add(f, g): """Add two multivariate polynomials ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_add(F, G, lev, dom)) def sub(f, g): """Subtract two multivariate polynomials ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_sub(F, G, lev, dom)) def mul(f, g): """Multiply two multivariate polynomials ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_mul(F, G, lev, dom)) def sqr(f): """Square a multivariate polynomial ``f``. """ return f.per(dmp_sqr(f.rep, f.lev, f.dom)) def pow(f, n): """Raise ``f`` to a non-negative power ``n``. """ if isinstance(n, int): return f.per(dmp_pow(f.rep, n, f.lev, f.dom)) else: raise TypeError("``int`` expected, got %s" % type(n)) def pdiv(f, g): """Polynomial pseudo-division of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) q, r = dmp_pdiv(F, G, lev, dom) return per(q), per(r) def prem(f, g): """Polynomial pseudo-remainder of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_prem(F, G, lev, dom)) def pquo(f, g): """Polynomial pseudo-quotient of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_pquo(F, G, lev, dom)) def pexquo(f, g): """Polynomial exact pseudo-quotient of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_pexquo(F, G, lev, dom)) def div(f, g): """Polynomial division with remainder of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) q, r = dmp_div(F, G, lev, dom) return per(q), per(r) def rem(f, g): """Computes polynomial remainder of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_rem(F, G, lev, dom)) def quo(f, g): """Computes polynomial quotient of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_quo(F, G, lev, dom)) def exquo(f, g): """Computes polynomial exact quotient of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) res = per(dmp_exquo(F, G, lev, dom)) if f.ring and res not in f.ring: from sympy.polys.polyerrors import ExactQuotientFailed raise ExactQuotientFailed(f, g, f.ring) return res def degree(f, j=0): """Returns the leading degree of ``f`` in ``x_j``. """ if isinstance(j, int): return dmp_degree_in(f.rep, j, f.lev) else: raise TypeError("``int`` expected, got %s" % type(j)) def degree_list(f): """Returns a list of degrees of ``f``. """ return dmp_degree_list(f.rep, f.lev) def total_degree(f): """Returns the total degree of ``f``. """ return max(sum(m) for m in f.monoms()) def homogenize(f, s): """Return homogeneous polynomial of ``f``""" td = f.total_degree() result = {} new_symbol = (s == len(f.terms()[0][0])) for term in f.terms(): d = sum(term[0]) if d < td: i = td - d else: i = 0 if new_symbol: result[term[0] + (i,)] = term[1] else: l = list(term[0]) l[s] += i result[tuple(l)] = term[1] return DMP(result, f.dom, f.lev + int(new_symbol), f.ring) def homogeneous_order(f): """Returns the homogeneous order of ``f``. """ if f.is_zero: return -oo monoms = f.monoms() tdeg = sum(monoms[0]) for monom in monoms: _tdeg = sum(monom) if _tdeg != tdeg: return None return tdeg def LC(f): """Returns the leading coefficient of ``f``. """ return dmp_ground_LC(f.rep, f.lev, f.dom) def TC(f): """Returns the trailing coefficient of ``f``. """ return dmp_ground_TC(f.rep, f.lev, f.dom) def nth(f, *N): """Returns the ``n``-th coefficient of ``f``. """ if all(isinstance(n, int) for n in N): return dmp_ground_nth(f.rep, N, f.lev, f.dom) else: raise TypeError("a sequence of integers expected") def max_norm(f): """Returns maximum norm of ``f``. """ return dmp_max_norm(f.rep, f.lev, f.dom) def l1_norm(f): """Returns l1 norm of ``f``. """ return dmp_l1_norm(f.rep, f.lev, f.dom) def clear_denoms(f): """Clear denominators, but keep the ground domain. """ coeff, F = dmp_clear_denoms(f.rep, f.lev, f.dom) return coeff, f.per(F) def integrate(f, m=1, j=0): """Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. """ if not isinstance(m, int): raise TypeError("``int`` expected, got %s" % type(m)) if not isinstance(j, int): raise TypeError("``int`` expected, got %s" % type(j)) return f.per(dmp_integrate_in(f.rep, m, j, f.lev, f.dom)) def diff(f, m=1, j=0): """Computes the ``m``-th order derivative of ``f`` in ``x_j``. """ if not isinstance(m, int): raise TypeError("``int`` expected, got %s" % type(m)) if not isinstance(j, int): raise TypeError("``int`` expected, got %s" % type(j)) return f.per(dmp_diff_in(f.rep, m, j, f.lev, f.dom)) def eval(f, a, j=0): """Evaluates ``f`` at the given point ``a`` in ``x_j``. """ if not isinstance(j, int): raise TypeError("``int`` expected, got %s" % type(j)) return f.per(dmp_eval_in(f.rep, f.dom.convert(a), j, f.lev, f.dom), kill=True) def half_gcdex(f, g): """Half extended Euclidean algorithm, if univariate. """ lev, dom, per, F, G = f.unify(g) if not lev: s, h = dup_half_gcdex(F, G, dom) return per(s), per(h) else: raise ValueError('univariate polynomial expected') def gcdex(f, g): """Extended Euclidean algorithm, if univariate. """ lev, dom, per, F, G = f.unify(g) if not lev: s, t, h = dup_gcdex(F, G, dom) return per(s), per(t), per(h) else: raise ValueError('univariate polynomial expected') def invert(f, g): """Invert ``f`` modulo ``g``, if possible. """ lev, dom, per, F, G = f.unify(g) if not lev: return per(dup_invert(F, G, dom)) else: raise ValueError('univariate polynomial expected') def revert(f, n): """Compute ``f**(-1)`` mod ``x**n``. """ if not f.lev: return f.per(dup_revert(f.rep, n, f.dom)) else: raise ValueError('univariate polynomial expected') def subresultants(f, g): """Computes subresultant PRS sequence of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) R = dmp_subresultants(F, G, lev, dom) return list(map(per, R)) def resultant(f, g, includePRS=False): """Computes resultant of ``f`` and ``g`` via PRS. """ lev, dom, per, F, G = f.unify(g) if includePRS: res, R = dmp_resultant(F, G, lev, dom, includePRS=includePRS) return per(res, kill=True), list(map(per, R)) return per(dmp_resultant(F, G, lev, dom), kill=True) def discriminant(f): """Computes discriminant of ``f``. """ return f.per(dmp_discriminant(f.rep, f.lev, f.dom), kill=True) def cofactors(f, g): """Returns GCD of ``f`` and ``g`` and their cofactors. """ lev, dom, per, F, G = f.unify(g) h, cff, cfg = dmp_inner_gcd(F, G, lev, dom) return per(h), per(cff), per(cfg) def gcd(f, g): """Returns polynomial GCD of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_gcd(F, G, lev, dom)) def lcm(f, g): """Returns polynomial LCM of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_lcm(F, G, lev, dom)) def cancel(f, g, include=True): """Cancel common factors in a rational function ``f/g``. """ lev, dom, per, F, G = f.unify(g) if include: F, G = dmp_cancel(F, G, lev, dom, include=True) else: cF, cG, F, G = dmp_cancel(F, G, lev, dom, include=False) F, G = per(F), per(G) if include: return F, G else: return cF, cG, F, G def trunc(f, p): """Reduce ``f`` modulo a constant ``p``. """ return f.per(dmp_ground_trunc(f.rep, f.dom.convert(p), f.lev, f.dom)) def monic(f): """Divides all coefficients by ``LC(f)``. """ return f.per(dmp_ground_monic(f.rep, f.lev, f.dom)) def content(f): """Returns GCD of polynomial coefficients. """ return dmp_ground_content(f.rep, f.lev, f.dom) def primitive(f): """Returns content and a primitive form of ``f``. """ cont, F = dmp_ground_primitive(f.rep, f.lev, f.dom) return cont, f.per(F) def compose(f, g): """Computes functional composition of ``f`` and ``g``. """ lev, dom, per, F, G = f.unify(g) return per(dmp_compose(F, G, lev, dom)) def decompose(f): """Computes functional decomposition of ``f``. """ if not f.lev: return list(map(f.per, dup_decompose(f.rep, f.dom))) else: raise ValueError('univariate polynomial expected') def shift(f, a): """Efficiently compute Taylor shift ``f(x + a)``. """ if not f.lev: return f.per(dup_shift(f.rep, f.dom.convert(a), f.dom)) else: raise ValueError('univariate polynomial expected') def transform(f, p, q): """Evaluate functional transformation ``q**n * f(p/q)``.""" if f.lev: raise ValueError('univariate polynomial expected') lev, dom, per, P, Q = p.unify(q) lev, dom, per, F, P = f.unify(per(P, dom, lev)) lev, dom, per, F, Q = per(F, dom, lev).unify(per(Q, dom, lev)) if not lev: return per(dup_transform(F, P, Q, dom)) else: raise ValueError('univariate polynomial expected') def sturm(f): """Computes the Sturm sequence of ``f``. """ if not f.lev: return list(map(f.per, dup_sturm(f.rep, f.dom))) else: raise ValueError('univariate polynomial expected') def gff_list(f): """Computes greatest factorial factorization of ``f``. """ if not f.lev: return [ (f.per(g), k) for g, k in dup_gff_list(f.rep, f.dom) ] else: raise ValueError('univariate polynomial expected') def norm(f): """Computes ``Norm(f)``.""" r = dmp_norm(f.rep, f.lev, f.dom) return f.per(r, dom=f.dom.dom) def sqf_norm(f): """Computes square-free norm of ``f``. """ s, g, r = dmp_sqf_norm(f.rep, f.lev, f.dom) return s, f.per(g), f.per(r, dom=f.dom.dom) def sqf_part(f): """Computes square-free part of ``f``. """ return f.per(dmp_sqf_part(f.rep, f.lev, f.dom)) def sqf_list(f, all=False): """Returns a list of square-free factors of ``f``. """ coeff, factors = dmp_sqf_list(f.rep, f.lev, f.dom, all) return coeff, [ (f.per(g), k) for g, k in factors ] def sqf_list_include(f, all=False): """Returns a list of square-free factors of ``f``. """ factors = dmp_sqf_list_include(f.rep, f.lev, f.dom, all) return [ (f.per(g), k) for g, k in factors ] def factor_list(f): """Returns a list of irreducible factors of ``f``. """ coeff, factors = dmp_factor_list(f.rep, f.lev, f.dom) return coeff, [ (f.per(g), k) for g, k in factors ] def factor_list_include(f): """Returns a list of irreducible factors of ``f``. """ factors = dmp_factor_list_include(f.rep, f.lev, f.dom) return [ (f.per(g), k) for g, k in factors ] def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False): """Compute isolating intervals for roots of ``f``. """ if not f.lev: if not all: if not sqf: return dup_isolate_real_roots(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) else: return dup_isolate_real_roots_sqf(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) else: if not sqf: return dup_isolate_all_roots(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) else: return dup_isolate_all_roots_sqf(f.rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) else: raise PolynomialError( "can't isolate roots of a multivariate polynomial") def refine_root(f, s, t, eps=None, steps=None, fast=False): """ Refine an isolating interval to the given precision. ``eps`` should be a rational number. """ if not f.lev: return dup_refine_real_root(f.rep, s, t, f.dom, eps=eps, steps=steps, fast=fast) else: raise PolynomialError( "can't refine a root of a multivariate polynomial") def count_real_roots(f, inf=None, sup=None): """Return the number of real roots of ``f`` in ``[inf, sup]``. """ return dup_count_real_roots(f.rep, f.dom, inf=inf, sup=sup) def count_complex_roots(f, inf=None, sup=None): """Return the number of complex roots of ``f`` in ``[inf, sup]``. """ return dup_count_complex_roots(f.rep, f.dom, inf=inf, sup=sup) @property def is_zero(f): """Returns ``True`` if ``f`` is a zero polynomial. """ return dmp_zero_p(f.rep, f.lev) @property def is_one(f): """Returns ``True`` if ``f`` is a unit polynomial. """ return dmp_one_p(f.rep, f.lev, f.dom) @property def is_ground(f): """Returns ``True`` if ``f`` is an element of the ground domain. """ return dmp_ground_p(f.rep, None, f.lev) @property def is_sqf(f): """Returns ``True`` if ``f`` is a square-free polynomial. """ return dmp_sqf_p(f.rep, f.lev, f.dom) @property def is_monic(f): """Returns ``True`` if the leading coefficient of ``f`` is one. """ return f.dom.is_one(dmp_ground_LC(f.rep, f.lev, f.dom)) @property def is_primitive(f): """Returns ``True`` if the GCD of the coefficients of ``f`` is one. """ return f.dom.is_one(dmp_ground_content(f.rep, f.lev, f.dom)) @property def is_linear(f): """Returns ``True`` if ``f`` is linear in all its variables. """ return all(sum(monom) <= 1 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys()) @property def is_quadratic(f): """Returns ``True`` if ``f`` is quadratic in all its variables. """ return all(sum(monom) <= 2 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys()) @property def is_monomial(f): """Returns ``True`` if ``f`` is zero or has only one term. """ return len(f.to_dict()) <= 1 @property def is_homogeneous(f): """Returns ``True`` if ``f`` is a homogeneous polynomial. """ return f.homogeneous_order() is not None @property def is_irreducible(f): """Returns ``True`` if ``f`` has no factors over its domain. """ return dmp_irreducible_p(f.rep, f.lev, f.dom) @property def is_cyclotomic(f): """Returns ``True`` if ``f`` is a cyclotomic polynomial. """ if not f.lev: return dup_cyclotomic_p(f.rep, f.dom) else: return False def __abs__(f): return f.abs() def __neg__(f): return f.neg() def __add__(f, g): if not isinstance(g, DMP): try: g = f.per(dmp_ground(f.dom.convert(g), f.lev)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: g = f.ring.convert(g) except (CoercionFailed, NotImplementedError): return NotImplemented return f.add(g) def __radd__(f, g): return f.__add__(g) def __sub__(f, g): if not isinstance(g, DMP): try: g = f.per(dmp_ground(f.dom.convert(g), f.lev)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: g = f.ring.convert(g) except (CoercionFailed, NotImplementedError): return NotImplemented return f.sub(g) def __rsub__(f, g): return (-f).__add__(g) def __mul__(f, g): if isinstance(g, DMP): return f.mul(g) else: try: return f.mul_ground(g) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.mul(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __truediv__(f, g): if isinstance(g, DMP): return f.exquo(g) else: try: return f.mul_ground(g) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.exquo(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __rtruediv__(f, g): if isinstance(g, DMP): return g.exquo(f) elif f.ring is not None: try: return f.ring.convert(g).exquo(f) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __rmul__(f, g): return f.__mul__(g) def __pow__(f, n): return f.pow(n) def __divmod__(f, g): return f.div(g) def __mod__(f, g): return f.rem(g) def __floordiv__(f, g): if isinstance(g, DMP): return f.quo(g) else: try: return f.quo_ground(g) except TypeError: return NotImplemented def __eq__(f, g): try: _, _, _, F, G = f.unify(g) if f.lev == g.lev: return F == G except UnificationFailed: pass return False def __ne__(f, g): return not f == g def eq(f, g, strict=False): if not strict: return f == g else: return f._strict_eq(g) def ne(f, g, strict=False): return not f.eq(g, strict=strict) def _strict_eq(f, g): return isinstance(g, f.__class__) and f.lev == g.lev \ and f.dom == g.dom \ and f.rep == g.rep def __lt__(f, g): _, _, _, F, G = f.unify(g) return F < G def __le__(f, g): _, _, _, F, G = f.unify(g) return F <= G def __gt__(f, g): _, _, _, F, G = f.unify(g) return F > G def __ge__(f, g): _, _, _, F, G = f.unify(g) return F >= G def __bool__(f): return not dmp_zero_p(f.rep, f.lev) def init_normal_DMF(num, den, lev, dom): return DMF(dmp_normal(num, lev, dom), dmp_normal(den, lev, dom), dom, lev) class DMF(PicklableWithSlots, CantSympify): """Dense Multivariate Fractions over `K`. """ __slots__ = ('num', 'den', 'lev', 'dom', 'ring') def __init__(self, rep, dom, lev=None, ring=None): num, den, lev = self._parse(rep, dom, lev) num, den = dmp_cancel(num, den, lev, dom) self.num = num self.den = den self.lev = lev self.dom = dom self.ring = ring @classmethod def new(cls, rep, dom, lev=None, ring=None): num, den, lev = cls._parse(rep, dom, lev) obj = object.__new__(cls) obj.num = num obj.den = den obj.lev = lev obj.dom = dom obj.ring = ring return obj @classmethod def _parse(cls, rep, dom, lev=None): if type(rep) is tuple: num, den = rep if lev is not None: if type(num) is dict: num = dmp_from_dict(num, lev, dom) if type(den) is dict: den = dmp_from_dict(den, lev, dom) else: num, num_lev = dmp_validate(num) den, den_lev = dmp_validate(den) if num_lev == den_lev: lev = num_lev else: raise ValueError('inconsistent number of levels') if dmp_zero_p(den, lev): raise ZeroDivisionError('fraction denominator') if dmp_zero_p(num, lev): den = dmp_one(lev, dom) else: if dmp_negative_p(den, lev, dom): num = dmp_neg(num, lev, dom) den = dmp_neg(den, lev, dom) else: num = rep if lev is not None: if type(num) is dict: num = dmp_from_dict(num, lev, dom) elif type(num) is not list: num = dmp_ground(dom.convert(num), lev) else: num, lev = dmp_validate(num) den = dmp_one(lev, dom) return num, den, lev def __repr__(f): return "%s((%s, %s), %s, %s)" % (f.__class__.__name__, f.num, f.den, f.dom, f.ring) def __hash__(f): return hash((f.__class__.__name__, dmp_to_tuple(f.num, f.lev), dmp_to_tuple(f.den, f.lev), f.lev, f.dom, f.ring)) def poly_unify(f, g): """Unify a multivariate fraction and a polynomial. """ if not isinstance(g, DMP) or f.lev != g.lev: raise UnificationFailed("can't unify %s with %s" % (f, g)) if f.dom == g.dom and f.ring == g.ring: return (f.lev, f.dom, f.per, (f.num, f.den), g.rep) else: lev, dom = f.lev, f.dom.unify(g.dom) ring = f.ring if g.ring is not None: if ring is not None: ring = ring.unify(g.ring) else: ring = g.ring F = (dmp_convert(f.num, lev, f.dom, dom), dmp_convert(f.den, lev, f.dom, dom)) G = dmp_convert(g.rep, lev, g.dom, dom) def per(num, den, cancel=True, kill=False, lev=lev): if kill: if not lev: return num/den else: lev = lev - 1 if cancel: num, den = dmp_cancel(num, den, lev, dom) return f.__class__.new((num, den), dom, lev, ring=ring) return lev, dom, per, F, G def frac_unify(f, g): """Unify representations of two multivariate fractions. """ if not isinstance(g, DMF) or f.lev != g.lev: raise UnificationFailed("can't unify %s with %s" % (f, g)) if f.dom == g.dom and f.ring == g.ring: return (f.lev, f.dom, f.per, (f.num, f.den), (g.num, g.den)) else: lev, dom = f.lev, f.dom.unify(g.dom) ring = f.ring if g.ring is not None: if ring is not None: ring = ring.unify(g.ring) else: ring = g.ring F = (dmp_convert(f.num, lev, f.dom, dom), dmp_convert(f.den, lev, f.dom, dom)) G = (dmp_convert(g.num, lev, g.dom, dom), dmp_convert(g.den, lev, g.dom, dom)) def per(num, den, cancel=True, kill=False, lev=lev): if kill: if not lev: return num/den else: lev = lev - 1 if cancel: num, den = dmp_cancel(num, den, lev, dom) return f.__class__.new((num, den), dom, lev, ring=ring) return lev, dom, per, F, G def per(f, num, den, cancel=True, kill=False, ring=None): """Create a DMF out of the given representation. """ lev, dom = f.lev, f.dom if kill: if not lev: return num/den else: lev -= 1 if cancel: num, den = dmp_cancel(num, den, lev, dom) if ring is None: ring = f.ring return f.__class__.new((num, den), dom, lev, ring=ring) def half_per(f, rep, kill=False): """Create a DMP out of the given representation. """ lev = f.lev if kill: if not lev: return rep else: lev -= 1 return DMP(rep, f.dom, lev) @classmethod def zero(cls, lev, dom, ring=None): return cls.new(0, dom, lev, ring=ring) @classmethod def one(cls, lev, dom, ring=None): return cls.new(1, dom, lev, ring=ring) def numer(f): """Returns the numerator of ``f``. """ return f.half_per(f.num) def denom(f): """Returns the denominator of ``f``. """ return f.half_per(f.den) def cancel(f): """Remove common factors from ``f.num`` and ``f.den``. """ return f.per(f.num, f.den) def neg(f): """Negate all coefficients in ``f``. """ return f.per(dmp_neg(f.num, f.lev, f.dom), f.den, cancel=False) def add(f, g): """Add two multivariate fractions ``f`` and ``g``. """ if isinstance(g, DMP): lev, dom, per, (F_num, F_den), G = f.poly_unify(g) num, den = dmp_add_mul(F_num, F_den, G, lev, dom), F_den else: lev, dom, per, F, G = f.frac_unify(g) (F_num, F_den), (G_num, G_den) = F, G num = dmp_add(dmp_mul(F_num, G_den, lev, dom), dmp_mul(F_den, G_num, lev, dom), lev, dom) den = dmp_mul(F_den, G_den, lev, dom) return per(num, den) def sub(f, g): """Subtract two multivariate fractions ``f`` and ``g``. """ if isinstance(g, DMP): lev, dom, per, (F_num, F_den), G = f.poly_unify(g) num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den else: lev, dom, per, F, G = f.frac_unify(g) (F_num, F_den), (G_num, G_den) = F, G num = dmp_sub(dmp_mul(F_num, G_den, lev, dom), dmp_mul(F_den, G_num, lev, dom), lev, dom) den = dmp_mul(F_den, G_den, lev, dom) return per(num, den) def mul(f, g): """Multiply two multivariate fractions ``f`` and ``g``. """ if isinstance(g, DMP): lev, dom, per, (F_num, F_den), G = f.poly_unify(g) num, den = dmp_mul(F_num, G, lev, dom), F_den else: lev, dom, per, F, G = f.frac_unify(g) (F_num, F_den), (G_num, G_den) = F, G num = dmp_mul(F_num, G_num, lev, dom) den = dmp_mul(F_den, G_den, lev, dom) return per(num, den) def pow(f, n): """Raise ``f`` to a non-negative power ``n``. """ if isinstance(n, int): return f.per(dmp_pow(f.num, n, f.lev, f.dom), dmp_pow(f.den, n, f.lev, f.dom), cancel=False) else: raise TypeError("``int`` expected, got %s" % type(n)) def quo(f, g): """Computes quotient of fractions ``f`` and ``g``. """ if isinstance(g, DMP): lev, dom, per, (F_num, F_den), G = f.poly_unify(g) num, den = F_num, dmp_mul(F_den, G, lev, dom) else: lev, dom, per, F, G = f.frac_unify(g) (F_num, F_den), (G_num, G_den) = F, G num = dmp_mul(F_num, G_den, lev, dom) den = dmp_mul(F_den, G_num, lev, dom) res = per(num, den) if f.ring is not None and res not in f.ring: from sympy.polys.polyerrors import ExactQuotientFailed raise ExactQuotientFailed(f, g, f.ring) return res exquo = quo def invert(f, check=True): """Computes inverse of a fraction ``f``. """ if check and f.ring is not None and not f.ring.is_unit(f): raise NotReversible(f, f.ring) res = f.per(f.den, f.num, cancel=False) return res @property def is_zero(f): """Returns ``True`` if ``f`` is a zero fraction. """ return dmp_zero_p(f.num, f.lev) @property def is_one(f): """Returns ``True`` if ``f`` is a unit fraction. """ return dmp_one_p(f.num, f.lev, f.dom) and \ dmp_one_p(f.den, f.lev, f.dom) def __neg__(f): return f.neg() def __add__(f, g): if isinstance(g, (DMP, DMF)): return f.add(g) try: return f.add(f.half_per(g)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.add(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __radd__(f, g): return f.__add__(g) def __sub__(f, g): if isinstance(g, (DMP, DMF)): return f.sub(g) try: return f.sub(f.half_per(g)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.sub(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __rsub__(f, g): return (-f).__add__(g) def __mul__(f, g): if isinstance(g, (DMP, DMF)): return f.mul(g) try: return f.mul(f.half_per(g)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.mul(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __rmul__(f, g): return f.__mul__(g) def __pow__(f, n): return f.pow(n) def __truediv__(f, g): if isinstance(g, (DMP, DMF)): return f.quo(g) try: return f.quo(f.half_per(g)) except TypeError: return NotImplemented except (CoercionFailed, NotImplementedError): if f.ring is not None: try: return f.quo(f.ring.convert(g)) except (CoercionFailed, NotImplementedError): pass return NotImplemented def __rtruediv__(self, g): r = self.invert(check=False)*g if self.ring and r not in self.ring: from sympy.polys.polyerrors import ExactQuotientFailed raise ExactQuotientFailed(g, self, self.ring) return r def __eq__(f, g): try: if isinstance(g, DMP): _, _, _, (F_num, F_den), G = f.poly_unify(g) if f.lev == g.lev: return dmp_one_p(F_den, f.lev, f.dom) and F_num == G else: _, _, _, F, G = f.frac_unify(g) if f.lev == g.lev: return F == G except UnificationFailed: pass return False def __ne__(f, g): try: if isinstance(g, DMP): _, _, _, (F_num, F_den), G = f.poly_unify(g) if f.lev == g.lev: return not (dmp_one_p(F_den, f.lev, f.dom) and F_num == G) else: _, _, _, F, G = f.frac_unify(g) if f.lev == g.lev: return F != G except UnificationFailed: pass return True def __lt__(f, g): _, _, _, F, G = f.frac_unify(g) return F < G def __le__(f, g): _, _, _, F, G = f.frac_unify(g) return F <= G def __gt__(f, g): _, _, _, F, G = f.frac_unify(g) return F > G def __ge__(f, g): _, _, _, F, G = f.frac_unify(g) return F >= G def __bool__(f): return not dmp_zero_p(f.num, f.lev) def init_normal_ANP(rep, mod, dom): return ANP(dup_normal(rep, dom), dup_normal(mod, dom), dom) class ANP(PicklableWithSlots, CantSympify): """Dense Algebraic Number Polynomials over a field. """ __slots__ = ('rep', 'mod', 'dom') def __init__(self, rep, mod, dom): if type(rep) is dict: self.rep = dup_from_dict(rep, dom) else: if type(rep) is not list: rep = [dom.convert(rep)] self.rep = dup_strip(rep) if isinstance(mod, DMP): self.mod = mod.rep else: if type(mod) is dict: self.mod = dup_from_dict(mod, dom) else: self.mod = dup_strip(mod) self.dom = dom def __repr__(f): return "%s(%s, %s, %s)" % (f.__class__.__name__, f.rep, f.mod, f.dom) def __hash__(f): return hash((f.__class__.__name__, f.to_tuple(), dmp_to_tuple(f.mod, 0), f.dom)) def unify(f, g): """Unify representations of two algebraic numbers. """ if not isinstance(g, ANP) or f.mod != g.mod: raise UnificationFailed("can't unify %s with %s" % (f, g)) if f.dom == g.dom: return f.dom, f.per, f.rep, g.rep, f.mod else: dom = f.dom.unify(g.dom) F = dup_convert(f.rep, f.dom, dom) G = dup_convert(g.rep, g.dom, dom) if dom != f.dom and dom != g.dom: mod = dup_convert(f.mod, f.dom, dom) else: if dom == f.dom: mod = f.mod else: mod = g.mod per = lambda rep: ANP(rep, mod, dom) return dom, per, F, G, mod def per(f, rep, mod=None, dom=None): return ANP(rep, mod or f.mod, dom or f.dom) @classmethod def zero(cls, mod, dom): return ANP(0, mod, dom) @classmethod def one(cls, mod, dom): return ANP(1, mod, dom) def to_dict(f): """Convert ``f`` to a dict representation with native coefficients. """ return dmp_to_dict(f.rep, 0, f.dom) def to_sympy_dict(f): """Convert ``f`` to a dict representation with SymPy coefficients. """ rep = dmp_to_dict(f.rep, 0, f.dom) for k, v in rep.items(): rep[k] = f.dom.to_sympy(v) return rep def to_list(f): """Convert ``f`` to a list representation with native coefficients. """ return f.rep def to_sympy_list(f): """Convert ``f`` to a list representation with SymPy coefficients. """ return [ f.dom.to_sympy(c) for c in f.rep ] def to_tuple(f): """ Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. """ return dmp_to_tuple(f.rep, 0) @classmethod def from_list(cls, rep, mod, dom): return ANP(dup_strip(list(map(dom.convert, rep))), mod, dom) def neg(f): return f.per(dup_neg(f.rep, f.dom)) def add(f, g): dom, per, F, G, mod = f.unify(g) return per(dup_add(F, G, dom)) def sub(f, g): dom, per, F, G, mod = f.unify(g) return per(dup_sub(F, G, dom)) def mul(f, g): dom, per, F, G, mod = f.unify(g) return per(dup_rem(dup_mul(F, G, dom), mod, dom)) def pow(f, n): """Raise ``f`` to a non-negative power ``n``. """ if isinstance(n, int): if n < 0: F, n = dup_invert(f.rep, f.mod, f.dom), -n else: F = f.rep return f.per(dup_rem(dup_pow(F, n, f.dom), f.mod, f.dom)) else: raise TypeError("``int`` expected, got %s" % type(n)) def div(f, g): dom, per, F, G, mod = f.unify(g) return (per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom)), f.zero(mod, dom)) def rem(f, g): dom, _, _, G, mod = f.unify(g) s, h = dup_half_gcdex(G, mod, dom) if h == [dom.one]: return f.zero(mod, dom) else: raise NotInvertible("zero divisor") def quo(f, g): dom, per, F, G, mod = f.unify(g) return per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom)) exquo = quo def LC(f): """Returns the leading coefficient of ``f``. """ return dup_LC(f.rep, f.dom) def TC(f): """Returns the trailing coefficient of ``f``. """ return dup_TC(f.rep, f.dom) @property def is_zero(f): """Returns ``True`` if ``f`` is a zero algebraic number. """ return not f @property def is_one(f): """Returns ``True`` if ``f`` is a unit algebraic number. """ return f.rep == [f.dom.one] @property def is_ground(f): """Returns ``True`` if ``f`` is an element of the ground domain. """ return not f.rep or len(f.rep) == 1 def __neg__(f): return f.neg() def __add__(f, g): if isinstance(g, ANP): return f.add(g) else: try: return f.add(f.per(g)) except (CoercionFailed, TypeError): return NotImplemented def __radd__(f, g): return f.__add__(g) def __sub__(f, g): if isinstance(g, ANP): return f.sub(g) else: try: return f.sub(f.per(g)) except (CoercionFailed, TypeError): return NotImplemented def __rsub__(f, g): return (-f).__add__(g) def __mul__(f, g): if isinstance(g, ANP): return f.mul(g) else: try: return f.mul(f.per(g)) except (CoercionFailed, TypeError): return NotImplemented def __rmul__(f, g): return f.__mul__(g) def __pow__(f, n): return f.pow(n) def __divmod__(f, g): return f.div(g) def __mod__(f, g): return f.rem(g) def __truediv__(f, g): if isinstance(g, ANP): return f.quo(g) else: try: return f.quo(f.per(g)) except (CoercionFailed, TypeError): return NotImplemented def __eq__(f, g): try: _, _, F, G, _ = f.unify(g) return F == G except UnificationFailed: return False def __ne__(f, g): try: _, _, F, G, _ = f.unify(g) return F != G except UnificationFailed: return True def __lt__(f, g): _, _, F, G, _ = f.unify(g) return F < G def __le__(f, g): _, _, F, G, _ = f.unify(g) return F <= G def __gt__(f, g): _, _, F, G, _ = f.unify(g) return F > G def __ge__(f, g): _, _, F, G, _ = f.unify(g) return F >= G def __bool__(f): return bool(f.rep)
10f80f4702d72ba8123d0d581f18d730e1e21ba9c79a588556aad90d2efc2f88
""" Solving solvable quintics - An implementation of DS Dummit's paper Paper : http://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf Mathematica notebook: http://www.emba.uvm.edu/~ddummit/quintics/quintics.nb """ from sympy.core import Symbol from sympy.core.evalf import N from sympy.core.numbers import I, Rational from sympy.functions import sqrt from sympy.polys.polytools import Poly from sympy.utilities import public x = Symbol('x') @public class PolyQuintic: """Special functions for solvable quintics""" def __init__(self, poly): _, _, self.p, self.q, self.r, self.s = poly.all_coeffs() self.zeta1 = Rational(-1, 4) + (sqrt(5)/4) + I*sqrt((sqrt(5)/8) + Rational(5, 8)) self.zeta2 = (-sqrt(5)/4) - Rational(1, 4) + I*sqrt((-sqrt(5)/8) + Rational(5, 8)) self.zeta3 = (-sqrt(5)/4) - Rational(1, 4) - I*sqrt((-sqrt(5)/8) + Rational(5, 8)) self.zeta4 = Rational(-1, 4) + (sqrt(5)/4) - I*sqrt((sqrt(5)/8) + Rational(5, 8)) @property def f20(self): p, q, r, s = self.p, self.q, self.r, self.s f20 = q**8 - 13*p*q**6*r + p**5*q**2*r**2 + 65*p**2*q**4*r**2 - 4*p**6*r**3 - 128*p**3*q**2*r**3 + 17*q**4*r**3 + 48*p**4*r**4 - 16*p*q**2*r**4 - 192*p**2*r**5 + 256*r**6 - 4*p**5*q**3*s - 12*p**2*q**5*s + 18*p**6*q*r*s + 12*p**3*q**3*r*s - 124*q**5*r*s + 196*p**4*q*r**2*s + 590*p*q**3*r**2*s - 160*p**2*q*r**3*s - 1600*q*r**4*s - 27*p**7*s**2 - 150*p**4*q**2*s**2 - 125*p*q**4*s**2 - 99*p**5*r*s**2 - 725*p**2*q**2*r*s**2 + 1200*p**3*r**2*s**2 + 3250*q**2*r**2*s**2 - 2000*p*r**3*s**2 - 1250*p*q*r*s**3 + 3125*p**2*s**4 - 9375*r*s**4-(2*p*q**6 - 19*p**2*q**4*r + 51*p**3*q**2*r**2 - 3*q**4*r**2 - 32*p**4*r**3 - 76*p*q**2*r**3 + 256*p**2*r**4 - 512*r**5 + 31*p**3*q**3*s + 58*q**5*s - 117*p**4*q*r*s - 105*p*q**3*r*s - 260*p**2*q*r**2*s + 2400*q*r**3*s + 108*p**5*s**2 + 325*p**2*q**2*s**2 - 525*p**3*r*s**2 - 2750*q**2*r*s**2 + 500*p*r**2*s**2 - 625*p*q*s**3 + 3125*s**4)*x+(p**2*q**4 - 6*p**3*q**2*r - 8*q**4*r + 9*p**4*r**2 + 76*p*q**2*r**2 - 136*p**2*r**3 + 400*r**4 - 50*p*q**3*s + 90*p**2*q*r*s - 1400*q*r**2*s + 625*q**2*s**2 + 500*p*r*s**2)*x**2-(2*q**4 - 21*p*q**2*r + 40*p**2*r**2 - 160*r**3 + 15*p**2*q*s + 400*q*r*s - 125*p*s**2)*x**3+(2*p*q**2 - 6*p**2*r + 40*r**2 - 50*q*s)*x**4 + 8*r*x**5 + x**6 return Poly(f20, x) @property def b(self): p, q, r, s = self.p, self.q, self.r, self.s b = ( [], [0,0,0,0,0,0], [0,0,0,0,0,0], [0,0,0,0,0,0], [0,0,0,0,0,0],) b[1][5] = 100*p**7*q**7 + 2175*p**4*q**9 + 10500*p*q**11 - 1100*p**8*q**5*r - 27975*p**5*q**7*r - 152950*p**2*q**9*r + 4125*p**9*q**3*r**2 + 128875*p**6*q**5*r**2 + 830525*p**3*q**7*r**2 - 59450*q**9*r**2 - 5400*p**10*q*r**3 - 243800*p**7*q**3*r**3 - 2082650*p**4*q**5*r**3 + 333925*p*q**7*r**3 + 139200*p**8*q*r**4 + 2406000*p**5*q**3*r**4 + 122600*p**2*q**5*r**4 - 1254400*p**6*q*r**5 - 3776000*p**3*q**3*r**5 - 1832000*q**5*r**5 + 4736000*p**4*q*r**6 + 6720000*p*q**3*r**6 - 6400000*p**2*q*r**7 + 900*p**9*q**4*s + 37400*p**6*q**6*s + 281625*p**3*q**8*s + 435000*q**10*s - 6750*p**10*q**2*r*s - 322300*p**7*q**4*r*s - 2718575*p**4*q**6*r*s - 4214250*p*q**8*r*s + 16200*p**11*r**2*s + 859275*p**8*q**2*r**2*s + 8925475*p**5*q**4*r**2*s + 14427875*p**2*q**6*r**2*s - 453600*p**9*r**3*s - 10038400*p**6*q**2*r**3*s - 17397500*p**3*q**4*r**3*s + 11333125*q**6*r**3*s + 4451200*p**7*r**4*s + 15850000*p**4*q**2*r**4*s - 34000000*p*q**4*r**4*s - 17984000*p**5*r**5*s + 10000000*p**2*q**2*r**5*s + 25600000*p**3*r**6*s + 8000000*q**2*r**6*s - 6075*p**11*q*s**2 + 83250*p**8*q**3*s**2 + 1282500*p**5*q**5*s**2 + 2862500*p**2*q**7*s**2 - 724275*p**9*q*r*s**2 - 9807250*p**6*q**3*r*s**2 - 28374375*p**3*q**5*r*s**2 - 22212500*q**7*r*s**2 + 8982000*p**7*q*r**2*s**2 + 39600000*p**4*q**3*r**2*s**2 + 61746875*p*q**5*r**2*s**2 + 1010000*p**5*q*r**3*s**2 + 1000000*p**2*q**3*r**3*s**2 - 78000000*p**3*q*r**4*s**2 - 30000000*q**3*r**4*s**2 - 80000000*p*q*r**5*s**2 + 759375*p**10*s**3 + 9787500*p**7*q**2*s**3 + 39062500*p**4*q**4*s**3 + 52343750*p*q**6*s**3 - 12301875*p**8*r*s**3 - 98175000*p**5*q**2*r*s**3 - 225078125*p**2*q**4*r*s**3 + 54900000*p**6*r**2*s**3 + 310000000*p**3*q**2*r**2*s**3 + 7890625*q**4*r**2*s**3 - 51250000*p**4*r**3*s**3 + 420000000*p*q**2*r**3*s**3 - 110000000*p**2*r**4*s**3 + 200000000*r**5*s**3 - 2109375*p**6*q*s**4 + 21093750*p**3*q**3*s**4 + 89843750*q**5*s**4 - 182343750*p**4*q*r*s**4 - 733203125*p*q**3*r*s**4 + 196875000*p**2*q*r**2*s**4 - 1125000000*q*r**3*s**4 + 158203125*p**5*s**5 + 566406250*p**2*q**2*s**5 - 101562500*p**3*r*s**5 + 1669921875*q**2*r*s**5 - 1250000000*p*r**2*s**5 + 1220703125*p*q*s**6 - 6103515625*s**7 b[1][4] = -1000*p**5*q**7 - 7250*p**2*q**9 + 10800*p**6*q**5*r + 96900*p**3*q**7*r + 52500*q**9*r - 37400*p**7*q**3*r**2 - 470850*p**4*q**5*r**2 - 640600*p*q**7*r**2 + 39600*p**8*q*r**3 + 983600*p**5*q**3*r**3 + 2848100*p**2*q**5*r**3 - 814400*p**6*q*r**4 - 6076000*p**3*q**3*r**4 - 2308000*q**5*r**4 + 5024000*p**4*q*r**5 + 9680000*p*q**3*r**5 - 9600000*p**2*q*r**6 - 13800*p**7*q**4*s - 94650*p**4*q**6*s + 26500*p*q**8*s + 86400*p**8*q**2*r*s + 816500*p**5*q**4*r*s + 257500*p**2*q**6*r*s - 91800*p**9*r**2*s - 1853700*p**6*q**2*r**2*s - 630000*p**3*q**4*r**2*s + 8971250*q**6*r**2*s + 2071200*p**7*r**3*s + 7240000*p**4*q**2*r**3*s - 29375000*p*q**4*r**3*s - 14416000*p**5*r**4*s + 5200000*p**2*q**2*r**4*s + 30400000*p**3*r**5*s + 12000000*q**2*r**5*s - 64800*p**9*q*s**2 - 567000*p**6*q**3*s**2 - 1655000*p**3*q**5*s**2 - 6987500*q**7*s**2 - 337500*p**7*q*r*s**2 - 8462500*p**4*q**3*r*s**2 + 5812500*p*q**5*r*s**2 + 24930000*p**5*q*r**2*s**2 + 69125000*p**2*q**3*r**2*s**2 - 103500000*p**3*q*r**3*s**2 - 30000000*q**3*r**3*s**2 - 90000000*p*q*r**4*s**2 + 708750*p**8*s**3 + 5400000*p**5*q**2*s**3 - 8906250*p**2*q**4*s**3 - 18562500*p**6*r*s**3 + 625000*p**3*q**2*r*s**3 - 29687500*q**4*r*s**3 + 75000000*p**4*r**2*s**3 + 416250000*p*q**2*r**2*s**3 - 60000000*p**2*r**3*s**3 + 300000000*r**4*s**3 - 71718750*p**4*q*s**4 - 189062500*p*q**3*s**4 - 210937500*p**2*q*r*s**4 - 1187500000*q*r**2*s**4 + 187500000*p**3*s**5 + 800781250*q**2*s**5 + 390625000*p*r*s**5 b[1][3] = 500*p**6*q**5 + 6350*p**3*q**7 + 19800*q**9 - 3750*p**7*q**3*r - 65100*p**4*q**5*r - 264950*p*q**7*r + 6750*p**8*q*r**2 + 209050*p**5*q**3*r**2 + 1217250*p**2*q**5*r**2 - 219000*p**6*q*r**3 - 2510000*p**3*q**3*r**3 - 1098500*q**5*r**3 + 2068000*p**4*q*r**4 + 5060000*p*q**3*r**4 - 5200000*p**2*q*r**5 + 6750*p**8*q**2*s + 96350*p**5*q**4*s + 346000*p**2*q**6*s - 20250*p**9*r*s - 459900*p**6*q**2*r*s - 1828750*p**3*q**4*r*s + 2930000*q**6*r*s + 594000*p**7*r**2*s + 4301250*p**4*q**2*r**2*s - 10906250*p*q**4*r**2*s - 5252000*p**5*r**3*s + 1450000*p**2*q**2*r**3*s + 12800000*p**3*r**4*s + 6500000*q**2*r**4*s - 74250*p**7*q*s**2 - 1418750*p**4*q**3*s**2 - 5956250*p*q**5*s**2 + 4297500*p**5*q*r*s**2 + 29906250*p**2*q**3*r*s**2 - 31500000*p**3*q*r**2*s**2 - 12500000*q**3*r**2*s**2 - 35000000*p*q*r**3*s**2 - 1350000*p**6*s**3 - 6093750*p**3*q**2*s**3 - 17500000*q**4*s**3 + 7031250*p**4*r*s**3 + 127812500*p*q**2*r*s**3 - 18750000*p**2*r**2*s**3 + 162500000*r**3*s**3 - 107812500*p**2*q*s**4 - 460937500*q*r*s**4 + 214843750*p*s**5 b[1][2] = -1950*p**4*q**5 - 14100*p*q**7 + 14350*p**5*q**3*r + 125600*p**2*q**5*r - 27900*p**6*q*r**2 - 402250*p**3*q**3*r**2 - 288250*q**5*r**2 + 436000*p**4*q*r**3 + 1345000*p*q**3*r**3 - 1400000*p**2*q*r**4 - 9450*p**6*q**2*s + 1250*p**3*q**4*s + 465000*q**6*s + 49950*p**7*r*s + 302500*p**4*q**2*r*s - 1718750*p*q**4*r*s - 834000*p**5*r**2*s - 437500*p**2*q**2*r**2*s + 3100000*p**3*r**3*s + 1750000*q**2*r**3*s + 292500*p**5*q*s**2 + 1937500*p**2*q**3*s**2 - 3343750*p**3*q*r*s**2 - 1875000*q**3*r*s**2 - 8125000*p*q*r**2*s**2 + 1406250*p**4*s**3 + 12343750*p*q**2*s**3 - 5312500*p**2*r*s**3 + 43750000*r**2*s**3 - 74218750*q*s**4 b[1][1] = 300*p**5*q**3 + 2150*p**2*q**5 - 1350*p**6*q*r - 21500*p**3*q**3*r - 61500*q**5*r + 42000*p**4*q*r**2 + 290000*p*q**3*r**2 - 300000*p**2*q*r**3 + 4050*p**7*s + 45000*p**4*q**2*s + 125000*p*q**4*s - 108000*p**5*r*s - 643750*p**2*q**2*r*s + 700000*p**3*r**2*s + 375000*q**2*r**2*s + 93750*p**3*q*s**2 + 312500*q**3*s**2 - 1875000*p*q*r*s**2 + 1406250*p**2*s**3 + 9375000*r*s**3 b[1][0] = -1250*p**3*q**3 - 9000*q**5 + 4500*p**4*q*r + 46250*p*q**3*r - 50000*p**2*q*r**2 - 6750*p**5*s - 43750*p**2*q**2*s + 75000*p**3*r*s + 62500*q**2*r*s - 156250*p*q*s**2 + 1562500*s**3 b[2][5] = 200*p**6*q**11 - 250*p**3*q**13 - 10800*q**15 - 3900*p**7*q**9*r - 3325*p**4*q**11*r + 181800*p*q**13*r + 26950*p**8*q**7*r**2 + 69625*p**5*q**9*r**2 - 1214450*p**2*q**11*r**2 - 78725*p**9*q**5*r**3 - 368675*p**6*q**7*r**3 + 4166325*p**3*q**9*r**3 + 1131100*q**11*r**3 + 73400*p**10*q**3*r**4 + 661950*p**7*q**5*r**4 - 9151950*p**4*q**7*r**4 - 16633075*p*q**9*r**4 + 36000*p**11*q*r**5 + 135600*p**8*q**3*r**5 + 17321400*p**5*q**5*r**5 + 85338300*p**2*q**7*r**5 - 832000*p**9*q*r**6 - 21379200*p**6*q**3*r**6 - 176044000*p**3*q**5*r**6 - 1410000*q**7*r**6 + 6528000*p**7*q*r**7 + 129664000*p**4*q**3*r**7 + 47344000*p*q**5*r**7 - 21504000*p**5*q*r**8 - 115200000*p**2*q**3*r**8 + 25600000*p**3*q*r**9 + 64000000*q**3*r**9 + 15700*p**8*q**8*s + 120525*p**5*q**10*s + 113250*p**2*q**12*s - 196900*p**9*q**6*r*s - 1776925*p**6*q**8*r*s - 3062475*p**3*q**10*r*s - 4153500*q**12*r*s + 857925*p**10*q**4*r**2*s + 10562775*p**7*q**6*r**2*s + 34866250*p**4*q**8*r**2*s + 73486750*p*q**10*r**2*s - 1333800*p**11*q**2*r**3*s - 29212625*p**8*q**4*r**3*s - 168729675*p**5*q**6*r**3*s - 427230750*p**2*q**8*r**3*s + 108000*p**12*r**4*s + 30384200*p**9*q**2*r**4*s + 324535100*p**6*q**4*r**4*s + 952666750*p**3*q**6*r**4*s - 38076875*q**8*r**4*s - 4296000*p**10*r**5*s - 213606400*p**7*q**2*r**5*s - 842060000*p**4*q**4*r**5*s - 95285000*p*q**6*r**5*s + 61184000*p**8*r**6*s + 567520000*p**5*q**2*r**6*s + 547000000*p**2*q**4*r**6*s - 390912000*p**6*r**7*s - 812800000*p**3*q**2*r**7*s - 924000000*q**4*r**7*s + 1152000000*p**4*r**8*s + 800000000*p*q**2*r**8*s - 1280000000*p**2*r**9*s + 141750*p**10*q**5*s**2 - 31500*p**7*q**7*s**2 - 11325000*p**4*q**9*s**2 - 31687500*p*q**11*s**2 - 1293975*p**11*q**3*r*s**2 - 4803800*p**8*q**5*r*s**2 + 71398250*p**5*q**7*r*s**2 + 227625000*p**2*q**9*r*s**2 + 3256200*p**12*q*r**2*s**2 + 43870125*p**9*q**3*r**2*s**2 + 64581500*p**6*q**5*r**2*s**2 + 56090625*p**3*q**7*r**2*s**2 + 260218750*q**9*r**2*s**2 - 74610000*p**10*q*r**3*s**2 - 662186500*p**7*q**3*r**3*s**2 - 1987747500*p**4*q**5*r**3*s**2 - 811928125*p*q**7*r**3*s**2 + 471286000*p**8*q*r**4*s**2 + 2106040000*p**5*q**3*r**4*s**2 + 792687500*p**2*q**5*r**4*s**2 - 135120000*p**6*q*r**5*s**2 + 2479000000*p**3*q**3*r**5*s**2 + 5242250000*q**5*r**5*s**2 - 6400000000*p**4*q*r**6*s**2 - 8620000000*p*q**3*r**6*s**2 + 13280000000*p**2*q*r**7*s**2 + 1600000000*q*r**8*s**2 + 273375*p**12*q**2*s**3 - 13612500*p**9*q**4*s**3 - 177250000*p**6*q**6*s**3 - 511015625*p**3*q**8*s**3 - 320937500*q**10*s**3 - 2770200*p**13*r*s**3 + 12595500*p**10*q**2*r*s**3 + 543950000*p**7*q**4*r*s**3 + 1612281250*p**4*q**6*r*s**3 + 968125000*p*q**8*r*s**3 + 77031000*p**11*r**2*s**3 + 373218750*p**8*q**2*r**2*s**3 + 1839765625*p**5*q**4*r**2*s**3 + 1818515625*p**2*q**6*r**2*s**3 - 776745000*p**9*r**3*s**3 - 6861075000*p**6*q**2*r**3*s**3 - 20014531250*p**3*q**4*r**3*s**3 - 13747812500*q**6*r**3*s**3 + 3768000000*p**7*r**4*s**3 + 35365000000*p**4*q**2*r**4*s**3 + 34441875000*p*q**4*r**4*s**3 - 9628000000*p**5*r**5*s**3 - 63230000000*p**2*q**2*r**5*s**3 + 13600000000*p**3*r**6*s**3 - 15000000000*q**2*r**6*s**3 - 10400000000*p*r**7*s**3 - 45562500*p**11*q*s**4 - 525937500*p**8*q**3*s**4 - 1364218750*p**5*q**5*s**4 - 1382812500*p**2*q**7*s**4 + 572062500*p**9*q*r*s**4 + 2473515625*p**6*q**3*r*s**4 + 13192187500*p**3*q**5*r*s**4 + 12703125000*q**7*r*s**4 - 451406250*p**7*q*r**2*s**4 - 18153906250*p**4*q**3*r**2*s**4 - 36908203125*p*q**5*r**2*s**4 - 9069375000*p**5*q*r**3*s**4 + 79957812500*p**2*q**3*r**3*s**4 + 5512500000*p**3*q*r**4*s**4 + 50656250000*q**3*r**4*s**4 + 74750000000*p*q*r**5*s**4 + 56953125*p**10*s**5 + 1381640625*p**7*q**2*s**5 - 781250000*p**4*q**4*s**5 + 878906250*p*q**6*s**5 - 2655703125*p**8*r*s**5 - 3223046875*p**5*q**2*r*s**5 - 35117187500*p**2*q**4*r*s**5 + 26573437500*p**6*r**2*s**5 + 14785156250*p**3*q**2*r**2*s**5 - 52050781250*q**4*r**2*s**5 - 103062500000*p**4*r**3*s**5 - 281796875000*p*q**2*r**3*s**5 + 146875000000*p**2*r**4*s**5 - 37500000000*r**5*s**5 - 8789062500*p**6*q*s**6 - 3906250000*p**3*q**3*s**6 + 1464843750*q**5*s**6 + 102929687500*p**4*q*r*s**6 + 297119140625*p*q**3*r*s**6 - 217773437500*p**2*q*r**2*s**6 + 167968750000*q*r**3*s**6 + 10986328125*p**5*s**7 + 98876953125*p**2*q**2*s**7 - 188964843750*p**3*r*s**7 - 278320312500*q**2*r*s**7 + 517578125000*p*r**2*s**7 - 610351562500*p*q*s**8 + 762939453125*s**9 b[2][4] = -200*p**7*q**9 + 1850*p**4*q**11 + 21600*p*q**13 + 3200*p**8*q**7*r - 19200*p**5*q**9*r - 316350*p**2*q**11*r - 19050*p**9*q**5*r**2 + 37400*p**6*q**7*r**2 + 1759250*p**3*q**9*r**2 + 440100*q**11*r**2 + 48750*p**10*q**3*r**3 + 190200*p**7*q**5*r**3 - 4604200*p**4*q**7*r**3 - 6072800*p*q**9*r**3 - 43200*p**11*q*r**4 - 834500*p**8*q**3*r**4 + 4916000*p**5*q**5*r**4 + 27926850*p**2*q**7*r**4 + 969600*p**9*q*r**5 + 2467200*p**6*q**3*r**5 - 45393200*p**3*q**5*r**5 - 5399500*q**7*r**5 - 7283200*p**7*q*r**6 + 10536000*p**4*q**3*r**6 + 41656000*p*q**5*r**6 + 22784000*p**5*q*r**7 - 35200000*p**2*q**3*r**7 - 25600000*p**3*q*r**8 + 96000000*q**3*r**8 - 3000*p**9*q**6*s + 40400*p**6*q**8*s + 136550*p**3*q**10*s - 1647000*q**12*s + 40500*p**10*q**4*r*s - 173600*p**7*q**6*r*s - 126500*p**4*q**8*r*s + 23969250*p*q**10*r*s - 153900*p**11*q**2*r**2*s - 486150*p**8*q**4*r**2*s - 4115800*p**5*q**6*r**2*s - 112653250*p**2*q**8*r**2*s + 129600*p**12*r**3*s + 2683350*p**9*q**2*r**3*s + 10906650*p**6*q**4*r**3*s + 187289500*p**3*q**6*r**3*s + 44098750*q**8*r**3*s - 4384800*p**10*r**4*s - 35660800*p**7*q**2*r**4*s - 175420000*p**4*q**4*r**4*s - 426538750*p*q**6*r**4*s + 60857600*p**8*r**5*s + 349436000*p**5*q**2*r**5*s + 900600000*p**2*q**4*r**5*s - 429568000*p**6*r**6*s - 1511200000*p**3*q**2*r**6*s - 1286000000*q**4*r**6*s + 1472000000*p**4*r**7*s + 1440000000*p*q**2*r**7*s - 1920000000*p**2*r**8*s - 36450*p**11*q**3*s**2 - 188100*p**8*q**5*s**2 - 5504750*p**5*q**7*s**2 - 37968750*p**2*q**9*s**2 + 255150*p**12*q*r*s**2 + 2754000*p**9*q**3*r*s**2 + 49196500*p**6*q**5*r*s**2 + 323587500*p**3*q**7*r*s**2 - 83250000*q**9*r*s**2 - 465750*p**10*q*r**2*s**2 - 31881500*p**7*q**3*r**2*s**2 - 415585000*p**4*q**5*r**2*s**2 + 1054775000*p*q**7*r**2*s**2 - 96823500*p**8*q*r**3*s**2 - 701490000*p**5*q**3*r**3*s**2 - 2953531250*p**2*q**5*r**3*s**2 + 1454560000*p**6*q*r**4*s**2 + 7670500000*p**3*q**3*r**4*s**2 + 5661062500*q**5*r**4*s**2 - 7785000000*p**4*q*r**5*s**2 - 9450000000*p*q**3*r**5*s**2 + 14000000000*p**2*q*r**6*s**2 + 2400000000*q*r**7*s**2 - 437400*p**13*s**3 - 10145250*p**10*q**2*s**3 - 121912500*p**7*q**4*s**3 - 576531250*p**4*q**6*s**3 - 528593750*p*q**8*s**3 + 12939750*p**11*r*s**3 + 313368750*p**8*q**2*r*s**3 + 2171812500*p**5*q**4*r*s**3 + 2381718750*p**2*q**6*r*s**3 - 124638750*p**9*r**2*s**3 - 3001575000*p**6*q**2*r**2*s**3 - 12259375000*p**3*q**4*r**2*s**3 - 9985312500*q**6*r**2*s**3 + 384000000*p**7*r**3*s**3 + 13997500000*p**4*q**2*r**3*s**3 + 20749531250*p*q**4*r**3*s**3 - 553500000*p**5*r**4*s**3 - 41835000000*p**2*q**2*r**4*s**3 + 5420000000*p**3*r**5*s**3 - 16300000000*q**2*r**5*s**3 - 17600000000*p*r**6*s**3 - 7593750*p**9*q*s**4 + 289218750*p**6*q**3*s**4 + 3591406250*p**3*q**5*s**4 + 5992187500*q**7*s**4 + 658125000*p**7*q*r*s**4 - 269531250*p**4*q**3*r*s**4 - 15882812500*p*q**5*r*s**4 - 4785000000*p**5*q*r**2*s**4 + 54375781250*p**2*q**3*r**2*s**4 - 5668750000*p**3*q*r**3*s**4 + 35867187500*q**3*r**3*s**4 + 113875000000*p*q*r**4*s**4 - 544218750*p**8*s**5 - 5407031250*p**5*q**2*s**5 - 14277343750*p**2*q**4*s**5 + 5421093750*p**6*r*s**5 - 24941406250*p**3*q**2*r*s**5 - 25488281250*q**4*r*s**5 - 11500000000*p**4*r**2*s**5 - 231894531250*p*q**2*r**2*s**5 - 6250000000*p**2*r**3*s**5 - 43750000000*r**4*s**5 + 35449218750*p**4*q*s**6 + 137695312500*p*q**3*s**6 + 34667968750*p**2*q*r*s**6 + 202148437500*q*r**2*s**6 - 33691406250*p**3*s**7 - 214843750000*q**2*s**7 - 31738281250*p*r*s**7 b[2][3] = -800*p**5*q**9 - 5400*p**2*q**11 + 5800*p**6*q**7*r + 48750*p**3*q**9*r + 16200*q**11*r - 3000*p**7*q**5*r**2 - 108350*p**4*q**7*r**2 - 263250*p*q**9*r**2 - 60700*p**8*q**3*r**3 - 386250*p**5*q**5*r**3 + 253100*p**2*q**7*r**3 + 127800*p**9*q*r**4 + 2326700*p**6*q**3*r**4 + 6565550*p**3*q**5*r**4 - 705750*q**7*r**4 - 2903200*p**7*q*r**5 - 21218000*p**4*q**3*r**5 + 1057000*p*q**5*r**5 + 20368000*p**5*q*r**6 + 33000000*p**2*q**3*r**6 - 43200000*p**3*q*r**7 + 52000000*q**3*r**7 + 6200*p**7*q**6*s + 188250*p**4*q**8*s + 931500*p*q**10*s - 73800*p**8*q**4*r*s - 1466850*p**5*q**6*r*s - 6894000*p**2*q**8*r*s + 315900*p**9*q**2*r**2*s + 4547000*p**6*q**4*r**2*s + 20362500*p**3*q**6*r**2*s + 15018750*q**8*r**2*s - 653400*p**10*r**3*s - 13897550*p**7*q**2*r**3*s - 76757500*p**4*q**4*r**3*s - 124207500*p*q**6*r**3*s + 18567600*p**8*r**4*s + 175911000*p**5*q**2*r**4*s + 253787500*p**2*q**4*r**4*s - 183816000*p**6*r**5*s - 706900000*p**3*q**2*r**5*s - 665750000*q**4*r**5*s + 740000000*p**4*r**6*s + 890000000*p*q**2*r**6*s - 1040000000*p**2*r**7*s - 763000*p**6*q**5*s**2 - 12375000*p**3*q**7*s**2 - 40500000*q**9*s**2 + 364500*p**10*q*r*s**2 + 15537000*p**7*q**3*r*s**2 + 154392500*p**4*q**5*r*s**2 + 372206250*p*q**7*r*s**2 - 25481250*p**8*q*r**2*s**2 - 386300000*p**5*q**3*r**2*s**2 - 996343750*p**2*q**5*r**2*s**2 + 459872500*p**6*q*r**3*s**2 + 2943937500*p**3*q**3*r**3*s**2 + 2437781250*q**5*r**3*s**2 - 2883750000*p**4*q*r**4*s**2 - 4343750000*p*q**3*r**4*s**2 + 5495000000*p**2*q*r**5*s**2 + 1300000000*q*r**6*s**2 - 364500*p**11*s**3 - 13668750*p**8*q**2*s**3 - 113406250*p**5*q**4*s**3 - 159062500*p**2*q**6*s**3 + 13972500*p**9*r*s**3 + 61537500*p**6*q**2*r*s**3 - 1622656250*p**3*q**4*r*s**3 - 2720625000*q**6*r*s**3 - 201656250*p**7*r**2*s**3 + 1949687500*p**4*q**2*r**2*s**3 + 4979687500*p*q**4*r**2*s**3 + 497125000*p**5*r**3*s**3 - 11150625000*p**2*q**2*r**3*s**3 + 2982500000*p**3*r**4*s**3 - 6612500000*q**2*r**4*s**3 - 10450000000*p*r**5*s**3 + 126562500*p**7*q*s**4 + 1443750000*p**4*q**3*s**4 + 281250000*p*q**5*s**4 - 1648125000*p**5*q*r*s**4 + 11271093750*p**2*q**3*r*s**4 - 4785156250*p**3*q*r**2*s**4 + 8808593750*q**3*r**2*s**4 + 52390625000*p*q*r**3*s**4 - 611718750*p**6*s**5 - 13027343750*p**3*q**2*s**5 - 1464843750*q**4*s**5 + 6492187500*p**4*r*s**5 - 65351562500*p*q**2*r*s**5 - 13476562500*p**2*r**2*s**5 - 24218750000*r**3*s**5 + 41992187500*p**2*q*s**6 + 69824218750*q*r*s**6 - 34179687500*p*s**7 b[2][2] = -1000*p**6*q**7 - 5150*p**3*q**9 + 10800*q**11 + 11000*p**7*q**5*r + 66450*p**4*q**7*r - 127800*p*q**9*r - 41250*p**8*q**3*r**2 - 368400*p**5*q**5*r**2 + 204200*p**2*q**7*r**2 + 54000*p**9*q*r**3 + 1040950*p**6*q**3*r**3 + 2096500*p**3*q**5*r**3 + 200000*q**7*r**3 - 1140000*p**7*q*r**4 - 7691000*p**4*q**3*r**4 - 2281000*p*q**5*r**4 + 7296000*p**5*q*r**5 + 13300000*p**2*q**3*r**5 - 14400000*p**3*q*r**6 + 14000000*q**3*r**6 - 9000*p**8*q**4*s + 52100*p**5*q**6*s + 710250*p**2*q**8*s + 67500*p**9*q**2*r*s - 256100*p**6*q**4*r*s - 5753000*p**3*q**6*r*s + 292500*q**8*r*s - 162000*p**10*r**2*s - 1432350*p**7*q**2*r**2*s + 5410000*p**4*q**4*r**2*s - 7408750*p*q**6*r**2*s + 4401000*p**8*r**3*s + 24185000*p**5*q**2*r**3*s + 20781250*p**2*q**4*r**3*s - 43012000*p**6*r**4*s - 146300000*p**3*q**2*r**4*s - 165875000*q**4*r**4*s + 182000000*p**4*r**5*s + 250000000*p*q**2*r**5*s - 280000000*p**2*r**6*s + 60750*p**10*q*s**2 + 2414250*p**7*q**3*s**2 + 15770000*p**4*q**5*s**2 + 15825000*p*q**7*s**2 - 6021000*p**8*q*r*s**2 - 62252500*p**5*q**3*r*s**2 - 74718750*p**2*q**5*r*s**2 + 90888750*p**6*q*r**2*s**2 + 471312500*p**3*q**3*r**2*s**2 + 525875000*q**5*r**2*s**2 - 539375000*p**4*q*r**3*s**2 - 1030000000*p*q**3*r**3*s**2 + 1142500000*p**2*q*r**4*s**2 + 350000000*q*r**5*s**2 - 303750*p**9*s**3 - 35943750*p**6*q**2*s**3 - 331875000*p**3*q**4*s**3 - 505937500*q**6*s**3 + 8437500*p**7*r*s**3 + 530781250*p**4*q**2*r*s**3 + 1150312500*p*q**4*r*s**3 - 154500000*p**5*r**2*s**3 - 2059062500*p**2*q**2*r**2*s**3 + 1150000000*p**3*r**3*s**3 - 1343750000*q**2*r**3*s**3 - 2900000000*p*r**4*s**3 + 30937500*p**5*q*s**4 + 1166406250*p**2*q**3*s**4 - 1496875000*p**3*q*r*s**4 + 1296875000*q**3*r*s**4 + 10640625000*p*q*r**2*s**4 - 281250000*p**4*s**5 - 9746093750*p*q**2*s**5 + 1269531250*p**2*r*s**5 - 7421875000*r**2*s**5 + 15625000000*q*s**6 b[2][1] = -1600*p**4*q**7 - 10800*p*q**9 + 9800*p**5*q**5*r + 80550*p**2*q**7*r - 4600*p**6*q**3*r**2 - 112700*p**3*q**5*r**2 + 40500*q**7*r**2 - 34200*p**7*q*r**3 - 279500*p**4*q**3*r**3 - 665750*p*q**5*r**3 + 632000*p**5*q*r**4 + 3200000*p**2*q**3*r**4 - 2800000*p**3*q*r**5 + 3000000*q**3*r**5 - 18600*p**6*q**4*s - 51750*p**3*q**6*s + 405000*q**8*s + 21600*p**7*q**2*r*s - 122500*p**4*q**4*r*s - 2891250*p*q**6*r*s + 156600*p**8*r**2*s + 1569750*p**5*q**2*r**2*s + 6943750*p**2*q**4*r**2*s - 3774000*p**6*r**3*s - 27100000*p**3*q**2*r**3*s - 30187500*q**4*r**3*s + 28000000*p**4*r**4*s + 52500000*p*q**2*r**4*s - 60000000*p**2*r**5*s - 81000*p**8*q*s**2 - 240000*p**5*q**3*s**2 + 937500*p**2*q**5*s**2 + 3273750*p**6*q*r*s**2 + 30406250*p**3*q**3*r*s**2 + 55687500*q**5*r*s**2 - 42187500*p**4*q*r**2*s**2 - 112812500*p*q**3*r**2*s**2 + 152500000*p**2*q*r**3*s**2 + 75000000*q*r**4*s**2 - 4218750*p**4*q**2*s**3 + 15156250*p*q**4*s**3 + 5906250*p**5*r*s**3 - 206562500*p**2*q**2*r*s**3 + 107500000*p**3*r**2*s**3 - 159375000*q**2*r**2*s**3 - 612500000*p*r**3*s**3 + 135937500*p**3*q*s**4 + 46875000*q**3*s**4 + 1175781250*p*q*r*s**4 - 292968750*p**2*s**5 - 1367187500*r*s**5 b[2][0] = -800*p**5*q**5 - 5400*p**2*q**7 + 6000*p**6*q**3*r + 51700*p**3*q**5*r + 27000*q**7*r - 10800*p**7*q*r**2 - 163250*p**4*q**3*r**2 - 285750*p*q**5*r**2 + 192000*p**5*q*r**3 + 1000000*p**2*q**3*r**3 - 800000*p**3*q*r**4 + 500000*q**3*r**4 - 10800*p**7*q**2*s - 57500*p**4*q**4*s + 67500*p*q**6*s + 32400*p**8*r*s + 279000*p**5*q**2*r*s - 131250*p**2*q**4*r*s - 729000*p**6*r**2*s - 4100000*p**3*q**2*r**2*s - 5343750*q**4*r**2*s + 5000000*p**4*r**3*s + 10000000*p*q**2*r**3*s - 10000000*p**2*r**4*s + 641250*p**6*q*s**2 + 5812500*p**3*q**3*s**2 + 10125000*q**5*s**2 - 7031250*p**4*q*r*s**2 - 20625000*p*q**3*r*s**2 + 17500000*p**2*q*r**2*s**2 + 12500000*q*r**3*s**2 - 843750*p**5*s**3 - 19375000*p**2*q**2*s**3 + 30000000*p**3*r*s**3 - 20312500*q**2*r*s**3 - 112500000*p*r**2*s**3 + 183593750*p*q*s**4 - 292968750*s**5 b[3][5] = 500*p**11*q**6 + 9875*p**8*q**8 + 42625*p**5*q**10 - 35000*p**2*q**12 - 4500*p**12*q**4*r - 108375*p**9*q**6*r - 516750*p**6*q**8*r + 1110500*p**3*q**10*r + 2730000*q**12*r + 10125*p**13*q**2*r**2 + 358250*p**10*q**4*r**2 + 1908625*p**7*q**6*r**2 - 11744250*p**4*q**8*r**2 - 43383250*p*q**10*r**2 - 313875*p**11*q**2*r**3 - 2074875*p**8*q**4*r**3 + 52094750*p**5*q**6*r**3 + 264567500*p**2*q**8*r**3 + 796125*p**9*q**2*r**4 - 92486250*p**6*q**4*r**4 - 757957500*p**3*q**6*r**4 - 29354375*q**8*r**4 + 60970000*p**7*q**2*r**5 + 1112462500*p**4*q**4*r**5 + 571094375*p*q**6*r**5 - 685290000*p**5*q**2*r**6 - 2037800000*p**2*q**4*r**6 + 2279600000*p**3*q**2*r**7 + 849000000*q**4*r**7 - 1480000000*p*q**2*r**8 + 13500*p**13*q**3*s + 363000*p**10*q**5*s + 2861250*p**7*q**7*s + 8493750*p**4*q**9*s + 17031250*p*q**11*s - 60750*p**14*q*r*s - 2319750*p**11*q**3*r*s - 22674250*p**8*q**5*r*s - 74368750*p**5*q**7*r*s - 170578125*p**2*q**9*r*s + 2760750*p**12*q*r**2*s + 46719000*p**9*q**3*r**2*s + 163356375*p**6*q**5*r**2*s + 360295625*p**3*q**7*r**2*s - 195990625*q**9*r**2*s - 37341750*p**10*q*r**3*s - 194739375*p**7*q**3*r**3*s - 105463125*p**4*q**5*r**3*s - 415825000*p*q**7*r**3*s + 90180000*p**8*q*r**4*s - 990552500*p**5*q**3*r**4*s + 3519212500*p**2*q**5*r**4*s + 1112220000*p**6*q*r**5*s - 4508750000*p**3*q**3*r**5*s - 8159500000*q**5*r**5*s - 4356000000*p**4*q*r**6*s + 14615000000*p*q**3*r**6*s - 2160000000*p**2*q*r**7*s + 91125*p**15*s**2 + 3290625*p**12*q**2*s**2 + 35100000*p**9*q**4*s**2 + 175406250*p**6*q**6*s**2 + 629062500*p**3*q**8*s**2 + 910937500*q**10*s**2 - 5710500*p**13*r*s**2 - 100423125*p**10*q**2*r*s**2 - 604743750*p**7*q**4*r*s**2 - 2954843750*p**4*q**6*r*s**2 - 4587578125*p*q**8*r*s**2 + 116194500*p**11*r**2*s**2 + 1280716250*p**8*q**2*r**2*s**2 + 7401190625*p**5*q**4*r**2*s**2 + 11619937500*p**2*q**6*r**2*s**2 - 952173125*p**9*r**3*s**2 - 6519712500*p**6*q**2*r**3*s**2 - 10238593750*p**3*q**4*r**3*s**2 + 29984609375*q**6*r**3*s**2 + 2558300000*p**7*r**4*s**2 + 16225000000*p**4*q**2*r**4*s**2 - 64994140625*p*q**4*r**4*s**2 + 4202250000*p**5*r**5*s**2 + 46925000000*p**2*q**2*r**5*s**2 - 28950000000*p**3*r**6*s**2 - 1000000000*q**2*r**6*s**2 + 37000000000*p*r**7*s**2 - 48093750*p**11*q*s**3 - 673359375*p**8*q**3*s**3 - 2170312500*p**5*q**5*s**3 - 2466796875*p**2*q**7*s**3 + 647578125*p**9*q*r*s**3 + 597031250*p**6*q**3*r*s**3 - 7542578125*p**3*q**5*r*s**3 - 41125000000*q**7*r*s**3 - 2175828125*p**7*q*r**2*s**3 - 7101562500*p**4*q**3*r**2*s**3 + 100596875000*p*q**5*r**2*s**3 - 8984687500*p**5*q*r**3*s**3 - 120070312500*p**2*q**3*r**3*s**3 + 57343750000*p**3*q*r**4*s**3 + 9500000000*q**3*r**4*s**3 - 342875000000*p*q*r**5*s**3 + 400781250*p**10*s**4 + 8531250000*p**7*q**2*s**4 + 34033203125*p**4*q**4*s**4 + 42724609375*p*q**6*s**4 - 6289453125*p**8*r*s**4 - 24037109375*p**5*q**2*r*s**4 - 62626953125*p**2*q**4*r*s**4 + 17299218750*p**6*r**2*s**4 + 108357421875*p**3*q**2*r**2*s**4 - 55380859375*q**4*r**2*s**4 + 105648437500*p**4*r**3*s**4 + 1204228515625*p*q**2*r**3*s**4 - 365000000000*p**2*r**4*s**4 + 184375000000*r**5*s**4 - 32080078125*p**6*q*s**5 - 98144531250*p**3*q**3*s**5 + 93994140625*q**5*s**5 - 178955078125*p**4*q*r*s**5 - 1299804687500*p*q**3*r*s**5 + 332421875000*p**2*q*r**2*s**5 - 1195312500000*q*r**3*s**5 + 72021484375*p**5*s**6 + 323486328125*p**2*q**2*s**6 + 682373046875*p**3*r*s**6 + 2447509765625*q**2*r*s**6 - 3011474609375*p*r**2*s**6 + 3051757812500*p*q*s**7 - 7629394531250*s**8 b[3][4] = 1500*p**9*q**6 + 69625*p**6*q**8 + 590375*p**3*q**10 + 1035000*q**12 - 13500*p**10*q**4*r - 760625*p**7*q**6*r - 7904500*p**4*q**8*r - 18169250*p*q**10*r + 30375*p**11*q**2*r**2 + 2628625*p**8*q**4*r**2 + 37879000*p**5*q**6*r**2 + 121367500*p**2*q**8*r**2 - 2699250*p**9*q**2*r**3 - 76776875*p**6*q**4*r**3 - 403583125*p**3*q**6*r**3 - 78865625*q**8*r**3 + 60907500*p**7*q**2*r**4 + 735291250*p**4*q**4*r**4 + 781142500*p*q**6*r**4 - 558270000*p**5*q**2*r**5 - 2150725000*p**2*q**4*r**5 + 2015400000*p**3*q**2*r**6 + 1181000000*q**4*r**6 - 2220000000*p*q**2*r**7 + 40500*p**11*q**3*s + 1376500*p**8*q**5*s + 9953125*p**5*q**7*s + 9765625*p**2*q**9*s - 182250*p**12*q*r*s - 8859000*p**9*q**3*r*s - 82854500*p**6*q**5*r*s - 71511250*p**3*q**7*r*s + 273631250*q**9*r*s + 10233000*p**10*q*r**2*s + 179627500*p**7*q**3*r**2*s + 25164375*p**4*q**5*r**2*s - 2927290625*p*q**7*r**2*s - 171305000*p**8*q*r**3*s - 544768750*p**5*q**3*r**3*s + 7583437500*p**2*q**5*r**3*s + 1139860000*p**6*q*r**4*s - 6489375000*p**3*q**3*r**4*s - 9625375000*q**5*r**4*s - 1838000000*p**4*q*r**5*s + 19835000000*p*q**3*r**5*s - 3240000000*p**2*q*r**6*s + 273375*p**13*s**2 + 9753750*p**10*q**2*s**2 + 82575000*p**7*q**4*s**2 + 202265625*p**4*q**6*s**2 + 556093750*p*q**8*s**2 - 11552625*p**11*r*s**2 - 115813125*p**8*q**2*r*s**2 + 630590625*p**5*q**4*r*s**2 + 1347015625*p**2*q**6*r*s**2 + 157578750*p**9*r**2*s**2 - 689206250*p**6*q**2*r**2*s**2 - 4299609375*p**3*q**4*r**2*s**2 + 23896171875*q**6*r**2*s**2 - 1022437500*p**7*r**3*s**2 + 6648125000*p**4*q**2*r**3*s**2 - 52895312500*p*q**4*r**3*s**2 + 4401750000*p**5*r**4*s**2 + 26500000000*p**2*q**2*r**4*s**2 - 22125000000*p**3*r**5*s**2 - 1500000000*q**2*r**5*s**2 + 55500000000*p*r**6*s**2 - 137109375*p**9*q*s**3 - 1955937500*p**6*q**3*s**3 - 6790234375*p**3*q**5*s**3 - 16996093750*q**7*s**3 + 2146218750*p**7*q*r*s**3 + 6570312500*p**4*q**3*r*s**3 + 39918750000*p*q**5*r*s**3 - 7673281250*p**5*q*r**2*s**3 - 52000000000*p**2*q**3*r**2*s**3 + 50796875000*p**3*q*r**3*s**3 + 18750000000*q**3*r**3*s**3 - 399875000000*p*q*r**4*s**3 + 780468750*p**8*s**4 + 14455078125*p**5*q**2*s**4 + 10048828125*p**2*q**4*s**4 - 15113671875*p**6*r*s**4 + 39298828125*p**3*q**2*r*s**4 - 52138671875*q**4*r*s**4 + 45964843750*p**4*r**2*s**4 + 914414062500*p*q**2*r**2*s**4 + 1953125000*p**2*r**3*s**4 + 334375000000*r**4*s**4 - 149169921875*p**4*q*s**5 - 459716796875*p*q**3*s**5 - 325585937500*p**2*q*r*s**5 - 1462890625000*q*r**2*s**5 + 296630859375*p**3*s**6 + 1324462890625*q**2*s**6 + 307617187500*p*r*s**6 b[3][3] = -20750*p**7*q**6 - 290125*p**4*q**8 - 993000*p*q**10 + 146125*p**8*q**4*r + 2721500*p**5*q**6*r + 11833750*p**2*q**8*r - 237375*p**9*q**2*r**2 - 8167500*p**6*q**4*r**2 - 54605625*p**3*q**6*r**2 - 23802500*q**8*r**2 + 8927500*p**7*q**2*r**3 + 131184375*p**4*q**4*r**3 + 254695000*p*q**6*r**3 - 121561250*p**5*q**2*r**4 - 728003125*p**2*q**4*r**4 + 702550000*p**3*q**2*r**5 + 597312500*q**4*r**5 - 1202500000*p*q**2*r**6 - 194625*p**9*q**3*s - 1568875*p**6*q**5*s + 9685625*p**3*q**7*s + 74662500*q**9*s + 327375*p**10*q*r*s + 1280000*p**7*q**3*r*s - 123703750*p**4*q**5*r*s - 850121875*p*q**7*r*s - 7436250*p**8*q*r**2*s + 164820000*p**5*q**3*r**2*s + 2336659375*p**2*q**5*r**2*s + 32202500*p**6*q*r**3*s - 2429765625*p**3*q**3*r**3*s - 4318609375*q**5*r**3*s + 148000000*p**4*q*r**4*s + 9902812500*p*q**3*r**4*s - 1755000000*p**2*q*r**5*s + 1154250*p**11*s**2 + 36821250*p**8*q**2*s**2 + 372825000*p**5*q**4*s**2 + 1170921875*p**2*q**6*s**2 - 38913750*p**9*r*s**2 - 797071875*p**6*q**2*r*s**2 - 2848984375*p**3*q**4*r*s**2 + 7651406250*q**6*r*s**2 + 415068750*p**7*r**2*s**2 + 3151328125*p**4*q**2*r**2*s**2 - 17696875000*p*q**4*r**2*s**2 - 725968750*p**5*r**3*s**2 + 5295312500*p**2*q**2*r**3*s**2 - 8581250000*p**3*r**4*s**2 - 812500000*q**2*r**4*s**2 + 30062500000*p*r**5*s**2 - 110109375*p**7*q*s**3 - 1976562500*p**4*q**3*s**3 - 6329296875*p*q**5*s**3 + 2256328125*p**5*q*r*s**3 + 8554687500*p**2*q**3*r*s**3 + 12947265625*p**3*q*r**2*s**3 + 7984375000*q**3*r**2*s**3 - 167039062500*p*q*r**3*s**3 + 1181250000*p**6*s**4 + 17873046875*p**3*q**2*s**4 - 20449218750*q**4*s**4 - 16265625000*p**4*r*s**4 + 260869140625*p*q**2*r*s**4 + 21025390625*p**2*r**2*s**4 + 207617187500*r**3*s**4 - 207177734375*p**2*q*s**5 - 615478515625*q*r*s**5 + 301513671875*p*s**6 b[3][2] = 53125*p**5*q**6 + 425000*p**2*q**8 - 394375*p**6*q**4*r - 4301875*p**3*q**6*r - 3225000*q**8*r + 851250*p**7*q**2*r**2 + 16910625*p**4*q**4*r**2 + 44210000*p*q**6*r**2 - 20474375*p**5*q**2*r**3 - 147190625*p**2*q**4*r**3 + 163975000*p**3*q**2*r**4 + 156812500*q**4*r**4 - 323750000*p*q**2*r**5 - 99375*p**7*q**3*s - 6395000*p**4*q**5*s - 49243750*p*q**7*s - 1164375*p**8*q*r*s + 4465625*p**5*q**3*r*s + 205546875*p**2*q**5*r*s + 12163750*p**6*q*r**2*s - 315546875*p**3*q**3*r**2*s - 946453125*q**5*r**2*s - 23500000*p**4*q*r**3*s + 2313437500*p*q**3*r**3*s - 472500000*p**2*q*r**4*s + 1316250*p**9*s**2 + 22715625*p**6*q**2*s**2 + 206953125*p**3*q**4*s**2 + 1220000000*q**6*s**2 - 20953125*p**7*r*s**2 - 277656250*p**4*q**2*r*s**2 - 3317187500*p*q**4*r*s**2 + 293734375*p**5*r**2*s**2 + 1351562500*p**2*q**2*r**2*s**2 - 2278125000*p**3*r**3*s**2 - 218750000*q**2*r**3*s**2 + 8093750000*p*r**4*s**2 - 9609375*p**5*q*s**3 + 240234375*p**2*q**3*s**3 + 2310546875*p**3*q*r*s**3 + 1171875000*q**3*r*s**3 - 33460937500*p*q*r**2*s**3 + 2185546875*p**4*s**4 + 32578125000*p*q**2*s**4 - 8544921875*p**2*r*s**4 + 58398437500*r**2*s**4 - 114013671875*q*s**5 b[3][1] = -16250*p**6*q**4 - 191875*p**3*q**6 - 495000*q**8 + 73125*p**7*q**2*r + 1437500*p**4*q**4*r + 5866250*p*q**6*r - 2043125*p**5*q**2*r**2 - 17218750*p**2*q**4*r**2 + 19106250*p**3*q**2*r**3 + 34015625*q**4*r**3 - 69375000*p*q**2*r**4 - 219375*p**8*q*s - 2846250*p**5*q**3*s - 8021875*p**2*q**5*s + 3420000*p**6*q*r*s - 1640625*p**3*q**3*r*s - 152468750*q**5*r*s + 3062500*p**4*q*r**2*s + 381171875*p*q**3*r**2*s - 101250000*p**2*q*r**3*s + 2784375*p**7*s**2 + 43515625*p**4*q**2*s**2 + 115625000*p*q**4*s**2 - 48140625*p**5*r*s**2 - 307421875*p**2*q**2*r*s**2 - 25781250*p**3*r**2*s**2 - 46875000*q**2*r**2*s**2 + 1734375000*p*r**3*s**2 - 128906250*p**3*q*s**3 + 339843750*q**3*s**3 - 4583984375*p*q*r*s**3 + 2236328125*p**2*s**4 + 12255859375*r*s**4 b[3][0] = 31875*p**4*q**4 + 255000*p*q**6 - 82500*p**5*q**2*r - 1106250*p**2*q**4*r + 1653125*p**3*q**2*r**2 + 5187500*q**4*r**2 - 11562500*p*q**2*r**3 - 118125*p**6*q*s - 3593750*p**3*q**3*s - 23812500*q**5*s + 4656250*p**4*q*r*s + 67109375*p*q**3*r*s - 16875000*p**2*q*r**2*s - 984375*p**5*s**2 - 19531250*p**2*q**2*s**2 - 37890625*p**3*r*s**2 - 7812500*q**2*r*s**2 + 289062500*p*r**2*s**2 - 529296875*p*q*s**3 + 2343750000*s**4 b[4][5] = 600*p**10*q**10 + 13850*p**7*q**12 + 106150*p**4*q**14 + 270000*p*q**16 - 9300*p**11*q**8*r - 234075*p**8*q**10*r - 1942825*p**5*q**12*r - 5319900*p**2*q**14*r + 52050*p**12*q**6*r**2 + 1481025*p**9*q**8*r**2 + 13594450*p**6*q**10*r**2 + 40062750*p**3*q**12*r**2 - 3569400*q**14*r**2 - 122175*p**13*q**4*r**3 - 4260350*p**10*q**6*r**3 - 45052375*p**7*q**8*r**3 - 142634900*p**4*q**10*r**3 + 54186350*p*q**12*r**3 + 97200*p**14*q**2*r**4 + 5284225*p**11*q**4*r**4 + 70389525*p**8*q**6*r**4 + 232732850*p**5*q**8*r**4 - 318849400*p**2*q**10*r**4 - 2046000*p**12*q**2*r**5 - 43874125*p**9*q**4*r**5 - 107411850*p**6*q**6*r**5 + 948310700*p**3*q**8*r**5 - 34763575*q**10*r**5 + 5915600*p**10*q**2*r**6 - 115887800*p**7*q**4*r**6 - 1649542400*p**4*q**6*r**6 + 224468875*p*q**8*r**6 + 120252800*p**8*q**2*r**7 + 1779902000*p**5*q**4*r**7 - 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3906250000*p**3*q**2*r*s**6 - 1275878906250*q**4*r*s**6 - 121093750000*p**4*r**2*s**6 - 3308593750000*p*q**2*r**2*s**6 + 18066406250*p**2*r**3*s**6 - 244140625000*r**4*s**6 + 327148437500*p**4*q*s**7 + 1672363281250*p*q**3*s**7 + 446777343750*p**2*q*r*s**7 + 1232910156250*q*r**2*s**7 - 274658203125*p**3*s**8 - 1068115234375*q**2*s**8 - 61035156250*p*r*s**8 b[4][3] = 200*p**9*q**8 + 7550*p**6*q**10 + 78650*p**3*q**12 + 248400*q**14 - 4800*p**10*q**6*r - 164300*p**7*q**8*r - 1709575*p**4*q**10*r - 5566500*p*q**12*r + 31050*p**11*q**4*r**2 + 1116175*p**8*q**6*r**2 + 12674650*p**5*q**8*r**2 + 45333850*p**2*q**10*r**2 - 60750*p**12*q**2*r**3 - 2872725*p**9*q**4*r**3 - 40403050*p**6*q**6*r**3 - 173564375*p**3*q**8*r**3 - 11242250*q**10*r**3 + 2174100*p**10*q**2*r**4 + 54010000*p**7*q**4*r**4 + 331074875*p**4*q**6*r**4 + 114173750*p*q**8*r**4 - 24858500*p**8*q**2*r**5 - 300875000*p**5*q**4*r**5 - 319430625*p**2*q**6*r**5 + 69810000*p**6*q**2*r**6 - 23900000*p**3*q**4*r**6 - 294662500*q**6*r**6 + 524200000*p**4*q**2*r**7 + 1432000000*p*q**4*r**7 - 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8743359375*p**3*q**6*r**2*s**2 - 4154375000*q**8*r**2*s**2 - 296559000*p**10*r**3*s**2 - 4065056250*p**7*q**2*r**3*s**2 - 186328125*p**4*q**4*r**3*s**2 + 19419453125*p*q**6*r**3*s**2 + 2326262500*p**8*r**4*s**2 + 21189375000*p**5*q**2*r**4*s**2 - 26301953125*p**2*q**4*r**4*s**2 - 10513250000*p**6*r**5*s**2 - 69937500000*p**3*q**2*r**5*s**2 - 42257812500*q**4*r**5*s**2 + 23375000000*p**4*r**6*s**2 + 40750000000*p*q**2*r**6*s**2 - 19500000000*p**2*r**7*s**2 + 4009500*p**12*q*s**3 + 36140625*p**9*q**3*s**3 - 335459375*p**6*q**5*s**3 - 2695312500*p**3*q**7*s**3 - 1486250000*q**9*s**3 + 102515625*p**10*q*r*s**3 + 4006812500*p**7*q**3*r*s**3 + 27589609375*p**4*q**5*r*s**3 + 20195312500*p*q**7*r*s**3 - 2792812500*p**8*q*r**2*s**3 - 44115156250*p**5*q**3*r**2*s**3 - 72609453125*p**2*q**5*r**2*s**3 + 18752500000*p**6*q*r**3*s**3 + 218140625000*p**3*q**3*r**3*s**3 + 109940234375*q**5*r**3*s**3 - 21893750000*p**4*q*r**4*s**3 - 65187500000*p*q**3*r**4*s**3 - 31000000000*p**2*q*r**5*s**3 + 97500000000*q*r**6*s**3 - 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174000*p**4*q**10 - 518400*p*q**12 + 5400*p**11*q**4*r + 197550*p**8*q**6*r + 2147775*p**5*q**8*r + 7219800*p**2*q**10*r - 12150*p**12*q**2*r**2 - 662200*p**9*q**4*r**2 - 9274775*p**6*q**6*r**2 - 38330625*p**3*q**8*r**2 - 5508000*q**10*r**2 + 656550*p**10*q**2*r**3 + 16233750*p**7*q**4*r**3 + 97335875*p**4*q**6*r**3 + 58271250*p*q**8*r**3 - 9845500*p**8*q**2*r**4 - 119464375*p**5*q**4*r**4 - 194431875*p**2*q**6*r**4 + 49465000*p**6*q**2*r**5 + 166000000*p**3*q**4*r**5 - 80793750*q**6*r**5 + 54400000*p**4*q**2*r**6 + 377750000*p*q**4*r**6 - 630000000*p**2*q**2*r**7 - 16200*p**12*q**3*s - 459300*p**9*q**5*s - 4207225*p**6*q**7*s - 10827500*p**3*q**9*s + 13635000*q**11*s + 72900*p**13*q*r*s + 2877300*p**10*q**3*r*s + 33239700*p**7*q**5*r*s + 107080625*p**4*q**7*r*s - 114975000*p*q**9*r*s - 3601800*p**11*q*r**2*s - 75214375*p**8*q**3*r**2*s - 387073250*p**5*q**5*r**2*s + 55540625*p**2*q**7*r**2*s + 53793000*p**9*q*r**3*s + 687176875*p**6*q**3*r**3*s + 1670018750*p**3*q**5*r**3*s + 665234375*q**7*r**3*s - 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11642812500*p**2*q**5*r*s**3 + 2553203125*p**6*q*r**2*s**3 + 37234375000*p**3*q**3*r**2*s**3 + 21871484375*q**5*r**2*s**3 + 2803125000*p**4*q*r**3*s**3 - 10796875000*p*q**3*r**3*s**3 - 16656250000*p**2*q*r**4*s**3 + 26250000000*q*r**5*s**3 - 75937500*p**9*s**4 - 704062500*p**6*q**2*s**4 - 8363281250*p**3*q**4*s**4 - 10398437500*q**6*s**4 + 197578125*p**7*r*s**4 - 16441406250*p**4*q**2*r*s**4 - 24277343750*p*q**4*r*s**4 - 5716015625*p**5*r**2*s**4 + 31728515625*p**2*q**2*r**2*s**4 + 27031250000*p**3*r**3*s**4 - 92285156250*q**2*r**3*s**4 - 33593750000*p*r**4*s**4 + 10394531250*p**5*q*s**5 + 38037109375*p**2*q**3*s**5 - 48144531250*p**3*q*r*s**5 + 74462890625*q**3*r*s**5 + 121093750000*p*q*r**2*s**5 - 2197265625*p**4*s**6 - 92529296875*p*q**2*s**6 + 15380859375*p**2*r*s**6 - 31738281250*r**2*s**6 + 54931640625*q*s**7 b[4][1] = 200*p**8*q**6 + 2950*p**5*q**8 + 10800*p**2*q**10 - 1800*p**9*q**4*r - 49650*p**6*q**6*r - 403375*p**3*q**8*r - 999000*q**10*r + 4050*p**10*q**2*r**2 + 236625*p**7*q**4*r**2 + 3109500*p**4*q**6*r**2 + 11463750*p*q**8*r**2 - 331500*p**8*q**2*r**3 - 7818125*p**5*q**4*r**3 - 41411250*p**2*q**6*r**3 + 4782500*p**6*q**2*r**4 + 47475000*p**3*q**4*r**4 - 16728125*q**6*r**4 - 8700000*p**4*q**2*r**5 + 81750000*p*q**4*r**5 - 135000000*p**2*q**2*r**6 + 5400*p**10*q**3*s + 144200*p**7*q**5*s + 939375*p**4*q**7*s + 1012500*p*q**9*s - 24300*p**11*q*r*s - 1169250*p**8*q**3*r*s - 14027250*p**5*q**5*r*s - 44446875*p**2*q**7*r*s + 2011500*p**9*q*r**2*s + 49330625*p**6*q**3*r**2*s + 272009375*p**3*q**5*r**2*s + 104062500*q**7*r**2*s - 34660000*p**7*q*r**3*s - 455062500*p**4*q**3*r**3*s - 625906250*p*q**5*r**3*s + 210200000*p**5*q*r**4*s + 1298750000*p**2*q**3*r**4*s - 240000000*p**3*q*r**5*s + 225000000*q**3*r**5*s + 36450*p**12*s**2 + 1231875*p**9*q**2*s**2 + 10712500*p**6*q**4*s**2 + 21718750*p**3*q**6*s**2 + 16875000*q**8*s**2 - 2814750*p**10*r*s**2 - 67612500*p**7*q**2*r*s**2 - 345156250*p**4*q**4*r*s**2 - 283125000*p*q**6*r*s**2 + 51300000*p**8*r**2*s**2 + 734531250*p**5*q**2*r**2*s**2 + 1267187500*p**2*q**4*r**2*s**2 - 384312500*p**6*r**3*s**2 - 3912500000*p**3*q**2*r**3*s**2 - 1822265625*q**4*r**3*s**2 + 1112500000*p**4*r**4*s**2 + 2437500000*p*q**2*r**4*s**2 - 1125000000*p**2*r**5*s**2 - 72578125*p**5*q**3*s**3 - 189296875*p**2*q**5*s**3 + 127265625*p**6*q*r*s**3 + 1415625000*p**3*q**3*r*s**3 + 1229687500*q**5*r*s**3 + 1448437500*p**4*q*r**2*s**3 + 2218750000*p*q**3*r**2*s**3 - 4031250000*p**2*q*r**3*s**3 + 5625000000*q*r**4*s**3 - 132890625*p**7*s**4 - 529296875*p**4*q**2*s**4 - 175781250*p*q**4*s**4 - 401953125*p**5*r*s**4 - 4482421875*p**2*q**2*r*s**4 + 4140625000*p**3*r**2*s**4 - 10498046875*q**2*r**2*s**4 - 7031250000*p*r**3*s**4 + 1220703125*p**3*q*s**5 + 1953125000*q**3*s**5 + 14160156250*p*q*r*s**5 - 1708984375*p**2*s**6 - 3662109375*r*s**6 b[4][0] = -4600*p**6*q**6 - 67850*p**3*q**8 - 248400*q**10 + 38900*p**7*q**4*r + 679575*p**4*q**6*r + 2866500*p*q**8*r - 81900*p**8*q**2*r**2 - 2009750*p**5*q**4*r**2 - 10783750*p**2*q**6*r**2 + 1478750*p**6*q**2*r**3 + 14165625*p**3*q**4*r**3 - 2743750*q**6*r**3 - 5450000*p**4*q**2*r**4 + 12687500*p*q**4*r**4 - 22500000*p**2*q**2*r**5 - 101700*p**8*q**3*s - 1700975*p**5*q**5*s - 7061250*p**2*q**7*s + 423900*p**9*q*r*s + 9292375*p**6*q**3*r*s + 50438750*p**3*q**5*r*s + 20475000*q**7*r*s - 7852500*p**7*q*r**2*s - 87765625*p**4*q**3*r**2*s - 121609375*p*q**5*r**2*s + 47700000*p**5*q*r**3*s + 264687500*p**2*q**3*r**3*s - 65000000*p**3*q*r**4*s + 37500000*q**3*r**4*s - 534600*p**10*s**2 - 10344375*p**7*q**2*s**2 - 54859375*p**4*q**4*s**2 - 40312500*p*q**6*s**2 + 10158750*p**8*r*s**2 + 117778125*p**5*q**2*r*s**2 + 192421875*p**2*q**4*r*s**2 - 70593750*p**6*r**2*s**2 - 685312500*p**3*q**2*r**2*s**2 - 334375000*q**4*r**2*s**2 + 193750000*p**4*r**3*s**2 + 500000000*p*q**2*r**3*s**2 - 187500000*p**2*r**4*s**2 + 8437500*p**6*q*s**3 + 159218750*p**3*q**3*s**3 + 220625000*q**5*s**3 + 353828125*p**4*q*r*s**3 + 412500000*p*q**3*r*s**3 - 1023437500*p**2*q*r**2*s**3 + 937500000*q*r**3*s**3 - 206015625*p**5*s**4 - 701171875*p**2*q**2*s**4 + 998046875*p**3*r*s**4 - 1308593750*q**2*r*s**4 - 1367187500*p*r**2*s**4 + 1708984375*p*q*s**5 - 976562500*s**6 return b @property def o(self): p, q, r, s = self.p, self.q, self.r, self.s o = [0]*6 o[5] = -1600*p**10*q**10 - 23600*p**7*q**12 - 86400*p**4*q**14 + 24800*p**11*q**8*r + 419200*p**8*q**10*r + 1850450*p**5*q**12*r + 896400*p**2*q**14*r - 138800*p**12*q**6*r**2 - 2921900*p**9*q**8*r**2 - 17295200*p**6*q**10*r**2 - 27127750*p**3*q**12*r**2 - 26076600*q**14*r**2 + 325800*p**13*q**4*r**3 + 9993850*p**10*q**6*r**3 + 88010500*p**7*q**8*r**3 + 274047650*p**4*q**10*r**3 + 410171400*p*q**12*r**3 - 259200*p**14*q**2*r**4 - 17147100*p**11*q**4*r**4 - 254289150*p**8*q**6*r**4 - 1318548225*p**5*q**8*r**4 - 2633598475*p**2*q**10*r**4 + 12636000*p**12*q**2*r**5 + 388911000*p**9*q**4*r**5 + 3269704725*p**6*q**6*r**5 + 8791192300*p**3*q**8*r**5 + 93560575*q**10*r**5 - 228361600*p**10*q**2*r**6 - 3951199200*p**7*q**4*r**6 - 16276981100*p**4*q**6*r**6 - 1597227000*p*q**8*r**6 + 1947899200*p**8*q**2*r**7 + 17037648000*p**5*q**4*r**7 + 8919740000*p**2*q**6*r**7 - 7672160000*p**6*q**2*r**8 - 15496000000*p**3*q**4*r**8 + 4224000000*q**6*r**8 + 9968000000*p**4*q**2*r**9 - 8640000000*p*q**4*r**9 + 4800000000*p**2*q**2*r**10 - 55200*p**12*q**7*s - 685600*p**9*q**9*s + 1028250*p**6*q**11*s + 37650000*p**3*q**13*s + 111375000*q**15*s + 583200*p**13*q**5*r*s + 9075600*p**10*q**7*r*s - 883150*p**7*q**9*r*s - 506830750*p**4*q**11*r*s - 1793137500*p*q**13*r*s - 1852200*p**14*q**3*r**2*s - 41435250*p**11*q**5*r**2*s - 80566700*p**8*q**7*r**2*s + 2485673600*p**5*q**9*r**2*s + 11442286125*p**2*q**11*r**2*s + 1555200*p**15*q*r**3*s + 80846100*p**12*q**3*r**3*s + 564906800*p**9*q**5*r**3*s - 4493012400*p**6*q**7*r**3*s - 35492391250*p**3*q**9*r**3*s - 789931875*q**11*r**3*s - 71766000*p**13*q*r**4*s - 1551149200*p**10*q**3*r**4*s - 1773437900*p**7*q**5*r**4*s + 51957593125*p**4*q**7*r**4*s + 14964765625*p*q**9*r**4*s + 1231569600*p**11*q*r**5*s + 12042977600*p**8*q**3*r**5*s - 27151011200*p**5*q**5*r**5*s - 88080610000*p**2*q**7*r**5*s - 9912995200*p**9*q*r**6*s - 29448104000*p**6*q**3*r**6*s + 144954840000*p**3*q**5*r**6*s - 44601300000*q**7*r**6*s + 35453760000*p**7*q*r**7*s - 63264000000*p**4*q**3*r**7*s + 60544000000*p*q**5*r**7*s - 30048000000*p**5*q*r**8*s + 37040000000*p**2*q**3*r**8*s - 60800000000*p**3*q*r**9*s - 48000000000*q**3*r**9*s - 615600*p**14*q**4*s**2 - 10524500*p**11*q**6*s**2 - 33831250*p**8*q**8*s**2 + 222806250*p**5*q**10*s**2 + 1099687500*p**2*q**12*s**2 + 3353400*p**15*q**2*r*s**2 + 74269350*p**12*q**4*r*s**2 + 276445750*p**9*q**6*r*s**2 - 2618600000*p**6*q**8*r*s**2 - 14473243750*p**3*q**10*r*s**2 + 1383750000*q**12*r*s**2 - 2332800*p**16*r**2*s**2 - 132750900*p**13*q**2*r**2*s**2 - 900775150*p**10*q**4*r**2*s**2 + 8249244500*p**7*q**6*r**2*s**2 + 59525796875*p**4*q**8*r**2*s**2 - 40292868750*p*q**10*r**2*s**2 + 128304000*p**14*r**3*s**2 + 3160232100*p**11*q**2*r**3*s**2 + 8329580000*p**8*q**4*r**3*s**2 - 45558458750*p**5*q**6*r**3*s**2 + 297252890625*p**2*q**8*r**3*s**2 - 2769854400*p**12*r**4*s**2 - 37065970000*p**9*q**2*r**4*s**2 - 90812546875*p**6*q**4*r**4*s**2 - 627902000000*p**3*q**6*r**4*s**2 + 181347421875*q**8*r**4*s**2 + 30946932800*p**10*r**5*s**2 + 249954680000*p**7*q**2*r**5*s**2 + 802954812500*p**4*q**4*r**5*s**2 - 80900000000*p*q**6*r**5*s**2 - 192137320000*p**8*r**6*s**2 - 932641600000*p**5*q**2*r**6*s**2 - 943242500000*p**2*q**4*r**6*s**2 + 658412000000*p**6*r**7*s**2 + 1930720000000*p**3*q**2*r**7*s**2 + 593800000000*q**4*r**7*s**2 - 1162800000000*p**4*r**8*s**2 - 280000000000*p*q**2*r**8*s**2 + 840000000000*p**2*r**9*s**2 - 2187000*p**16*q*s**3 - 47418750*p**13*q**3*s**3 - 180618750*p**10*q**5*s**3 + 2231250000*p**7*q**7*s**3 + 17857734375*p**4*q**9*s**3 + 29882812500*p*q**11*s**3 + 24664500*p**14*q*r*s**3 - 853368750*p**11*q**3*r*s**3 - 25939693750*p**8*q**5*r*s**3 - 177541562500*p**5*q**7*r*s**3 - 297978828125*p**2*q**9*r*s**3 - 153468000*p**12*q*r**2*s**3 + 30188125000*p**9*q**3*r**2*s**3 + 344049821875*p**6*q**5*r**2*s**3 + 534026875000*p**3*q**7*r**2*s**3 - 340726484375*q**9*r**2*s**3 - 9056190000*p**10*q*r**3*s**3 - 322314687500*p**7*q**3*r**3*s**3 - 769632109375*p**4*q**5*r**3*s**3 - 83276875000*p*q**7*r**3*s**3 + 164061000000*p**8*q*r**4*s**3 + 1381358750000*p**5*q**3*r**4*s**3 + 3088020000000*p**2*q**5*r**4*s**3 - 1267655000000*p**6*q*r**5*s**3 - 7642630000000*p**3*q**3*r**5*s**3 - 2759877500000*q**5*r**5*s**3 + 4597760000000*p**4*q*r**6*s**3 + 1846200000000*p*q**3*r**6*s**3 - 7006000000000*p**2*q*r**7*s**3 - 1200000000000*q*r**8*s**3 + 18225000*p**15*s**4 + 1328906250*p**12*q**2*s**4 + 24729140625*p**9*q**4*s**4 + 169467187500*p**6*q**6*s**4 + 413281250000*p**3*q**8*s**4 + 223828125000*q**10*s**4 + 710775000*p**13*r*s**4 - 18611015625*p**10*q**2*r*s**4 - 314344375000*p**7*q**4*r*s**4 - 828439843750*p**4*q**6*r*s**4 + 460937500000*p*q**8*r*s**4 - 25674975000*p**11*r**2*s**4 - 52223515625*p**8*q**2*r**2*s**4 - 387160000000*p**5*q**4*r**2*s**4 - 4733680078125*p**2*q**6*r**2*s**4 + 343911875000*p**9*r**3*s**4 + 3328658359375*p**6*q**2*r**3*s**4 + 16532406250000*p**3*q**4*r**3*s**4 + 5980613281250*q**6*r**3*s**4 - 2295497500000*p**7*r**4*s**4 - 14809820312500*p**4*q**2*r**4*s**4 - 6491406250000*p*q**4*r**4*s**4 + 7768470000000*p**5*r**5*s**4 + 34192562500000*p**2*q**2*r**5*s**4 - 11859000000000*p**3*r**6*s**4 + 10530000000000*q**2*r**6*s**4 + 6000000000000*p*r**7*s**4 + 11453906250*p**11*q*s**5 + 149765625000*p**8*q**3*s**5 + 545537109375*p**5*q**5*s**5 + 527343750000*p**2*q**7*s**5 - 371313281250*p**9*q*r*s**5 - 3461455078125*p**6*q**3*r*s**5 - 7920878906250*p**3*q**5*r*s**5 - 4747314453125*q**7*r*s**5 + 2417815625000*p**7*q*r**2*s**5 + 5465576171875*p**4*q**3*r**2*s**5 + 5937128906250*p*q**5*r**2*s**5 - 10661156250000*p**5*q*r**3*s**5 - 63574218750000*p**2*q**3*r**3*s**5 + 24059375000000*p**3*q*r**4*s**5 - 33023437500000*q**3*r**4*s**5 - 43125000000000*p*q*r**5*s**5 + 94394531250*p**10*s**6 + 1097167968750*p**7*q**2*s**6 + 2829833984375*p**4*q**4*s**6 - 1525878906250*p*q**6*s**6 + 2724609375*p**8*r*s**6 + 13998535156250*p**5*q**2*r*s**6 + 57094482421875*p**2*q**4*r*s**6 - 8512509765625*p**6*r**2*s**6 - 37941406250000*p**3*q**2*r**2*s**6 + 33191894531250*q**4*r**2*s**6 + 50534179687500*p**4*r**3*s**6 + 156656250000000*p*q**2*r**3*s**6 - 85023437500000*p**2*r**4*s**6 + 10125000000000*r**5*s**6 - 2717285156250*p**6*q*s**7 - 11352539062500*p**3*q**3*s**7 - 2593994140625*q**5*s**7 - 47154541015625*p**4*q*r*s**7 - 160644531250000*p*q**3*r*s**7 + 142500000000000*p**2*q*r**2*s**7 - 26757812500000*q*r**3*s**7 - 4364013671875*p**5*s**8 - 94604492187500*p**2*q**2*s**8 + 114379882812500*p**3*r*s**8 + 51116943359375*q**2*r*s**8 - 346435546875000*p*r**2*s**8 + 476837158203125*p*q*s**9 - 476837158203125*s**10 o[4] = 1600*p**11*q**8 + 20800*p**8*q**10 + 45100*p**5*q**12 - 151200*p**2*q**14 - 19200*p**12*q**6*r - 293200*p**9*q**8*r - 794600*p**6*q**10*r + 2634675*p**3*q**12*r + 2640600*q**14*r + 75600*p**13*q**4*r**2 + 1529100*p**10*q**6*r**2 + 6233350*p**7*q**8*r**2 - 12013350*p**4*q**10*r**2 - 29069550*p*q**12*r**2 - 97200*p**14*q**2*r**3 - 3562500*p**11*q**4*r**3 - 26984900*p**8*q**6*r**3 - 15900325*p**5*q**8*r**3 + 76267100*p**2*q**10*r**3 + 3272400*p**12*q**2*r**4 + 59486850*p**9*q**4*r**4 + 221270075*p**6*q**6*r**4 + 74065250*p**3*q**8*r**4 - 300564375*q**10*r**4 - 45569400*p**10*q**2*r**5 - 438666000*p**7*q**4*r**5 - 444821250*p**4*q**6*r**5 + 2448256250*p*q**8*r**5 + 290640000*p**8*q**2*r**6 + 855850000*p**5*q**4*r**6 - 5741875000*p**2*q**6*r**6 - 644000000*p**6*q**2*r**7 + 5574000000*p**3*q**4*r**7 + 4643000000*q**6*r**7 - 1696000000*p**4*q**2*r**8 - 12660000000*p*q**4*r**8 + 7200000000*p**2*q**2*r**9 + 43200*p**13*q**5*s + 572000*p**10*q**7*s - 59800*p**7*q**9*s - 24174625*p**4*q**11*s - 74587500*p*q**13*s - 324000*p**14*q**3*r*s - 5531400*p**11*q**5*r*s - 3712100*p**8*q**7*r*s + 293009275*p**5*q**9*r*s + 1115548875*p**2*q**11*r*s + 583200*p**15*q*r**2*s + 18343800*p**12*q**3*r**2*s + 77911100*p**9*q**5*r**2*s - 957488825*p**6*q**7*r**2*s - 5449661250*p**3*q**9*r**2*s + 960120000*q**11*r**2*s - 23684400*p**13*q*r**3*s - 373761900*p**10*q**3*r**3*s - 27944975*p**7*q**5*r**3*s + 10375740625*p**4*q**7*r**3*s - 4649093750*p*q**9*r**3*s + 395816400*p**11*q*r**4*s + 2910968000*p**8*q**3*r**4*s - 9126162500*p**5*q**5*r**4*s - 11696118750*p**2*q**7*r**4*s - 3028640000*p**9*q*r**5*s - 3251550000*p**6*q**3*r**5*s + 47914250000*p**3*q**5*r**5*s - 30255625000*q**7*r**5*s + 9304000000*p**7*q*r**6*s - 42970000000*p**4*q**3*r**6*s + 31475000000*p*q**5*r**6*s + 2176000000*p**5*q*r**7*s + 62100000000*p**2*q**3*r**7*s - 43200000000*p**3*q*r**8*s - 72000000000*q**3*r**8*s + 291600*p**15*q**2*s**2 + 2702700*p**12*q**4*s**2 - 38692250*p**9*q**6*s**2 - 538903125*p**6*q**8*s**2 - 1613112500*p**3*q**10*s**2 + 320625000*q**12*s**2 - 874800*p**16*r*s**2 - 14166900*p**13*q**2*r*s**2 + 193284900*p**10*q**4*r*s**2 + 3688520500*p**7*q**6*r*s**2 + 11613390625*p**4*q**8*r*s**2 - 15609881250*p*q**10*r*s**2 + 44031600*p**14*r**2*s**2 + 482345550*p**11*q**2*r**2*s**2 - 2020881875*p**8*q**4*r**2*s**2 - 7407026250*p**5*q**6*r**2*s**2 + 136175750000*p**2*q**8*r**2*s**2 - 1000884600*p**12*r**3*s**2 - 8888950000*p**9*q**2*r**3*s**2 - 30101703125*p**6*q**4*r**3*s**2 - 319761000000*p**3*q**6*r**3*s**2 + 51519218750*q**8*r**3*s**2 + 12622395000*p**10*r**4*s**2 + 97032450000*p**7*q**2*r**4*s**2 + 469929218750*p**4*q**4*r**4*s**2 + 291342187500*p*q**6*r**4*s**2 - 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168750000*q**8*s**2 - 2551500*p**10*r*s**2 - 5062500*p**7*q**2*r*s**2 + 712343750*p**4*q**4*r*s**2 + 4788281250*p*q**6*r*s**2 - 256837500*p**8*r**2*s**2 - 3574812500*p**5*q**2*r**2*s**2 - 14967968750*p**2*q**4*r**2*s**2 + 4040937500*p**6*r**3*s**2 + 26400000000*p**3*q**2*r**3*s**2 + 17083984375*q**4*r**3*s**2 - 21812500000*p**4*r**4*s**2 - 24375000000*p*q**2*r**4*s**2 + 39375000000*p**2*r**5*s**2 - 127265625*p**5*q**3*s**3 - 680234375*p**2*q**5*s**3 - 2048203125*p**6*q*r*s**3 - 18794531250*p**3*q**3*r*s**3 - 25050000000*q**5*r*s**3 + 26621875000*p**4*q*r**2*s**3 + 37007812500*p*q**3*r**2*s**3 - 105468750000*p**2*q*r**3*s**3 - 56250000000*q*r**4*s**3 + 1124296875*p**7*s**4 + 9251953125*p**4*q**2*s**4 - 8007812500*p*q**4*s**4 - 4004296875*p**5*r*s**4 + 179931640625*p**2*q**2*r*s**4 - 75703125000*p**3*r**2*s**4 + 133447265625*q**2*r**2*s**4 + 363281250000*p*r**3*s**4 - 91552734375*p**3*q*s**5 - 19531250000*q**3*s**5 - 751953125000*p*q*r*s**5 + 157958984375*p**2*s**6 + 748291015625*r*s**6 o[0] = -14400*p**6*q**6 - 212400*p**3*q**8 - 777600*q**10 + 92100*p**7*q**4*r + 1689675*p**4*q**6*r + 7371000*p*q**8*r - 122850*p**8*q**2*r**2 - 3735250*p**5*q**4*r**2 - 22432500*p**2*q**6*r**2 + 2298750*p**6*q**2*r**3 + 29390625*p**3*q**4*r**3 + 18000000*q**6*r**3 - 17750000*p**4*q**2*r**4 - 62812500*p*q**4*r**4 + 37500000*p**2*q**2*r**5 - 51300*p**8*q**3*s - 768025*p**5*q**5*s - 2801250*p**2*q**7*s - 275400*p**9*q*r*s - 5479875*p**6*q**3*r*s - 35538750*p**3*q**5*r*s - 68850000*q**7*r*s + 12757500*p**7*q*r**2*s + 133640625*p**4*q**3*r**2*s + 222609375*p*q**5*r**2*s - 108500000*p**5*q*r**3*s - 290312500*p**2*q**3*r**3*s + 275000000*p**3*q*r**4*s - 375000000*q**3*r**4*s + 1931850*p**10*s**2 + 40213125*p**7*q**2*s**2 + 253921875*p**4*q**4*s**2 + 464062500*p*q**6*s**2 - 71077500*p**8*r*s**2 - 818746875*p**5*q**2*r*s**2 - 1882265625*p**2*q**4*r*s**2 + 826031250*p**6*r**2*s**2 + 4369687500*p**3*q**2*r**2*s**2 + 3107812500*q**4*r**2*s**2 - 3943750000*p**4*r**3*s**2 - 5000000000*p*q**2*r**3*s**2 + 6562500000*p**2*r**4*s**2 - 295312500*p**6*q*s**3 - 2938906250*p**3*q**3*s**3 - 4848750000*q**5*s**3 + 3791484375*p**4*q*r*s**3 + 7556250000*p*q**3*r*s**3 - 11960937500*p**2*q*r**2*s**3 - 9375000000*q*r**3*s**3 + 1668515625*p**5*s**4 + 20447265625*p**2*q**2*s**4 - 21955078125*p**3*r*s**4 + 18984375000*q**2*r*s**4 + 67382812500*p*r**2*s**4 - 120849609375*p*q*s**5 + 157226562500*s**6 return o @property def a(self): p, q, r, s = self.p, self.q, self.r, self.s a = [0]*6 a[5] = -100*p**7*q**7 - 2175*p**4*q**9 - 10500*p*q**11 + 1100*p**8*q**5*r + 27975*p**5*q**7*r + 152950*p**2*q**9*r - 4125*p**9*q**3*r**2 - 128875*p**6*q**5*r**2 - 830525*p**3*q**7*r**2 + 59450*q**9*r**2 + 5400*p**10*q*r**3 + 243800*p**7*q**3*r**3 + 2082650*p**4*q**5*r**3 - 333925*p*q**7*r**3 - 139200*p**8*q*r**4 - 2406000*p**5*q**3*r**4 - 122600*p**2*q**5*r**4 + 1254400*p**6*q*r**5 + 3776000*p**3*q**3*r**5 + 1832000*q**5*r**5 - 4736000*p**4*q*r**6 - 6720000*p*q**3*r**6 + 6400000*p**2*q*r**7 - 900*p**9*q**4*s - 37400*p**6*q**6*s - 281625*p**3*q**8*s - 435000*q**10*s + 6750*p**10*q**2*r*s + 322300*p**7*q**4*r*s + 2718575*p**4*q**6*r*s + 4214250*p*q**8*r*s - 16200*p**11*r**2*s - 859275*p**8*q**2*r**2*s - 8925475*p**5*q**4*r**2*s - 14427875*p**2*q**6*r**2*s + 453600*p**9*r**3*s + 10038400*p**6*q**2*r**3*s + 17397500*p**3*q**4*r**3*s - 11333125*q**6*r**3*s - 4451200*p**7*r**4*s - 15850000*p**4*q**2*r**4*s + 34000000*p*q**4*r**4*s + 17984000*p**5*r**5*s - 10000000*p**2*q**2*r**5*s - 25600000*p**3*r**6*s - 8000000*q**2*r**6*s + 6075*p**11*q*s**2 - 83250*p**8*q**3*s**2 - 1282500*p**5*q**5*s**2 - 2862500*p**2*q**7*s**2 + 724275*p**9*q*r*s**2 + 9807250*p**6*q**3*r*s**2 + 28374375*p**3*q**5*r*s**2 + 22212500*q**7*r*s**2 - 8982000*p**7*q*r**2*s**2 - 39600000*p**4*q**3*r**2*s**2 - 61746875*p*q**5*r**2*s**2 - 1010000*p**5*q*r**3*s**2 - 1000000*p**2*q**3*r**3*s**2 + 78000000*p**3*q*r**4*s**2 + 30000000*q**3*r**4*s**2 + 80000000*p*q*r**5*s**2 - 759375*p**10*s**3 - 9787500*p**7*q**2*s**3 - 39062500*p**4*q**4*s**3 - 52343750*p*q**6*s**3 + 12301875*p**8*r*s**3 + 98175000*p**5*q**2*r*s**3 + 225078125*p**2*q**4*r*s**3 - 54900000*p**6*r**2*s**3 - 310000000*p**3*q**2*r**2*s**3 - 7890625*q**4*r**2*s**3 + 51250000*p**4*r**3*s**3 - 420000000*p*q**2*r**3*s**3 + 110000000*p**2*r**4*s**3 - 200000000*r**5*s**3 + 2109375*p**6*q*s**4 - 21093750*p**3*q**3*s**4 - 89843750*q**5*s**4 + 182343750*p**4*q*r*s**4 + 733203125*p*q**3*r*s**4 - 196875000*p**2*q*r**2*s**4 + 1125000000*q*r**3*s**4 - 158203125*p**5*s**5 - 566406250*p**2*q**2*s**5 + 101562500*p**3*r*s**5 - 1669921875*q**2*r*s**5 + 1250000000*p*r**2*s**5 - 1220703125*p*q*s**6 + 6103515625*s**7 a[4] = 1000*p**5*q**7 + 7250*p**2*q**9 - 10800*p**6*q**5*r - 96900*p**3*q**7*r - 52500*q**9*r + 37400*p**7*q**3*r**2 + 470850*p**4*q**5*r**2 + 640600*p*q**7*r**2 - 39600*p**8*q*r**3 - 983600*p**5*q**3*r**3 - 2848100*p**2*q**5*r**3 + 814400*p**6*q*r**4 + 6076000*p**3*q**3*r**4 + 2308000*q**5*r**4 - 5024000*p**4*q*r**5 - 9680000*p*q**3*r**5 + 9600000*p**2*q*r**6 + 13800*p**7*q**4*s + 94650*p**4*q**6*s - 26500*p*q**8*s - 86400*p**8*q**2*r*s - 816500*p**5*q**4*r*s - 257500*p**2*q**6*r*s + 91800*p**9*r**2*s + 1853700*p**6*q**2*r**2*s + 630000*p**3*q**4*r**2*s - 8971250*q**6*r**2*s - 2071200*p**7*r**3*s - 7240000*p**4*q**2*r**3*s + 29375000*p*q**4*r**3*s + 14416000*p**5*r**4*s - 5200000*p**2*q**2*r**4*s - 30400000*p**3*r**5*s - 12000000*q**2*r**5*s + 64800*p**9*q*s**2 + 567000*p**6*q**3*s**2 + 1655000*p**3*q**5*s**2 + 6987500*q**7*s**2 + 337500*p**7*q*r*s**2 + 8462500*p**4*q**3*r*s**2 - 5812500*p*q**5*r*s**2 - 24930000*p**5*q*r**2*s**2 - 69125000*p**2*q**3*r**2*s**2 + 103500000*p**3*q*r**3*s**2 + 30000000*q**3*r**3*s**2 + 90000000*p*q*r**4*s**2 - 708750*p**8*s**3 - 5400000*p**5*q**2*s**3 + 8906250*p**2*q**4*s**3 + 18562500*p**6*r*s**3 - 625000*p**3*q**2*r*s**3 + 29687500*q**4*r*s**3 - 75000000*p**4*r**2*s**3 - 416250000*p*q**2*r**2*s**3 + 60000000*p**2*r**3*s**3 - 300000000*r**4*s**3 + 71718750*p**4*q*s**4 + 189062500*p*q**3*s**4 + 210937500*p**2*q*r*s**4 + 1187500000*q*r**2*s**4 - 187500000*p**3*s**5 - 800781250*q**2*s**5 - 390625000*p*r*s**5 a[3] = -500*p**6*q**5 - 6350*p**3*q**7 - 19800*q**9 + 3750*p**7*q**3*r + 65100*p**4*q**5*r + 264950*p*q**7*r - 6750*p**8*q*r**2 - 209050*p**5*q**3*r**2 - 1217250*p**2*q**5*r**2 + 219000*p**6*q*r**3 + 2510000*p**3*q**3*r**3 + 1098500*q**5*r**3 - 2068000*p**4*q*r**4 - 5060000*p*q**3*r**4 + 5200000*p**2*q*r**5 - 6750*p**8*q**2*s - 96350*p**5*q**4*s - 346000*p**2*q**6*s + 20250*p**9*r*s + 459900*p**6*q**2*r*s + 1828750*p**3*q**4*r*s - 2930000*q**6*r*s - 594000*p**7*r**2*s - 4301250*p**4*q**2*r**2*s + 10906250*p*q**4*r**2*s + 5252000*p**5*r**3*s - 1450000*p**2*q**2*r**3*s - 12800000*p**3*r**4*s - 6500000*q**2*r**4*s + 74250*p**7*q*s**2 + 1418750*p**4*q**3*s**2 + 5956250*p*q**5*s**2 - 4297500*p**5*q*r*s**2 - 29906250*p**2*q**3*r*s**2 + 31500000*p**3*q*r**2*s**2 + 12500000*q**3*r**2*s**2 + 35000000*p*q*r**3*s**2 + 1350000*p**6*s**3 + 6093750*p**3*q**2*s**3 + 17500000*q**4*s**3 - 7031250*p**4*r*s**3 - 127812500*p*q**2*r*s**3 + 18750000*p**2*r**2*s**3 - 162500000*r**3*s**3 + 107812500*p**2*q*s**4 + 460937500*q*r*s**4 - 214843750*p*s**5 a[2] = 1950*p**4*q**5 + 14100*p*q**7 - 14350*p**5*q**3*r - 125600*p**2*q**5*r + 27900*p**6*q*r**2 + 402250*p**3*q**3*r**2 + 288250*q**5*r**2 - 436000*p**4*q*r**3 - 1345000*p*q**3*r**3 + 1400000*p**2*q*r**4 + 9450*p**6*q**2*s - 1250*p**3*q**4*s - 465000*q**6*s - 49950*p**7*r*s - 302500*p**4*q**2*r*s + 1718750*p*q**4*r*s + 834000*p**5*r**2*s + 437500*p**2*q**2*r**2*s - 3100000*p**3*r**3*s - 1750000*q**2*r**3*s - 292500*p**5*q*s**2 - 1937500*p**2*q**3*s**2 + 3343750*p**3*q*r*s**2 + 1875000*q**3*r*s**2 + 8125000*p*q*r**2*s**2 - 1406250*p**4*s**3 - 12343750*p*q**2*s**3 + 5312500*p**2*r*s**3 - 43750000*r**2*s**3 + 74218750*q*s**4 a[1] = -300*p**5*q**3 - 2150*p**2*q**5 + 1350*p**6*q*r + 21500*p**3*q**3*r + 61500*q**5*r - 42000*p**4*q*r**2 - 290000*p*q**3*r**2 + 300000*p**2*q*r**3 - 4050*p**7*s - 45000*p**4*q**2*s - 125000*p*q**4*s + 108000*p**5*r*s + 643750*p**2*q**2*r*s - 700000*p**3*r**2*s - 375000*q**2*r**2*s - 93750*p**3*q*s**2 - 312500*q**3*s**2 + 1875000*p*q*r*s**2 - 1406250*p**2*s**3 - 9375000*r*s**3 a[0] = 1250*p**3*q**3 + 9000*q**5 - 4500*p**4*q*r - 46250*p*q**3*r + 50000*p**2*q*r**2 + 6750*p**5*s + 43750*p**2*q**2*s - 75000*p**3*r*s - 62500*q**2*r*s + 156250*p*q*s**2 - 1562500*s**3 return a @property def c(self): p, q, r, s = self.p, self.q, self.r, self.s c = [0]*6 c[5] = -40*p**5*q**11 - 270*p**2*q**13 + 700*p**6*q**9*r + 5165*p**3*q**11*r + 540*q**13*r - 4230*p**7*q**7*r**2 - 31845*p**4*q**9*r**2 + 20880*p*q**11*r**2 + 9645*p**8*q**5*r**3 + 57615*p**5*q**7*r**3 - 358255*p**2*q**9*r**3 - 1880*p**9*q**3*r**4 + 114020*p**6*q**5*r**4 + 2012190*p**3*q**7*r**4 - 26855*q**9*r**4 - 14400*p**10*q*r**5 - 470400*p**7*q**3*r**5 - 5088640*p**4*q**5*r**5 + 920*p*q**7*r**5 + 332800*p**8*q*r**6 + 5797120*p**5*q**3*r**6 + 1608000*p**2*q**5*r**6 - 2611200*p**6*q*r**7 - 7424000*p**3*q**3*r**7 - 2323200*q**5*r**7 + 8601600*p**4*q*r**8 + 9472000*p*q**3*r**8 - 10240000*p**2*q*r**9 - 3060*p**7*q**8*s - 39085*p**4*q**10*s - 132300*p*q**12*s + 36580*p**8*q**6*r*s + 520185*p**5*q**8*r*s + 1969860*p**2*q**10*r*s - 144045*p**9*q**4*r**2*s - 2438425*p**6*q**6*r**2*s - 10809475*p**3*q**8*r**2*s + 518850*q**10*r**2*s + 182520*p**10*q**2*r**3*s + 4533930*p**7*q**4*r**3*s + 26196770*p**4*q**6*r**3*s - 4542325*p*q**8*r**3*s + 21600*p**11*r**4*s - 2208080*p**8*q**2*r**4*s - 24787960*p**5*q**4*r**4*s + 10813900*p**2*q**6*r**4*s - 499200*p**9*r**5*s + 3827840*p**6*q**2*r**5*s + 9596000*p**3*q**4*r**5*s + 22662000*q**6*r**5*s + 3916800*p**7*r**6*s - 29952000*p**4*q**2*r**6*s - 90800000*p*q**4*r**6*s - 12902400*p**5*r**7*s + 87040000*p**2*q**2*r**7*s + 15360000*p**3*r**8*s + 12800000*q**2*r**8*s - 38070*p**9*q**5*s**2 - 566700*p**6*q**7*s**2 - 2574375*p**3*q**9*s**2 - 1822500*q**11*s**2 + 292815*p**10*q**3*r*s**2 + 5170280*p**7*q**5*r*s**2 + 27918125*p**4*q**7*r*s**2 + 21997500*p*q**9*r*s**2 - 573480*p**11*q*r**2*s**2 - 14566350*p**8*q**3*r**2*s**2 - 104851575*p**5*q**5*r**2*s**2 - 96448750*p**2*q**7*r**2*s**2 + 11001240*p**9*q*r**3*s**2 + 147798600*p**6*q**3*r**3*s**2 + 158632750*p**3*q**5*r**3*s**2 - 78222500*q**7*r**3*s**2 - 62819200*p**7*q*r**4*s**2 - 136160000*p**4*q**3*r**4*s**2 + 317555000*p*q**5*r**4*s**2 + 160224000*p**5*q*r**5*s**2 - 267600000*p**2*q**3*r**5*s**2 - 153600000*p**3*q*r**6*s**2 - 120000000*q**3*r**6*s**2 - 32000000*p*q*r**7*s**2 - 127575*p**11*q**2*s**3 - 2148750*p**8*q**4*s**3 - 13652500*p**5*q**6*s**3 - 19531250*p**2*q**8*s**3 + 495720*p**12*r*s**3 + 11856375*p**9*q**2*r*s**3 + 107807500*p**6*q**4*r*s**3 + 222334375*p**3*q**6*r*s**3 + 105062500*q**8*r*s**3 - 11566800*p**10*r**2*s**3 - 216787500*p**7*q**2*r**2*s**3 - 633437500*p**4*q**4*r**2*s**3 - 504484375*p*q**6*r**2*s**3 + 90918000*p**8*r**3*s**3 + 567080000*p**5*q**2*r**3*s**3 + 692937500*p**2*q**4*r**3*s**3 - 326640000*p**6*r**4*s**3 - 339000000*p**3*q**2*r**4*s**3 + 369250000*q**4*r**4*s**3 + 560000000*p**4*r**5*s**3 + 508000000*p*q**2*r**5*s**3 - 480000000*p**2*r**6*s**3 + 320000000*r**7*s**3 - 455625*p**10*q*s**4 - 27562500*p**7*q**3*s**4 - 120593750*p**4*q**5*s**4 - 60312500*p*q**7*s**4 + 110615625*p**8*q*r*s**4 + 662984375*p**5*q**3*r*s**4 + 528515625*p**2*q**5*r*s**4 - 541687500*p**6*q*r**2*s**4 - 1262343750*p**3*q**3*r**2*s**4 - 466406250*q**5*r**2*s**4 + 633000000*p**4*q*r**3*s**4 - 1264375000*p*q**3*r**3*s**4 + 1085000000*p**2*q*r**4*s**4 - 2700000000*q*r**5*s**4 - 68343750*p**9*s**5 - 478828125*p**6*q**2*s**5 - 355468750*p**3*q**4*s**5 - 11718750*q**6*s**5 + 718031250*p**7*r*s**5 + 1658593750*p**4*q**2*r*s**5 + 2212890625*p*q**4*r*s**5 - 2855625000*p**5*r**2*s**5 - 4273437500*p**2*q**2*r**2*s**5 + 4537500000*p**3*r**3*s**5 + 8031250000*q**2*r**3*s**5 - 1750000000*p*r**4*s**5 + 1353515625*p**5*q*s**6 + 1562500000*p**2*q**3*s**6 - 3964843750*p**3*q*r*s**6 - 7226562500*q**3*r*s**6 + 1953125000*p*q*r**2*s**6 - 1757812500*p**4*s**7 - 3173828125*p*q**2*s**7 + 6445312500*p**2*r*s**7 - 3906250000*r**2*s**7 + 6103515625*q*s**8 c[4] = 40*p**6*q**9 + 110*p**3*q**11 - 1080*q**13 - 560*p**7*q**7*r - 1780*p**4*q**9*r + 17370*p*q**11*r + 2850*p**8*q**5*r**2 + 10520*p**5*q**7*r**2 - 115910*p**2*q**9*r**2 - 6090*p**9*q**3*r**3 - 25330*p**6*q**5*r**3 + 448740*p**3*q**7*r**3 + 128230*q**9*r**3 + 4320*p**10*q*r**4 + 16960*p**7*q**3*r**4 - 1143600*p**4*q**5*r**4 - 1410310*p*q**7*r**4 + 3840*p**8*q*r**5 + 1744480*p**5*q**3*r**5 + 5619520*p**2*q**5*r**5 - 1198080*p**6*q*r**6 - 10579200*p**3*q**3*r**6 - 2940800*q**5*r**6 + 8294400*p**4*q*r**7 + 13568000*p*q**3*r**7 - 15360000*p**2*q*r**8 + 840*p**8*q**6*s + 7580*p**5*q**8*s + 24420*p**2*q**10*s - 8100*p**9*q**4*r*s - 94100*p**6*q**6*r*s - 473000*p**3*q**8*r*s - 473400*q**10*r*s + 22680*p**10*q**2*r**2*s + 374370*p**7*q**4*r**2*s + 2888020*p**4*q**6*r**2*s + 5561050*p*q**8*r**2*s - 12960*p**11*r**3*s - 485820*p**8*q**2*r**3*s - 6723440*p**5*q**4*r**3*s - 23561400*p**2*q**6*r**3*s + 190080*p**9*r**4*s + 5894880*p**6*q**2*r**4*s + 50882000*p**3*q**4*r**4*s + 22411500*q**6*r**4*s - 258560*p**7*r**5*s - 46248000*p**4*q**2*r**5*s - 103800000*p*q**4*r**5*s - 3737600*p**5*r**6*s + 119680000*p**2*q**2*r**6*s + 10240000*p**3*r**7*s + 19200000*q**2*r**7*s + 7290*p**10*q**3*s**2 + 117360*p**7*q**5*s**2 + 691250*p**4*q**7*s**2 - 198750*p*q**9*s**2 - 36450*p**11*q*r*s**2 - 854550*p**8*q**3*r*s**2 - 7340700*p**5*q**5*r*s**2 - 2028750*p**2*q**7*r*s**2 + 995490*p**9*q*r**2*s**2 + 18896600*p**6*q**3*r**2*s**2 + 5026500*p**3*q**5*r**2*s**2 - 52272500*q**7*r**2*s**2 - 16636800*p**7*q*r**3*s**2 - 43200000*p**4*q**3*r**3*s**2 + 223426250*p*q**5*r**3*s**2 + 112068000*p**5*q*r**4*s**2 - 177000000*p**2*q**3*r**4*s**2 - 244000000*p**3*q*r**5*s**2 - 156000000*q**3*r**5*s**2 + 43740*p**12*s**3 + 1032750*p**9*q**2*s**3 + 8602500*p**6*q**4*s**3 + 15606250*p**3*q**6*s**3 + 39625000*q**8*s**3 - 1603800*p**10*r*s**3 - 26932500*p**7*q**2*r*s**3 - 19562500*p**4*q**4*r*s**3 - 152000000*p*q**6*r*s**3 + 25555500*p**8*r**2*s**3 + 16230000*p**5*q**2*r**2*s**3 + 42187500*p**2*q**4*r**2*s**3 - 165660000*p**6*r**3*s**3 + 373500000*p**3*q**2*r**3*s**3 + 332937500*q**4*r**3*s**3 + 465000000*p**4*r**4*s**3 + 586000000*p*q**2*r**4*s**3 - 592000000*p**2*r**5*s**3 + 480000000*r**6*s**3 - 1518750*p**8*q*s**4 - 62531250*p**5*q**3*s**4 + 7656250*p**2*q**5*s**4 + 184781250*p**6*q*r*s**4 - 15781250*p**3*q**3*r*s**4 - 135156250*q**5*r*s**4 - 1148250000*p**4*q*r**2*s**4 - 2121406250*p*q**3*r**2*s**4 + 1990000000*p**2*q*r**3*s**4 - 3150000000*q*r**4*s**4 - 2531250*p**7*s**5 + 660937500*p**4*q**2*s**5 + 1339843750*p*q**4*s**5 - 33750000*p**5*r*s**5 - 679687500*p**2*q**2*r*s**5 + 6250000*p**3*r**2*s**5 + 6195312500*q**2*r**2*s**5 + 1125000000*p*r**3*s**5 - 996093750*p**3*q*s**6 - 3125000000*q**3*s**6 - 3222656250*p*q*r*s**6 + 1171875000*p**2*s**7 + 976562500*r*s**7 c[3] = 80*p**4*q**9 + 540*p*q**11 - 600*p**5*q**7*r - 4770*p**2*q**9*r + 1230*p**6*q**5*r**2 + 20900*p**3*q**7*r**2 + 47250*q**9*r**2 - 710*p**7*q**3*r**3 - 84950*p**4*q**5*r**3 - 526310*p*q**7*r**3 + 720*p**8*q*r**4 + 216280*p**5*q**3*r**4 + 2068020*p**2*q**5*r**4 - 198080*p**6*q*r**5 - 3703200*p**3*q**3*r**5 - 1423600*q**5*r**5 + 2860800*p**4*q*r**6 + 7056000*p*q**3*r**6 - 8320000*p**2*q*r**7 - 2720*p**6*q**6*s - 46350*p**3*q**8*s - 178200*q**10*s + 25740*p**7*q**4*r*s + 489490*p**4*q**6*r*s + 2152350*p*q**8*r*s - 61560*p**8*q**2*r**2*s - 1568150*p**5*q**4*r**2*s - 9060500*p**2*q**6*r**2*s + 24840*p**9*r**3*s + 1692380*p**6*q**2*r**3*s + 18098250*p**3*q**4*r**3*s + 9387750*q**6*r**3*s - 382560*p**7*r**4*s - 16818000*p**4*q**2*r**4*s - 49325000*p*q**4*r**4*s + 1212800*p**5*r**5*s + 64840000*p**2*q**2*r**5*s - 320000*p**3*r**6*s + 10400000*q**2*r**6*s - 36450*p**8*q**3*s**2 - 588350*p**5*q**5*s**2 - 2156250*p**2*q**7*s**2 + 123930*p**9*q*r*s**2 + 2879700*p**6*q**3*r*s**2 + 12548000*p**3*q**5*r*s**2 - 14445000*q**7*r*s**2 - 3233250*p**7*q*r**2*s**2 - 28485000*p**4*q**3*r**2*s**2 + 72231250*p*q**5*r**2*s**2 + 32093000*p**5*q*r**3*s**2 - 61275000*p**2*q**3*r**3*s**2 - 107500000*p**3*q*r**4*s**2 - 78500000*q**3*r**4*s**2 + 22000000*p*q*r**5*s**2 - 72900*p**10*s**3 - 1215000*p**7*q**2*s**3 - 2937500*p**4*q**4*s**3 + 9156250*p*q**6*s**3 + 2612250*p**8*r*s**3 + 16560000*p**5*q**2*r*s**3 - 75468750*p**2*q**4*r*s**3 - 32737500*p**6*r**2*s**3 + 169062500*p**3*q**2*r**2*s**3 + 121718750*q**4*r**2*s**3 + 160250000*p**4*r**3*s**3 + 219750000*p*q**2*r**3*s**3 - 317000000*p**2*r**4*s**3 + 260000000*r**5*s**3 + 2531250*p**6*q*s**4 + 22500000*p**3*q**3*s**4 + 39843750*q**5*s**4 - 266343750*p**4*q*r*s**4 - 776406250*p*q**3*r*s**4 + 789062500*p**2*q*r**2*s**4 - 1368750000*q*r**3*s**4 + 67500000*p**5*s**5 + 441406250*p**2*q**2*s**5 - 311718750*p**3*r*s**5 + 1785156250*q**2*r*s**5 + 546875000*p*r**2*s**5 - 1269531250*p*q*s**6 + 488281250*s**7 c[2] = 120*p**5*q**7 + 810*p**2*q**9 - 1280*p**6*q**5*r - 9160*p**3*q**7*r + 3780*q**9*r + 4530*p**7*q**3*r**2 + 36640*p**4*q**5*r**2 - 45270*p*q**7*r**2 - 5400*p**8*q*r**3 - 60920*p**5*q**3*r**3 + 200050*p**2*q**5*r**3 + 31200*p**6*q*r**4 - 476000*p**3*q**3*r**4 - 378200*q**5*r**4 + 521600*p**4*q*r**5 + 1872000*p*q**3*r**5 - 2240000*p**2*q*r**6 + 1440*p**7*q**4*s + 15310*p**4*q**6*s + 59400*p*q**8*s - 9180*p**8*q**2*r*s - 115240*p**5*q**4*r*s - 589650*p**2*q**6*r*s + 16200*p**9*r**2*s + 316710*p**6*q**2*r**2*s + 2547750*p**3*q**4*r**2*s + 2178000*q**6*r**2*s - 259200*p**7*r**3*s - 4123000*p**4*q**2*r**3*s - 11700000*p*q**4*r**3*s + 937600*p**5*r**4*s + 16340000*p**2*q**2*r**4*s - 640000*p**3*r**5*s + 2800000*q**2*r**5*s - 2430*p**9*q*s**2 - 54450*p**6*q**3*s**2 - 285500*p**3*q**5*s**2 - 2767500*q**7*s**2 + 43200*p**7*q*r*s**2 - 916250*p**4*q**3*r*s**2 + 14482500*p*q**5*r*s**2 + 4806000*p**5*q*r**2*s**2 - 13212500*p**2*q**3*r**2*s**2 - 25400000*p**3*q*r**3*s**2 - 18750000*q**3*r**3*s**2 + 8000000*p*q*r**4*s**2 + 121500*p**8*s**3 + 2058750*p**5*q**2*s**3 - 6656250*p**2*q**4*s**3 - 6716250*p**6*r*s**3 + 24125000*p**3*q**2*r*s**3 + 23875000*q**4*r*s**3 + 43125000*p**4*r**2*s**3 + 45750000*p*q**2*r**2*s**3 - 87500000*p**2*r**3*s**3 + 70000000*r**4*s**3 - 44437500*p**4*q*s**4 - 107968750*p*q**3*s**4 + 159531250*p**2*q*r*s**4 - 284375000*q*r**2*s**4 + 7031250*p**3*s**5 + 265625000*q**2*s**5 + 31250000*p*r*s**5 c[1] = 160*p**3*q**7 + 1080*q**9 - 1080*p**4*q**5*r - 8730*p*q**7*r + 1510*p**5*q**3*r**2 + 20420*p**2*q**5*r**2 + 720*p**6*q*r**3 - 23200*p**3*q**3*r**3 - 79900*q**5*r**3 + 35200*p**4*q*r**4 + 404000*p*q**3*r**4 - 480000*p**2*q*r**5 + 960*p**5*q**4*s + 2850*p**2*q**6*s + 540*p**6*q**2*r*s + 63500*p**3*q**4*r*s + 319500*q**6*r*s - 7560*p**7*r**2*s - 253500*p**4*q**2*r**2*s - 1806250*p*q**4*r**2*s + 91200*p**5*r**3*s + 2600000*p**2*q**2*r**3*s - 80000*p**3*r**4*s + 600000*q**2*r**4*s - 4050*p**7*q*s**2 - 120000*p**4*q**3*s**2 - 273750*p*q**5*s**2 + 425250*p**5*q*r*s**2 + 2325000*p**2*q**3*r*s**2 - 5400000*p**3*q*r**2*s**2 - 2875000*q**3*r**2*s**2 + 1500000*p*q*r**3*s**2 - 303750*p**6*s**3 - 843750*p**3*q**2*s**3 - 812500*q**4*s**3 + 5062500*p**4*r*s**3 + 13312500*p*q**2*r*s**3 - 14500000*p**2*r**2*s**3 + 15000000*r**3*s**3 - 3750000*p**2*q*s**4 - 35937500*q*r*s**4 + 11718750*p*s**5 c[0] = 80*p**4*q**5 + 540*p*q**7 - 600*p**5*q**3*r - 4770*p**2*q**5*r + 1080*p**6*q*r**2 + 11200*p**3*q**3*r**2 - 12150*q**5*r**2 - 4800*p**4*q*r**3 + 64000*p*q**3*r**3 - 80000*p**2*q*r**4 + 1080*p**6*q**2*s + 13250*p**3*q**4*s + 54000*q**6*s - 3240*p**7*r*s - 56250*p**4*q**2*r*s - 337500*p*q**4*r*s + 43200*p**5*r**2*s + 560000*p**2*q**2*r**2*s - 80000*p**3*r**3*s + 100000*q**2*r**3*s + 6750*p**5*q*s**2 + 225000*p**2*q**3*s**2 - 900000*p**3*q*r*s**2 - 562500*q**3*r*s**2 + 500000*p*q*r**2*s**2 + 843750*p**4*s**3 + 1937500*p*q**2*s**3 - 3000000*p**2*r*s**3 + 2500000*r**2*s**3 - 5468750*q*s**4 return c @property def F(self): p, q, r, s = self.p, self.q, self.r, self.s F = 4*p**6*q**6 + 59*p**3*q**8 + 216*q**10 - 36*p**7*q**4*r - 623*p**4*q**6*r - 2610*p*q**8*r + 81*p**8*q**2*r**2 + 2015*p**5*q**4*r**2 + 10825*p**2*q**6*r**2 - 1800*p**6*q**2*r**3 - 17500*p**3*q**4*r**3 + 625*q**6*r**3 + 10000*p**4*q**2*r**4 + 108*p**8*q**3*s + 1584*p**5*q**5*s + 5700*p**2*q**7*s - 486*p**9*q*r*s - 9720*p**6*q**3*r*s - 45050*p**3*q**5*r*s - 9000*q**7*r*s + 10800*p**7*q*r**2*s + 92500*p**4*q**3*r**2*s + 32500*p*q**5*r**2*s - 60000*p**5*q*r**3*s - 50000*p**2*q**3*r**3*s + 729*p**10*s**2 + 12150*p**7*q**2*s**2 + 60000*p**4*q**4*s**2 + 93750*p*q**6*s**2 - 18225*p**8*r*s**2 - 175500*p**5*q**2*r*s**2 - 478125*p**2*q**4*r*s**2 + 135000*p**6*r**2*s**2 + 850000*p**3*q**2*r**2*s**2 + 15625*q**4*r**2*s**2 - 250000*p**4*r**3*s**2 + 225000*p**3*q**3*s**3 + 175000*q**5*s**3 - 1012500*p**4*q*r*s**3 - 1187500*p*q**3*r*s**3 + 1250000*p**2*q*r**2*s**3 + 928125*p**5*s**4 + 1875000*p**2*q**2*s**4 - 2812500*p**3*r*s**4 - 390625*q**2*r*s**4 - 9765625*s**6 return F def l0(self, theta): F = self.F a = self.a l0 = Poly(a, x).eval(theta)/F return l0 def T(self, theta, d): F = self.F T = [0]*5 b = self.b # Note that the order of sublists of the b's has been reversed compared to the paper T[1] = -Poly(b[1], x).eval(theta)/(2*F) T[2] = Poly(b[2], x).eval(theta)/(2*d*F) T[3] = Poly(b[3], x).eval(theta)/(2*F) T[4] = Poly(b[4], x).eval(theta)/(2*d*F) return T def order(self, theta, d): F = self.F o = self.o order = Poly(o, x).eval(theta)/(d*F) return N(order) def uv(self, theta, d): c = self.c u = self.q*Rational(-25, 2) v = Poly(c, x).eval(theta)/(2*d*self.F) return N(u), N(v) @property def zeta(self): return [self.zeta1, self.zeta2, self.zeta3, self.zeta4]
b33258c0f63c0c21ce03928f6d03006e55744fa71584cbfccbd8b0e0bad25b10
"""Implementation of RootOf class and related tools. """ from sympy import Basic from sympy.core import (S, Expr, Integer, Float, I, oo, Add, Lambda, symbols, sympify, Rational, Dummy) from sympy.core.cache import cacheit from sympy.core.compatibility import ordered from sympy.core.relational import is_le from sympy.polys.domains import QQ from sympy.polys.polyerrors import ( MultivariatePolynomialError, GeneratorsNeeded, PolynomialError, DomainError) from sympy.polys.polyfuncs import symmetrize, viete from sympy.polys.polyroots import ( roots_linear, roots_quadratic, roots_binomial, preprocess_roots, roots) from sympy.polys.polytools import Poly, PurePoly, factor from sympy.polys.rationaltools import together from sympy.polys.rootisolation import ( dup_isolate_complex_roots_sqf, dup_isolate_real_roots_sqf) from sympy.utilities import lambdify, public, sift, numbered_symbols from mpmath import mpf, mpc, findroot, workprec from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps from sympy.multipledispatch import dispatch from itertools import chain __all__ = ['CRootOf'] class _pure_key_dict: """A minimal dictionary that makes sure that the key is a univariate PurePoly instance. Examples ======== Only the following actions are guaranteed: >>> from sympy.polys.rootoftools import _pure_key_dict >>> from sympy import PurePoly >>> from sympy.abc import x, y 1) creation >>> P = _pure_key_dict() 2) assignment for a PurePoly or univariate polynomial >>> P[x] = 1 >>> P[PurePoly(x - y, x)] = 2 3) retrieval based on PurePoly key comparison (use this instead of the get method) >>> P[y] 1 4) KeyError when trying to retrieve a nonexisting key >>> P[y + 1] Traceback (most recent call last): ... KeyError: PurePoly(y + 1, y, domain='ZZ') 5) ability to query with ``in`` >>> x + 1 in P False NOTE: this is a *not* a dictionary. It is a very basic object for internal use that makes sure to always address its cache via PurePoly instances. It does not, for example, implement ``get`` or ``setdefault``. """ def __init__(self): self._dict = {} def __getitem__(self, k): if not isinstance(k, PurePoly): if not (isinstance(k, Expr) and len(k.free_symbols) == 1): raise KeyError k = PurePoly(k, expand=False) return self._dict[k] def __setitem__(self, k, v): if not isinstance(k, PurePoly): if not (isinstance(k, Expr) and len(k.free_symbols) == 1): raise ValueError('expecting univariate expression') k = PurePoly(k, expand=False) self._dict[k] = v def __contains__(self, k): try: self[k] return True except KeyError: return False _reals_cache = _pure_key_dict() _complexes_cache = _pure_key_dict() def _pure_factors(poly): _, factors = poly.factor_list() return [(PurePoly(f, expand=False), m) for f, m in factors] def _imag_count_of_factor(f): """Return the number of imaginary roots for irreducible univariate polynomial ``f``. """ terms = [(i, j) for (i,), j in f.terms()] if any(i % 2 for i, j in terms): return 0 # update signs even = [(i, I**i*j) for i, j in terms] even = Poly.from_dict(dict(even), Dummy('x')) return int(even.count_roots(-oo, oo)) @public def rootof(f, x, index=None, radicals=True, expand=True): """An indexed root of a univariate polynomial. Returns either a :obj:`ComplexRootOf` object or an explicit expression involving radicals. Parameters ========== f : Expr Univariate polynomial. x : Symbol, optional Generator for ``f``. index : int or Integer radicals : bool Return a radical expression if possible. expand : bool Expand ``f``. """ return CRootOf(f, x, index=index, radicals=radicals, expand=expand) @public class RootOf(Expr): """Represents a root of a univariate polynomial. Base class for roots of different kinds of polynomials. Only complex roots are currently supported. """ __slots__ = ('poly',) def __new__(cls, f, x, index=None, radicals=True, expand=True): """Construct a new ``CRootOf`` object for ``k``-th root of ``f``.""" return rootof(f, x, index=index, radicals=radicals, expand=expand) @public class ComplexRootOf(RootOf): """Represents an indexed complex root of a polynomial. Roots of a univariate polynomial separated into disjoint real or complex intervals and indexed in a fixed order. Currently only rational coefficients are allowed. Can be imported as ``CRootOf``. To avoid confusion, the generator must be a Symbol. Examples ======== >>> from sympy import CRootOf, rootof >>> from sympy.abc import x CRootOf is a way to reference a particular root of a polynomial. If there is a rational root, it will be returned: >>> CRootOf.clear_cache() # for doctest reproducibility >>> CRootOf(x**2 - 4, 0) -2 Whether roots involving radicals are returned or not depends on whether the ``radicals`` flag is true (which is set to True with rootof): >>> CRootOf(x**2 - 3, 0) CRootOf(x**2 - 3, 0) >>> CRootOf(x**2 - 3, 0, radicals=True) -sqrt(3) >>> rootof(x**2 - 3, 0) -sqrt(3) The following cannot be expressed in terms of radicals: >>> r = rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0); r CRootOf(4*x**5 + 16*x**3 + 12*x**2 + 7, 0) The root bounds can be seen, however, and they are used by the evaluation methods to get numerical approximations for the root. >>> interval = r._get_interval(); interval (-1, 0) >>> r.evalf(2) -0.98 The evalf method refines the width of the root bounds until it guarantees that any decimal approximation within those bounds will satisfy the desired precision. It then stores the refined interval so subsequent requests at or below the requested precision will not have to recompute the root bounds and will return very quickly. Before evaluation above, the interval was >>> interval (-1, 0) After evaluation it is now >>> r._get_interval() # doctest: +SKIP (-165/169, -206/211) To reset all intervals for a given polynomial, the :meth:`_reset` method can be called from any CRootOf instance of the polynomial: >>> r._reset() >>> r._get_interval() (-1, 0) The :meth:`eval_approx` method will also find the root to a given precision but the interval is not modified unless the search for the root fails to converge within the root bounds. And the secant method is used to find the root. (The ``evalf`` method uses bisection and will always update the interval.) >>> r.eval_approx(2) -0.98 The interval needed to be slightly updated to find that root: >>> r._get_interval() (-1, -1/2) The ``evalf_rational`` will compute a rational approximation of the root to the desired accuracy or precision. >>> r.eval_rational(n=2) -69629/71318 >>> t = CRootOf(x**3 + 10*x + 1, 1) >>> t.eval_rational(1e-1) 15/256 - 805*I/256 >>> t.eval_rational(1e-1, 1e-4) 3275/65536 - 414645*I/131072 >>> t.eval_rational(1e-4, 1e-4) 6545/131072 - 414645*I/131072 >>> t.eval_rational(n=2) 104755/2097152 - 6634255*I/2097152 Notes ===== Although a PurePoly can be constructed from a non-symbol generator RootOf instances of non-symbols are disallowed to avoid confusion over what root is being represented. >>> from sympy import exp, PurePoly >>> PurePoly(x) == PurePoly(exp(x)) True >>> CRootOf(x - 1, 0) 1 >>> CRootOf(exp(x) - 1, 0) # would correspond to x == 0 Traceback (most recent call last): ... sympy.polys.polyerrors.PolynomialError: generator must be a Symbol See Also ======== eval_approx eval_rational """ __slots__ = ('index',) is_complex = True is_number = True is_finite = True def __new__(cls, f, x, index=None, radicals=False, expand=True): """ Construct an indexed complex root of a polynomial. See ``rootof`` for the parameters. The default value of ``radicals`` is ``False`` to satisfy ``eval(srepr(expr) == expr``. """ x = sympify(x) if index is None and x.is_Integer: x, index = None, x else: index = sympify(index) if index is not None and index.is_Integer: index = int(index) else: raise ValueError("expected an integer root index, got %s" % index) poly = PurePoly(f, x, greedy=False, expand=expand) if not poly.is_univariate: raise PolynomialError("only univariate polynomials are allowed") if not poly.gen.is_Symbol: # PurePoly(sin(x) + 1) == PurePoly(x + 1) but the roots of # x for each are not the same: issue 8617 raise PolynomialError("generator must be a Symbol") degree = poly.degree() if degree <= 0: raise PolynomialError("can't construct CRootOf object for %s" % f) if index < -degree or index >= degree: raise IndexError("root index out of [%d, %d] range, got %d" % (-degree, degree - 1, index)) elif index < 0: index += degree dom = poly.get_domain() if not dom.is_Exact: poly = poly.to_exact() roots = cls._roots_trivial(poly, radicals) if roots is not None: return roots[index] coeff, poly = preprocess_roots(poly) dom = poly.get_domain() if not dom.is_ZZ: raise NotImplementedError("CRootOf is not supported over %s" % dom) root = cls._indexed_root(poly, index) return coeff * cls._postprocess_root(root, radicals) @classmethod def _new(cls, poly, index): """Construct new ``CRootOf`` object from raw data. """ obj = Expr.__new__(cls) obj.poly = PurePoly(poly) obj.index = index try: _reals_cache[obj.poly] = _reals_cache[poly] _complexes_cache[obj.poly] = _complexes_cache[poly] except KeyError: pass return obj def _hashable_content(self): return (self.poly, self.index) @property def expr(self): return self.poly.as_expr() @property def args(self): return (self.expr, Integer(self.index)) @property def free_symbols(self): # CRootOf currently only works with univariate expressions # whose poly attribute should be a PurePoly with no free # symbols return set() def _eval_is_real(self): """Return ``True`` if the root is real. """ return self.index < len(_reals_cache[self.poly]) def _eval_is_imaginary(self): """Return ``True`` if the root is imaginary. """ if self.index >= len(_reals_cache[self.poly]): ivl = self._get_interval() return ivl.ax*ivl.bx <= 0 # all others are on one side or the other return False # XXX is this necessary? @classmethod def real_roots(cls, poly, radicals=True): """Get real roots of a polynomial. """ return cls._get_roots("_real_roots", poly, radicals) @classmethod def all_roots(cls, poly, radicals=True): """Get real and complex roots of a polynomial. """ return cls._get_roots("_all_roots", poly, radicals) @classmethod def _get_reals_sqf(cls, currentfactor, use_cache=True): """Get real root isolating intervals for a square-free factor.""" if use_cache and currentfactor in _reals_cache: real_part = _reals_cache[currentfactor] else: _reals_cache[currentfactor] = real_part = \ dup_isolate_real_roots_sqf( currentfactor.rep.rep, currentfactor.rep.dom, blackbox=True) return real_part @classmethod def _get_complexes_sqf(cls, currentfactor, use_cache=True): """Get complex root isolating intervals for a square-free factor.""" if use_cache and currentfactor in _complexes_cache: complex_part = _complexes_cache[currentfactor] else: _complexes_cache[currentfactor] = complex_part = \ dup_isolate_complex_roots_sqf( currentfactor.rep.rep, currentfactor.rep.dom, blackbox=True) return complex_part @classmethod def _get_reals(cls, factors, use_cache=True): """Compute real root isolating intervals for a list of factors. """ reals = [] for currentfactor, k in factors: try: if not use_cache: raise KeyError r = _reals_cache[currentfactor] reals.extend([(i, currentfactor, k) for i in r]) except KeyError: real_part = cls._get_reals_sqf(currentfactor, use_cache) new = [(root, currentfactor, k) for root in real_part] reals.extend(new) reals = cls._reals_sorted(reals) return reals @classmethod def _get_complexes(cls, factors, use_cache=True): """Compute complex root isolating intervals for a list of factors. """ complexes = [] for currentfactor, k in ordered(factors): try: if not use_cache: raise KeyError c = _complexes_cache[currentfactor] complexes.extend([(i, currentfactor, k) for i in c]) except KeyError: complex_part = cls._get_complexes_sqf(currentfactor, use_cache) new = [(root, currentfactor, k) for root in complex_part] complexes.extend(new) complexes = cls._complexes_sorted(complexes) return complexes @classmethod def _reals_sorted(cls, reals): """Make real isolating intervals disjoint and sort roots. """ cache = {} for i, (u, f, k) in enumerate(reals): for j, (v, g, m) in enumerate(reals[i + 1:]): u, v = u.refine_disjoint(v) reals[i + j + 1] = (v, g, m) reals[i] = (u, f, k) reals = sorted(reals, key=lambda r: r[0].a) for root, currentfactor, _ in reals: if currentfactor in cache: cache[currentfactor].append(root) else: cache[currentfactor] = [root] for currentfactor, root in cache.items(): _reals_cache[currentfactor] = root return reals @classmethod def _refine_imaginary(cls, complexes): sifted = sift(complexes, lambda c: c[1]) complexes = [] for f in ordered(sifted): nimag = _imag_count_of_factor(f) if nimag == 0: # refine until xbounds are neg or pos for u, f, k in sifted[f]: while u.ax*u.bx <= 0: u = u._inner_refine() complexes.append((u, f, k)) else: # refine until all but nimag xbounds are neg or pos potential_imag = list(range(len(sifted[f]))) while True: assert len(potential_imag) > 1 for i in list(potential_imag): u, f, k = sifted[f][i] if u.ax*u.bx > 0: potential_imag.remove(i) elif u.ax != u.bx: u = u._inner_refine() sifted[f][i] = u, f, k if len(potential_imag) == nimag: break complexes.extend(sifted[f]) return complexes @classmethod def _refine_complexes(cls, complexes): """return complexes such that no bounding rectangles of non-conjugate roots would intersect. In addition, assure that neither ay nor by is 0 to guarantee that non-real roots are distinct from real roots in terms of the y-bounds. """ # get the intervals pairwise-disjoint. # If rectangles were drawn around the coordinates of the bounding # rectangles, no rectangles would intersect after this procedure. for i, (u, f, k) in enumerate(complexes): for j, (v, g, m) in enumerate(complexes[i + 1:]): u, v = u.refine_disjoint(v) complexes[i + j + 1] = (v, g, m) complexes[i] = (u, f, k) # refine until the x-bounds are unambiguously positive or negative # for non-imaginary roots complexes = cls._refine_imaginary(complexes) # make sure that all y bounds are off the real axis # and on the same side of the axis for i, (u, f, k) in enumerate(complexes): while u.ay*u.by <= 0: u = u.refine() complexes[i] = u, f, k return complexes @classmethod def _complexes_sorted(cls, complexes): """Make complex isolating intervals disjoint and sort roots. """ complexes = cls._refine_complexes(complexes) # XXX don't sort until you are sure that it is compatible # with the indexing method but assert that the desired state # is not broken C, F = 0, 1 # location of ComplexInterval and factor fs = {i[F] for i in complexes} for i in range(1, len(complexes)): if complexes[i][F] != complexes[i - 1][F]: # if this fails the factors of a root were not # contiguous because a discontinuity should only # happen once fs.remove(complexes[i - 1][F]) for i in range(len(complexes)): # negative im part (conj=True) comes before # positive im part (conj=False) assert complexes[i][C].conj is (i % 2 == 0) # update cache cache = {} # -- collate for root, currentfactor, _ in complexes: cache.setdefault(currentfactor, []).append(root) # -- store for currentfactor, root in cache.items(): _complexes_cache[currentfactor] = root return complexes @classmethod def _reals_index(cls, reals, index): """ Map initial real root index to an index in a factor where the root belongs. """ i = 0 for j, (_, currentfactor, k) in enumerate(reals): if index < i + k: poly, index = currentfactor, 0 for _, currentfactor, _ in reals[:j]: if currentfactor == poly: index += 1 return poly, index else: i += k @classmethod def _complexes_index(cls, complexes, index): """ Map initial complex root index to an index in a factor where the root belongs. """ i = 0 for j, (_, currentfactor, k) in enumerate(complexes): if index < i + k: poly, index = currentfactor, 0 for _, currentfactor, _ in complexes[:j]: if currentfactor == poly: index += 1 index += len(_reals_cache[poly]) return poly, index else: i += k @classmethod def _count_roots(cls, roots): """Count the number of real or complex roots with multiplicities.""" return sum([k for _, _, k in roots]) @classmethod def _indexed_root(cls, poly, index): """Get a root of a composite polynomial by index. """ factors = _pure_factors(poly) reals = cls._get_reals(factors) reals_count = cls._count_roots(reals) if index < reals_count: return cls._reals_index(reals, index) else: complexes = cls._get_complexes(factors) return cls._complexes_index(complexes, index - reals_count) @classmethod def _real_roots(cls, poly): """Get real roots of a composite polynomial. """ factors = _pure_factors(poly) reals = cls._get_reals(factors) reals_count = cls._count_roots(reals) roots = [] for index in range(0, reals_count): roots.append(cls._reals_index(reals, index)) return roots def _reset(self): """ Reset all intervals """ self._all_roots(self.poly, use_cache=False) @classmethod def _all_roots(cls, poly, use_cache=True): """Get real and complex roots of a composite polynomial. """ factors = _pure_factors(poly) reals = cls._get_reals(factors, use_cache=use_cache) reals_count = cls._count_roots(reals) roots = [] for index in range(0, reals_count): roots.append(cls._reals_index(reals, index)) complexes = cls._get_complexes(factors, use_cache=use_cache) complexes_count = cls._count_roots(complexes) for index in range(0, complexes_count): roots.append(cls._complexes_index(complexes, index)) return roots @classmethod @cacheit def _roots_trivial(cls, poly, radicals): """Compute roots in linear, quadratic and binomial cases. """ if poly.degree() == 1: return roots_linear(poly) if not radicals: return None if poly.degree() == 2: return roots_quadratic(poly) elif poly.length() == 2 and poly.TC(): return roots_binomial(poly) else: return None @classmethod def _preprocess_roots(cls, poly): """Take heroic measures to make ``poly`` compatible with ``CRootOf``.""" dom = poly.get_domain() if not dom.is_Exact: poly = poly.to_exact() coeff, poly = preprocess_roots(poly) dom = poly.get_domain() if not dom.is_ZZ: raise NotImplementedError( "sorted roots not supported over %s" % dom) return coeff, poly @classmethod def _postprocess_root(cls, root, radicals): """Return the root if it is trivial or a ``CRootOf`` object. """ poly, index = root roots = cls._roots_trivial(poly, radicals) if roots is not None: return roots[index] else: return cls._new(poly, index) @classmethod def _get_roots(cls, method, poly, radicals): """Return postprocessed roots of specified kind. """ if not poly.is_univariate: raise PolynomialError("only univariate polynomials are allowed") # get rid of gen and it's free symbol d = Dummy() poly = poly.subs(poly.gen, d) x = symbols('x') # see what others are left and select x or a numbered x # that doesn't clash free_names = {str(i) for i in poly.free_symbols} for x in chain((symbols('x'),), numbered_symbols('x')): if x.name not in free_names: poly = poly.xreplace({d: x}) break coeff, poly = cls._preprocess_roots(poly) roots = [] for root in getattr(cls, method)(poly): roots.append(coeff*cls._postprocess_root(root, radicals)) return roots @classmethod def clear_cache(cls): """Reset cache for reals and complexes. The intervals used to approximate a root instance are updated as needed. When a request is made to see the intervals, the most current values are shown. `clear_cache` will reset all CRootOf instances back to their original state. See Also ======== _reset """ global _reals_cache, _complexes_cache _reals_cache = _pure_key_dict() _complexes_cache = _pure_key_dict() def _get_interval(self): """Internal function for retrieving isolation interval from cache. """ if self.is_real: return _reals_cache[self.poly][self.index] else: reals_count = len(_reals_cache[self.poly]) return _complexes_cache[self.poly][self.index - reals_count] def _set_interval(self, interval): """Internal function for updating isolation interval in cache. """ if self.is_real: _reals_cache[self.poly][self.index] = interval else: reals_count = len(_reals_cache[self.poly]) _complexes_cache[self.poly][self.index - reals_count] = interval def _eval_subs(self, old, new): # don't allow subs to change anything return self def _eval_conjugate(self): if self.is_real: return self expr, i = self.args return self.func(expr, i + (1 if self._get_interval().conj else -1)) def eval_approx(self, n): """Evaluate this complex root to the given precision. This uses secant method and root bounds are used to both generate an initial guess and to check that the root returned is valid. If ever the method converges outside the root bounds, the bounds will be made smaller and updated. """ prec = dps_to_prec(n) with workprec(prec): g = self.poly.gen if not g.is_Symbol: d = Dummy('x') if self.is_imaginary: d *= I func = lambdify(d, self.expr.subs(g, d)) else: expr = self.expr if self.is_imaginary: expr = self.expr.subs(g, I*g) func = lambdify(g, expr) interval = self._get_interval() while True: if self.is_real: a = mpf(str(interval.a)) b = mpf(str(interval.b)) if a == b: root = a break x0 = mpf(str(interval.center)) x1 = x0 + mpf(str(interval.dx))/4 elif self.is_imaginary: a = mpf(str(interval.ay)) b = mpf(str(interval.by)) if a == b: root = mpc(mpf('0'), a) break x0 = mpf(str(interval.center[1])) x1 = x0 + mpf(str(interval.dy))/4 else: ax = mpf(str(interval.ax)) bx = mpf(str(interval.bx)) ay = mpf(str(interval.ay)) by = mpf(str(interval.by)) if ax == bx and ay == by: root = mpc(ax, ay) break x0 = mpc(*map(str, interval.center)) x1 = x0 + mpc(*map(str, (interval.dx, interval.dy)))/4 try: # without a tolerance, this will return when (to within # the given precision) x_i == x_{i-1} root = findroot(func, (x0, x1)) # If the (real or complex) root is not in the 'interval', # then keep refining the interval. This happens if findroot # accidentally finds a different root outside of this # interval because our initial estimate 'x0' was not close # enough. It is also possible that the secant method will # get trapped by a max/min in the interval; the root # verification by findroot will raise a ValueError in this # case and the interval will then be tightened -- and # eventually the root will be found. # # It is also possible that findroot will not have any # successful iterations to process (in which case it # will fail to initialize a variable that is tested # after the iterations and raise an UnboundLocalError). if self.is_real or self.is_imaginary: if not bool(root.imag) == self.is_real and ( a <= root <= b): if self.is_imaginary: root = mpc(mpf('0'), root.real) break elif (ax <= root.real <= bx and ay <= root.imag <= by): break except (UnboundLocalError, ValueError): pass interval = interval.refine() # update the interval so we at least (for this precision or # less) don't have much work to do to recompute the root self._set_interval(interval) return (Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)) def _eval_evalf(self, prec, **kwargs): """Evaluate this complex root to the given precision.""" # all kwargs are ignored return self.eval_rational(n=prec_to_dps(prec))._evalf(prec) def eval_rational(self, dx=None, dy=None, n=15): """ Return a Rational approximation of ``self`` that has real and imaginary component approximations that are within ``dx`` and ``dy`` of the true values, respectively. Alternatively, ``n`` digits of precision can be specified. The interval is refined with bisection and is sure to converge. The root bounds are updated when the refinement is complete so recalculation at the same or lesser precision will not have to repeat the refinement and should be much faster. The following example first obtains Rational approximation to 1e-8 accuracy for all roots of the 4-th order Legendre polynomial. Since the roots are all less than 1, this will ensure the decimal representation of the approximation will be correct (including rounding) to 6 digits: >>> from sympy import legendre_poly, Symbol >>> x = Symbol("x") >>> p = legendre_poly(4, x, polys=True) >>> r = p.real_roots()[-1] >>> r.eval_rational(10**-8).n(6) 0.861136 It is not necessary to a two-step calculation, however: the decimal representation can be computed directly: >>> r.evalf(17) 0.86113631159405258 """ dy = dy or dx if dx: rtol = None dx = dx if isinstance(dx, Rational) else Rational(str(dx)) dy = dy if isinstance(dy, Rational) else Rational(str(dy)) else: # 5 binary (or 2 decimal) digits are needed to ensure that # a given digit is correctly rounded # prec_to_dps(dps_to_prec(n) + 5) - n <= 2 (tested for # n in range(1000000) rtol = S(10)**-(n + 2) # +2 for guard digits interval = self._get_interval() while True: if self.is_real: if rtol: dx = abs(interval.center*rtol) interval = interval.refine_size(dx=dx) c = interval.center real = Rational(c) imag = S.Zero if not rtol or interval.dx < abs(c*rtol): break elif self.is_imaginary: if rtol: dy = abs(interval.center[1]*rtol) dx = 1 interval = interval.refine_size(dx=dx, dy=dy) c = interval.center[1] imag = Rational(c) real = S.Zero if not rtol or interval.dy < abs(c*rtol): break else: if rtol: dx = abs(interval.center[0]*rtol) dy = abs(interval.center[1]*rtol) interval = interval.refine_size(dx, dy) c = interval.center real, imag = map(Rational, c) if not rtol or ( interval.dx < abs(c[0]*rtol) and interval.dy < abs(c[1]*rtol)): break # update the interval so we at least (for this precision or # less) don't have much work to do to recompute the root self._set_interval(interval) return real + I*imag CRootOf = ComplexRootOf @dispatch(ComplexRootOf, ComplexRootOf) def _eval_is_eq(lhs, rhs): # noqa:F811 # if we use is_eq to check here, we get infinite recurion return lhs == rhs @dispatch(ComplexRootOf, Basic) # type:ignore def _eval_is_eq(lhs, rhs): # noqa:F811 # CRootOf represents a Root, so if rhs is that root, it should set # the expression to zero *and* it should be in the interval of the # CRootOf instance. It must also be a number that agrees with the # is_real value of the CRootOf instance. if not rhs.is_number: return None if not rhs.is_finite: return False z = lhs.expr.subs(lhs.expr.free_symbols.pop(), rhs).is_zero if z is False: # all roots will make z True but we don't know # whether this is the right root if z is True return False o = rhs.is_real, rhs.is_imaginary s = lhs.is_real, lhs.is_imaginary assert None not in s # this is part of initial refinement if o != s and None not in o: return False re, im = rhs.as_real_imag() if lhs.is_real: if im: return False i = lhs._get_interval() a, b = [Rational(str(_)) for _ in (i.a, i.b)] return sympify(a <= rhs and rhs <= b) i = lhs._get_interval() r1, r2, i1, i2 = [Rational(str(j)) for j in ( i.ax, i.bx, i.ay, i.by)] return is_le(r1, re) and is_le(re,r2) and is_le(i1,im) and is_le(im,i2) @public class RootSum(Expr): """Represents a sum of all roots of a univariate polynomial. """ __slots__ = ('poly', 'fun', 'auto') def __new__(cls, expr, func=None, x=None, auto=True, quadratic=False): """Construct a new ``RootSum`` instance of roots of a polynomial.""" coeff, poly = cls._transform(expr, x) if not poly.is_univariate: raise MultivariatePolynomialError( "only univariate polynomials are allowed") if func is None: func = Lambda(poly.gen, poly.gen) else: is_func = getattr(func, 'is_Function', False) if is_func and 1 in func.nargs: if not isinstance(func, Lambda): func = Lambda(poly.gen, func(poly.gen)) else: raise ValueError( "expected a univariate function, got %s" % func) var, expr = func.variables[0], func.expr if coeff is not S.One: expr = expr.subs(var, coeff*var) deg = poly.degree() if not expr.has(var): return deg*expr if expr.is_Add: add_const, expr = expr.as_independent(var) else: add_const = S.Zero if expr.is_Mul: mul_const, expr = expr.as_independent(var) else: mul_const = S.One func = Lambda(var, expr) rational = cls._is_func_rational(poly, func) factors, terms = _pure_factors(poly), [] for poly, k in factors: if poly.is_linear: term = func(roots_linear(poly)[0]) elif quadratic and poly.is_quadratic: term = sum(map(func, roots_quadratic(poly))) else: if not rational or not auto: term = cls._new(poly, func, auto) else: term = cls._rational_case(poly, func) terms.append(k*term) return mul_const*Add(*terms) + deg*add_const @classmethod def _new(cls, poly, func, auto=True): """Construct new raw ``RootSum`` instance. """ obj = Expr.__new__(cls) obj.poly = poly obj.fun = func obj.auto = auto return obj @classmethod def new(cls, poly, func, auto=True): """Construct new ``RootSum`` instance. """ if not func.expr.has(*func.variables): return func.expr rational = cls._is_func_rational(poly, func) if not rational or not auto: return cls._new(poly, func, auto) else: return cls._rational_case(poly, func) @classmethod def _transform(cls, expr, x): """Transform an expression to a polynomial. """ poly = PurePoly(expr, x, greedy=False) return preprocess_roots(poly) @classmethod def _is_func_rational(cls, poly, func): """Check if a lambda is a rational function. """ var, expr = func.variables[0], func.expr return expr.is_rational_function(var) @classmethod def _rational_case(cls, poly, func): """Handle the rational function case. """ roots = symbols('r:%d' % poly.degree()) var, expr = func.variables[0], func.expr f = sum(expr.subs(var, r) for r in roots) p, q = together(f).as_numer_denom() domain = QQ[roots] p = p.expand() q = q.expand() try: p = Poly(p, domain=domain, expand=False) except GeneratorsNeeded: p, p_coeff = None, (p,) else: p_monom, p_coeff = zip(*p.terms()) try: q = Poly(q, domain=domain, expand=False) except GeneratorsNeeded: q, q_coeff = None, (q,) else: q_monom, q_coeff = zip(*q.terms()) coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True) formulas, values = viete(poly, roots), [] for (sym, _), (_, val) in zip(mapping, formulas): values.append((sym, val)) for i, (coeff, _) in enumerate(coeffs): coeffs[i] = coeff.subs(values) n = len(p_coeff) p_coeff = coeffs[:n] q_coeff = coeffs[n:] if p is not None: p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr() else: (p,) = p_coeff if q is not None: q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr() else: (q,) = q_coeff return factor(p/q) def _hashable_content(self): return (self.poly, self.fun) @property def expr(self): return self.poly.as_expr() @property def args(self): return (self.expr, self.fun, self.poly.gen) @property def free_symbols(self): return self.poly.free_symbols | self.fun.free_symbols @property def is_commutative(self): return True def doit(self, **hints): if not hints.get('roots', True): return self _roots = roots(self.poly, multiple=True) if len(_roots) < self.poly.degree(): return self else: return Add(*[self.fun(r) for r in _roots]) def _eval_evalf(self, prec): try: _roots = self.poly.nroots(n=prec_to_dps(prec)) except (DomainError, PolynomialError): return self else: return Add(*[self.fun(r) for r in _roots]) def _eval_derivative(self, x): var, expr = self.fun.args func = Lambda(var, expr.diff(x)) return self.new(self.poly, func, self.auto)
d8676ff6df5336075b9dcfbb5eba448606097d2bda707532b84d5a396a3d0d75
"""Square-free decomposition algorithms and related tools. """ from sympy.polys.densearith import ( dup_neg, dmp_neg, dup_sub, dmp_sub, dup_mul, dup_quo, dmp_quo, dup_mul_ground, dmp_mul_ground) from sympy.polys.densebasic import ( dup_strip, dup_LC, dmp_ground_LC, dmp_zero_p, dmp_ground, dup_degree, dmp_degree, dmp_raise, dmp_inject, dup_convert) from sympy.polys.densetools import ( dup_diff, dmp_diff, dmp_diff_in, dup_shift, dmp_compose, dup_monic, dmp_ground_monic, dup_primitive, dmp_ground_primitive) from sympy.polys.euclidtools import ( dup_inner_gcd, dmp_inner_gcd, dup_gcd, dmp_gcd, dmp_resultant) from sympy.polys.galoistools import ( gf_sqf_list, gf_sqf_part) from sympy.polys.polyerrors import ( MultivariatePolynomialError, DomainError) def dup_sqf_p(f, K): """ Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqf_p(x**2 - 2*x + 1) False >>> R.dup_sqf_p(x**2 - 1) True """ if not f: return True else: return not dup_degree(dup_gcd(f, dup_diff(f, 1, K), K)) def dmp_sqf_p(f, u, K): """ Return ``True`` if ``f`` is a square-free polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqf_p(x**2 + 2*x*y + y**2) False >>> R.dmp_sqf_p(x**2 + y**2) True """ if dmp_zero_p(f, u): return True else: return not dmp_degree(dmp_gcd(f, dmp_diff(f, 1, u, K), u, K), u) def dup_sqf_norm(f, K): """ Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains. Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))`` is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``. Examples ======== >>> from sympy.polys import ring, QQ >>> from sympy import sqrt >>> K = QQ.algebraic_field(sqrt(3)) >>> R, x = ring("x", K) >>> _, X = ring("x", QQ) >>> s, f, r = R.dup_sqf_norm(x**2 - 2) >>> s == 1 True >>> f == x**2 + K([QQ(-2), QQ(0)])*x + 1 True >>> r == X**4 - 10*X**2 + 1 True """ if not K.is_Algebraic: raise DomainError("ground domain must be algebraic") s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom) while True: h, _ = dmp_inject(f, 0, K, front=True) r = dmp_resultant(g, h, 1, K.dom) if dup_sqf_p(r, K.dom): break else: f, s = dup_shift(f, -K.unit, K), s + 1 return s, f, r def dmp_sqf_norm(f, u, K): """ Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains. Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))`` is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``. Examples ======== >>> from sympy.polys import ring, QQ >>> from sympy import I >>> K = QQ.algebraic_field(I) >>> R, x, y = ring("x,y", K) >>> _, X, Y = ring("x,y", QQ) >>> s, f, r = R.dmp_sqf_norm(x*y + y**2) >>> s == 1 True >>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y True >>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2 True """ if not u: return dup_sqf_norm(f, K) if not K.is_Algebraic: raise DomainError("ground domain must be algebraic") g = dmp_raise(K.mod.rep, u + 1, 0, K.dom) F = dmp_raise([K.one, -K.unit], u, 0, K) s = 0 while True: h, _ = dmp_inject(f, u, K, front=True) r = dmp_resultant(g, h, u + 1, K.dom) if dmp_sqf_p(r, u, K.dom): break else: f, s = dmp_compose(f, F, u, K), s + 1 return s, f, r def dmp_norm(f, u, K): """ Norm of ``f`` in ``K[X1, ..., Xn]``, often not square-free. """ if not K.is_Algebraic: raise DomainError("ground domain must be algebraic") g = dmp_raise(K.mod.rep, u + 1, 0, K.dom) h, _ = dmp_inject(f, u, K, front=True) return dmp_resultant(g, h, u + 1, K.dom) def dup_gf_sqf_part(f, K): """Compute square-free part of ``f`` in ``GF(p)[x]``. """ f = dup_convert(f, K, K.dom) g = gf_sqf_part(f, K.mod, K.dom) return dup_convert(g, K.dom, K) def dmp_gf_sqf_part(f, u, K): """Compute square-free part of ``f`` in ``GF(p)[X]``. """ raise NotImplementedError('multivariate polynomials over finite fields') def dup_sqf_part(f, K): """ Returns square-free part of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqf_part(x**3 - 3*x - 2) x**2 - x - 2 """ if K.is_FiniteField: return dup_gf_sqf_part(f, K) if not f: return f if K.is_negative(dup_LC(f, K)): f = dup_neg(f, K) gcd = dup_gcd(f, dup_diff(f, 1, K), K) sqf = dup_quo(f, gcd, K) if K.is_Field: return dup_monic(sqf, K) else: return dup_primitive(sqf, K)[1] def dmp_sqf_part(f, u, K): """ Returns square-free part of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqf_part(x**3 + 2*x**2*y + x*y**2) x**2 + x*y """ if not u: return dup_sqf_part(f, K) if K.is_FiniteField: return dmp_gf_sqf_part(f, u, K) if dmp_zero_p(f, u): return f if K.is_negative(dmp_ground_LC(f, u, K)): f = dmp_neg(f, u, K) gcd = f for i in range(u+1): gcd = dmp_gcd(gcd, dmp_diff_in(f, 1, i, u, K), u, K) sqf = dmp_quo(f, gcd, u, K) if K.is_Field: return dmp_ground_monic(sqf, u, K) else: return dmp_ground_primitive(sqf, u, K)[1] def dup_gf_sqf_list(f, K, all=False): """Compute square-free decomposition of ``f`` in ``GF(p)[x]``. """ f = dup_convert(f, K, K.dom) coeff, factors = gf_sqf_list(f, K.mod, K.dom, all=all) for i, (f, k) in enumerate(factors): factors[i] = (dup_convert(f, K.dom, K), k) return K.convert(coeff, K.dom), factors def dmp_gf_sqf_list(f, u, K, all=False): """Compute square-free decomposition of ``f`` in ``GF(p)[X]``. """ raise NotImplementedError('multivariate polynomials over finite fields') def dup_sqf_list(f, K, all=False): """ Return square-free decomposition of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 >>> R.dup_sqf_list(f) (2, [(x + 1, 2), (x + 2, 3)]) >>> R.dup_sqf_list(f, all=True) (2, [(1, 1), (x + 1, 2), (x + 2, 3)]) """ if K.is_FiniteField: return dup_gf_sqf_list(f, K, all=all) if K.is_Field: coeff = dup_LC(f, K) f = dup_monic(f, K) else: coeff, f = dup_primitive(f, K) if K.is_negative(dup_LC(f, K)): f = dup_neg(f, K) coeff = -coeff if dup_degree(f) <= 0: return coeff, [] result, i = [], 1 h = dup_diff(f, 1, K) g, p, q = dup_inner_gcd(f, h, K) while True: d = dup_diff(p, 1, K) h = dup_sub(q, d, K) if not h: result.append((p, i)) break g, p, q = dup_inner_gcd(p, h, K) if all or dup_degree(g) > 0: result.append((g, i)) i += 1 return coeff, result def dup_sqf_list_include(f, K, all=False): """ Return square-free decomposition of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 >>> R.dup_sqf_list_include(f) [(2, 1), (x + 1, 2), (x + 2, 3)] >>> R.dup_sqf_list_include(f, all=True) [(2, 1), (x + 1, 2), (x + 2, 3)] """ coeff, factors = dup_sqf_list(f, K, all=all) if factors and factors[0][1] == 1: g = dup_mul_ground(factors[0][0], coeff, K) return [(g, 1)] + factors[1:] else: g = dup_strip([coeff]) return [(g, 1)] + factors def dmp_sqf_list(f, u, K, all=False): """ Return square-free decomposition of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**5 + 2*x**4*y + x**3*y**2 >>> R.dmp_sqf_list(f) (1, [(x + y, 2), (x, 3)]) >>> R.dmp_sqf_list(f, all=True) (1, [(1, 1), (x + y, 2), (x, 3)]) """ if not u: return dup_sqf_list(f, K, all=all) if K.is_FiniteField: return dmp_gf_sqf_list(f, u, K, all=all) if K.is_Field: coeff = dmp_ground_LC(f, u, K) f = dmp_ground_monic(f, u, K) else: coeff, f = dmp_ground_primitive(f, u, K) if K.is_negative(dmp_ground_LC(f, u, K)): f = dmp_neg(f, u, K) coeff = -coeff if dmp_degree(f, u) <= 0: return coeff, [] result, i = [], 1 h = dmp_diff(f, 1, u, K) g, p, q = dmp_inner_gcd(f, h, u, K) while True: d = dmp_diff(p, 1, u, K) h = dmp_sub(q, d, u, K) if dmp_zero_p(h, u): result.append((p, i)) break g, p, q = dmp_inner_gcd(p, h, u, K) if all or dmp_degree(g, u) > 0: result.append((g, i)) i += 1 return coeff, result def dmp_sqf_list_include(f, u, K, all=False): """ Return square-free decomposition of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**5 + 2*x**4*y + x**3*y**2 >>> R.dmp_sqf_list_include(f) [(1, 1), (x + y, 2), (x, 3)] >>> R.dmp_sqf_list_include(f, all=True) [(1, 1), (x + y, 2), (x, 3)] """ if not u: return dup_sqf_list_include(f, K, all=all) coeff, factors = dmp_sqf_list(f, u, K, all=all) if factors and factors[0][1] == 1: g = dmp_mul_ground(factors[0][0], coeff, u, K) return [(g, 1)] + factors[1:] else: g = dmp_ground(coeff, u) return [(g, 1)] + factors def dup_gff_list(f, K): """ Compute greatest factorial factorization of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2) [(x, 1), (x + 2, 4)] """ if not f: raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial") f = dup_monic(f, K) if not dup_degree(f): return [] else: g = dup_gcd(f, dup_shift(f, K.one, K), K) H = dup_gff_list(g, K) for i, (h, k) in enumerate(H): g = dup_mul(g, dup_shift(h, -K(k), K), K) H[i] = (h, k + 1) f = dup_quo(f, g, K) if not dup_degree(f): return H else: return [(f, 1)] + H def dmp_gff_list(f, u, K): """ Compute greatest factorial factorization of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) """ if not u: return dup_gff_list(f, K) else: raise MultivariatePolynomialError(f)
d9a3ad18b8f453c29016497a73972110b21bc0de2728c6cf7cd6e053f5a0c8a2
"""Advanced tools for dense recursive polynomials in ``K[x]`` or ``K[X]``. """ from sympy.polys.densearith import ( dup_add_term, dmp_add_term, dup_lshift, dup_add, dmp_add, dup_sub, dmp_sub, dup_mul, dmp_mul, dup_sqr, dup_div, dup_rem, dmp_rem, dmp_expand, dup_mul_ground, dmp_mul_ground, dup_quo_ground, dmp_quo_ground, dup_exquo_ground, dmp_exquo_ground, ) from sympy.polys.densebasic import ( dup_strip, dmp_strip, dup_convert, dmp_convert, dup_degree, dmp_degree, dmp_to_dict, dmp_from_dict, dup_LC, dmp_LC, dmp_ground_LC, dup_TC, dmp_TC, dmp_zero, dmp_ground, dmp_zero_p, dup_to_raw_dict, dup_from_raw_dict, dmp_zeros ) from sympy.polys.polyerrors import ( MultivariatePolynomialError, DomainError ) from sympy.utilities import variations from math import ceil as _ceil, log as _log def dup_integrate(f, m, K): """ Computes the indefinite integral of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_integrate(x**2 + 2*x, 1) 1/3*x**3 + x**2 >>> R.dup_integrate(x**2 + 2*x, 2) 1/12*x**4 + 1/3*x**3 """ if m <= 0 or not f: return f g = [K.zero]*m for i, c in enumerate(reversed(f)): n = i + 1 for j in range(1, m): n *= i + j + 1 g.insert(0, K.exquo(c, K(n))) return g def dmp_integrate(f, m, u, K): """ Computes the indefinite integral of ``f`` in ``x_0`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_integrate(x + 2*y, 1) 1/2*x**2 + 2*x*y >>> R.dmp_integrate(x + 2*y, 2) 1/6*x**3 + x**2*y """ if not u: return dup_integrate(f, m, K) if m <= 0 or dmp_zero_p(f, u): return f g, v = dmp_zeros(m, u - 1, K), u - 1 for i, c in enumerate(reversed(f)): n = i + 1 for j in range(1, m): n *= i + j + 1 g.insert(0, dmp_quo_ground(c, K(n), v, K)) return g def _rec_integrate_in(g, m, v, i, j, K): """Recursive helper for :func:`dmp_integrate_in`.""" if i == j: return dmp_integrate(g, m, v, K) w, i = v - 1, i + 1 return dmp_strip([ _rec_integrate_in(c, m, w, i, j, K) for c in g ], v) def dmp_integrate_in(f, m, j, u, K): """ Computes the indefinite integral of ``f`` in ``x_j`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_integrate_in(x + 2*y, 1, 0) 1/2*x**2 + 2*x*y >>> R.dmp_integrate_in(x + 2*y, 1, 1) x*y + y**2 """ if j < 0 or j > u: raise IndexError("0 <= j <= u expected, got u = %d, j = %d" % (u, j)) return _rec_integrate_in(f, m, u, 0, j, K) def dup_diff(f, m, K): """ ``m``-th order derivative of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 1) 3*x**2 + 4*x + 3 >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 2) 6*x + 4 """ if m <= 0: return f n = dup_degree(f) if n < m: return [] deriv = [] if m == 1: for coeff in f[:-m]: deriv.append(K(n)*coeff) n -= 1 else: for coeff in f[:-m]: k = n for i in range(n - 1, n - m, -1): k *= i deriv.append(K(k)*coeff) n -= 1 return dup_strip(deriv) def dmp_diff(f, m, u, K): """ ``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff(f, 1) y**2 + 2*y + 3 >>> R.dmp_diff(f, 2) 0 """ if not u: return dup_diff(f, m, K) if m <= 0: return f n = dmp_degree(f, u) if n < m: return dmp_zero(u) deriv, v = [], u - 1 if m == 1: for coeff in f[:-m]: deriv.append(dmp_mul_ground(coeff, K(n), v, K)) n -= 1 else: for coeff in f[:-m]: k = n for i in range(n - 1, n - m, -1): k *= i deriv.append(dmp_mul_ground(coeff, K(k), v, K)) n -= 1 return dmp_strip(deriv, u) def _rec_diff_in(g, m, v, i, j, K): """Recursive helper for :func:`dmp_diff_in`.""" if i == j: return dmp_diff(g, m, v, K) w, i = v - 1, i + 1 return dmp_strip([ _rec_diff_in(c, m, w, i, j, K) for c in g ], v) def dmp_diff_in(f, m, j, u, K): """ ``m``-th order derivative in ``x_j`` of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff_in(f, 1, 0) y**2 + 2*y + 3 >>> R.dmp_diff_in(f, 1, 1) 2*x*y + 2*x + 4*y + 3 """ if j < 0 or j > u: raise IndexError("0 <= j <= %s expected, got %s" % (u, j)) return _rec_diff_in(f, m, u, 0, j, K) def dup_eval(f, a, K): """ Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_eval(x**2 + 2*x + 3, 2) 11 """ if not a: return dup_TC(f, K) result = K.zero for c in f: result *= a result += c return result def dmp_eval(f, a, u, K): """ Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_eval(2*x*y + 3*x + y + 2, 2) 5*y + 8 """ if not u: return dup_eval(f, a, K) if not a: return dmp_TC(f, K) result, v = dmp_LC(f, K), u - 1 for coeff in f[1:]: result = dmp_mul_ground(result, a, v, K) result = dmp_add(result, coeff, v, K) return result def _rec_eval_in(g, a, v, i, j, K): """Recursive helper for :func:`dmp_eval_in`.""" if i == j: return dmp_eval(g, a, v, K) v, i = v - 1, i + 1 return dmp_strip([ _rec_eval_in(c, a, v, i, j, K) for c in g ], v) def dmp_eval_in(f, a, j, u, K): """ Evaluate a polynomial at ``x_j = a`` in ``K[X]`` using the Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 3*x + y + 2 >>> R.dmp_eval_in(f, 2, 0) 5*y + 8 >>> R.dmp_eval_in(f, 2, 1) 7*x + 4 """ if j < 0 or j > u: raise IndexError("0 <= j <= %s expected, got %s" % (u, j)) return _rec_eval_in(f, a, u, 0, j, K) def _rec_eval_tail(g, i, A, u, K): """Recursive helper for :func:`dmp_eval_tail`.""" if i == u: return dup_eval(g, A[-1], K) else: h = [ _rec_eval_tail(c, i + 1, A, u, K) for c in g ] if i < u - len(A) + 1: return h else: return dup_eval(h, A[-u + i - 1], K) def dmp_eval_tail(f, A, u, K): """ Evaluate a polynomial at ``x_j = a_j, ...`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 3*x + y + 2 >>> R.dmp_eval_tail(f, [2]) 7*x + 4 >>> R.dmp_eval_tail(f, [2, 2]) 18 """ if not A: return f if dmp_zero_p(f, u): return dmp_zero(u - len(A)) e = _rec_eval_tail(f, 0, A, u, K) if u == len(A) - 1: return e else: return dmp_strip(e, u - len(A)) def _rec_diff_eval(g, m, a, v, i, j, K): """Recursive helper for :func:`dmp_diff_eval`.""" if i == j: return dmp_eval(dmp_diff(g, m, v, K), a, v, K) v, i = v - 1, i + 1 return dmp_strip([ _rec_diff_eval(c, m, a, v, i, j, K) for c in g ], v) def dmp_diff_eval_in(f, m, a, j, u, K): """ Differentiate and evaluate a polynomial in ``x_j`` at ``a`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff_eval_in(f, 1, 2, 0) y**2 + 2*y + 3 >>> R.dmp_diff_eval_in(f, 1, 2, 1) 6*x + 11 """ if j > u: raise IndexError("-%s <= j < %s expected, got %s" % (u, u, j)) if not j: return dmp_eval(dmp_diff(f, m, u, K), a, u, K) return _rec_diff_eval(f, m, a, u, 0, j, K) def dup_trunc(f, p, K): """ Reduce a ``K[x]`` polynomial modulo a constant ``p`` in ``K``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_trunc(2*x**3 + 3*x**2 + 5*x + 7, ZZ(3)) -x**3 - x + 1 """ if K.is_ZZ: g = [] for c in f: c = c % p if c > p // 2: g.append(c - p) else: g.append(c) else: g = [ c % p for c in f ] return dup_strip(g) def dmp_trunc(f, p, u, K): """ Reduce a ``K[X]`` polynomial modulo a polynomial ``p`` in ``K[Y]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> g = (y - 1).drop(x) >>> R.dmp_trunc(f, g) 11*x**2 + 11*x + 5 """ return dmp_strip([ dmp_rem(c, p, u - 1, K) for c in f ], u) def dmp_ground_trunc(f, p, u, K): """ Reduce a ``K[X]`` polynomial modulo a constant ``p`` in ``K``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> R.dmp_ground_trunc(f, ZZ(3)) -x**2 - x*y - y """ if not u: return dup_trunc(f, p, K) v = u - 1 return dmp_strip([ dmp_ground_trunc(c, p, v, K) for c in f ], u) def dup_monic(f, K): """ Divide all coefficients by ``LC(f)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_monic(3*x**2 + 6*x + 9) x**2 + 2*x + 3 >>> R, x = ring("x", QQ) >>> R.dup_monic(3*x**2 + 4*x + 2) x**2 + 4/3*x + 2/3 """ if not f: return f lc = dup_LC(f, K) if K.is_one(lc): return f else: return dup_exquo_ground(f, lc, K) def dmp_ground_monic(f, u, K): """ Divide all coefficients by ``LC(f)`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 2*x**2 + x*y + 3*y + 1 >>> R, x,y = ring("x,y", QQ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1 """ if not u: return dup_monic(f, K) if dmp_zero_p(f, u): return f lc = dmp_ground_LC(f, u, K) if K.is_one(lc): return f else: return dmp_exquo_ground(f, lc, u, K) def dup_content(f, K): """ Compute the GCD of coefficients of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_content(f) 2 >>> R, x = ring("x", QQ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_content(f) 2 """ from sympy.polys.domains import QQ if not f: return K.zero cont = K.zero if K == QQ: for c in f: cont = K.gcd(cont, c) else: for c in f: cont = K.gcd(cont, c) if K.is_one(cont): break return cont def dmp_ground_content(f, u, K): """ Compute the GCD of coefficients of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_content(f) 2 >>> R, x,y = ring("x,y", QQ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_content(f) 2 """ from sympy.polys.domains import QQ if not u: return dup_content(f, K) if dmp_zero_p(f, u): return K.zero cont, v = K.zero, u - 1 if K == QQ: for c in f: cont = K.gcd(cont, dmp_ground_content(c, v, K)) else: for c in f: cont = K.gcd(cont, dmp_ground_content(c, v, K)) if K.is_one(cont): break return cont def dup_primitive(f, K): """ Compute content and the primitive form of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_primitive(f) (2, 3*x**2 + 4*x + 6) >>> R, x = ring("x", QQ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_primitive(f) (2, 3*x**2 + 4*x + 6) """ if not f: return K.zero, f cont = dup_content(f, K) if K.is_one(cont): return cont, f else: return cont, dup_quo_ground(f, cont, K) def dmp_ground_primitive(f, u, K): """ Compute content and the primitive form of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_primitive(f) (2, x*y + 3*x + 2*y + 6) >>> R, x,y = ring("x,y", QQ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_primitive(f) (2, x*y + 3*x + 2*y + 6) """ if not u: return dup_primitive(f, K) if dmp_zero_p(f, u): return K.zero, f cont = dmp_ground_content(f, u, K) if K.is_one(cont): return cont, f else: return cont, dmp_quo_ground(f, cont, u, K) def dup_extract(f, g, K): """ Extract common content from a pair of polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_extract(6*x**2 + 12*x + 18, 4*x**2 + 8*x + 12) (2, 3*x**2 + 6*x + 9, 2*x**2 + 4*x + 6) """ fc = dup_content(f, K) gc = dup_content(g, K) gcd = K.gcd(fc, gc) if not K.is_one(gcd): f = dup_quo_ground(f, gcd, K) g = dup_quo_ground(g, gcd, K) return gcd, f, g def dmp_ground_extract(f, g, u, K): """ Extract common content from a pair of polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_ground_extract(6*x*y + 12*x + 18, 4*x*y + 8*x + 12) (2, 3*x*y + 6*x + 9, 2*x*y + 4*x + 6) """ fc = dmp_ground_content(f, u, K) gc = dmp_ground_content(g, u, K) gcd = K.gcd(fc, gc) if not K.is_one(gcd): f = dmp_quo_ground(f, gcd, u, K) g = dmp_quo_ground(g, gcd, u, K) return gcd, f, g def dup_real_imag(f, K): """ Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dup_real_imag(x**3 + x**2 + x + 1) (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y) """ if not K.is_ZZ and not K.is_QQ: raise DomainError("computing real and imaginary parts is not supported over %s" % K) f1 = dmp_zero(1) f2 = dmp_zero(1) if not f: return f1, f2 g = [[[K.one, K.zero]], [[K.one], []]] h = dmp_ground(f[0], 2) for c in f[1:]: h = dmp_mul(h, g, 2, K) h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K) H = dup_to_raw_dict(h) for k, h in H.items(): m = k % 4 if not m: f1 = dmp_add(f1, h, 1, K) elif m == 1: f2 = dmp_add(f2, h, 1, K) elif m == 2: f1 = dmp_sub(f1, h, 1, K) else: f2 = dmp_sub(f2, h, 1, K) return f1, f2 def dup_mirror(f, K): """ Evaluate efficiently the composition ``f(-x)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mirror(x**3 + 2*x**2 - 4*x + 2) -x**3 + 2*x**2 + 4*x + 2 """ f = list(f) for i in range(len(f) - 2, -1, -2): f[i] = -f[i] return f def dup_scale(f, a, K): """ Evaluate efficiently composition ``f(a*x)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_scale(x**2 - 2*x + 1, ZZ(2)) 4*x**2 - 4*x + 1 """ f, n, b = list(f), len(f) - 1, a for i in range(n - 1, -1, -1): f[i], b = b*f[i], b*a return f def dup_shift(f, a, K): """ Evaluate efficiently Taylor shift ``f(x + a)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_shift(x**2 - 2*x + 1, ZZ(2)) x**2 + 2*x + 1 """ f, n = list(f), len(f) - 1 for i in range(n, 0, -1): for j in range(0, i): f[j + 1] += a*f[j] return f def dup_transform(f, p, q, K): """ Evaluate functional transformation ``q**n * f(p/q)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_transform(x**2 - 2*x + 1, x**2 + 1, x - 1) x**4 - 2*x**3 + 5*x**2 - 4*x + 4 """ if not f: return [] n = len(f) - 1 h, Q = [f[0]], [[K.one]] for i in range(0, n): Q.append(dup_mul(Q[-1], q, K)) for c, q in zip(f[1:], Q[1:]): h = dup_mul(h, p, K) q = dup_mul_ground(q, c, K) h = dup_add(h, q, K) return h def dup_compose(f, g, K): """ Evaluate functional composition ``f(g)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_compose(x**2 + x, x - 1) x**2 - x """ if len(g) <= 1: return dup_strip([dup_eval(f, dup_LC(g, K), K)]) if not f: return [] h = [f[0]] for c in f[1:]: h = dup_mul(h, g, K) h = dup_add_term(h, c, 0, K) return h def dmp_compose(f, g, u, K): """ Evaluate functional composition ``f(g)`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_compose(x*y + 2*x + y, y) y**2 + 3*y """ if not u: return dup_compose(f, g, K) if dmp_zero_p(f, u): return f h = [f[0]] for c in f[1:]: h = dmp_mul(h, g, u, K) h = dmp_add_term(h, c, 0, u, K) return h def _dup_right_decompose(f, s, K): """Helper function for :func:`_dup_decompose`.""" n = len(f) - 1 lc = dup_LC(f, K) f = dup_to_raw_dict(f) g = { s: K.one } r = n // s for i in range(1, s): coeff = K.zero for j in range(0, i): if not n + j - i in f: continue if not s - j in g: continue fc, gc = f[n + j - i], g[s - j] coeff += (i - r*j)*fc*gc g[s - i] = K.quo(coeff, i*r*lc) return dup_from_raw_dict(g, K) def _dup_left_decompose(f, h, K): """Helper function for :func:`_dup_decompose`.""" g, i = {}, 0 while f: q, r = dup_div(f, h, K) if dup_degree(r) > 0: return None else: g[i] = dup_LC(r, K) f, i = q, i + 1 return dup_from_raw_dict(g, K) def _dup_decompose(f, K): """Helper function for :func:`dup_decompose`.""" df = len(f) - 1 for s in range(2, df): if df % s != 0: continue h = _dup_right_decompose(f, s, K) if h is not None: g = _dup_left_decompose(f, h, K) if g is not None: return g, h return None def dup_decompose(f, K): """ Computes functional decomposition of ``f`` in ``K[x]``. Given a univariate polynomial ``f`` with coefficients in a field of characteristic zero, returns list ``[f_1, f_2, ..., f_n]``, where:: f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n)) and ``f_2, ..., f_n`` are monic and homogeneous polynomials of at least second degree. Unlike factorization, complete functional decompositions of polynomials are not unique, consider examples: 1. ``f o g = f(x + b) o (g - b)`` 2. ``x**n o x**m = x**m o x**n`` 3. ``T_n o T_m = T_m o T_n`` where ``T_n`` and ``T_m`` are Chebyshev polynomials. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_decompose(x**4 - 2*x**3 + x**2) [x**2, x**2 - x] References ========== .. [1] [Kozen89]_ """ F = [] while True: result = _dup_decompose(f, K) if result is not None: f, h = result F = [h] + F else: break return [f] + F def dmp_lift(f, u, K): """ Convert algebraic coefficients to integers in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> from sympy import I >>> K = QQ.algebraic_field(I) >>> R, x = ring("x", K) >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)]) >>> R.dmp_lift(f) x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16 """ if K.is_GaussianField: K1 = K.as_AlgebraicField() f = dmp_convert(f, u, K, K1) K = K1 if not K.is_Algebraic: raise DomainError( 'computation can be done only in an algebraic domain') F, monoms, polys = dmp_to_dict(f, u), [], [] for monom, coeff in F.items(): if not coeff.is_ground: monoms.append(monom) perms = variations([-1, 1], len(monoms), repetition=True) for perm in perms: G = dict(F) for sign, monom in zip(perm, monoms): if sign == -1: G[monom] = -G[monom] polys.append(dmp_from_dict(G, u, K)) return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom) def dup_sign_variations(f, K): """ Compute the number of sign variations of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sign_variations(x**4 - x**2 - x + 1) 2 """ prev, k = K.zero, 0 for coeff in f: if K.is_negative(coeff*prev): k += 1 if coeff: prev = coeff return k def dup_clear_denoms(f, K0, K1=None, convert=False): """ Clear denominators, i.e. transform ``K_0`` to ``K_1``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = QQ(1,2)*x + QQ(1,3) >>> R.dup_clear_denoms(f, convert=False) (6, 3*x + 2) >>> R.dup_clear_denoms(f, convert=True) (6, 3*x + 2) """ if K1 is None: if K0.has_assoc_Ring: K1 = K0.get_ring() else: K1 = K0 common = K1.one for c in f: common = K1.lcm(common, K0.denom(c)) if not K1.is_one(common): f = dup_mul_ground(f, common, K0) if not convert: return common, f else: return common, dup_convert(f, K0, K1) def _rec_clear_denoms(g, v, K0, K1): """Recursive helper for :func:`dmp_clear_denoms`.""" common = K1.one if not v: for c in g: common = K1.lcm(common, K0.denom(c)) else: w = v - 1 for c in g: common = K1.lcm(common, _rec_clear_denoms(c, w, K0, K1)) return common def dmp_clear_denoms(f, u, K0, K1=None, convert=False): """ Clear denominators, i.e. transform ``K_0`` to ``K_1``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> f = QQ(1,2)*x + QQ(1,3)*y + 1 >>> R.dmp_clear_denoms(f, convert=False) (6, 3*x + 2*y + 6) >>> R.dmp_clear_denoms(f, convert=True) (6, 3*x + 2*y + 6) """ if not u: return dup_clear_denoms(f, K0, K1, convert=convert) if K1 is None: if K0.has_assoc_Ring: K1 = K0.get_ring() else: K1 = K0 common = _rec_clear_denoms(f, u, K0, K1) if not K1.is_one(common): f = dmp_mul_ground(f, common, u, K0) if not convert: return common, f else: return common, dmp_convert(f, u, K0, K1) def dup_revert(f, n, K): """ Compute ``f**(-1)`` mod ``x**n`` using Newton iteration. This function computes first ``2**n`` terms of a polynomial that is a result of inversion of a polynomial modulo ``x**n``. This is useful to efficiently compute series expansion of ``1/f``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1 >>> R.dup_revert(f, 8) 61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1 """ g = [K.revert(dup_TC(f, K))] h = [K.one, K.zero, K.zero] N = int(_ceil(_log(n, 2))) for i in range(1, N + 1): a = dup_mul_ground(g, K(2), K) b = dup_mul(f, dup_sqr(g, K), K) g = dup_rem(dup_sub(a, b, K), h, K) h = dup_lshift(h, dup_degree(h), K) return g def dmp_revert(f, g, u, K): """ Compute ``f**(-1)`` mod ``x**n`` using Newton iteration. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) """ if not u: return dup_revert(f, g, K) else: raise MultivariatePolynomialError(f, g)
8b75b4ea99832884fcd53c252bf7d0a200ddc483e65151000a4ce5e063c74bdf
"""User-friendly public interface to polynomial functions. """ from functools import wraps, reduce from operator import mul from sympy.core import ( S, Basic, Expr, I, Integer, Add, Mul, Dummy, Tuple ) from sympy.core.basic import preorder_traversal from sympy.core.compatibility import iterable, ordered from sympy.core.decorators import _sympifyit from sympy.core.evalf import pure_complex from sympy.core.function import Derivative from sympy.core.mul import _keep_coeff from sympy.core.relational import Relational from sympy.core.symbol import Symbol from sympy.core.sympify import sympify, _sympify from sympy.logic.boolalg import BooleanAtom from sympy.polys import polyoptions as options from sympy.polys.constructor import construct_domain from sympy.polys.domains import FF, QQ, ZZ from sympy.polys.domains.domainelement import DomainElement from sympy.polys.fglmtools import matrix_fglm from sympy.polys.groebnertools import groebner as _groebner from sympy.polys.monomials import Monomial from sympy.polys.orderings import monomial_key from sympy.polys.polyclasses import DMP, DMF, ANP from sympy.polys.polyerrors import ( OperationNotSupported, DomainError, CoercionFailed, UnificationFailed, GeneratorsNeeded, PolynomialError, MultivariatePolynomialError, ExactQuotientFailed, PolificationFailed, ComputationFailed, GeneratorsError, ) from sympy.polys.polyutils import ( basic_from_dict, _sort_gens, _unify_gens, _dict_reorder, _dict_from_expr, _parallel_dict_from_expr, ) from sympy.polys.rationaltools import together from sympy.polys.rootisolation import dup_isolate_real_roots_list from sympy.utilities import group, sift, public, filldedent from sympy.utilities.exceptions import SymPyDeprecationWarning # Required to avoid errors import sympy.polys import mpmath from mpmath.libmp.libhyper import NoConvergence def _polifyit(func): @wraps(func) def wrapper(f, g): g = _sympify(g) if isinstance(g, Poly): return func(f, g) elif isinstance(g, Expr): try: g = f.from_expr(g, *f.gens) except PolynomialError: if g.is_Matrix: return NotImplemented expr_method = getattr(f.as_expr(), func.__name__) result = expr_method(g) if result is not NotImplemented: SymPyDeprecationWarning( feature="Mixing Poly with non-polynomial expressions in binary operations", issue=18613, deprecated_since_version="1.6", useinstead="the as_expr or as_poly method to convert types").warn() return result else: return func(f, g) else: return NotImplemented return wrapper @public class Poly(Basic): """ Generic class for representing and operating on polynomial expressions. Poly is a subclass of Basic rather than Expr but instances can be converted to Expr with the ``as_expr`` method. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y Create a univariate polynomial: >>> Poly(x*(x**2 + x - 1)**2) Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ') Create a univariate polynomial with specific domain: >>> from sympy import sqrt >>> Poly(x**2 + 2*x + sqrt(3), domain='R') Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR') Create a multivariate polynomial: >>> Poly(y*x**2 + x*y + 1) Poly(x**2*y + x*y + 1, x, y, domain='ZZ') Create a univariate polynomial, where y is a constant: >>> Poly(y*x**2 + x*y + 1,x) Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]') You can evaluate the above polynomial as a function of y: >>> Poly(y*x**2 + x*y + 1,x).eval(2) 6*y + 1 See Also ======== sympy.core.expr.Expr """ __slots__ = ('rep', 'gens') is_commutative = True is_Poly = True _op_priority = 10.001 def __new__(cls, rep, *gens, **args): """Create a new polynomial instance out of something useful. """ opt = options.build_options(gens, args) if 'order' in opt: raise NotImplementedError("'order' keyword is not implemented yet") if isinstance(rep, (DMP, DMF, ANP, DomainElement)): return cls._from_domain_element(rep, opt) elif iterable(rep, exclude=str): if isinstance(rep, dict): return cls._from_dict(rep, opt) else: return cls._from_list(list(rep), opt) else: rep = sympify(rep) if rep.is_Poly: return cls._from_poly(rep, opt) else: return cls._from_expr(rep, opt) # Poly does not pass its args to Basic.__new__ to be stored in _args so we # have to emulate them here with an args property that derives from rep # and gens which are instance attributes. This also means we need to # define _hashable_content. The _hashable_content is rep and gens but args # uses expr instead of rep (expr is the Basic version of rep). Passing # expr in args means that Basic methods like subs should work. Using rep # otherwise means that Poly can remain more efficient than Basic by # avoiding creating a Basic instance just to be hashable. @classmethod def new(cls, rep, *gens): """Construct :class:`Poly` instance from raw representation. """ if not isinstance(rep, DMP): raise PolynomialError( "invalid polynomial representation: %s" % rep) elif rep.lev != len(gens) - 1: raise PolynomialError("invalid arguments: %s, %s" % (rep, gens)) obj = Basic.__new__(cls) obj.rep = rep obj.gens = gens return obj @property def expr(self): return basic_from_dict(self.rep.to_sympy_dict(), *self.gens) @property def args(self): return (self.expr,) + self.gens def _hashable_content(self): return (self.rep,) + self.gens @classmethod def from_dict(cls, rep, *gens, **args): """Construct a polynomial from a ``dict``. """ opt = options.build_options(gens, args) return cls._from_dict(rep, opt) @classmethod def from_list(cls, rep, *gens, **args): """Construct a polynomial from a ``list``. """ opt = options.build_options(gens, args) return cls._from_list(rep, opt) @classmethod def from_poly(cls, rep, *gens, **args): """Construct a polynomial from a polynomial. """ opt = options.build_options(gens, args) return cls._from_poly(rep, opt) @classmethod def from_expr(cls, rep, *gens, **args): """Construct a polynomial from an expression. """ opt = options.build_options(gens, args) return cls._from_expr(rep, opt) @classmethod def _from_dict(cls, rep, opt): """Construct a polynomial from a ``dict``. """ gens = opt.gens if not gens: raise GeneratorsNeeded( "can't initialize from 'dict' without generators") level = len(gens) - 1 domain = opt.domain if domain is None: domain, rep = construct_domain(rep, opt=opt) else: for monom, coeff in rep.items(): rep[monom] = domain.convert(coeff) return cls.new(DMP.from_dict(rep, level, domain), *gens) @classmethod def _from_list(cls, rep, opt): """Construct a polynomial from a ``list``. """ gens = opt.gens if not gens: raise GeneratorsNeeded( "can't initialize from 'list' without generators") elif len(gens) != 1: raise MultivariatePolynomialError( "'list' representation not supported") level = len(gens) - 1 domain = opt.domain if domain is None: domain, rep = construct_domain(rep, opt=opt) else: rep = list(map(domain.convert, rep)) return cls.new(DMP.from_list(rep, level, domain), *gens) @classmethod def _from_poly(cls, rep, opt): """Construct a polynomial from a polynomial. """ if cls != rep.__class__: rep = cls.new(rep.rep, *rep.gens) gens = opt.gens field = opt.field domain = opt.domain if gens and rep.gens != gens: if set(rep.gens) != set(gens): return cls._from_expr(rep.as_expr(), opt) else: rep = rep.reorder(*gens) if 'domain' in opt and domain: rep = rep.set_domain(domain) elif field is True: rep = rep.to_field() return rep @classmethod def _from_expr(cls, rep, opt): """Construct a polynomial from an expression. """ rep, opt = _dict_from_expr(rep, opt) return cls._from_dict(rep, opt) @classmethod def _from_domain_element(cls, rep, opt): gens = opt.gens domain = opt.domain level = len(gens) - 1 rep = [domain.convert(rep)] return cls.new(DMP.from_list(rep, level, domain), *gens) def __hash__(self): return super().__hash__() @property def free_symbols(self): """ Free symbols of a polynomial expression. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> Poly(x**2 + 1).free_symbols {x} >>> Poly(x**2 + y).free_symbols {x, y} >>> Poly(x**2 + y, x).free_symbols {x, y} >>> Poly(x**2 + y, x, z).free_symbols {x, y} """ symbols = set() gens = self.gens for i in range(len(gens)): for monom in self.monoms(): if monom[i]: symbols |= gens[i].free_symbols break return symbols | self.free_symbols_in_domain @property def free_symbols_in_domain(self): """ Free symbols of the domain of ``self``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 1).free_symbols_in_domain set() >>> Poly(x**2 + y).free_symbols_in_domain set() >>> Poly(x**2 + y, x).free_symbols_in_domain {y} """ domain, symbols = self.rep.dom, set() if domain.is_Composite: for gen in domain.symbols: symbols |= gen.free_symbols elif domain.is_EX: for coeff in self.coeffs(): symbols |= coeff.free_symbols return symbols @property def gen(self): """ Return the principal generator. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).gen x """ return self.gens[0] @property def domain(self): """Get the ground domain of ``self``. """ return self.get_domain() @property def zero(self): """Return zero polynomial with ``self``'s properties. """ return self.new(self.rep.zero(self.rep.lev, self.rep.dom), *self.gens) @property def one(self): """Return one polynomial with ``self``'s properties. """ return self.new(self.rep.one(self.rep.lev, self.rep.dom), *self.gens) @property def unit(self): """Return unit polynomial with ``self``'s properties. """ return self.new(self.rep.unit(self.rep.lev, self.rep.dom), *self.gens) def unify(f, g): """ Make ``f`` and ``g`` belong to the same domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f, g = Poly(x/2 + 1), Poly(2*x + 1) >>> f Poly(1/2*x + 1, x, domain='QQ') >>> g Poly(2*x + 1, x, domain='ZZ') >>> F, G = f.unify(g) >>> F Poly(1/2*x + 1, x, domain='QQ') >>> G Poly(2*x + 1, x, domain='QQ') """ _, per, F, G = f._unify(g) return per(F), per(G) def _unify(f, g): g = sympify(g) if not g.is_Poly: try: return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g)) except CoercionFailed: raise UnificationFailed("can't unify %s with %s" % (f, g)) if isinstance(f.rep, DMP) and isinstance(g.rep, DMP): gens = _unify_gens(f.gens, g.gens) dom, lev = f.rep.dom.unify(g.rep.dom, gens), len(gens) - 1 if f.gens != gens: f_monoms, f_coeffs = _dict_reorder( f.rep.to_dict(), f.gens, gens) if f.rep.dom != dom: f_coeffs = [dom.convert(c, f.rep.dom) for c in f_coeffs] F = DMP(dict(list(zip(f_monoms, f_coeffs))), dom, lev) else: F = f.rep.convert(dom) if g.gens != gens: g_monoms, g_coeffs = _dict_reorder( g.rep.to_dict(), g.gens, gens) if g.rep.dom != dom: g_coeffs = [dom.convert(c, g.rep.dom) for c in g_coeffs] G = DMP(dict(list(zip(g_monoms, g_coeffs))), dom, lev) else: G = g.rep.convert(dom) else: raise UnificationFailed("can't unify %s with %s" % (f, g)) cls = f.__class__ def per(rep, dom=dom, gens=gens, remove=None): if remove is not None: gens = gens[:remove] + gens[remove + 1:] if not gens: return dom.to_sympy(rep) return cls.new(rep, *gens) return dom, per, F, G def per(f, rep, gens=None, remove=None): """ Create a Poly out of the given representation. Examples ======== >>> from sympy import Poly, ZZ >>> from sympy.abc import x, y >>> from sympy.polys.polyclasses import DMP >>> a = Poly(x**2 + 1) >>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y]) Poly(y + 1, y, domain='ZZ') """ if gens is None: gens = f.gens if remove is not None: gens = gens[:remove] + gens[remove + 1:] if not gens: return f.rep.dom.to_sympy(rep) return f.__class__.new(rep, *gens) def set_domain(f, domain): """Set the ground domain of ``f``. """ opt = options.build_options(f.gens, {'domain': domain}) return f.per(f.rep.convert(opt.domain)) def get_domain(f): """Get the ground domain of ``f``. """ return f.rep.dom def set_modulus(f, modulus): """ Set the modulus of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2) Poly(x**2 + 1, x, modulus=2) """ modulus = options.Modulus.preprocess(modulus) return f.set_domain(FF(modulus)) def get_modulus(f): """ Get the modulus of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, modulus=2).get_modulus() 2 """ domain = f.get_domain() if domain.is_FiniteField: return Integer(domain.characteristic()) else: raise PolynomialError("not a polynomial over a Galois field") def _eval_subs(f, old, new): """Internal implementation of :func:`subs`. """ if old in f.gens: if new.is_number: return f.eval(old, new) else: try: return f.replace(old, new) except PolynomialError: pass return f.as_expr().subs(old, new) def exclude(f): """ Remove unnecessary generators from ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import a, b, c, d, x >>> Poly(a + x, a, b, c, d, x).exclude() Poly(a + x, a, x, domain='ZZ') """ J, new = f.rep.exclude() gens = [] for j in range(len(f.gens)): if j not in J: gens.append(f.gens[j]) return f.per(new, gens=gens) def replace(f, x, y=None, *_ignore): # XXX this does not match Basic's signature """ Replace ``x`` with ``y`` in generators list. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 1, x).replace(x, y) Poly(y**2 + 1, y, domain='ZZ') """ if y is None: if f.is_univariate: x, y = f.gen, x else: raise PolynomialError( "syntax supported only in univariate case") if x == y or x not in f.gens: return f if x in f.gens and y not in f.gens: dom = f.get_domain() if not dom.is_Composite or y not in dom.symbols: gens = list(f.gens) gens[gens.index(x)] = y return f.per(f.rep, gens=gens) raise PolynomialError("can't replace %s with %s in %s" % (x, y, f)) def match(f, *args, **kwargs): """Match expression from Poly. See Basic.match()""" return f.as_expr().match(*args, **kwargs) def reorder(f, *gens, **args): """ Efficiently apply new order of generators. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + x*y**2, x, y).reorder(y, x) Poly(y**2*x + x**2, y, x, domain='ZZ') """ opt = options.Options((), args) if not gens: gens = _sort_gens(f.gens, opt=opt) elif set(f.gens) != set(gens): raise PolynomialError( "generators list can differ only up to order of elements") rep = dict(list(zip(*_dict_reorder(f.rep.to_dict(), f.gens, gens)))) return f.per(DMP(rep, f.rep.dom, len(gens) - 1), gens=gens) def ltrim(f, gen): """ Remove dummy generators from ``f`` that are to the left of specified ``gen`` in the generators as ordered. When ``gen`` is an integer, it refers to the generator located at that position within the tuple of generators of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> Poly(y**2 + y*z**2, x, y, z).ltrim(y) Poly(y**2 + y*z**2, y, z, domain='ZZ') >>> Poly(z, x, y, z).ltrim(-1) Poly(z, z, domain='ZZ') """ rep = f.as_dict(native=True) j = f._gen_to_level(gen) terms = {} for monom, coeff in rep.items(): if any(monom[:j]): # some generator is used in the portion to be trimmed raise PolynomialError("can't left trim %s" % f) terms[monom[j:]] = coeff gens = f.gens[j:] return f.new(DMP.from_dict(terms, len(gens) - 1, f.rep.dom), *gens) def has_only_gens(f, *gens): """ Return ``True`` if ``Poly(f, *gens)`` retains ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> Poly(x*y + 1, x, y, z).has_only_gens(x, y) True >>> Poly(x*y + z, x, y, z).has_only_gens(x, y) False """ indices = set() for gen in gens: try: index = f.gens.index(gen) except ValueError: raise GeneratorsError( "%s doesn't have %s as generator" % (f, gen)) else: indices.add(index) for monom in f.monoms(): for i, elt in enumerate(monom): if i not in indices and elt: return False return True def to_ring(f): """ Make the ground domain a ring. Examples ======== >>> from sympy import Poly, QQ >>> from sympy.abc import x >>> Poly(x**2 + 1, domain=QQ).to_ring() Poly(x**2 + 1, x, domain='ZZ') """ if hasattr(f.rep, 'to_ring'): result = f.rep.to_ring() else: # pragma: no cover raise OperationNotSupported(f, 'to_ring') return f.per(result) def to_field(f): """ Make the ground domain a field. Examples ======== >>> from sympy import Poly, ZZ >>> from sympy.abc import x >>> Poly(x**2 + 1, x, domain=ZZ).to_field() Poly(x**2 + 1, x, domain='QQ') """ if hasattr(f.rep, 'to_field'): result = f.rep.to_field() else: # pragma: no cover raise OperationNotSupported(f, 'to_field') return f.per(result) def to_exact(f): """ Make the ground domain exact. Examples ======== >>> from sympy import Poly, RR >>> from sympy.abc import x >>> Poly(x**2 + 1.0, x, domain=RR).to_exact() Poly(x**2 + 1, x, domain='QQ') """ if hasattr(f.rep, 'to_exact'): result = f.rep.to_exact() else: # pragma: no cover raise OperationNotSupported(f, 'to_exact') return f.per(result) def retract(f, field=None): """ Recalculate the ground domain of a polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = Poly(x**2 + 1, x, domain='QQ[y]') >>> f Poly(x**2 + 1, x, domain='QQ[y]') >>> f.retract() Poly(x**2 + 1, x, domain='ZZ') >>> f.retract(field=True) Poly(x**2 + 1, x, domain='QQ') """ dom, rep = construct_domain(f.as_dict(zero=True), field=field, composite=f.domain.is_Composite or None) return f.from_dict(rep, f.gens, domain=dom) def slice(f, x, m, n=None): """Take a continuous subsequence of terms of ``f``. """ if n is None: j, m, n = 0, x, m else: j = f._gen_to_level(x) m, n = int(m), int(n) if hasattr(f.rep, 'slice'): result = f.rep.slice(m, n, j) else: # pragma: no cover raise OperationNotSupported(f, 'slice') return f.per(result) def coeffs(f, order=None): """ Returns all non-zero coefficients from ``f`` in lex order. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x + 3, x).coeffs() [1, 2, 3] See Also ======== all_coeffs coeff_monomial nth """ return [f.rep.dom.to_sympy(c) for c in f.rep.coeffs(order=order)] def monoms(f, order=None): """ Returns all non-zero monomials from ``f`` in lex order. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms() [(2, 0), (1, 2), (1, 1), (0, 1)] See Also ======== all_monoms """ return f.rep.monoms(order=order) def terms(f, order=None): """ Returns all non-zero terms from ``f`` in lex order. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms() [((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)] See Also ======== all_terms """ return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.terms(order=order)] def all_coeffs(f): """ Returns all coefficients from a univariate polynomial ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x - 1, x).all_coeffs() [1, 0, 2, -1] """ return [f.rep.dom.to_sympy(c) for c in f.rep.all_coeffs()] def all_monoms(f): """ Returns all monomials from a univariate polynomial ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x - 1, x).all_monoms() [(3,), (2,), (1,), (0,)] See Also ======== all_terms """ return f.rep.all_monoms() def all_terms(f): """ Returns all terms from a univariate polynomial ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x - 1, x).all_terms() [((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)] """ return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.all_terms()] def termwise(f, func, *gens, **args): """ Apply a function to all terms of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> def func(k, coeff): ... k = k[0] ... return coeff//10**(2-k) >>> Poly(x**2 + 20*x + 400).termwise(func) Poly(x**2 + 2*x + 4, x, domain='ZZ') """ terms = {} for monom, coeff in f.terms(): result = func(monom, coeff) if isinstance(result, tuple): monom, coeff = result else: coeff = result if coeff: if monom not in terms: terms[monom] = coeff else: raise PolynomialError( "%s monomial was generated twice" % monom) return f.from_dict(terms, *(gens or f.gens), **args) def length(f): """ Returns the number of non-zero terms in ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 2*x - 1).length() 3 """ return len(f.as_dict()) def as_dict(f, native=False, zero=False): """ Switch to a ``dict`` representation. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict() {(0, 1): -1, (1, 2): 2, (2, 0): 1} """ if native: return f.rep.to_dict(zero=zero) else: return f.rep.to_sympy_dict(zero=zero) def as_list(f, native=False): """Switch to a ``list`` representation. """ if native: return f.rep.to_list() else: return f.rep.to_sympy_list() def as_expr(f, *gens): """ Convert a Poly instance to an Expr instance. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = Poly(x**2 + 2*x*y**2 - y, x, y) >>> f.as_expr() x**2 + 2*x*y**2 - y >>> f.as_expr({x: 5}) 10*y**2 - y + 25 >>> f.as_expr(5, 6) 379 """ if not gens: return f.expr if len(gens) == 1 and isinstance(gens[0], dict): mapping = gens[0] gens = list(f.gens) for gen, value in mapping.items(): try: index = gens.index(gen) except ValueError: raise GeneratorsError( "%s doesn't have %s as generator" % (f, gen)) else: gens[index] = value return basic_from_dict(f.rep.to_sympy_dict(), *gens) def as_poly(self, *gens, **args): """Converts ``self`` to a polynomial or returns ``None``. >>> from sympy import sin >>> from sympy.abc import x, y >>> print((x**2 + x*y).as_poly()) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + x*y).as_poly(x, y)) Poly(x**2 + x*y, x, y, domain='ZZ') >>> print((x**2 + sin(y)).as_poly(x, y)) None """ try: poly = Poly(self, *gens, **args) if not poly.is_Poly: return None else: return poly except PolynomialError: return None def lift(f): """ Convert algebraic coefficients to rationals. Examples ======== >>> from sympy import Poly, I >>> from sympy.abc import x >>> Poly(x**2 + I*x + 1, x, extension=I).lift() Poly(x**4 + 3*x**2 + 1, x, domain='QQ') """ if hasattr(f.rep, 'lift'): result = f.rep.lift() else: # pragma: no cover raise OperationNotSupported(f, 'lift') return f.per(result) def deflate(f): """ Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate() ((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ')) """ if hasattr(f.rep, 'deflate'): J, result = f.rep.deflate() else: # pragma: no cover raise OperationNotSupported(f, 'deflate') return J, f.per(result) def inject(f, front=False): """ Inject ground domain generators into ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x) >>> f.inject() Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ') >>> f.inject(front=True) Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ') """ dom = f.rep.dom if dom.is_Numerical: return f elif not dom.is_Poly: raise DomainError("can't inject generators over %s" % dom) if hasattr(f.rep, 'inject'): result = f.rep.inject(front=front) else: # pragma: no cover raise OperationNotSupported(f, 'inject') if front: gens = dom.symbols + f.gens else: gens = f.gens + dom.symbols return f.new(result, *gens) def eject(f, *gens): """ Eject selected generators into the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y) >>> f.eject(x) Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]') >>> f.eject(y) Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]') """ dom = f.rep.dom if not dom.is_Numerical: raise DomainError("can't eject generators over %s" % dom) k = len(gens) if f.gens[:k] == gens: _gens, front = f.gens[k:], True elif f.gens[-k:] == gens: _gens, front = f.gens[:-k], False else: raise NotImplementedError( "can only eject front or back generators") dom = dom.inject(*gens) if hasattr(f.rep, 'eject'): result = f.rep.eject(dom, front=front) else: # pragma: no cover raise OperationNotSupported(f, 'eject') return f.new(result, *_gens) def terms_gcd(f): """ Remove GCD of terms from the polynomial ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd() ((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ')) """ if hasattr(f.rep, 'terms_gcd'): J, result = f.rep.terms_gcd() else: # pragma: no cover raise OperationNotSupported(f, 'terms_gcd') return J, f.per(result) def add_ground(f, coeff): """ Add an element of the ground domain to ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x + 1).add_ground(2) Poly(x + 3, x, domain='ZZ') """ if hasattr(f.rep, 'add_ground'): result = f.rep.add_ground(coeff) else: # pragma: no cover raise OperationNotSupported(f, 'add_ground') return f.per(result) def sub_ground(f, coeff): """ Subtract an element of the ground domain from ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x + 1).sub_ground(2) Poly(x - 1, x, domain='ZZ') """ if hasattr(f.rep, 'sub_ground'): result = f.rep.sub_ground(coeff) else: # pragma: no cover raise OperationNotSupported(f, 'sub_ground') return f.per(result) def mul_ground(f, coeff): """ Multiply ``f`` by a an element of the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x + 1).mul_ground(2) Poly(2*x + 2, x, domain='ZZ') """ if hasattr(f.rep, 'mul_ground'): result = f.rep.mul_ground(coeff) else: # pragma: no cover raise OperationNotSupported(f, 'mul_ground') return f.per(result) def quo_ground(f, coeff): """ Quotient of ``f`` by a an element of the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x + 4).quo_ground(2) Poly(x + 2, x, domain='ZZ') >>> Poly(2*x + 3).quo_ground(2) Poly(x + 1, x, domain='ZZ') """ if hasattr(f.rep, 'quo_ground'): result = f.rep.quo_ground(coeff) else: # pragma: no cover raise OperationNotSupported(f, 'quo_ground') return f.per(result) def exquo_ground(f, coeff): """ Exact quotient of ``f`` by a an element of the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x + 4).exquo_ground(2) Poly(x + 2, x, domain='ZZ') >>> Poly(2*x + 3).exquo_ground(2) Traceback (most recent call last): ... ExactQuotientFailed: 2 does not divide 3 in ZZ """ if hasattr(f.rep, 'exquo_ground'): result = f.rep.exquo_ground(coeff) else: # pragma: no cover raise OperationNotSupported(f, 'exquo_ground') return f.per(result) def abs(f): """ Make all coefficients in ``f`` positive. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).abs() Poly(x**2 + 1, x, domain='ZZ') """ if hasattr(f.rep, 'abs'): result = f.rep.abs() else: # pragma: no cover raise OperationNotSupported(f, 'abs') return f.per(result) def neg(f): """ Negate all coefficients in ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).neg() Poly(-x**2 + 1, x, domain='ZZ') >>> -Poly(x**2 - 1, x) Poly(-x**2 + 1, x, domain='ZZ') """ if hasattr(f.rep, 'neg'): result = f.rep.neg() else: # pragma: no cover raise OperationNotSupported(f, 'neg') return f.per(result) def add(f, g): """ Add two polynomials ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).add(Poly(x - 2, x)) Poly(x**2 + x - 1, x, domain='ZZ') >>> Poly(x**2 + 1, x) + Poly(x - 2, x) Poly(x**2 + x - 1, x, domain='ZZ') """ g = sympify(g) if not g.is_Poly: return f.add_ground(g) _, per, F, G = f._unify(g) if hasattr(f.rep, 'add'): result = F.add(G) else: # pragma: no cover raise OperationNotSupported(f, 'add') return per(result) def sub(f, g): """ Subtract two polynomials ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).sub(Poly(x - 2, x)) Poly(x**2 - x + 3, x, domain='ZZ') >>> Poly(x**2 + 1, x) - Poly(x - 2, x) Poly(x**2 - x + 3, x, domain='ZZ') """ g = sympify(g) if not g.is_Poly: return f.sub_ground(g) _, per, F, G = f._unify(g) if hasattr(f.rep, 'sub'): result = F.sub(G) else: # pragma: no cover raise OperationNotSupported(f, 'sub') return per(result) def mul(f, g): """ Multiply two polynomials ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).mul(Poly(x - 2, x)) Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ') >>> Poly(x**2 + 1, x)*Poly(x - 2, x) Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ') """ g = sympify(g) if not g.is_Poly: return f.mul_ground(g) _, per, F, G = f._unify(g) if hasattr(f.rep, 'mul'): result = F.mul(G) else: # pragma: no cover raise OperationNotSupported(f, 'mul') return per(result) def sqr(f): """ Square a polynomial ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x - 2, x).sqr() Poly(x**2 - 4*x + 4, x, domain='ZZ') >>> Poly(x - 2, x)**2 Poly(x**2 - 4*x + 4, x, domain='ZZ') """ if hasattr(f.rep, 'sqr'): result = f.rep.sqr() else: # pragma: no cover raise OperationNotSupported(f, 'sqr') return f.per(result) def pow(f, n): """ Raise ``f`` to a non-negative power ``n``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x - 2, x).pow(3) Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ') >>> Poly(x - 2, x)**3 Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ') """ n = int(n) if hasattr(f.rep, 'pow'): result = f.rep.pow(n) else: # pragma: no cover raise OperationNotSupported(f, 'pow') return f.per(result) def pdiv(f, g): """ Polynomial pseudo-division of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x)) (Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ')) """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'pdiv'): q, r = F.pdiv(G) else: # pragma: no cover raise OperationNotSupported(f, 'pdiv') return per(q), per(r) def prem(f, g): """ Polynomial pseudo-remainder of ``f`` by ``g``. Caveat: The function prem(f, g, x) can be safely used to compute in Z[x] _only_ subresultant polynomial remainder sequences (prs's). To safely compute Euclidean and Sturmian prs's in Z[x] employ anyone of the corresponding functions found in the module sympy.polys.subresultants_qq_zz. The functions in the module with suffix _pg compute prs's in Z[x] employing rem(f, g, x), whereas the functions with suffix _amv compute prs's in Z[x] employing rem_z(f, g, x). The function rem_z(f, g, x) differs from prem(f, g, x) in that to compute the remainder polynomials in Z[x] it premultiplies the divident times the absolute value of the leading coefficient of the divisor raised to the power degree(f, x) - degree(g, x) + 1. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x)) Poly(20, x, domain='ZZ') """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'prem'): result = F.prem(G) else: # pragma: no cover raise OperationNotSupported(f, 'prem') return per(result) def pquo(f, g): """ Polynomial pseudo-quotient of ``f`` by ``g``. See the Caveat note in the function prem(f, g). Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x)) Poly(2*x + 4, x, domain='ZZ') >>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x)) Poly(2*x + 2, x, domain='ZZ') """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'pquo'): result = F.pquo(G) else: # pragma: no cover raise OperationNotSupported(f, 'pquo') return per(result) def pexquo(f, g): """ Polynomial exact pseudo-quotient of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x)) Poly(2*x + 2, x, domain='ZZ') >>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x)) Traceback (most recent call last): ... ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'pexquo'): try: result = F.pexquo(G) except ExactQuotientFailed as exc: raise exc.new(f.as_expr(), g.as_expr()) else: # pragma: no cover raise OperationNotSupported(f, 'pexquo') return per(result) def div(f, g, auto=True): """ Polynomial division with remainder of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x)) (Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ')) >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False) (Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ')) """ dom, per, F, G = f._unify(g) retract = False if auto and dom.is_Ring and not dom.is_Field: F, G = F.to_field(), G.to_field() retract = True if hasattr(f.rep, 'div'): q, r = F.div(G) else: # pragma: no cover raise OperationNotSupported(f, 'div') if retract: try: Q, R = q.to_ring(), r.to_ring() except CoercionFailed: pass else: q, r = Q, R return per(q), per(r) def rem(f, g, auto=True): """ Computes the polynomial remainder of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x)) Poly(5, x, domain='ZZ') >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False) Poly(x**2 + 1, x, domain='ZZ') """ dom, per, F, G = f._unify(g) retract = False if auto and dom.is_Ring and not dom.is_Field: F, G = F.to_field(), G.to_field() retract = True if hasattr(f.rep, 'rem'): r = F.rem(G) else: # pragma: no cover raise OperationNotSupported(f, 'rem') if retract: try: r = r.to_ring() except CoercionFailed: pass return per(r) def quo(f, g, auto=True): """ Computes polynomial quotient of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x)) Poly(1/2*x + 1, x, domain='QQ') >>> Poly(x**2 - 1, x).quo(Poly(x - 1, x)) Poly(x + 1, x, domain='ZZ') """ dom, per, F, G = f._unify(g) retract = False if auto and dom.is_Ring and not dom.is_Field: F, G = F.to_field(), G.to_field() retract = True if hasattr(f.rep, 'quo'): q = F.quo(G) else: # pragma: no cover raise OperationNotSupported(f, 'quo') if retract: try: q = q.to_ring() except CoercionFailed: pass return per(q) def exquo(f, g, auto=True): """ Computes polynomial exact quotient of ``f`` by ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x)) Poly(x + 1, x, domain='ZZ') >>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x)) Traceback (most recent call last): ... ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 """ dom, per, F, G = f._unify(g) retract = False if auto and dom.is_Ring and not dom.is_Field: F, G = F.to_field(), G.to_field() retract = True if hasattr(f.rep, 'exquo'): try: q = F.exquo(G) except ExactQuotientFailed as exc: raise exc.new(f.as_expr(), g.as_expr()) else: # pragma: no cover raise OperationNotSupported(f, 'exquo') if retract: try: q = q.to_ring() except CoercionFailed: pass return per(q) def _gen_to_level(f, gen): """Returns level associated with the given generator. """ if isinstance(gen, int): length = len(f.gens) if -length <= gen < length: if gen < 0: return length + gen else: return gen else: raise PolynomialError("-%s <= gen < %s expected, got %s" % (length, length, gen)) else: try: return f.gens.index(sympify(gen)) except ValueError: raise PolynomialError( "a valid generator expected, got %s" % gen) def degree(f, gen=0): """ Returns degree of ``f`` in ``x_j``. The degree of 0 is negative infinity. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + y*x + 1, x, y).degree() 2 >>> Poly(x**2 + y*x + y, x, y).degree(y) 1 >>> Poly(0, x).degree() -oo """ j = f._gen_to_level(gen) if hasattr(f.rep, 'degree'): return f.rep.degree(j) else: # pragma: no cover raise OperationNotSupported(f, 'degree') def degree_list(f): """ Returns a list of degrees of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + y*x + 1, x, y).degree_list() (2, 1) """ if hasattr(f.rep, 'degree_list'): return f.rep.degree_list() else: # pragma: no cover raise OperationNotSupported(f, 'degree_list') def total_degree(f): """ Returns the total degree of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + y*x + 1, x, y).total_degree() 2 >>> Poly(x + y**5, x, y).total_degree() 5 """ if hasattr(f.rep, 'total_degree'): return f.rep.total_degree() else: # pragma: no cover raise OperationNotSupported(f, 'total_degree') def homogenize(f, s): """ Returns the homogeneous polynomial of ``f``. A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. If you only want to check if a polynomial is homogeneous, then use :func:`Poly.is_homogeneous`. If you want not only to check if a polynomial is homogeneous but also compute its homogeneous order, then use :func:`Poly.homogeneous_order`. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3) >>> f.homogenize(z) Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ') """ if not isinstance(s, Symbol): raise TypeError("``Symbol`` expected, got %s" % type(s)) if s in f.gens: i = f.gens.index(s) gens = f.gens else: i = len(f.gens) gens = f.gens + (s,) if hasattr(f.rep, 'homogenize'): return f.per(f.rep.homogenize(i), gens=gens) raise OperationNotSupported(f, 'homogeneous_order') def homogeneous_order(f): """ Returns the homogeneous order of ``f``. A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. This degree is the homogeneous order of ``f``. If you only want to check if a polynomial is homogeneous, then use :func:`Poly.is_homogeneous`. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4) >>> f.homogeneous_order() 5 """ if hasattr(f.rep, 'homogeneous_order'): return f.rep.homogeneous_order() else: # pragma: no cover raise OperationNotSupported(f, 'homogeneous_order') def LC(f, order=None): """ Returns the leading coefficient of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC() 4 """ if order is not None: return f.coeffs(order)[0] if hasattr(f.rep, 'LC'): result = f.rep.LC() else: # pragma: no cover raise OperationNotSupported(f, 'LC') return f.rep.dom.to_sympy(result) def TC(f): """ Returns the trailing coefficient of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x**2 + 3*x, x).TC() 0 """ if hasattr(f.rep, 'TC'): result = f.rep.TC() else: # pragma: no cover raise OperationNotSupported(f, 'TC') return f.rep.dom.to_sympy(result) def EC(f, order=None): """ Returns the last non-zero coefficient of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 + 2*x**2 + 3*x, x).EC() 3 """ if hasattr(f.rep, 'coeffs'): return f.coeffs(order)[-1] else: # pragma: no cover raise OperationNotSupported(f, 'EC') def coeff_monomial(f, monom): """ Returns the coefficient of ``monom`` in ``f`` if there, else None. Examples ======== >>> from sympy import Poly, exp >>> from sympy.abc import x, y >>> p = Poly(24*x*y*exp(8) + 23*x, x, y) >>> p.coeff_monomial(x) 23 >>> p.coeff_monomial(y) 0 >>> p.coeff_monomial(x*y) 24*exp(8) Note that ``Expr.coeff()`` behaves differently, collecting terms if possible; the Poly must be converted to an Expr to use that method, however: >>> p.as_expr().coeff(x) 24*y*exp(8) + 23 >>> p.as_expr().coeff(y) 24*x*exp(8) >>> p.as_expr().coeff(x*y) 24*exp(8) See Also ======== nth: more efficient query using exponents of the monomial's generators """ return f.nth(*Monomial(monom, f.gens).exponents) def nth(f, *N): """ Returns the ``n``-th coefficient of ``f`` where ``N`` are the exponents of the generators in the term of interest. Examples ======== >>> from sympy import Poly, sqrt >>> from sympy.abc import x, y >>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2) 2 >>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2) 2 >>> Poly(4*sqrt(x)*y) Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ') >>> _.nth(1, 1) 4 See Also ======== coeff_monomial """ if hasattr(f.rep, 'nth'): if len(N) != len(f.gens): raise ValueError('exponent of each generator must be specified') result = f.rep.nth(*list(map(int, N))) else: # pragma: no cover raise OperationNotSupported(f, 'nth') return f.rep.dom.to_sympy(result) def coeff(f, x, n=1, right=False): # the semantics of coeff_monomial and Expr.coeff are different; # if someone is working with a Poly, they should be aware of the # differences and chose the method best suited for the query. # Alternatively, a pure-polys method could be written here but # at this time the ``right`` keyword would be ignored because Poly # doesn't work with non-commutatives. raise NotImplementedError( 'Either convert to Expr with `as_expr` method ' 'to use Expr\'s coeff method or else use the ' '`coeff_monomial` method of Polys.') def LM(f, order=None): """ Returns the leading monomial of ``f``. The Leading monomial signifies the monomial having the highest power of the principal generator in the expression f. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM() x**2*y**0 """ return Monomial(f.monoms(order)[0], f.gens) def EM(f, order=None): """ Returns the last non-zero monomial of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM() x**0*y**1 """ return Monomial(f.monoms(order)[-1], f.gens) def LT(f, order=None): """ Returns the leading term of ``f``. The Leading term signifies the term having the highest power of the principal generator in the expression f along with its coefficient. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT() (x**2*y**0, 4) """ monom, coeff = f.terms(order)[0] return Monomial(monom, f.gens), coeff def ET(f, order=None): """ Returns the last non-zero term of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET() (x**0*y**1, 3) """ monom, coeff = f.terms(order)[-1] return Monomial(monom, f.gens), coeff def max_norm(f): """ Returns maximum norm of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(-x**2 + 2*x - 3, x).max_norm() 3 """ if hasattr(f.rep, 'max_norm'): result = f.rep.max_norm() else: # pragma: no cover raise OperationNotSupported(f, 'max_norm') return f.rep.dom.to_sympy(result) def l1_norm(f): """ Returns l1 norm of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(-x**2 + 2*x - 3, x).l1_norm() 6 """ if hasattr(f.rep, 'l1_norm'): result = f.rep.l1_norm() else: # pragma: no cover raise OperationNotSupported(f, 'l1_norm') return f.rep.dom.to_sympy(result) def clear_denoms(self, convert=False): """ Clear denominators, but keep the ground domain. Examples ======== >>> from sympy import Poly, S, QQ >>> from sympy.abc import x >>> f = Poly(x/2 + S(1)/3, x, domain=QQ) >>> f.clear_denoms() (6, Poly(3*x + 2, x, domain='QQ')) >>> f.clear_denoms(convert=True) (6, Poly(3*x + 2, x, domain='ZZ')) """ f = self if not f.rep.dom.is_Field: return S.One, f dom = f.get_domain() if dom.has_assoc_Ring: dom = f.rep.dom.get_ring() if hasattr(f.rep, 'clear_denoms'): coeff, result = f.rep.clear_denoms() else: # pragma: no cover raise OperationNotSupported(f, 'clear_denoms') coeff, f = dom.to_sympy(coeff), f.per(result) if not convert or not dom.has_assoc_Ring: return coeff, f else: return coeff, f.to_ring() def rat_clear_denoms(self, g): """ Clear denominators in a rational function ``f/g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = Poly(x**2/y + 1, x) >>> g = Poly(x**3 + y, x) >>> p, q = f.rat_clear_denoms(g) >>> p Poly(x**2 + y, x, domain='ZZ[y]') >>> q Poly(y*x**3 + y**2, x, domain='ZZ[y]') """ f = self dom, per, f, g = f._unify(g) f = per(f) g = per(g) if not (dom.is_Field and dom.has_assoc_Ring): return f, g a, f = f.clear_denoms(convert=True) b, g = g.clear_denoms(convert=True) f = f.mul_ground(b) g = g.mul_ground(a) return f, g def integrate(self, *specs, **args): """ Computes indefinite integral of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 2*x + 1, x).integrate() Poly(1/3*x**3 + x**2 + x, x, domain='QQ') >>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0)) Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ') """ f = self if args.get('auto', True) and f.rep.dom.is_Ring: f = f.to_field() if hasattr(f.rep, 'integrate'): if not specs: return f.per(f.rep.integrate(m=1)) rep = f.rep for spec in specs: if type(spec) is tuple: gen, m = spec else: gen, m = spec, 1 rep = rep.integrate(int(m), f._gen_to_level(gen)) return f.per(rep) else: # pragma: no cover raise OperationNotSupported(f, 'integrate') def diff(f, *specs, **kwargs): """ Computes partial derivative of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + 2*x + 1, x).diff() Poly(2*x + 2, x, domain='ZZ') >>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1)) Poly(2*x*y, x, y, domain='ZZ') """ if not kwargs.get('evaluate', True): return Derivative(f, *specs, **kwargs) if hasattr(f.rep, 'diff'): if not specs: return f.per(f.rep.diff(m=1)) rep = f.rep for spec in specs: if type(spec) is tuple: gen, m = spec else: gen, m = spec, 1 rep = rep.diff(int(m), f._gen_to_level(gen)) return f.per(rep) else: # pragma: no cover raise OperationNotSupported(f, 'diff') _eval_derivative = diff def eval(self, x, a=None, auto=True): """ Evaluate ``f`` at ``a`` in the given variable. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> Poly(x**2 + 2*x + 3, x).eval(2) 11 >>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2) Poly(5*y + 8, y, domain='ZZ') >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z) >>> f.eval({x: 2}) Poly(5*y + 2*z + 6, y, z, domain='ZZ') >>> f.eval({x: 2, y: 5}) Poly(2*z + 31, z, domain='ZZ') >>> f.eval({x: 2, y: 5, z: 7}) 45 >>> f.eval((2, 5)) Poly(2*z + 31, z, domain='ZZ') >>> f(2, 5) Poly(2*z + 31, z, domain='ZZ') """ f = self if a is None: if isinstance(x, dict): mapping = x for gen, value in mapping.items(): f = f.eval(gen, value) return f elif isinstance(x, (tuple, list)): values = x if len(values) > len(f.gens): raise ValueError("too many values provided") for gen, value in zip(f.gens, values): f = f.eval(gen, value) return f else: j, a = 0, x else: j = f._gen_to_level(x) if not hasattr(f.rep, 'eval'): # pragma: no cover raise OperationNotSupported(f, 'eval') try: result = f.rep.eval(a, j) except CoercionFailed: if not auto: raise DomainError("can't evaluate at %s in %s" % (a, f.rep.dom)) else: a_domain, [a] = construct_domain([a]) new_domain = f.get_domain().unify_with_symbols(a_domain, f.gens) f = f.set_domain(new_domain) a = new_domain.convert(a, a_domain) result = f.rep.eval(a, j) return f.per(result, remove=j) def __call__(f, *values): """ Evaluate ``f`` at the give values. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y, z >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z) >>> f(2) Poly(5*y + 2*z + 6, y, z, domain='ZZ') >>> f(2, 5) Poly(2*z + 31, z, domain='ZZ') >>> f(2, 5, 7) 45 """ return f.eval(values) def half_gcdex(f, g, auto=True): """ Half extended Euclidean algorithm of ``f`` and ``g``. Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 >>> g = x**3 + x**2 - 4*x - 4 >>> Poly(f).half_gcdex(Poly(g)) (Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ')) """ dom, per, F, G = f._unify(g) if auto and dom.is_Ring: F, G = F.to_field(), G.to_field() if hasattr(f.rep, 'half_gcdex'): s, h = F.half_gcdex(G) else: # pragma: no cover raise OperationNotSupported(f, 'half_gcdex') return per(s), per(h) def gcdex(f, g, auto=True): """ Extended Euclidean algorithm of ``f`` and ``g``. Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 >>> g = x**3 + x**2 - 4*x - 4 >>> Poly(f).gcdex(Poly(g)) (Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'), Poly(x + 1, x, domain='QQ')) """ dom, per, F, G = f._unify(g) if auto and dom.is_Ring: F, G = F.to_field(), G.to_field() if hasattr(f.rep, 'gcdex'): s, t, h = F.gcdex(G) else: # pragma: no cover raise OperationNotSupported(f, 'gcdex') return per(s), per(t), per(h) def invert(f, g, auto=True): """ Invert ``f`` modulo ``g`` when possible. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x)) Poly(-4/3, x, domain='QQ') >>> Poly(x**2 - 1, x).invert(Poly(x - 1, x)) Traceback (most recent call last): ... NotInvertible: zero divisor """ dom, per, F, G = f._unify(g) if auto and dom.is_Ring: F, G = F.to_field(), G.to_field() if hasattr(f.rep, 'invert'): result = F.invert(G) else: # pragma: no cover raise OperationNotSupported(f, 'invert') return per(result) def revert(f, n): """ Compute ``f**(-1)`` mod ``x**n``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(1, x).revert(2) Poly(1, x, domain='ZZ') >>> Poly(1 + x, x).revert(1) Poly(1, x, domain='ZZ') >>> Poly(x**2 - 1, x).revert(1) Traceback (most recent call last): ... NotReversible: only unity is reversible in a ring >>> Poly(1/x, x).revert(1) Traceback (most recent call last): ... PolynomialError: 1/x contains an element of the generators set """ if hasattr(f.rep, 'revert'): result = f.rep.revert(int(n)) else: # pragma: no cover raise OperationNotSupported(f, 'revert') return f.per(result) def subresultants(f, g): """ Computes the subresultant PRS of ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x)) [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'), Poly(-2, x, domain='ZZ')] """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'subresultants'): result = F.subresultants(G) else: # pragma: no cover raise OperationNotSupported(f, 'subresultants') return list(map(per, result)) def resultant(f, g, includePRS=False): """ Computes the resultant of ``f`` and ``g`` via PRS. If includePRS=True, it includes the subresultant PRS in the result. Because the PRS is used to calculate the resultant, this is more efficient than calling :func:`subresultants` separately. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = Poly(x**2 + 1, x) >>> f.resultant(Poly(x**2 - 1, x)) 4 >>> f.resultant(Poly(x**2 - 1, x), includePRS=True) (4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'), Poly(-2, x, domain='ZZ')]) """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'resultant'): if includePRS: result, R = F.resultant(G, includePRS=includePRS) else: result = F.resultant(G) else: # pragma: no cover raise OperationNotSupported(f, 'resultant') if includePRS: return (per(result, remove=0), list(map(per, R))) return per(result, remove=0) def discriminant(f): """ Computes the discriminant of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + 2*x + 3, x).discriminant() -8 """ if hasattr(f.rep, 'discriminant'): result = f.rep.discriminant() else: # pragma: no cover raise OperationNotSupported(f, 'discriminant') return f.per(result, remove=0) def dispersionset(f, g=None): r"""Compute the *dispersion set* of two polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as: .. math:: \operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\} For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersion References ========== 1. [ManWright94]_ 2. [Koepf98]_ 3. [Abramov71]_ 4. [Man93]_ """ from sympy.polys.dispersion import dispersionset return dispersionset(f, g) def dispersion(f, g=None): r"""Compute the *dispersion* of polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as: .. math:: \operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \} and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersionset References ========== 1. [ManWright94]_ 2. [Koepf98]_ 3. [Abramov71]_ 4. [Man93]_ """ from sympy.polys.dispersion import dispersion return dispersion(f, g) def cofactors(f, g): """ Returns the GCD of ``f`` and ``g`` and their cofactors. Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors of ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x)) (Poly(x - 1, x, domain='ZZ'), Poly(x + 1, x, domain='ZZ'), Poly(x - 2, x, domain='ZZ')) """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'cofactors'): h, cff, cfg = F.cofactors(G) else: # pragma: no cover raise OperationNotSupported(f, 'cofactors') return per(h), per(cff), per(cfg) def gcd(f, g): """ Returns the polynomial GCD of ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x)) Poly(x - 1, x, domain='ZZ') """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'gcd'): result = F.gcd(G) else: # pragma: no cover raise OperationNotSupported(f, 'gcd') return per(result) def lcm(f, g): """ Returns polynomial LCM of ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x)) Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ') """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'lcm'): result = F.lcm(G) else: # pragma: no cover raise OperationNotSupported(f, 'lcm') return per(result) def trunc(f, p): """ Reduce ``f`` modulo a constant ``p``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3) Poly(-x**3 - x + 1, x, domain='ZZ') """ p = f.rep.dom.convert(p) if hasattr(f.rep, 'trunc'): result = f.rep.trunc(p) else: # pragma: no cover raise OperationNotSupported(f, 'trunc') return f.per(result) def monic(self, auto=True): """ Divides all coefficients by ``LC(f)``. Examples ======== >>> from sympy import Poly, ZZ >>> from sympy.abc import x >>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic() Poly(x**2 + 2*x + 3, x, domain='QQ') >>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic() Poly(x**2 + 4/3*x + 2/3, x, domain='QQ') """ f = self if auto and f.rep.dom.is_Ring: f = f.to_field() if hasattr(f.rep, 'monic'): result = f.rep.monic() else: # pragma: no cover raise OperationNotSupported(f, 'monic') return f.per(result) def content(f): """ Returns the GCD of polynomial coefficients. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(6*x**2 + 8*x + 12, x).content() 2 """ if hasattr(f.rep, 'content'): result = f.rep.content() else: # pragma: no cover raise OperationNotSupported(f, 'content') return f.rep.dom.to_sympy(result) def primitive(f): """ Returns the content and a primitive form of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**2 + 8*x + 12, x).primitive() (2, Poly(x**2 + 4*x + 6, x, domain='ZZ')) """ if hasattr(f.rep, 'primitive'): cont, result = f.rep.primitive() else: # pragma: no cover raise OperationNotSupported(f, 'primitive') return f.rep.dom.to_sympy(cont), f.per(result) def compose(f, g): """ Computes the functional composition of ``f`` and ``g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + x, x).compose(Poly(x - 1, x)) Poly(x**2 - x, x, domain='ZZ') """ _, per, F, G = f._unify(g) if hasattr(f.rep, 'compose'): result = F.compose(G) else: # pragma: no cover raise OperationNotSupported(f, 'compose') return per(result) def decompose(f): """ Computes a functional decomposition of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose() [Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')] """ if hasattr(f.rep, 'decompose'): result = f.rep.decompose() else: # pragma: no cover raise OperationNotSupported(f, 'decompose') return list(map(f.per, result)) def shift(f, a): """ Efficiently compute Taylor shift ``f(x + a)``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 2*x + 1, x).shift(2) Poly(x**2 + 2*x + 1, x, domain='ZZ') """ if hasattr(f.rep, 'shift'): result = f.rep.shift(a) else: # pragma: no cover raise OperationNotSupported(f, 'shift') return f.per(result) def transform(f, p, q): """ Efficiently evaluate the functional transformation ``q**n * f(p/q)``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x)) Poly(4, x, domain='ZZ') """ P, Q = p.unify(q) F, P = f.unify(P) F, Q = F.unify(Q) if hasattr(F.rep, 'transform'): result = F.rep.transform(P.rep, Q.rep) else: # pragma: no cover raise OperationNotSupported(F, 'transform') return F.per(result) def sturm(self, auto=True): """ Computes the Sturm sequence of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 - 2*x**2 + x - 3, x).sturm() [Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'), Poly(3*x**2 - 4*x + 1, x, domain='QQ'), Poly(2/9*x + 25/9, x, domain='QQ'), Poly(-2079/4, x, domain='QQ')] """ f = self if auto and f.rep.dom.is_Ring: f = f.to_field() if hasattr(f.rep, 'sturm'): result = f.rep.sturm() else: # pragma: no cover raise OperationNotSupported(f, 'sturm') return list(map(f.per, result)) def gff_list(f): """ Computes greatest factorial factorization of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = x**5 + 2*x**4 - x**3 - 2*x**2 >>> Poly(f).gff_list() [(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)] """ if hasattr(f.rep, 'gff_list'): result = f.rep.gff_list() else: # pragma: no cover raise OperationNotSupported(f, 'gff_list') return [(f.per(g), k) for g, k in result] def norm(f): """ Computes the product, ``Norm(f)``, of the conjugates of a polynomial ``f`` defined over a number field ``K``. Examples ======== >>> from sympy import Poly, sqrt >>> from sympy.abc import x >>> a, b = sqrt(2), sqrt(3) A polynomial over a quadratic extension. Two conjugates x - a and x + a. >>> f = Poly(x - a, x, extension=a) >>> f.norm() Poly(x**2 - 2, x, domain='QQ') A polynomial over a quartic extension. Four conjugates x - a, x - a, x + a and x + a. >>> f = Poly(x - a, x, extension=(a, b)) >>> f.norm() Poly(x**4 - 4*x**2 + 4, x, domain='QQ') """ if hasattr(f.rep, 'norm'): r = f.rep.norm() else: # pragma: no cover raise OperationNotSupported(f, 'norm') return f.per(r) def sqf_norm(f): """ Computes square-free norm of ``f``. Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``, where ``a`` is the algebraic extension of the ground domain. Examples ======== >>> from sympy import Poly, sqrt >>> from sympy.abc import x >>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm() >>> s 1 >>> f Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ<sqrt(3)>') >>> r Poly(x**4 - 4*x**2 + 16, x, domain='QQ') """ if hasattr(f.rep, 'sqf_norm'): s, g, r = f.rep.sqf_norm() else: # pragma: no cover raise OperationNotSupported(f, 'sqf_norm') return s, f.per(g), f.per(r) def sqf_part(f): """ Computes square-free part of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**3 - 3*x - 2, x).sqf_part() Poly(x**2 - x - 2, x, domain='ZZ') """ if hasattr(f.rep, 'sqf_part'): result = f.rep.sqf_part() else: # pragma: no cover raise OperationNotSupported(f, 'sqf_part') return f.per(result) def sqf_list(f, all=False): """ Returns a list of square-free factors of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 >>> Poly(f).sqf_list() (2, [(Poly(x + 1, x, domain='ZZ'), 2), (Poly(x + 2, x, domain='ZZ'), 3)]) >>> Poly(f).sqf_list(all=True) (2, [(Poly(1, x, domain='ZZ'), 1), (Poly(x + 1, x, domain='ZZ'), 2), (Poly(x + 2, x, domain='ZZ'), 3)]) """ if hasattr(f.rep, 'sqf_list'): coeff, factors = f.rep.sqf_list(all) else: # pragma: no cover raise OperationNotSupported(f, 'sqf_list') return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors] def sqf_list_include(f, all=False): """ Returns a list of square-free factors of ``f``. Examples ======== >>> from sympy import Poly, expand >>> from sympy.abc import x >>> f = expand(2*(x + 1)**3*x**4) >>> f 2*x**7 + 6*x**6 + 6*x**5 + 2*x**4 >>> Poly(f).sqf_list_include() [(Poly(2, x, domain='ZZ'), 1), (Poly(x + 1, x, domain='ZZ'), 3), (Poly(x, x, domain='ZZ'), 4)] >>> Poly(f).sqf_list_include(all=True) [(Poly(2, x, domain='ZZ'), 1), (Poly(1, x, domain='ZZ'), 2), (Poly(x + 1, x, domain='ZZ'), 3), (Poly(x, x, domain='ZZ'), 4)] """ if hasattr(f.rep, 'sqf_list_include'): factors = f.rep.sqf_list_include(all) else: # pragma: no cover raise OperationNotSupported(f, 'sqf_list_include') return [(f.per(g), k) for g, k in factors] def factor_list(f): """ Returns a list of irreducible factors of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y >>> Poly(f).factor_list() (2, [(Poly(x + y, x, y, domain='ZZ'), 1), (Poly(x**2 + 1, x, y, domain='ZZ'), 2)]) """ if hasattr(f.rep, 'factor_list'): try: coeff, factors = f.rep.factor_list() except DomainError: return S.One, [(f, 1)] else: # pragma: no cover raise OperationNotSupported(f, 'factor_list') return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors] def factor_list_include(f): """ Returns a list of irreducible factors of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y >>> Poly(f).factor_list_include() [(Poly(2*x + 2*y, x, y, domain='ZZ'), 1), (Poly(x**2 + 1, x, y, domain='ZZ'), 2)] """ if hasattr(f.rep, 'factor_list_include'): try: factors = f.rep.factor_list_include() except DomainError: return [(f, 1)] else: # pragma: no cover raise OperationNotSupported(f, 'factor_list_include') return [(f.per(g), k) for g, k in factors] def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False): """ Compute isolating intervals for roots of ``f``. For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used. References ========== .. [#] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. .. [#] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 3, x).intervals() [((-2, -1), 1), ((1, 2), 1)] >>> Poly(x**2 - 3, x).intervals(eps=1e-2) [((-26/15, -19/11), 1), ((19/11, 26/15), 1)] """ if eps is not None: eps = QQ.convert(eps) if eps <= 0: raise ValueError("'eps' must be a positive rational") if inf is not None: inf = QQ.convert(inf) if sup is not None: sup = QQ.convert(sup) if hasattr(f.rep, 'intervals'): result = f.rep.intervals( all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf) else: # pragma: no cover raise OperationNotSupported(f, 'intervals') if sqf: def _real(interval): s, t = interval return (QQ.to_sympy(s), QQ.to_sympy(t)) if not all: return list(map(_real, result)) def _complex(rectangle): (u, v), (s, t) = rectangle return (QQ.to_sympy(u) + I*QQ.to_sympy(v), QQ.to_sympy(s) + I*QQ.to_sympy(t)) real_part, complex_part = result return list(map(_real, real_part)), list(map(_complex, complex_part)) else: def _real(interval): (s, t), k = interval return ((QQ.to_sympy(s), QQ.to_sympy(t)), k) if not all: return list(map(_real, result)) def _complex(rectangle): ((u, v), (s, t)), k = rectangle return ((QQ.to_sympy(u) + I*QQ.to_sympy(v), QQ.to_sympy(s) + I*QQ.to_sympy(t)), k) real_part, complex_part = result return list(map(_real, real_part)), list(map(_complex, complex_part)) def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False): """ Refine an isolating interval of a root to the given precision. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2) (19/11, 26/15) """ if check_sqf and not f.is_sqf: raise PolynomialError("only square-free polynomials supported") s, t = QQ.convert(s), QQ.convert(t) if eps is not None: eps = QQ.convert(eps) if eps <= 0: raise ValueError("'eps' must be a positive rational") if steps is not None: steps = int(steps) elif eps is None: steps = 1 if hasattr(f.rep, 'refine_root'): S, T = f.rep.refine_root(s, t, eps=eps, steps=steps, fast=fast) else: # pragma: no cover raise OperationNotSupported(f, 'refine_root') return QQ.to_sympy(S), QQ.to_sympy(T) def count_roots(f, inf=None, sup=None): """ Return the number of roots of ``f`` in ``[inf, sup]`` interval. Examples ======== >>> from sympy import Poly, I >>> from sympy.abc import x >>> Poly(x**4 - 4, x).count_roots(-3, 3) 2 >>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I) 1 """ inf_real, sup_real = True, True if inf is not None: inf = sympify(inf) if inf is S.NegativeInfinity: inf = None else: re, im = inf.as_real_imag() if not im: inf = QQ.convert(inf) else: inf, inf_real = list(map(QQ.convert, (re, im))), False if sup is not None: sup = sympify(sup) if sup is S.Infinity: sup = None else: re, im = sup.as_real_imag() if not im: sup = QQ.convert(sup) else: sup, sup_real = list(map(QQ.convert, (re, im))), False if inf_real and sup_real: if hasattr(f.rep, 'count_real_roots'): count = f.rep.count_real_roots(inf=inf, sup=sup) else: # pragma: no cover raise OperationNotSupported(f, 'count_real_roots') else: if inf_real and inf is not None: inf = (inf, QQ.zero) if sup_real and sup is not None: sup = (sup, QQ.zero) if hasattr(f.rep, 'count_complex_roots'): count = f.rep.count_complex_roots(inf=inf, sup=sup) else: # pragma: no cover raise OperationNotSupported(f, 'count_complex_roots') return Integer(count) def root(f, index, radicals=True): """ Get an indexed root of a polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4) >>> f.root(0) -1/2 >>> f.root(1) 2 >>> f.root(2) 2 >>> f.root(3) Traceback (most recent call last): ... IndexError: root index out of [-3, 2] range, got 3 >>> Poly(x**5 + x + 1).root(0) CRootOf(x**3 - x**2 + 1, 0) """ return sympy.polys.rootoftools.rootof(f, index, radicals=radicals) def real_roots(f, multiple=True, radicals=True): """ Return a list of real roots with multiplicities. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots() [-1/2, 2, 2] >>> Poly(x**3 + x + 1).real_roots() [CRootOf(x**3 + x + 1, 0)] """ reals = sympy.polys.rootoftools.CRootOf.real_roots(f, radicals=radicals) if multiple: return reals else: return group(reals, multiple=False) def all_roots(f, multiple=True, radicals=True): """ Return a list of real and complex roots with multiplicities. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots() [-1/2, 2, 2] >>> Poly(x**3 + x + 1).all_roots() [CRootOf(x**3 + x + 1, 0), CRootOf(x**3 + x + 1, 1), CRootOf(x**3 + x + 1, 2)] """ roots = sympy.polys.rootoftools.CRootOf.all_roots(f, radicals=radicals) if multiple: return roots else: return group(roots, multiple=False) def nroots(f, n=15, maxsteps=50, cleanup=True): """ Compute numerical approximations of roots of ``f``. Parameters ========== n ... the number of digits to calculate maxsteps ... the maximum number of iterations to do If the accuracy `n` cannot be reached in `maxsteps`, it will raise an exception. You need to rerun with higher maxsteps. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 3).nroots(n=15) [-1.73205080756888, 1.73205080756888] >>> Poly(x**2 - 3).nroots(n=30) [-1.73205080756887729352744634151, 1.73205080756887729352744634151] """ from sympy.functions.elementary.complexes import sign if f.is_multivariate: raise MultivariatePolynomialError( "can't compute numerical roots of %s" % f) if f.degree() <= 0: return [] # For integer and rational coefficients, convert them to integers only # (for accuracy). Otherwise just try to convert the coefficients to # mpmath.mpc and raise an exception if the conversion fails. if f.rep.dom is ZZ: coeffs = [int(coeff) for coeff in f.all_coeffs()] elif f.rep.dom is QQ: denoms = [coeff.q for coeff in f.all_coeffs()] from sympy.core.numbers import ilcm fac = ilcm(*denoms) coeffs = [int(coeff*fac) for coeff in f.all_coeffs()] else: coeffs = [coeff.evalf(n=n).as_real_imag() for coeff in f.all_coeffs()] try: coeffs = [mpmath.mpc(*coeff) for coeff in coeffs] except TypeError: raise DomainError("Numerical domain expected, got %s" % \ f.rep.dom) dps = mpmath.mp.dps mpmath.mp.dps = n try: # We need to add extra precision to guard against losing accuracy. # 10 times the degree of the polynomial seems to work well. roots = mpmath.polyroots(coeffs, maxsteps=maxsteps, cleanup=cleanup, error=False, extraprec=f.degree()*10) # Mpmath puts real roots first, then complex ones (as does all_roots) # so we make sure this convention holds here, too. roots = list(map(sympify, sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag))))) except NoConvergence: raise NoConvergence( 'convergence to root failed; try n < %s or maxsteps > %s' % ( n, maxsteps)) finally: mpmath.mp.dps = dps return roots def ground_roots(f): """ Compute roots of ``f`` by factorization in the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots() {0: 2, 1: 2} """ if f.is_multivariate: raise MultivariatePolynomialError( "can't compute ground roots of %s" % f) roots = {} for factor, k in f.factor_list()[1]: if factor.is_linear: a, b = factor.all_coeffs() roots[-b/a] = k return roots def nth_power_roots_poly(f, n): """ Construct a polynomial with n-th powers of roots of ``f``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = Poly(x**4 - x**2 + 1) >>> f.nth_power_roots_poly(2) Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ') >>> f.nth_power_roots_poly(3) Poly(x**4 + 2*x**2 + 1, x, domain='ZZ') >>> f.nth_power_roots_poly(4) Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ') >>> f.nth_power_roots_poly(12) Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ') """ if f.is_multivariate: raise MultivariatePolynomialError( "must be a univariate polynomial") N = sympify(n) if N.is_Integer and N >= 1: n = int(N) else: raise ValueError("'n' must an integer and n >= 1, got %s" % n) x = f.gen t = Dummy('t') r = f.resultant(f.__class__.from_expr(x**n - t, x, t)) return r.replace(t, x) def cancel(f, g, include=False): """ Cancel common factors in a rational function ``f/g``. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x)) (1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ')) >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True) (Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ')) """ dom, per, F, G = f._unify(g) if hasattr(F, 'cancel'): result = F.cancel(G, include=include) else: # pragma: no cover raise OperationNotSupported(f, 'cancel') if not include: if dom.has_assoc_Ring: dom = dom.get_ring() cp, cq, p, q = result cp = dom.to_sympy(cp) cq = dom.to_sympy(cq) return cp/cq, per(p), per(q) else: return tuple(map(per, result)) @property def is_zero(f): """ Returns ``True`` if ``f`` is a zero polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(0, x).is_zero True >>> Poly(1, x).is_zero False """ return f.rep.is_zero @property def is_one(f): """ Returns ``True`` if ``f`` is a unit polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(0, x).is_one False >>> Poly(1, x).is_one True """ return f.rep.is_one @property def is_sqf(f): """ Returns ``True`` if ``f`` is a square-free polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 - 2*x + 1, x).is_sqf False >>> Poly(x**2 - 1, x).is_sqf True """ return f.rep.is_sqf @property def is_monic(f): """ Returns ``True`` if the leading coefficient of ``f`` is one. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x + 2, x).is_monic True >>> Poly(2*x + 2, x).is_monic False """ return f.rep.is_monic @property def is_primitive(f): """ Returns ``True`` if GCD of the coefficients of ``f`` is one. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(2*x**2 + 6*x + 12, x).is_primitive False >>> Poly(x**2 + 3*x + 6, x).is_primitive True """ return f.rep.is_primitive @property def is_ground(f): """ Returns ``True`` if ``f`` is an element of the ground domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x, x).is_ground False >>> Poly(2, x).is_ground True >>> Poly(y, x).is_ground True """ return f.rep.is_ground @property def is_linear(f): """ Returns ``True`` if ``f`` is linear in all its variables. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x + y + 2, x, y).is_linear True >>> Poly(x*y + 2, x, y).is_linear False """ return f.rep.is_linear @property def is_quadratic(f): """ Returns ``True`` if ``f`` is quadratic in all its variables. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x*y + 2, x, y).is_quadratic True >>> Poly(x*y**2 + 2, x, y).is_quadratic False """ return f.rep.is_quadratic @property def is_monomial(f): """ Returns ``True`` if ``f`` is zero or has only one term. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(3*x**2, x).is_monomial True >>> Poly(3*x**2 + 1, x).is_monomial False """ return f.rep.is_monomial @property def is_homogeneous(f): """ Returns ``True`` if ``f`` is a homogeneous polynomial. A homogeneous polynomial is a polynomial whose all monomials with non-zero coefficients have the same total degree. If you want not only to check if a polynomial is homogeneous but also compute its homogeneous order, then use :func:`Poly.homogeneous_order`. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + x*y, x, y).is_homogeneous True >>> Poly(x**3 + x*y, x, y).is_homogeneous False """ return f.rep.is_homogeneous @property def is_irreducible(f): """ Returns ``True`` if ``f`` has no factors over its domain. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible True >>> Poly(x**2 + 1, x, modulus=2).is_irreducible False """ return f.rep.is_irreducible @property def is_univariate(f): """ Returns ``True`` if ``f`` is a univariate polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + x + 1, x).is_univariate True >>> Poly(x*y**2 + x*y + 1, x, y).is_univariate False >>> Poly(x*y**2 + x*y + 1, x).is_univariate True >>> Poly(x**2 + x + 1, x, y).is_univariate False """ return len(f.gens) == 1 @property def is_multivariate(f): """ Returns ``True`` if ``f`` is a multivariate polynomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x, y >>> Poly(x**2 + x + 1, x).is_multivariate False >>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate True >>> Poly(x*y**2 + x*y + 1, x).is_multivariate False >>> Poly(x**2 + x + 1, x, y).is_multivariate True """ return len(f.gens) != 1 @property def is_cyclotomic(f): """ Returns ``True`` if ``f`` is a cyclotomic polnomial. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 >>> Poly(f).is_cyclotomic False >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 >>> Poly(g).is_cyclotomic True """ return f.rep.is_cyclotomic def __abs__(f): return f.abs() def __neg__(f): return f.neg() @_polifyit def __add__(f, g): return f.add(g) @_polifyit def __radd__(f, g): return g.add(f) @_polifyit def __sub__(f, g): return f.sub(g) @_polifyit def __rsub__(f, g): return g.sub(f) @_polifyit def __mul__(f, g): return f.mul(g) @_polifyit def __rmul__(f, g): return g.mul(f) @_sympifyit('n', NotImplemented) def __pow__(f, n): if n.is_Integer and n >= 0: return f.pow(n) else: return NotImplemented @_polifyit def __divmod__(f, g): return f.div(g) @_polifyit def __rdivmod__(f, g): return g.div(f) @_polifyit def __mod__(f, g): return f.rem(g) @_polifyit def __rmod__(f, g): return g.rem(f) @_polifyit def __floordiv__(f, g): return f.quo(g) @_polifyit def __rfloordiv__(f, g): return g.quo(f) @_sympifyit('g', NotImplemented) def __truediv__(f, g): return f.as_expr()/g.as_expr() @_sympifyit('g', NotImplemented) def __rtruediv__(f, g): return g.as_expr()/f.as_expr() @_sympifyit('other', NotImplemented) def __eq__(self, other): f, g = self, other if not g.is_Poly: try: g = f.__class__(g, f.gens, domain=f.get_domain()) except (PolynomialError, DomainError, CoercionFailed): return False if f.gens != g.gens: return False if f.rep.dom != g.rep.dom: return False return f.rep == g.rep @_sympifyit('g', NotImplemented) def __ne__(f, g): return not f == g def __bool__(f): return not f.is_zero def eq(f, g, strict=False): if not strict: return f == g else: return f._strict_eq(sympify(g)) def ne(f, g, strict=False): return not f.eq(g, strict=strict) def _strict_eq(f, g): return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True) @public class PurePoly(Poly): """Class for representing pure polynomials. """ def _hashable_content(self): """Allow SymPy to hash Poly instances. """ return (self.rep,) def __hash__(self): return super().__hash__() @property def free_symbols(self): """ Free symbols of a polynomial. Examples ======== >>> from sympy import PurePoly >>> from sympy.abc import x, y >>> PurePoly(x**2 + 1).free_symbols set() >>> PurePoly(x**2 + y).free_symbols set() >>> PurePoly(x**2 + y, x).free_symbols {y} """ return self.free_symbols_in_domain @_sympifyit('other', NotImplemented) def __eq__(self, other): f, g = self, other if not g.is_Poly: try: g = f.__class__(g, f.gens, domain=f.get_domain()) except (PolynomialError, DomainError, CoercionFailed): return False if len(f.gens) != len(g.gens): return False if f.rep.dom != g.rep.dom: try: dom = f.rep.dom.unify(g.rep.dom, f.gens) except UnificationFailed: return False f = f.set_domain(dom) g = g.set_domain(dom) return f.rep == g.rep def _strict_eq(f, g): return isinstance(g, f.__class__) and f.rep.eq(g.rep, strict=True) def _unify(f, g): g = sympify(g) if not g.is_Poly: try: return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g)) except CoercionFailed: raise UnificationFailed("can't unify %s with %s" % (f, g)) if len(f.gens) != len(g.gens): raise UnificationFailed("can't unify %s with %s" % (f, g)) if not (isinstance(f.rep, DMP) and isinstance(g.rep, DMP)): raise UnificationFailed("can't unify %s with %s" % (f, g)) cls = f.__class__ gens = f.gens dom = f.rep.dom.unify(g.rep.dom, gens) F = f.rep.convert(dom) G = g.rep.convert(dom) def per(rep, dom=dom, gens=gens, remove=None): if remove is not None: gens = gens[:remove] + gens[remove + 1:] if not gens: return dom.to_sympy(rep) return cls.new(rep, *gens) return dom, per, F, G @public def poly_from_expr(expr, *gens, **args): """Construct a polynomial from an expression. """ opt = options.build_options(gens, args) return _poly_from_expr(expr, opt) def _poly_from_expr(expr, opt): """Construct a polynomial from an expression. """ orig, expr = expr, sympify(expr) if not isinstance(expr, Basic): raise PolificationFailed(opt, orig, expr) elif expr.is_Poly: poly = expr.__class__._from_poly(expr, opt) opt.gens = poly.gens opt.domain = poly.domain if opt.polys is None: opt.polys = True return poly, opt elif opt.expand: expr = expr.expand() rep, opt = _dict_from_expr(expr, opt) if not opt.gens: raise PolificationFailed(opt, orig, expr) monoms, coeffs = list(zip(*list(rep.items()))) domain = opt.domain if domain is None: opt.domain, coeffs = construct_domain(coeffs, opt=opt) else: coeffs = list(map(domain.from_sympy, coeffs)) rep = dict(list(zip(monoms, coeffs))) poly = Poly._from_dict(rep, opt) if opt.polys is None: opt.polys = False return poly, opt @public def parallel_poly_from_expr(exprs, *gens, **args): """Construct polynomials from expressions. """ opt = options.build_options(gens, args) return _parallel_poly_from_expr(exprs, opt) def _parallel_poly_from_expr(exprs, opt): """Construct polynomials from expressions. """ from sympy.functions.elementary.piecewise import Piecewise if len(exprs) == 2: f, g = exprs if isinstance(f, Poly) and isinstance(g, Poly): f = f.__class__._from_poly(f, opt) g = g.__class__._from_poly(g, opt) f, g = f.unify(g) opt.gens = f.gens opt.domain = f.domain if opt.polys is None: opt.polys = True return [f, g], opt origs, exprs = list(exprs), [] _exprs, _polys = [], [] failed = False for i, expr in enumerate(origs): expr = sympify(expr) if isinstance(expr, Basic): if expr.is_Poly: _polys.append(i) else: _exprs.append(i) if opt.expand: expr = expr.expand() else: failed = True exprs.append(expr) if failed: raise PolificationFailed(opt, origs, exprs, True) if _polys: # XXX: this is a temporary solution for i in _polys: exprs[i] = exprs[i].as_expr() reps, opt = _parallel_dict_from_expr(exprs, opt) if not opt.gens: raise PolificationFailed(opt, origs, exprs, True) for k in opt.gens: if isinstance(k, Piecewise): raise PolynomialError("Piecewise generators do not make sense") coeffs_list, lengths = [], [] all_monoms = [] all_coeffs = [] for rep in reps: monoms, coeffs = list(zip(*list(rep.items()))) coeffs_list.extend(coeffs) all_monoms.append(monoms) lengths.append(len(coeffs)) domain = opt.domain if domain is None: opt.domain, coeffs_list = construct_domain(coeffs_list, opt=opt) else: coeffs_list = list(map(domain.from_sympy, coeffs_list)) for k in lengths: all_coeffs.append(coeffs_list[:k]) coeffs_list = coeffs_list[k:] polys = [] for monoms, coeffs in zip(all_monoms, all_coeffs): rep = dict(list(zip(monoms, coeffs))) poly = Poly._from_dict(rep, opt) polys.append(poly) if opt.polys is None: opt.polys = bool(_polys) return polys, opt def _update_args(args, key, value): """Add a new ``(key, value)`` pair to arguments ``dict``. """ args = dict(args) if key not in args: args[key] = value return args @public def degree(f, gen=0): """ Return the degree of ``f`` in the given variable. The degree of 0 is negative infinity. Examples ======== >>> from sympy import degree >>> from sympy.abc import x, y >>> degree(x**2 + y*x + 1, gen=x) 2 >>> degree(x**2 + y*x + 1, gen=y) 1 >>> degree(0, x) -oo See also ======== sympy.polys.polytools.Poly.total_degree degree_list """ f = sympify(f, strict=True) gen_is_Num = sympify(gen, strict=True).is_Number if f.is_Poly: p = f isNum = p.as_expr().is_Number else: isNum = f.is_Number if not isNum: if gen_is_Num: p, _ = poly_from_expr(f) else: p, _ = poly_from_expr(f, gen) if isNum: return S.Zero if f else S.NegativeInfinity if not gen_is_Num: if f.is_Poly and gen not in p.gens: # try recast without explicit gens p, _ = poly_from_expr(f.as_expr()) if gen not in p.gens: return S.Zero elif not f.is_Poly and len(f.free_symbols) > 1: raise TypeError(filldedent(''' A symbolic generator of interest is required for a multivariate expression like func = %s, e.g. degree(func, gen = %s) instead of degree(func, gen = %s). ''' % (f, next(ordered(f.free_symbols)), gen))) return Integer(p.degree(gen)) @public def total_degree(f, *gens): """ Return the total_degree of ``f`` in the given variables. Examples ======== >>> from sympy import total_degree, Poly >>> from sympy.abc import x, y >>> total_degree(1) 0 >>> total_degree(x + x*y) 2 >>> total_degree(x + x*y, x) 1 If the expression is a Poly and no variables are given then the generators of the Poly will be used: >>> p = Poly(x + x*y, y) >>> total_degree(p) 1 To deal with the underlying expression of the Poly, convert it to an Expr: >>> total_degree(p.as_expr()) 2 This is done automatically if any variables are given: >>> total_degree(p, x) 1 See also ======== degree """ p = sympify(f) if p.is_Poly: p = p.as_expr() if p.is_Number: rv = 0 else: if f.is_Poly: gens = gens or f.gens rv = Poly(p, gens).total_degree() return Integer(rv) @public def degree_list(f, *gens, **args): """ Return a list of degrees of ``f`` in all variables. Examples ======== >>> from sympy import degree_list >>> from sympy.abc import x, y >>> degree_list(x**2 + y*x + 1) (2, 1) """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('degree_list', 1, exc) degrees = F.degree_list() return tuple(map(Integer, degrees)) @public def LC(f, *gens, **args): """ Return the leading coefficient of ``f``. Examples ======== >>> from sympy import LC >>> from sympy.abc import x, y >>> LC(4*x**2 + 2*x*y**2 + x*y + 3*y) 4 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('LC', 1, exc) return F.LC(order=opt.order) @public def LM(f, *gens, **args): """ Return the leading monomial of ``f``. Examples ======== >>> from sympy import LM >>> from sympy.abc import x, y >>> LM(4*x**2 + 2*x*y**2 + x*y + 3*y) x**2 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('LM', 1, exc) monom = F.LM(order=opt.order) return monom.as_expr() @public def LT(f, *gens, **args): """ Return the leading term of ``f``. Examples ======== >>> from sympy import LT >>> from sympy.abc import x, y >>> LT(4*x**2 + 2*x*y**2 + x*y + 3*y) 4*x**2 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('LT', 1, exc) monom, coeff = F.LT(order=opt.order) return coeff*monom.as_expr() @public def pdiv(f, g, *gens, **args): """ Compute polynomial pseudo-division of ``f`` and ``g``. Examples ======== >>> from sympy import pdiv >>> from sympy.abc import x >>> pdiv(x**2 + 1, 2*x - 4) (2*x + 4, 20) """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('pdiv', 2, exc) q, r = F.pdiv(G) if not opt.polys: return q.as_expr(), r.as_expr() else: return q, r @public def prem(f, g, *gens, **args): """ Compute polynomial pseudo-remainder of ``f`` and ``g``. Examples ======== >>> from sympy import prem >>> from sympy.abc import x >>> prem(x**2 + 1, 2*x - 4) 20 """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('prem', 2, exc) r = F.prem(G) if not opt.polys: return r.as_expr() else: return r @public def pquo(f, g, *gens, **args): """ Compute polynomial pseudo-quotient of ``f`` and ``g``. Examples ======== >>> from sympy import pquo >>> from sympy.abc import x >>> pquo(x**2 + 1, 2*x - 4) 2*x + 4 >>> pquo(x**2 - 1, 2*x - 1) 2*x + 1 """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('pquo', 2, exc) try: q = F.pquo(G) except ExactQuotientFailed: raise ExactQuotientFailed(f, g) if not opt.polys: return q.as_expr() else: return q @public def pexquo(f, g, *gens, **args): """ Compute polynomial exact pseudo-quotient of ``f`` and ``g``. Examples ======== >>> from sympy import pexquo >>> from sympy.abc import x >>> pexquo(x**2 - 1, 2*x - 2) 2*x + 2 >>> pexquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('pexquo', 2, exc) q = F.pexquo(G) if not opt.polys: return q.as_expr() else: return q @public def div(f, g, *gens, **args): """ Compute polynomial division of ``f`` and ``g``. Examples ======== >>> from sympy import div, ZZ, QQ >>> from sympy.abc import x >>> div(x**2 + 1, 2*x - 4, domain=ZZ) (0, x**2 + 1) >>> div(x**2 + 1, 2*x - 4, domain=QQ) (x/2 + 1, 5) """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('div', 2, exc) q, r = F.div(G, auto=opt.auto) if not opt.polys: return q.as_expr(), r.as_expr() else: return q, r @public def rem(f, g, *gens, **args): """ Compute polynomial remainder of ``f`` and ``g``. Examples ======== >>> from sympy import rem, ZZ, QQ >>> from sympy.abc import x >>> rem(x**2 + 1, 2*x - 4, domain=ZZ) x**2 + 1 >>> rem(x**2 + 1, 2*x - 4, domain=QQ) 5 """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('rem', 2, exc) r = F.rem(G, auto=opt.auto) if not opt.polys: return r.as_expr() else: return r @public def quo(f, g, *gens, **args): """ Compute polynomial quotient of ``f`` and ``g``. Examples ======== >>> from sympy import quo >>> from sympy.abc import x >>> quo(x**2 + 1, 2*x - 4) x/2 + 1 >>> quo(x**2 - 1, x - 1) x + 1 """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('quo', 2, exc) q = F.quo(G, auto=opt.auto) if not opt.polys: return q.as_expr() else: return q @public def exquo(f, g, *gens, **args): """ Compute polynomial exact quotient of ``f`` and ``g``. Examples ======== >>> from sympy import exquo >>> from sympy.abc import x >>> exquo(x**2 - 1, x - 1) x + 1 >>> exquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('exquo', 2, exc) q = F.exquo(G, auto=opt.auto) if not opt.polys: return q.as_expr() else: return q @public def half_gcdex(f, g, *gens, **args): """ Half extended Euclidean algorithm of ``f`` and ``g``. Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``. Examples ======== >>> from sympy import half_gcdex >>> from sympy.abc import x >>> half_gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4) (3/5 - x/5, x + 1) """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: s, h = domain.half_gcdex(a, b) except NotImplementedError: raise ComputationFailed('half_gcdex', 2, exc) else: return domain.to_sympy(s), domain.to_sympy(h) s, h = F.half_gcdex(G, auto=opt.auto) if not opt.polys: return s.as_expr(), h.as_expr() else: return s, h @public def gcdex(f, g, *gens, **args): """ Extended Euclidean algorithm of ``f`` and ``g``. Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. Examples ======== >>> from sympy import gcdex >>> from sympy.abc import x >>> gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4) (3/5 - x/5, x**2/5 - 6*x/5 + 2, x + 1) """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: s, t, h = domain.gcdex(a, b) except NotImplementedError: raise ComputationFailed('gcdex', 2, exc) else: return domain.to_sympy(s), domain.to_sympy(t), domain.to_sympy(h) s, t, h = F.gcdex(G, auto=opt.auto) if not opt.polys: return s.as_expr(), t.as_expr(), h.as_expr() else: return s, t, h @public def invert(f, g, *gens, **args): """ Invert ``f`` modulo ``g`` when possible. Examples ======== >>> from sympy import invert, S >>> from sympy.core.numbers import mod_inverse >>> from sympy.abc import x >>> invert(x**2 - 1, 2*x - 1) -4/3 >>> invert(x**2 - 1, x - 1) Traceback (most recent call last): ... NotInvertible: zero divisor For more efficient inversion of Rationals, use the :obj:`~.mod_inverse` function: >>> mod_inverse(3, 5) 2 >>> (S(2)/5).invert(S(7)/3) 5/2 See Also ======== sympy.core.numbers.mod_inverse """ options.allowed_flags(args, ['auto', 'polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: return domain.to_sympy(domain.invert(a, b)) except NotImplementedError: raise ComputationFailed('invert', 2, exc) h = F.invert(G, auto=opt.auto) if not opt.polys: return h.as_expr() else: return h @public def subresultants(f, g, *gens, **args): """ Compute subresultant PRS of ``f`` and ``g``. Examples ======== >>> from sympy import subresultants >>> from sympy.abc import x >>> subresultants(x**2 + 1, x**2 - 1) [x**2 + 1, x**2 - 1, -2] """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('subresultants', 2, exc) result = F.subresultants(G) if not opt.polys: return [r.as_expr() for r in result] else: return result @public def resultant(f, g, *gens, includePRS=False, **args): """ Compute resultant of ``f`` and ``g``. Examples ======== >>> from sympy import resultant >>> from sympy.abc import x >>> resultant(x**2 + 1, x**2 - 1) 4 """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('resultant', 2, exc) if includePRS: result, R = F.resultant(G, includePRS=includePRS) else: result = F.resultant(G) if not opt.polys: if includePRS: return result.as_expr(), [r.as_expr() for r in R] return result.as_expr() else: if includePRS: return result, R return result @public def discriminant(f, *gens, **args): """ Compute discriminant of ``f``. Examples ======== >>> from sympy import discriminant >>> from sympy.abc import x >>> discriminant(x**2 + 2*x + 3) -8 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('discriminant', 1, exc) result = F.discriminant() if not opt.polys: return result.as_expr() else: return result @public def cofactors(f, g, *gens, **args): """ Compute GCD and cofactors of ``f`` and ``g``. Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors of ``f`` and ``g``. Examples ======== >>> from sympy import cofactors >>> from sympy.abc import x >>> cofactors(x**2 - 1, x**2 - 3*x + 2) (x - 1, x + 1, x - 2) """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: h, cff, cfg = domain.cofactors(a, b) except NotImplementedError: raise ComputationFailed('cofactors', 2, exc) else: return domain.to_sympy(h), domain.to_sympy(cff), domain.to_sympy(cfg) h, cff, cfg = F.cofactors(G) if not opt.polys: return h.as_expr(), cff.as_expr(), cfg.as_expr() else: return h, cff, cfg @public def gcd_list(seq, *gens, **args): """ Compute GCD of a list of polynomials. Examples ======== >>> from sympy import gcd_list >>> from sympy.abc import x >>> gcd_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2]) x - 1 """ seq = sympify(seq) def try_non_polynomial_gcd(seq): if not gens and not args: domain, numbers = construct_domain(seq) if not numbers: return domain.zero elif domain.is_Numerical: result, numbers = numbers[0], numbers[1:] for number in numbers: result = domain.gcd(result, number) if domain.is_one(result): break return domain.to_sympy(result) return None result = try_non_polynomial_gcd(seq) if result is not None: return result options.allowed_flags(args, ['polys']) try: polys, opt = parallel_poly_from_expr(seq, *gens, **args) # gcd for domain Q[irrational] (purely algebraic irrational) if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq): a = seq[-1] lst = [ (a/elt).ratsimp() for elt in seq[:-1] ] if all(frc.is_rational for frc in lst): lc = 1 for frc in lst: lc = lcm(lc, frc.as_numer_denom()[0]) # abs ensures that the gcd is always non-negative return abs(a/lc) except PolificationFailed as exc: result = try_non_polynomial_gcd(exc.exprs) if result is not None: return result else: raise ComputationFailed('gcd_list', len(seq), exc) if not polys: if not opt.polys: return S.Zero else: return Poly(0, opt=opt) result, polys = polys[0], polys[1:] for poly in polys: result = result.gcd(poly) if result.is_one: break if not opt.polys: return result.as_expr() else: return result @public def gcd(f, g=None, *gens, **args): """ Compute GCD of ``f`` and ``g``. Examples ======== >>> from sympy import gcd >>> from sympy.abc import x >>> gcd(x**2 - 1, x**2 - 3*x + 2) x - 1 """ if hasattr(f, '__iter__'): if g is not None: gens = (g,) + gens return gcd_list(f, *gens, **args) elif g is None: raise TypeError("gcd() takes 2 arguments or a sequence of arguments") options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) # gcd for domain Q[irrational] (purely algebraic irrational) a, b = map(sympify, (f, g)) if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational: frc = (a/b).ratsimp() if frc.is_rational: # abs ensures that the returned gcd is always non-negative return abs(a/frc.as_numer_denom()[0]) except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: return domain.to_sympy(domain.gcd(a, b)) except NotImplementedError: raise ComputationFailed('gcd', 2, exc) result = F.gcd(G) if not opt.polys: return result.as_expr() else: return result @public def lcm_list(seq, *gens, **args): """ Compute LCM of a list of polynomials. Examples ======== >>> from sympy import lcm_list >>> from sympy.abc import x >>> lcm_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2]) x**5 - x**4 - 2*x**3 - x**2 + x + 2 """ seq = sympify(seq) def try_non_polynomial_lcm(seq): if not gens and not args: domain, numbers = construct_domain(seq) if not numbers: return domain.one elif domain.is_Numerical: result, numbers = numbers[0], numbers[1:] for number in numbers: result = domain.lcm(result, number) return domain.to_sympy(result) return None result = try_non_polynomial_lcm(seq) if result is not None: return result options.allowed_flags(args, ['polys']) try: polys, opt = parallel_poly_from_expr(seq, *gens, **args) # lcm for domain Q[irrational] (purely algebraic irrational) if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq): a = seq[-1] lst = [ (a/elt).ratsimp() for elt in seq[:-1] ] if all(frc.is_rational for frc in lst): lc = 1 for frc in lst: lc = lcm(lc, frc.as_numer_denom()[1]) return a*lc except PolificationFailed as exc: result = try_non_polynomial_lcm(exc.exprs) if result is not None: return result else: raise ComputationFailed('lcm_list', len(seq), exc) if not polys: if not opt.polys: return S.One else: return Poly(1, opt=opt) result, polys = polys[0], polys[1:] for poly in polys: result = result.lcm(poly) if not opt.polys: return result.as_expr() else: return result @public def lcm(f, g=None, *gens, **args): """ Compute LCM of ``f`` and ``g``. Examples ======== >>> from sympy import lcm >>> from sympy.abc import x >>> lcm(x**2 - 1, x**2 - 3*x + 2) x**3 - 2*x**2 - x + 2 """ if hasattr(f, '__iter__'): if g is not None: gens = (g,) + gens return lcm_list(f, *gens, **args) elif g is None: raise TypeError("lcm() takes 2 arguments or a sequence of arguments") options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) # lcm for domain Q[irrational] (purely algebraic irrational) a, b = map(sympify, (f, g)) if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational: frc = (a/b).ratsimp() if frc.is_rational: return a*frc.as_numer_denom()[1] except PolificationFailed as exc: domain, (a, b) = construct_domain(exc.exprs) try: return domain.to_sympy(domain.lcm(a, b)) except NotImplementedError: raise ComputationFailed('lcm', 2, exc) result = F.lcm(G) if not opt.polys: return result.as_expr() else: return result @public def terms_gcd(f, *gens, **args): """ Remove GCD of terms from ``f``. If the ``deep`` flag is True, then the arguments of ``f`` will have terms_gcd applied to them. If a fraction is factored out of ``f`` and ``f`` is an Add, then an unevaluated Mul will be returned so that automatic simplification does not redistribute it. The hint ``clear``, when set to False, can be used to prevent such factoring when all coefficients are not fractions. Examples ======== >>> from sympy import terms_gcd, cos >>> from sympy.abc import x, y >>> terms_gcd(x**6*y**2 + x**3*y, x, y) x**3*y*(x**3*y + 1) The default action of polys routines is to expand the expression given to them. terms_gcd follows this behavior: >>> terms_gcd((3+3*x)*(x+x*y)) 3*x*(x*y + x + y + 1) If this is not desired then the hint ``expand`` can be set to False. In this case the expression will be treated as though it were comprised of one or more terms: >>> terms_gcd((3+3*x)*(x+x*y), expand=False) (3*x + 3)*(x*y + x) In order to traverse factors of a Mul or the arguments of other functions, the ``deep`` hint can be used: >>> terms_gcd((3 + 3*x)*(x + x*y), expand=False, deep=True) 3*x*(x + 1)*(y + 1) >>> terms_gcd(cos(x + x*y), deep=True) cos(x*(y + 1)) Rationals are factored out by default: >>> terms_gcd(x + y/2) (2*x + y)/2 Only the y-term had a coefficient that was a fraction; if one does not want to factor out the 1/2 in cases like this, the flag ``clear`` can be set to False: >>> terms_gcd(x + y/2, clear=False) x + y/2 >>> terms_gcd(x*y/2 + y**2, clear=False) y*(x/2 + y) The ``clear`` flag is ignored if all coefficients are fractions: >>> terms_gcd(x/3 + y/2, clear=False) (2*x + 3*y)/6 See Also ======== sympy.core.exprtools.gcd_terms, sympy.core.exprtools.factor_terms """ from sympy.core.relational import Equality orig = sympify(f) if isinstance(f, Equality): return Equality(*(terms_gcd(s, *gens, **args) for s in [f.lhs, f.rhs])) elif isinstance(f, Relational): raise TypeError("Inequalities can not be used with terms_gcd. Found: %s" %(f,)) if not isinstance(f, Expr) or f.is_Atom: return orig if args.get('deep', False): new = f.func(*[terms_gcd(a, *gens, **args) for a in f.args]) args.pop('deep') args['expand'] = False return terms_gcd(new, *gens, **args) clear = args.pop('clear', True) options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: return exc.expr J, f = F.terms_gcd() if opt.domain.is_Ring: if opt.domain.is_Field: denom, f = f.clear_denoms(convert=True) coeff, f = f.primitive() if opt.domain.is_Field: coeff /= denom else: coeff = S.One term = Mul(*[x**j for x, j in zip(f.gens, J)]) if coeff == 1: coeff = S.One if term == 1: return orig if clear: return _keep_coeff(coeff, term*f.as_expr()) # base the clearing on the form of the original expression, not # the (perhaps) Mul that we have now coeff, f = _keep_coeff(coeff, f.as_expr(), clear=False).as_coeff_Mul() return _keep_coeff(coeff, term*f, clear=False) @public def trunc(f, p, *gens, **args): """ Reduce ``f`` modulo a constant ``p``. Examples ======== >>> from sympy import trunc >>> from sympy.abc import x >>> trunc(2*x**3 + 3*x**2 + 5*x + 7, 3) -x**3 - x + 1 """ options.allowed_flags(args, ['auto', 'polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('trunc', 1, exc) result = F.trunc(sympify(p)) if not opt.polys: return result.as_expr() else: return result @public def monic(f, *gens, **args): """ Divide all coefficients of ``f`` by ``LC(f)``. Examples ======== >>> from sympy import monic >>> from sympy.abc import x >>> monic(3*x**2 + 4*x + 2) x**2 + 4*x/3 + 2/3 """ options.allowed_flags(args, ['auto', 'polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('monic', 1, exc) result = F.monic(auto=opt.auto) if not opt.polys: return result.as_expr() else: return result @public def content(f, *gens, **args): """ Compute GCD of coefficients of ``f``. Examples ======== >>> from sympy import content >>> from sympy.abc import x >>> content(6*x**2 + 8*x + 12) 2 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('content', 1, exc) return F.content() @public def primitive(f, *gens, **args): """ Compute content and the primitive form of ``f``. Examples ======== >>> from sympy.polys.polytools import primitive >>> from sympy.abc import x >>> primitive(6*x**2 + 8*x + 12) (2, 3*x**2 + 4*x + 6) >>> eq = (2 + 2*x)*x + 2 Expansion is performed by default: >>> primitive(eq) (2, x**2 + x + 1) Set ``expand`` to False to shut this off. Note that the extraction will not be recursive; use the as_content_primitive method for recursive, non-destructive Rational extraction. >>> primitive(eq, expand=False) (1, x*(2*x + 2) + 2) >>> eq.as_content_primitive() (2, x*(x + 1) + 1) """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('primitive', 1, exc) cont, result = F.primitive() if not opt.polys: return cont, result.as_expr() else: return cont, result @public def compose(f, g, *gens, **args): """ Compute functional composition ``f(g)``. Examples ======== >>> from sympy import compose >>> from sympy.abc import x >>> compose(x**2 + x, x - 1) x**2 - x """ options.allowed_flags(args, ['polys']) try: (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('compose', 2, exc) result = F.compose(G) if not opt.polys: return result.as_expr() else: return result @public def decompose(f, *gens, **args): """ Compute functional decomposition of ``f``. Examples ======== >>> from sympy import decompose >>> from sympy.abc import x >>> decompose(x**4 + 2*x**3 - x - 1) [x**2 - x - 1, x**2 + x] """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('decompose', 1, exc) result = F.decompose() if not opt.polys: return [r.as_expr() for r in result] else: return result @public def sturm(f, *gens, **args): """ Compute Sturm sequence of ``f``. Examples ======== >>> from sympy import sturm >>> from sympy.abc import x >>> sturm(x**3 - 2*x**2 + x - 3) [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4] """ options.allowed_flags(args, ['auto', 'polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('sturm', 1, exc) result = F.sturm(auto=opt.auto) if not opt.polys: return [r.as_expr() for r in result] else: return result @public def gff_list(f, *gens, **args): """ Compute a list of greatest factorial factors of ``f``. Note that the input to ff() and rf() should be Poly instances to use the definitions here. Examples ======== >>> from sympy import gff_list, ff, Poly >>> from sympy.abc import x >>> f = Poly(x**5 + 2*x**4 - x**3 - 2*x**2, x) >>> gff_list(f) [(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)] >>> (ff(Poly(x), 1)*ff(Poly(x + 2), 4)) == f True >>> f = Poly(x**12 + 6*x**11 - 11*x**10 - 56*x**9 + 220*x**8 + 208*x**7 - \ 1401*x**6 + 1090*x**5 + 2715*x**4 - 6720*x**3 - 1092*x**2 + 5040*x, x) >>> gff_list(f) [(Poly(x**3 + 7, x, domain='ZZ'), 2), (Poly(x**2 + 5*x, x, domain='ZZ'), 3)] >>> ff(Poly(x**3 + 7, x), 2)*ff(Poly(x**2 + 5*x, x), 3) == f True """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('gff_list', 1, exc) factors = F.gff_list() if not opt.polys: return [(g.as_expr(), k) for g, k in factors] else: return factors @public def gff(f, *gens, **args): """Compute greatest factorial factorization of ``f``. """ raise NotImplementedError('symbolic falling factorial') @public def sqf_norm(f, *gens, **args): """ Compute square-free norm of ``f``. Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``, where ``a`` is the algebraic extension of the ground domain. Examples ======== >>> from sympy import sqf_norm, sqrt >>> from sympy.abc import x >>> sqf_norm(x**2 + 1, extension=[sqrt(3)]) (1, x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16) """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('sqf_norm', 1, exc) s, g, r = F.sqf_norm() if not opt.polys: return Integer(s), g.as_expr(), r.as_expr() else: return Integer(s), g, r @public def sqf_part(f, *gens, **args): """ Compute square-free part of ``f``. Examples ======== >>> from sympy import sqf_part >>> from sympy.abc import x >>> sqf_part(x**3 - 3*x - 2) x**2 - x - 2 """ options.allowed_flags(args, ['polys']) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('sqf_part', 1, exc) result = F.sqf_part() if not opt.polys: return result.as_expr() else: return result def _sorted_factors(factors, method): """Sort a list of ``(expr, exp)`` pairs. """ if method == 'sqf': def key(obj): poly, exp = obj rep = poly.rep.rep return (exp, len(rep), len(poly.gens), rep) else: def key(obj): poly, exp = obj rep = poly.rep.rep return (len(rep), len(poly.gens), exp, rep) return sorted(factors, key=key) def _factors_product(factors): """Multiply a list of ``(expr, exp)`` pairs. """ return Mul(*[f.as_expr()**k for f, k in factors]) def _symbolic_factor_list(expr, opt, method): """Helper function for :func:`_symbolic_factor`. """ coeff, factors = S.One, [] args = [i._eval_factor() if hasattr(i, '_eval_factor') else i for i in Mul.make_args(expr)] for arg in args: if arg.is_Number or (isinstance(arg, Expr) and pure_complex(arg)): coeff *= arg continue elif arg.is_Pow: base, exp = arg.args if base.is_Number and exp.is_Number: coeff *= arg continue if base.is_Number: factors.append((base, exp)) continue else: base, exp = arg, S.One try: poly, _ = _poly_from_expr(base, opt) except PolificationFailed as exc: factors.append((exc.expr, exp)) else: func = getattr(poly, method + '_list') _coeff, _factors = func() if _coeff is not S.One: if exp.is_Integer: coeff *= _coeff**exp elif _coeff.is_positive: factors.append((_coeff, exp)) else: _factors.append((_coeff, S.One)) if exp is S.One: factors.extend(_factors) elif exp.is_integer: factors.extend([(f, k*exp) for f, k in _factors]) else: other = [] for f, k in _factors: if f.as_expr().is_positive: factors.append((f, k*exp)) else: other.append((f, k)) factors.append((_factors_product(other), exp)) if method == 'sqf': factors = [(reduce(mul, (f for f, _ in factors if _ == k)), k) for k in {i for _, i in factors}] return coeff, factors def _symbolic_factor(expr, opt, method): """Helper function for :func:`_factor`. """ if isinstance(expr, Expr): if hasattr(expr,'_eval_factor'): return expr._eval_factor() coeff, factors = _symbolic_factor_list(together(expr, fraction=opt['fraction']), opt, method) return _keep_coeff(coeff, _factors_product(factors)) elif hasattr(expr, 'args'): return expr.func(*[_symbolic_factor(arg, opt, method) for arg in expr.args]) elif hasattr(expr, '__iter__'): return expr.__class__([_symbolic_factor(arg, opt, method) for arg in expr]) else: return expr def _generic_factor_list(expr, gens, args, method): """Helper function for :func:`sqf_list` and :func:`factor_list`. """ options.allowed_flags(args, ['frac', 'polys']) opt = options.build_options(gens, args) expr = sympify(expr) if isinstance(expr, (Expr, Poly)): if isinstance(expr, Poly): numer, denom = expr, 1 else: numer, denom = together(expr).as_numer_denom() cp, fp = _symbolic_factor_list(numer, opt, method) cq, fq = _symbolic_factor_list(denom, opt, method) if fq and not opt.frac: raise PolynomialError("a polynomial expected, got %s" % expr) _opt = opt.clone(dict(expand=True)) for factors in (fp, fq): for i, (f, k) in enumerate(factors): if not f.is_Poly: f, _ = _poly_from_expr(f, _opt) factors[i] = (f, k) fp = _sorted_factors(fp, method) fq = _sorted_factors(fq, method) if not opt.polys: fp = [(f.as_expr(), k) for f, k in fp] fq = [(f.as_expr(), k) for f, k in fq] coeff = cp/cq if not opt.frac: return coeff, fp else: return coeff, fp, fq else: raise PolynomialError("a polynomial expected, got %s" % expr) def _generic_factor(expr, gens, args, method): """Helper function for :func:`sqf` and :func:`factor`. """ fraction = args.pop('fraction', True) options.allowed_flags(args, []) opt = options.build_options(gens, args) opt['fraction'] = fraction return _symbolic_factor(sympify(expr), opt, method) def to_rational_coeffs(f): """ try to transform a polynomial to have rational coefficients try to find a transformation ``x = alpha*y`` ``f(x) = lc*alpha**n * g(y)`` where ``g`` is a polynomial with rational coefficients, ``lc`` the leading coefficient. If this fails, try ``x = y + beta`` ``f(x) = g(y)`` Returns ``None`` if ``g`` not found; ``(lc, alpha, None, g)`` in case of rescaling ``(None, None, beta, g)`` in case of translation Notes ===== Currently it transforms only polynomials without roots larger than 2. Examples ======== >>> from sympy import sqrt, Poly, simplify >>> from sympy.polys.polytools import to_rational_coeffs >>> from sympy.abc import x >>> p = Poly(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}), x, domain='EX') >>> lc, r, _, g = to_rational_coeffs(p) >>> lc, r (7 + 5*sqrt(2), 2 - 2*sqrt(2)) >>> g Poly(x**3 + x**2 - 1/4*x - 1/4, x, domain='QQ') >>> r1 = simplify(1/r) >>> Poly(lc*r**3*(g.as_expr()).subs({x:x*r1}), x, domain='EX') == p True """ from sympy.simplify.simplify import simplify def _try_rescale(f, f1=None): """ try rescaling ``x -> alpha*x`` to convert f to a polynomial with rational coefficients. Returns ``alpha, f``; if the rescaling is successful, ``alpha`` is the rescaling factor, and ``f`` is the rescaled polynomial; else ``alpha`` is ``None``. """ from sympy.core.add import Add if not len(f.gens) == 1 or not (f.gens[0]).is_Atom: return None, f n = f.degree() lc = f.LC() f1 = f1 or f1.monic() coeffs = f1.all_coeffs()[1:] coeffs = [simplify(coeffx) for coeffx in coeffs] if coeffs[-2]: rescale1_x = simplify(coeffs[-2]/coeffs[-1]) coeffs1 = [] for i in range(len(coeffs)): coeffx = simplify(coeffs[i]*rescale1_x**(i + 1)) if not coeffx.is_rational: break coeffs1.append(coeffx) else: rescale_x = simplify(1/rescale1_x) x = f.gens[0] v = [x**n] for i in range(1, n + 1): v.append(coeffs1[i - 1]*x**(n - i)) f = Add(*v) f = Poly(f) return lc, rescale_x, f return None def _try_translate(f, f1=None): """ try translating ``x -> x + alpha`` to convert f to a polynomial with rational coefficients. Returns ``alpha, f``; if the translating is successful, ``alpha`` is the translating factor, and ``f`` is the shifted polynomial; else ``alpha`` is ``None``. """ from sympy.core.add import Add if not len(f.gens) == 1 or not (f.gens[0]).is_Atom: return None, f n = f.degree() f1 = f1 or f1.monic() coeffs = f1.all_coeffs()[1:] c = simplify(coeffs[0]) if c and not c.is_rational: func = Add if c.is_Add: args = c.args func = c.func else: args = [c] c1, c2 = sift(args, lambda z: z.is_rational, binary=True) alpha = -func(*c2)/n f2 = f1.shift(alpha) return alpha, f2 return None def _has_square_roots(p): """ Return True if ``f`` is a sum with square roots but no other root """ from sympy.core.exprtools import Factors coeffs = p.coeffs() has_sq = False for y in coeffs: for x in Add.make_args(y): f = Factors(x).factors r = [wx.q for b, wx in f.items() if b.is_number and wx.is_Rational and wx.q >= 2] if not r: continue if min(r) == 2: has_sq = True if max(r) > 2: return False return has_sq if f.get_domain().is_EX and _has_square_roots(f): f1 = f.monic() r = _try_rescale(f, f1) if r: return r[0], r[1], None, r[2] else: r = _try_translate(f, f1) if r: return None, None, r[0], r[1] return None def _torational_factor_list(p, x): """ helper function to factor polynomial using to_rational_coeffs Examples ======== >>> from sympy.polys.polytools import _torational_factor_list >>> from sympy.abc import x >>> from sympy import sqrt, expand, Mul >>> p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))})) >>> factors = _torational_factor_list(p, x); factors (-2, [(-x*(1 + sqrt(2))/2 + 1, 1), (-x*(1 + sqrt(2)) - 1, 1), (-x*(1 + sqrt(2)) + 1, 1)]) >>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p True >>> p = expand(((x**2-1)*(x-2)).subs({x:x + sqrt(2)})) >>> factors = _torational_factor_list(p, x); factors (1, [(x - 2 + sqrt(2), 1), (x - 1 + sqrt(2), 1), (x + 1 + sqrt(2), 1)]) >>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p True """ from sympy.simplify.simplify import simplify p1 = Poly(p, x, domain='EX') n = p1.degree() res = to_rational_coeffs(p1) if not res: return None lc, r, t, g = res factors = factor_list(g.as_expr()) if lc: c = simplify(factors[0]*lc*r**n) r1 = simplify(1/r) a = [] for z in factors[1:][0]: a.append((simplify(z[0].subs({x: x*r1})), z[1])) else: c = factors[0] a = [] for z in factors[1:][0]: a.append((z[0].subs({x: x - t}), z[1])) return (c, a) @public def sqf_list(f, *gens, **args): """ Compute a list of square-free factors of ``f``. Examples ======== >>> from sympy import sqf_list >>> from sympy.abc import x >>> sqf_list(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16) (2, [(x + 1, 2), (x + 2, 3)]) """ return _generic_factor_list(f, gens, args, method='sqf') @public def sqf(f, *gens, **args): """ Compute square-free factorization of ``f``. Examples ======== >>> from sympy import sqf >>> from sympy.abc import x >>> sqf(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16) 2*(x + 1)**2*(x + 2)**3 """ return _generic_factor(f, gens, args, method='sqf') @public def factor_list(f, *gens, **args): """ Compute a list of irreducible factors of ``f``. Examples ======== >>> from sympy import factor_list >>> from sympy.abc import x, y >>> factor_list(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y) (2, [(x + y, 1), (x**2 + 1, 2)]) """ return _generic_factor_list(f, gens, args, method='factor') @public def factor(f, *gens, deep=False, **args): """ Compute the factorization of expression, ``f``, into irreducibles. (To factor an integer into primes, use ``factorint``.) There two modes implemented: symbolic and formal. If ``f`` is not an instance of :class:`Poly` and generators are not specified, then the former mode is used. Otherwise, the formal mode is used. In symbolic mode, :func:`factor` will traverse the expression tree and factor its components without any prior expansion, unless an instance of :class:`~.Add` is encountered (in this case formal factorization is used). This way :func:`factor` can handle large or symbolic exponents. By default, the factorization is computed over the rationals. To factor over other domain, e.g. an algebraic or finite field, use appropriate options: ``extension``, ``modulus`` or ``domain``. Examples ======== >>> from sympy import factor, sqrt, exp >>> from sympy.abc import x, y >>> factor(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y) 2*(x + y)*(x**2 + 1)**2 >>> factor(x**2 + 1) x**2 + 1 >>> factor(x**2 + 1, modulus=2) (x + 1)**2 >>> factor(x**2 + 1, gaussian=True) (x - I)*(x + I) >>> factor(x**2 - 2, extension=sqrt(2)) (x - sqrt(2))*(x + sqrt(2)) >>> factor((x**2 - 1)/(x**2 + 4*x + 4)) (x - 1)*(x + 1)/(x + 2)**2 >>> factor((x**2 + 4*x + 4)**10000000*(x**2 + 1)) (x + 2)**20000000*(x**2 + 1) By default, factor deals with an expression as a whole: >>> eq = 2**(x**2 + 2*x + 1) >>> factor(eq) 2**(x**2 + 2*x + 1) If the ``deep`` flag is True then subexpressions will be factored: >>> factor(eq, deep=True) 2**((x + 1)**2) If the ``fraction`` flag is False then rational expressions won't be combined. By default it is True. >>> factor(5*x + 3*exp(2 - 7*x), deep=True) (5*x*exp(7*x) + 3*exp(2))*exp(-7*x) >>> factor(5*x + 3*exp(2 - 7*x), deep=True, fraction=False) 5*x + 3*exp(2)*exp(-7*x) See Also ======== sympy.ntheory.factor_.factorint """ f = sympify(f) if deep: from sympy.simplify.simplify import bottom_up def _try_factor(expr): """ Factor, but avoid changing the expression when unable to. """ fac = factor(expr, *gens, **args) if fac.is_Mul or fac.is_Pow: return fac return expr f = bottom_up(f, _try_factor) # clean up any subexpressions that may have been expanded # while factoring out a larger expression partials = {} muladd = f.atoms(Mul, Add) for p in muladd: fac = factor(p, *gens, **args) if (fac.is_Mul or fac.is_Pow) and fac != p: partials[p] = fac return f.xreplace(partials) try: return _generic_factor(f, gens, args, method='factor') except PolynomialError as msg: if not f.is_commutative: from sympy.core.exprtools import factor_nc return factor_nc(f) else: raise PolynomialError(msg) @public def intervals(F, all=False, eps=None, inf=None, sup=None, strict=False, fast=False, sqf=False): """ Compute isolating intervals for roots of ``f``. Examples ======== >>> from sympy import intervals >>> from sympy.abc import x >>> intervals(x**2 - 3) [((-2, -1), 1), ((1, 2), 1)] >>> intervals(x**2 - 3, eps=1e-2) [((-26/15, -19/11), 1), ((19/11, 26/15), 1)] """ if not hasattr(F, '__iter__'): try: F = Poly(F) except GeneratorsNeeded: return [] return F.intervals(all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf) else: polys, opt = parallel_poly_from_expr(F, domain='QQ') if len(opt.gens) > 1: raise MultivariatePolynomialError for i, poly in enumerate(polys): polys[i] = poly.rep.rep if eps is not None: eps = opt.domain.convert(eps) if eps <= 0: raise ValueError("'eps' must be a positive rational") if inf is not None: inf = opt.domain.convert(inf) if sup is not None: sup = opt.domain.convert(sup) intervals = dup_isolate_real_roots_list(polys, opt.domain, eps=eps, inf=inf, sup=sup, strict=strict, fast=fast) result = [] for (s, t), indices in intervals: s, t = opt.domain.to_sympy(s), opt.domain.to_sympy(t) result.append(((s, t), indices)) return result @public def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False): """ Refine an isolating interval of a root to the given precision. Examples ======== >>> from sympy import refine_root >>> from sympy.abc import x >>> refine_root(x**2 - 3, 1, 2, eps=1e-2) (19/11, 26/15) """ try: F = Poly(f) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except GeneratorsNeeded: raise PolynomialError( "can't refine a root of %s, not a polynomial" % f) return F.refine_root(s, t, eps=eps, steps=steps, fast=fast, check_sqf=check_sqf) @public def count_roots(f, inf=None, sup=None): """ Return the number of roots of ``f`` in ``[inf, sup]`` interval. If one of ``inf`` or ``sup`` is complex, it will return the number of roots in the complex rectangle with corners at ``inf`` and ``sup``. Examples ======== >>> from sympy import count_roots, I >>> from sympy.abc import x >>> count_roots(x**4 - 4, -3, 3) 2 >>> count_roots(x**4 - 4, 0, 1 + 3*I) 1 """ try: F = Poly(f, greedy=False) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except GeneratorsNeeded: raise PolynomialError("can't count roots of %s, not a polynomial" % f) return F.count_roots(inf=inf, sup=sup) @public def real_roots(f, multiple=True): """ Return a list of real roots with multiplicities of ``f``. Examples ======== >>> from sympy import real_roots >>> from sympy.abc import x >>> real_roots(2*x**3 - 7*x**2 + 4*x + 4) [-1/2, 2, 2] """ try: F = Poly(f, greedy=False) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except GeneratorsNeeded: raise PolynomialError( "can't compute real roots of %s, not a polynomial" % f) return F.real_roots(multiple=multiple) @public def nroots(f, n=15, maxsteps=50, cleanup=True): """ Compute numerical approximations of roots of ``f``. Examples ======== >>> from sympy import nroots >>> from sympy.abc import x >>> nroots(x**2 - 3, n=15) [-1.73205080756888, 1.73205080756888] >>> nroots(x**2 - 3, n=30) [-1.73205080756887729352744634151, 1.73205080756887729352744634151] """ try: F = Poly(f, greedy=False) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except GeneratorsNeeded: raise PolynomialError( "can't compute numerical roots of %s, not a polynomial" % f) return F.nroots(n=n, maxsteps=maxsteps, cleanup=cleanup) @public def ground_roots(f, *gens, **args): """ Compute roots of ``f`` by factorization in the ground domain. Examples ======== >>> from sympy import ground_roots >>> from sympy.abc import x >>> ground_roots(x**6 - 4*x**4 + 4*x**3 - x**2) {0: 2, 1: 2} """ options.allowed_flags(args, []) try: F, opt = poly_from_expr(f, *gens, **args) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except PolificationFailed as exc: raise ComputationFailed('ground_roots', 1, exc) return F.ground_roots() @public def nth_power_roots_poly(f, n, *gens, **args): """ Construct a polynomial with n-th powers of roots of ``f``. Examples ======== >>> from sympy import nth_power_roots_poly, factor, roots >>> from sympy.abc import x >>> f = x**4 - x**2 + 1 >>> g = factor(nth_power_roots_poly(f, 2)) >>> g (x**2 - x + 1)**2 >>> R_f = [ (r**2).expand() for r in roots(f) ] >>> R_g = roots(g).keys() >>> set(R_f) == set(R_g) True """ options.allowed_flags(args, []) try: F, opt = poly_from_expr(f, *gens, **args) if not isinstance(f, Poly) and not F.gen.is_Symbol: # root of sin(x) + 1 is -1 but when someone # passes an Expr instead of Poly they may not expect # that the generator will be sin(x), not x raise PolynomialError("generator must be a Symbol") except PolificationFailed as exc: raise ComputationFailed('nth_power_roots_poly', 1, exc) result = F.nth_power_roots_poly(n) if not opt.polys: return result.as_expr() else: return result @public def cancel(f, *gens, **args): """ Cancel common factors in a rational function ``f``. Examples ======== >>> from sympy import cancel, sqrt, Symbol, together >>> from sympy.abc import x >>> A = Symbol('A', commutative=False) >>> cancel((2*x**2 - 2)/(x**2 - 2*x + 1)) (2*x + 2)/(x - 1) >>> cancel((sqrt(3) + sqrt(15)*A)/(sqrt(2) + sqrt(10)*A)) sqrt(6)/2 Note: due to automatic distribution of Rationals, a sum divided by an integer will appear as a sum. To recover a rational form use `together` on the result: >>> cancel(x/2 + 1) x/2 + 1 >>> together(_) (x + 2)/2 """ from sympy.core.exprtools import factor_terms from sympy.functions.elementary.piecewise import Piecewise from sympy.polys.rings import sring options.allowed_flags(args, ['polys']) f = sympify(f) opt = {} if 'polys' in args: opt['polys'] = args['polys'] if not isinstance(f, (tuple, Tuple)): if f.is_Number or isinstance(f, Relational) or not isinstance(f, Expr): return f f = factor_terms(f, radical=True) p, q = f.as_numer_denom() elif len(f) == 2: p, q = f if isinstance(p, Poly) and isinstance(q, Poly): opt['gens'] = p.gens opt['domain'] = p.domain opt['polys'] = opt.get('polys', True) p, q = p.as_expr(), q.as_expr() elif isinstance(f, Tuple): return factor_terms(f) else: raise ValueError('unexpected argument: %s' % f) try: if f.has(Piecewise): raise PolynomialError() R, (F, G) = sring((p, q), *gens, **args) if not R.ngens: if not isinstance(f, (tuple, Tuple)): return f.expand() else: return S.One, p, q except PolynomialError as msg: if f.is_commutative and not f.has(Piecewise): raise PolynomialError(msg) # Handling of noncommutative and/or piecewise expressions if f.is_Add or f.is_Mul: c, nc = sift(f.args, lambda x: x.is_commutative is True and not x.has(Piecewise), binary=True) nc = [cancel(i) for i in nc] return f.func(cancel(f.func(*c)), *nc) else: reps = [] pot = preorder_traversal(f) next(pot) for e in pot: # XXX: This should really skip anything that's not Expr. if isinstance(e, (tuple, Tuple, BooleanAtom)): continue try: reps.append((e, cancel(e))) pot.skip() # this was handled successfully except NotImplementedError: pass return f.xreplace(dict(reps)) c, (P, Q) = 1, F.cancel(G) if opt.get('polys', False) and not 'gens' in opt: opt['gens'] = R.symbols if not isinstance(f, (tuple, Tuple)): return c*(P.as_expr()/Q.as_expr()) else: P, Q = P.as_expr(), Q.as_expr() if not opt.get('polys', False): return c, P, Q else: return c, Poly(P, *gens, **opt), Poly(Q, *gens, **opt) @public def reduced(f, G, *gens, **args): """ Reduces a polynomial ``f`` modulo a set of polynomials ``G``. Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``, computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r`` such that ``f = q_1*g_1 + ... + q_n*g_n + r``, where ``r`` vanishes or ``r`` is a completely reduced polynomial with respect to ``G``. Examples ======== >>> from sympy import reduced >>> from sympy.abc import x, y >>> reduced(2*x**4 + y**2 - x**2 + y**3, [x**3 - x, y**3 - y]) ([2*x, 1], x**2 + y**2 + y) """ options.allowed_flags(args, ['polys', 'auto']) try: polys, opt = parallel_poly_from_expr([f] + list(G), *gens, **args) except PolificationFailed as exc: raise ComputationFailed('reduced', 0, exc) domain = opt.domain retract = False if opt.auto and domain.is_Ring and not domain.is_Field: opt = opt.clone(dict(domain=domain.get_field())) retract = True from sympy.polys.rings import xring _ring, _ = xring(opt.gens, opt.domain, opt.order) for i, poly in enumerate(polys): poly = poly.set_domain(opt.domain).rep.to_dict() polys[i] = _ring.from_dict(poly) Q, r = polys[0].div(polys[1:]) Q = [Poly._from_dict(dict(q), opt) for q in Q] r = Poly._from_dict(dict(r), opt) if retract: try: _Q, _r = [q.to_ring() for q in Q], r.to_ring() except CoercionFailed: pass else: Q, r = _Q, _r if not opt.polys: return [q.as_expr() for q in Q], r.as_expr() else: return Q, r @public def groebner(F, *gens, **args): """ Computes the reduced Groebner basis for a set of polynomials. Use the ``order`` argument to set the monomial ordering that will be used to compute the basis. Allowed orders are ``lex``, ``grlex`` and ``grevlex``. If no order is specified, it defaults to ``lex``. For more information on Groebner bases, see the references and the docstring of :func:`~.solve_poly_system`. Examples ======== Example taken from [1]. >>> from sympy import groebner >>> from sympy.abc import x, y >>> F = [x*y - 2*y, 2*y**2 - x**2] >>> groebner(F, x, y, order='lex') GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y, domain='ZZ', order='lex') >>> groebner(F, x, y, order='grlex') GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y, domain='ZZ', order='grlex') >>> groebner(F, x, y, order='grevlex') GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y, domain='ZZ', order='grevlex') By default, an improved implementation of the Buchberger algorithm is used. Optionally, an implementation of the F5B algorithm can be used. The algorithm can be set using the ``method`` flag or with the :func:`sympy.polys.polyconfig.setup` function. >>> F = [x**2 - x - 1, (2*x - 1) * y - (x**10 - (1 - x)**10)] >>> groebner(F, x, y, method='buchberger') GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex') >>> groebner(F, x, y, method='f5b') GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex') References ========== 1. [Buchberger01]_ 2. [Cox97]_ """ return GroebnerBasis(F, *gens, **args) @public def is_zero_dimensional(F, *gens, **args): """ Checks if the ideal generated by a Groebner basis is zero-dimensional. The algorithm checks if the set of monomials not divisible by the leading monomial of any element of ``F`` is bounded. References ========== David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and Algorithms, 3rd edition, p. 230 """ return GroebnerBasis(F, *gens, **args).is_zero_dimensional @public class GroebnerBasis(Basic): """Represents a reduced Groebner basis. """ def __new__(cls, F, *gens, **args): """Compute a reduced Groebner basis for a system of polynomials. """ options.allowed_flags(args, ['polys', 'method']) try: polys, opt = parallel_poly_from_expr(F, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('groebner', len(F), exc) from sympy.polys.rings import PolyRing ring = PolyRing(opt.gens, opt.domain, opt.order) polys = [ring.from_dict(poly.rep.to_dict()) for poly in polys if poly] G = _groebner(polys, ring, method=opt.method) G = [Poly._from_dict(g, opt) for g in G] return cls._new(G, opt) @classmethod def _new(cls, basis, options): obj = Basic.__new__(cls) obj._basis = tuple(basis) obj._options = options return obj @property def args(self): basis = (p.as_expr() for p in self._basis) return (Tuple(*basis), Tuple(*self._options.gens)) @property def exprs(self): return [poly.as_expr() for poly in self._basis] @property def polys(self): return list(self._basis) @property def gens(self): return self._options.gens @property def domain(self): return self._options.domain @property def order(self): return self._options.order def __len__(self): return len(self._basis) def __iter__(self): if self._options.polys: return iter(self.polys) else: return iter(self.exprs) def __getitem__(self, item): if self._options.polys: basis = self.polys else: basis = self.exprs return basis[item] def __hash__(self): return hash((self._basis, tuple(self._options.items()))) def __eq__(self, other): if isinstance(other, self.__class__): return self._basis == other._basis and self._options == other._options elif iterable(other): return self.polys == list(other) or self.exprs == list(other) else: return False def __ne__(self, other): return not self == other @property def is_zero_dimensional(self): """ Checks if the ideal generated by a Groebner basis is zero-dimensional. The algorithm checks if the set of monomials not divisible by the leading monomial of any element of ``F`` is bounded. References ========== David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and Algorithms, 3rd edition, p. 230 """ def single_var(monomial): return sum(map(bool, monomial)) == 1 exponents = Monomial([0]*len(self.gens)) order = self._options.order for poly in self.polys: monomial = poly.LM(order=order) if single_var(monomial): exponents *= monomial # If any element of the exponents vector is zero, then there's # a variable for which there's no degree bound and the ideal # generated by this Groebner basis isn't zero-dimensional. return all(exponents) def fglm(self, order): """ Convert a Groebner basis from one ordering to another. The FGLM algorithm converts reduced Groebner bases of zero-dimensional ideals from one ordering to another. This method is often used when it is infeasible to compute a Groebner basis with respect to a particular ordering directly. Examples ======== >>> from sympy.abc import x, y >>> from sympy import groebner >>> F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1] >>> G = groebner(F, x, y, order='grlex') >>> list(G.fglm('lex')) [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7] >>> list(groebner(F, x, y, order='lex')) [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7] References ========== .. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient Computation of Zero-dimensional Groebner Bases by Change of Ordering """ opt = self._options src_order = opt.order dst_order = monomial_key(order) if src_order == dst_order: return self if not self.is_zero_dimensional: raise NotImplementedError("can't convert Groebner bases of ideals with positive dimension") polys = list(self._basis) domain = opt.domain opt = opt.clone(dict( domain=domain.get_field(), order=dst_order, )) from sympy.polys.rings import xring _ring, _ = xring(opt.gens, opt.domain, src_order) for i, poly in enumerate(polys): poly = poly.set_domain(opt.domain).rep.to_dict() polys[i] = _ring.from_dict(poly) G = matrix_fglm(polys, _ring, dst_order) G = [Poly._from_dict(dict(g), opt) for g in G] if not domain.is_Field: G = [g.clear_denoms(convert=True)[1] for g in G] opt.domain = domain return self._new(G, opt) def reduce(self, expr, auto=True): """ Reduces a polynomial modulo a Groebner basis. Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``, computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r`` such that ``f = q_1*f_1 + ... + q_n*f_n + r``, where ``r`` vanishes or ``r`` is a completely reduced polynomial with respect to ``G``. Examples ======== >>> from sympy import groebner, expand >>> from sympy.abc import x, y >>> f = 2*x**4 - x**2 + y**3 + y**2 >>> G = groebner([x**3 - x, y**3 - y]) >>> G.reduce(f) ([2*x, 1], x**2 + y**2 + y) >>> Q, r = _ >>> expand(sum(q*g for q, g in zip(Q, G)) + r) 2*x**4 - x**2 + y**3 + y**2 >>> _ == f True """ poly = Poly._from_expr(expr, self._options) polys = [poly] + list(self._basis) opt = self._options domain = opt.domain retract = False if auto and domain.is_Ring and not domain.is_Field: opt = opt.clone(dict(domain=domain.get_field())) retract = True from sympy.polys.rings import xring _ring, _ = xring(opt.gens, opt.domain, opt.order) for i, poly in enumerate(polys): poly = poly.set_domain(opt.domain).rep.to_dict() polys[i] = _ring.from_dict(poly) Q, r = polys[0].div(polys[1:]) Q = [Poly._from_dict(dict(q), opt) for q in Q] r = Poly._from_dict(dict(r), opt) if retract: try: _Q, _r = [q.to_ring() for q in Q], r.to_ring() except CoercionFailed: pass else: Q, r = _Q, _r if not opt.polys: return [q.as_expr() for q in Q], r.as_expr() else: return Q, r def contains(self, poly): """ Check if ``poly`` belongs the ideal generated by ``self``. Examples ======== >>> from sympy import groebner >>> from sympy.abc import x, y >>> f = 2*x**3 + y**3 + 3*y >>> G = groebner([x**2 + y**2 - 1, x*y - 2]) >>> G.contains(f) True >>> G.contains(f + 1) False """ return self.reduce(poly)[1] == 0 @public def poly(expr, *gens, **args): """ Efficiently transform an expression into a polynomial. Examples ======== >>> from sympy import poly >>> from sympy.abc import x >>> poly(x*(x**2 + x - 1)**2) Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ') """ options.allowed_flags(args, []) def _poly(expr, opt): terms, poly_terms = [], [] for term in Add.make_args(expr): factors, poly_factors = [], [] for factor in Mul.make_args(term): if factor.is_Add: poly_factors.append(_poly(factor, opt)) elif factor.is_Pow and factor.base.is_Add and \ factor.exp.is_Integer and factor.exp >= 0: poly_factors.append( _poly(factor.base, opt).pow(factor.exp)) else: factors.append(factor) if not poly_factors: terms.append(term) else: product = poly_factors[0] for factor in poly_factors[1:]: product = product.mul(factor) if factors: factor = Mul(*factors) if factor.is_Number: product = product.mul(factor) else: product = product.mul(Poly._from_expr(factor, opt)) poly_terms.append(product) if not poly_terms: result = Poly._from_expr(expr, opt) else: result = poly_terms[0] for term in poly_terms[1:]: result = result.add(term) if terms: term = Add(*terms) if term.is_Number: result = result.add(term) else: result = result.add(Poly._from_expr(term, opt)) return result.reorder(*opt.get('gens', ()), **args) expr = sympify(expr) if expr.is_Poly: return Poly(expr, *gens, **args) if 'expand' not in args: args['expand'] = False opt = options.build_options(gens, args) return _poly(expr, opt)
b1577e4e12364dc3fccbe2214d1245a5477920e573abc266bfefd3e38f130b5b
"""Functions for generating interesting polynomials, e.g. for benchmarking. """ from sympy.core import Add, Mul, Symbol, sympify, Dummy, symbols from sympy.core.containers import Tuple from sympy.core.singleton import S from sympy.functions.elementary.miscellaneous import sqrt from sympy.ntheory import nextprime from sympy.polys.densearith import ( dmp_add_term, dmp_neg, dmp_mul, dmp_sqr ) from sympy.polys.densebasic import ( dmp_zero, dmp_one, dmp_ground, dup_from_raw_dict, dmp_raise, dup_random ) from sympy.polys.domains import ZZ from sympy.polys.factortools import dup_zz_cyclotomic_poly from sympy.polys.polyclasses import DMP from sympy.polys.polytools import Poly, PurePoly from sympy.polys.polyutils import _analyze_gens from sympy.utilities import subsets, public, filldedent @public def swinnerton_dyer_poly(n, x=None, polys=False): """Generates n-th Swinnerton-Dyer polynomial in `x`. Parameters ---------- n : int `n` decides the order of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ from .numberfields import minimal_polynomial if n <= 0: raise ValueError( "can't generate Swinnerton-Dyer polynomial of order %s" % n) if x is not None: sympify(x) else: x = Dummy('x') if n > 3: p = 2 a = [sqrt(2)] for i in range(2, n + 1): p = nextprime(p) a.append(sqrt(p)) return minimal_polynomial(Add(*a), x, polys=polys) if n == 1: ex = x**2 - 2 elif n == 2: ex = x**4 - 10*x**2 + 1 elif n == 3: ex = x**8 - 40*x**6 + 352*x**4 - 960*x**2 + 576 return PurePoly(ex, x) if polys else ex @public def cyclotomic_poly(n, x=None, polys=False): """Generates cyclotomic polynomial of order `n` in `x`. Parameters ---------- n : int `n` decides the order of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n <= 0: raise ValueError( "can't generate cyclotomic polynomial of order %s" % n) poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() @public def symmetric_poly(n, *gens, **args): """Generates symmetric polynomial of order `n`. Returns a Poly object when ``polys=True``, otherwise (default) returns an expression. """ # TODO: use an explicit keyword argument when Python 2 support is dropped gens = _analyze_gens(gens) if n < 0 or n > len(gens) or not gens: raise ValueError("can't generate symmetric polynomial of order %s for %s" % (n, gens)) elif not n: poly = S.One else: poly = Add(*[Mul(*s) for s in subsets(gens, int(n))]) if not args.get('polys', False): return poly else: return Poly(poly, *gens) @public def random_poly(x, n, inf, sup, domain=ZZ, polys=False): """Generates a polynomial of degree ``n`` with coefficients in ``[inf, sup]``. Parameters ---------- x `x` is the independent term of polynomial n : int `n` decides the order of polynomial inf Lower limit of range in which coefficients lie sup Upper limit of range in which coefficients lie domain : optional Decides what ring the coefficients are supposed to belong. Default is set to Integers. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ poly = Poly(dup_random(n, inf, sup, domain), x, domain=domain) return poly if polys else poly.as_expr() @public def interpolating_poly(n, x, X='x', Y='y'): """Construct Lagrange interpolating polynomial for ``n`` data points. If a sequence of values are given for ``X`` and ``Y`` then the first ``n`` values will be used. """ ok = getattr(x, 'free_symbols', None) if isinstance(X, str): X = symbols("%s:%s" % (X, n)) elif ok and ok & Tuple(*X).free_symbols: ok = False if isinstance(Y, str): Y = symbols("%s:%s" % (Y, n)) elif ok and ok & Tuple(*Y).free_symbols: ok = False if not ok: raise ValueError(filldedent(''' Expecting symbol for x that does not appear in X or Y. Use `interpolate(list(zip(X, Y)), x)` instead.''')) coeffs = [] numert = Mul(*[x - X[i] for i in range(n)]) for i in range(n): numer = numert/(x - X[i]) denom = Mul(*[(X[i] - X[j]) for j in range(n) if i != j]) coeffs.append(numer/denom) return Add(*[coeff*y for coeff, y in zip(coeffs, Y)]) def fateman_poly_F_1(n): """Fateman's GCD benchmark: trivial GCD """ Y = [Symbol('y_' + str(i)) for i in range(n + 1)] y_0, y_1 = Y[0], Y[1] u = y_0 + Add(*[y for y in Y[1:]]) v = y_0**2 + Add(*[y**2 for y in Y[1:]]) F = ((u + 1)*(u + 2)).as_poly(*Y) G = ((v + 1)*(-3*y_1*y_0**2 + y_1**2 - 1)).as_poly(*Y) H = Poly(1, *Y) return F, G, H def dmp_fateman_poly_F_1(n, K): """Fateman's GCD benchmark: trivial GCD """ u = [K(1), K(0)] for i in range(n): u = [dmp_one(i, K), u] v = [K(1), K(0), K(0)] for i in range(0, n): v = [dmp_one(i, K), dmp_zero(i), v] m = n - 1 U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K) V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K) f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]] W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K) Y = dmp_raise(f, m, 1, K) F = dmp_mul(U, V, n, K) G = dmp_mul(W, Y, n, K) H = dmp_one(n, K) return F, G, H def fateman_poly_F_2(n): """Fateman's GCD benchmark: linearly dense quartic inputs """ Y = [Symbol('y_' + str(i)) for i in range(n + 1)] y_0 = Y[0] u = Add(*[y for y in Y[1:]]) H = Poly((y_0 + u + 1)**2, *Y) F = Poly((y_0 - u - 2)**2, *Y) G = Poly((y_0 + u + 2)**2, *Y) return H*F, H*G, H def dmp_fateman_poly_F_2(n, K): """Fateman's GCD benchmark: linearly dense quartic inputs """ u = [K(1), K(0)] for i in range(n - 1): u = [dmp_one(i, K), u] m = n - 1 v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K) f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K) g = dmp_sqr([dmp_one(m, K), v], n, K) v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K) h = dmp_sqr([dmp_one(m, K), v], n, K) return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h def fateman_poly_F_3(n): """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """ Y = [Symbol('y_' + str(i)) for i in range(n + 1)] y_0 = Y[0] u = Add(*[y**(n + 1) for y in Y[1:]]) H = Poly((y_0**(n + 1) + u + 1)**2, *Y) F = Poly((y_0**(n + 1) - u - 2)**2, *Y) G = Poly((y_0**(n + 1) + u + 2)**2, *Y) return H*F, H*G, H def dmp_fateman_poly_F_3(n, K): """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """ u = dup_from_raw_dict({n + 1: K.one}, K) for i in range(0, n - 1): u = dmp_add_term([u], dmp_one(i, K), n + 1, i + 1, K) v = dmp_add_term(u, dmp_ground(K(2), n - 2), 0, n, K) f = dmp_sqr( dmp_add_term([dmp_neg(v, n - 1, K)], dmp_one(n - 1, K), n + 1, n, K), n, K) g = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) v = dmp_add_term(u, dmp_one(n - 2, K), 0, n - 1, K) h = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h # A few useful polynomials from Wang's paper ('78). from sympy.polys.rings import ring def _f_0(): R, x, y, z = ring("x,y,z", ZZ) return x**2*y*z**2 + 2*x**2*y*z + 3*x**2*y + 2*x**2 + 3*x + 4*y**2*z**2 + 5*y**2*z + 6*y**2 + y*z**2 + 2*y*z + y + 1 def _f_1(): R, x, y, z = ring("x,y,z", ZZ) return x**3*y*z + x**2*y**2*z**2 + x**2*y**2 + 20*x**2*y*z + 30*x**2*y + x**2*z**2 + 10*x**2*z + x*y**3*z + 30*x*y**2*z + 20*x*y**2 + x*y*z**3 + 10*x*y*z**2 + x*y*z + 610*x*y + 20*x*z**2 + 230*x*z + 300*x + y**2*z**2 + 10*y**2*z + 30*y*z**2 + 320*y*z + 200*y + 600*z + 6000 def _f_2(): R, x, y, z = ring("x,y,z", ZZ) return x**5*y**3 + x**5*y**2*z + x**5*y*z**2 + x**5*z**3 + x**3*y**2 + x**3*y*z + 90*x**3*y + 90*x**3*z + x**2*y**2*z - 11*x**2*y**2 + x**2*z**3 - 11*x**2*z**2 + y*z - 11*y + 90*z - 990 def _f_3(): R, x, y, z = ring("x,y,z", ZZ) return x**5*y**2 + x**4*z**4 + x**4 + x**3*y**3*z + x**3*z + x**2*y**4 + x**2*y**3*z**3 + x**2*y*z**5 + x**2*y*z + x*y**2*z**4 + x*y**2 + x*y*z**7 + x*y*z**3 + x*y*z**2 + y**2*z + y*z**4 def _f_4(): R, x, y, z = ring("x,y,z", ZZ) return -x**9*y**8*z - x**8*y**5*z**3 - x**7*y**12*z**2 - 5*x**7*y**8 - x**6*y**9*z**4 + x**6*y**7*z**3 + 3*x**6*y**7*z - 5*x**6*y**5*z**2 - x**6*y**4*z**3 + x**5*y**4*z**5 + 3*x**5*y**4*z**3 - x**5*y*z**5 + x**4*y**11*z**4 + 3*x**4*y**11*z**2 - x**4*y**8*z**4 + 5*x**4*y**7*z**2 + 15*x**4*y**7 - 5*x**4*y**4*z**2 + x**3*y**8*z**6 + 3*x**3*y**8*z**4 - x**3*y**5*z**6 + 5*x**3*y**4*z**4 + 15*x**3*y**4*z**2 + x**3*y**3*z**5 + 3*x**3*y**3*z**3 - 5*x**3*y*z**4 + x**2*z**7 + 3*x**2*z**5 + x*y**7*z**6 + 3*x*y**7*z**4 + 5*x*y**3*z**4 + 15*x*y**3*z**2 + y**4*z**8 + 3*y**4*z**6 + 5*z**6 + 15*z**4 def _f_5(): R, x, y, z = ring("x,y,z", ZZ) return -x**3 - 3*x**2*y + 3*x**2*z - 3*x*y**2 + 6*x*y*z - 3*x*z**2 - y**3 + 3*y**2*z - 3*y*z**2 + z**3 def _f_6(): R, x, y, z, t = ring("x,y,z,t", ZZ) return 2115*x**4*y + 45*x**3*z**3*t**2 - 45*x**3*t**2 - 423*x*y**4 - 47*x*y**3 + 141*x*y*z**3 + 94*x*y*z*t - 9*y**3*z**3*t**2 + 9*y**3*t**2 - y**2*z**3*t**2 + y**2*t**2 + 3*z**6*t**2 + 2*z**4*t**3 - 3*z**3*t**2 - 2*z*t**3 def _w_1(): R, x, y, z = ring("x,y,z", ZZ) return 4*x**6*y**4*z**2 + 4*x**6*y**3*z**3 - 4*x**6*y**2*z**4 - 4*x**6*y*z**5 + x**5*y**4*z**3 + 12*x**5*y**3*z - x**5*y**2*z**5 + 12*x**5*y**2*z**2 - 12*x**5*y*z**3 - 12*x**5*z**4 + 8*x**4*y**4 + 6*x**4*y**3*z**2 + 8*x**4*y**3*z - 4*x**4*y**2*z**4 + 4*x**4*y**2*z**3 - 8*x**4*y**2*z**2 - 4*x**4*y*z**5 - 2*x**4*y*z**4 - 8*x**4*y*z**3 + 2*x**3*y**4*z + x**3*y**3*z**3 - x**3*y**2*z**5 - 2*x**3*y**2*z**3 + 9*x**3*y**2*z - 12*x**3*y*z**3 + 12*x**3*y*z**2 - 12*x**3*z**4 + 3*x**3*z**3 + 6*x**2*y**3 - 6*x**2*y**2*z**2 + 8*x**2*y**2*z - 2*x**2*y*z**4 - 8*x**2*y*z**3 + 2*x**2*y*z**2 + 2*x*y**3*z - 2*x*y**2*z**3 - 3*x*y*z + 3*x*z**3 - 2*y**2 + 2*y*z**2 def _w_2(): R, x, y = ring("x,y", ZZ) return 24*x**8*y**3 + 48*x**8*y**2 + 24*x**7*y**5 - 72*x**7*y**2 + 25*x**6*y**4 + 2*x**6*y**3 + 4*x**6*y + 8*x**6 + x**5*y**6 + x**5*y**3 - 12*x**5 + x**4*y**5 - x**4*y**4 - 2*x**4*y**3 + 292*x**4*y**2 - x**3*y**6 + 3*x**3*y**3 - x**2*y**5 + 12*x**2*y**3 + 48*x**2 - 12*y**3 def f_polys(): return _f_0(), _f_1(), _f_2(), _f_3(), _f_4(), _f_5(), _f_6() def w_polys(): return _w_1(), _w_2()
8a4766efd977d27c95acdfd70050e4000cd545eb35bb62cb1927ca79dfbd88dd
"""Implementation of matrix FGLM Groebner basis conversion algorithm. """ from sympy.polys.monomials import monomial_mul, monomial_div def matrix_fglm(F, ring, O_to): """ Converts the reduced Groebner basis ``F`` of a zero-dimensional ideal w.r.t. ``O_from`` to a reduced Groebner basis w.r.t. ``O_to``. References ========== .. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient Computation of Zero-dimensional Groebner Bases by Change of Ordering """ domain = ring.domain ngens = ring.ngens ring_to = ring.clone(order=O_to) old_basis = _basis(F, ring) M = _representing_matrices(old_basis, F, ring) # V contains the normalforms (wrt O_from) of S S = [ring.zero_monom] V = [[domain.one] + [domain.zero] * (len(old_basis) - 1)] G = [] L = [(i, 0) for i in range(ngens)] # (i, j) corresponds to x_i * S[j] L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True) t = L.pop() P = _identity_matrix(len(old_basis), domain) while True: s = len(S) v = _matrix_mul(M[t[0]], V[t[1]]) _lambda = _matrix_mul(P, v) if all(_lambda[i] == domain.zero for i in range(s, len(old_basis))): # there is a linear combination of v by V lt = ring.term_new(_incr_k(S[t[1]], t[0]), domain.one) rest = ring.from_dict({S[i]: _lambda[i] for i in range(s)}) g = (lt - rest).set_ring(ring_to) if g: G.append(g) else: # v is linearly independent from V P = _update(s, _lambda, P) S.append(_incr_k(S[t[1]], t[0])) V.append(v) L.extend([(i, s) for i in range(ngens)]) L = list(set(L)) L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True) L = [(k, l) for (k, l) in L if all(monomial_div(_incr_k(S[l], k), g.LM) is None for g in G)] if not L: G = [ g.monic() for g in G ] return sorted(G, key=lambda g: O_to(g.LM), reverse=True) t = L.pop() def _incr_k(m, k): return tuple(list(m[:k]) + [m[k] + 1] + list(m[k + 1:])) def _identity_matrix(n, domain): M = [[domain.zero]*n for _ in range(n)] for i in range(n): M[i][i] = domain.one return M def _matrix_mul(M, v): return [sum([row[i] * v[i] for i in range(len(v))]) for row in M] def _update(s, _lambda, P): """ Update ``P`` such that for the updated `P'` `P' v = e_{s}`. """ k = min([j for j in range(s, len(_lambda)) if _lambda[j] != 0]) for r in range(len(_lambda)): if r != k: P[r] = [P[r][j] - (P[k][j] * _lambda[r]) / _lambda[k] for j in range(len(P[r]))] P[k] = [P[k][j] / _lambda[k] for j in range(len(P[k]))] P[k], P[s] = P[s], P[k] return P def _representing_matrices(basis, G, ring): r""" Compute the matrices corresponding to the linear maps `m \mapsto x_i m` for all variables `x_i`. """ domain = ring.domain u = ring.ngens-1 def var(i): return tuple([0] * i + [1] + [0] * (u - i)) def representing_matrix(m): M = [[domain.zero] * len(basis) for _ in range(len(basis))] for i, v in enumerate(basis): r = ring.term_new(monomial_mul(m, v), domain.one).rem(G) for monom, coeff in r.terms(): j = basis.index(monom) M[j][i] = coeff return M return [representing_matrix(var(i)) for i in range(u + 1)] def _basis(G, ring): r""" Computes a list of monomials which are not divisible by the leading monomials wrt to ``O`` of ``G``. These monomials are a basis of `K[X_1, \ldots, X_n]/(G)`. """ order = ring.order leading_monomials = [g.LM for g in G] candidates = [ring.zero_monom] basis = [] while candidates: t = candidates.pop() basis.append(t) new_candidates = [_incr_k(t, k) for k in range(ring.ngens) if all(monomial_div(_incr_k(t, k), lmg) is None for lmg in leading_monomials)] candidates.extend(new_candidates) candidates.sort(key=lambda m: order(m), reverse=True) basis = list(set(basis)) return sorted(basis, key=lambda m: order(m))
d93c4a290e0f849234b26d0318741e9e0646c03279c25e8b26c08c5c7046e395
"""Basic tools for dense recursive polynomials in ``K[x]`` or ``K[X]``. """ from sympy import oo from sympy.core import igcd from sympy.polys.monomials import monomial_min, monomial_div from sympy.polys.orderings import monomial_key import random def poly_LC(f, K): """ Return leading coefficient of ``f``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import poly_LC >>> poly_LC([], ZZ) 0 >>> poly_LC([ZZ(1), ZZ(2), ZZ(3)], ZZ) 1 """ if not f: return K.zero else: return f[0] def poly_TC(f, K): """ Return trailing coefficient of ``f``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import poly_TC >>> poly_TC([], ZZ) 0 >>> poly_TC([ZZ(1), ZZ(2), ZZ(3)], ZZ) 3 """ if not f: return K.zero else: return f[-1] dup_LC = dmp_LC = poly_LC dup_TC = dmp_TC = poly_TC def dmp_ground_LC(f, u, K): """ Return the ground leading coefficient. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_ground_LC >>> f = ZZ.map([[[1], [2, 3]]]) >>> dmp_ground_LC(f, 2, ZZ) 1 """ while u: f = dmp_LC(f, K) u -= 1 return dup_LC(f, K) def dmp_ground_TC(f, u, K): """ Return the ground trailing coefficient. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_ground_TC >>> f = ZZ.map([[[1], [2, 3]]]) >>> dmp_ground_TC(f, 2, ZZ) 3 """ while u: f = dmp_TC(f, K) u -= 1 return dup_TC(f, K) def dmp_true_LT(f, u, K): """ Return the leading term ``c * x_1**n_1 ... x_k**n_k``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_true_LT >>> f = ZZ.map([[4], [2, 0], [3, 0, 0]]) >>> dmp_true_LT(f, 1, ZZ) ((2, 0), 4) """ monom = [] while u: monom.append(len(f) - 1) f, u = f[0], u - 1 if not f: monom.append(0) else: monom.append(len(f) - 1) return tuple(monom), dup_LC(f, K) def dup_degree(f): """ Return the leading degree of ``f`` in ``K[x]``. Note that the degree of 0 is negative infinity (the SymPy object -oo). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_degree >>> f = ZZ.map([1, 2, 0, 3]) >>> dup_degree(f) 3 """ if not f: return -oo return len(f) - 1 def dmp_degree(f, u): """ Return the leading degree of ``f`` in ``x_0`` in ``K[X]``. Note that the degree of 0 is negative infinity (the SymPy object -oo). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_degree >>> dmp_degree([[[]]], 2) -oo >>> f = ZZ.map([[2], [1, 2, 3]]) >>> dmp_degree(f, 1) 1 """ if dmp_zero_p(f, u): return -oo else: return len(f) - 1 def _rec_degree_in(g, v, i, j): """Recursive helper function for :func:`dmp_degree_in`.""" if i == j: return dmp_degree(g, v) v, i = v - 1, i + 1 return max([ _rec_degree_in(c, v, i, j) for c in g ]) def dmp_degree_in(f, j, u): """ Return the leading degree of ``f`` in ``x_j`` in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_degree_in >>> f = ZZ.map([[2], [1, 2, 3]]) >>> dmp_degree_in(f, 0, 1) 1 >>> dmp_degree_in(f, 1, 1) 2 """ if not j: return dmp_degree(f, u) if j < 0 or j > u: raise IndexError("0 <= j <= %s expected, got %s" % (u, j)) return _rec_degree_in(f, u, 0, j) def _rec_degree_list(g, v, i, degs): """Recursive helper for :func:`dmp_degree_list`.""" degs[i] = max(degs[i], dmp_degree(g, v)) if v > 0: v, i = v - 1, i + 1 for c in g: _rec_degree_list(c, v, i, degs) def dmp_degree_list(f, u): """ Return a list of degrees of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_degree_list >>> f = ZZ.map([[1], [1, 2, 3]]) >>> dmp_degree_list(f, 1) (1, 2) """ degs = [-oo]*(u + 1) _rec_degree_list(f, u, 0, degs) return tuple(degs) def dup_strip(f): """ Remove leading zeros from ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.densebasic import dup_strip >>> dup_strip([0, 0, 1, 2, 3, 0]) [1, 2, 3, 0] """ if not f or f[0]: return f i = 0 for cf in f: if cf: break else: i += 1 return f[i:] def dmp_strip(f, u): """ Remove leading zeros from ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys.densebasic import dmp_strip >>> dmp_strip([[], [0, 1, 2], [1]], 1) [[0, 1, 2], [1]] """ if not u: return dup_strip(f) if dmp_zero_p(f, u): return f i, v = 0, u - 1 for c in f: if not dmp_zero_p(c, v): break else: i += 1 if i == len(f): return dmp_zero(u) else: return f[i:] def _rec_validate(f, g, i, K): """Recursive helper for :func:`dmp_validate`.""" if type(g) is not list: if K is not None and not K.of_type(g): raise TypeError("%s in %s in not of type %s" % (g, f, K.dtype)) return {i - 1} elif not g: return {i} else: levels = set() for c in g: levels |= _rec_validate(f, c, i + 1, K) return levels def _rec_strip(g, v): """Recursive helper for :func:`_rec_strip`.""" if not v: return dup_strip(g) w = v - 1 return dmp_strip([ _rec_strip(c, w) for c in g ], v) def dmp_validate(f, K=None): """ Return the number of levels in ``f`` and recursively strip it. Examples ======== >>> from sympy.polys.densebasic import dmp_validate >>> dmp_validate([[], [0, 1, 2], [1]]) ([[1, 2], [1]], 1) >>> dmp_validate([[1], 1]) Traceback (most recent call last): ... ValueError: invalid data structure for a multivariate polynomial """ levels = _rec_validate(f, f, 0, K) u = levels.pop() if not levels: return _rec_strip(f, u), u else: raise ValueError( "invalid data structure for a multivariate polynomial") def dup_reverse(f): """ Compute ``x**n * f(1/x)``, i.e.: reverse ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_reverse >>> f = ZZ.map([1, 2, 3, 0]) >>> dup_reverse(f) [3, 2, 1] """ return dup_strip(list(reversed(f))) def dup_copy(f): """ Create a new copy of a polynomial ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_copy >>> f = ZZ.map([1, 2, 3, 0]) >>> dup_copy([1, 2, 3, 0]) [1, 2, 3, 0] """ return list(f) def dmp_copy(f, u): """ Create a new copy of a polynomial ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_copy >>> f = ZZ.map([[1], [1, 2]]) >>> dmp_copy(f, 1) [[1], [1, 2]] """ if not u: return list(f) v = u - 1 return [ dmp_copy(c, v) for c in f ] def dup_to_tuple(f): """ Convert `f` into a tuple. This is needed for hashing. This is similar to dup_copy(). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_copy >>> f = ZZ.map([1, 2, 3, 0]) >>> dup_copy([1, 2, 3, 0]) [1, 2, 3, 0] """ return tuple(f) def dmp_to_tuple(f, u): """ Convert `f` into a nested tuple of tuples. This is needed for hashing. This is similar to dmp_copy(). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_to_tuple >>> f = ZZ.map([[1], [1, 2]]) >>> dmp_to_tuple(f, 1) ((1,), (1, 2)) """ if not u: return tuple(f) v = u - 1 return tuple(dmp_to_tuple(c, v) for c in f) def dup_normal(f, K): """ Normalize univariate polynomial in the given domain. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_normal >>> dup_normal([0, 1.5, 2, 3], ZZ) [1, 2, 3] """ return dup_strip([ K.normal(c) for c in f ]) def dmp_normal(f, u, K): """ Normalize a multivariate polynomial in the given domain. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_normal >>> dmp_normal([[], [0, 1.5, 2]], 1, ZZ) [[1, 2]] """ if not u: return dup_normal(f, K) v = u - 1 return dmp_strip([ dmp_normal(c, v, K) for c in f ], u) def dup_convert(f, K0, K1): """ Convert the ground domain of ``f`` from ``K0`` to ``K1``. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_convert >>> R, x = ring("x", ZZ) >>> dup_convert([R(1), R(2)], R.to_domain(), ZZ) [1, 2] >>> dup_convert([ZZ(1), ZZ(2)], ZZ, R.to_domain()) [1, 2] """ if K0 is not None and K0 == K1: return f else: return dup_strip([ K1.convert(c, K0) for c in f ]) def dmp_convert(f, u, K0, K1): """ Convert the ground domain of ``f`` from ``K0`` to ``K1``. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_convert >>> R, x = ring("x", ZZ) >>> dmp_convert([[R(1)], [R(2)]], 1, R.to_domain(), ZZ) [[1], [2]] >>> dmp_convert([[ZZ(1)], [ZZ(2)]], 1, ZZ, R.to_domain()) [[1], [2]] """ if not u: return dup_convert(f, K0, K1) if K0 is not None and K0 == K1: return f v = u - 1 return dmp_strip([ dmp_convert(c, v, K0, K1) for c in f ], u) def dup_from_sympy(f, K): """ Convert the ground domain of ``f`` from SymPy to ``K``. Examples ======== >>> from sympy import S >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_from_sympy >>> dup_from_sympy([S(1), S(2)], ZZ) == [ZZ(1), ZZ(2)] True """ return dup_strip([ K.from_sympy(c) for c in f ]) def dmp_from_sympy(f, u, K): """ Convert the ground domain of ``f`` from SymPy to ``K``. Examples ======== >>> from sympy import S >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_from_sympy >>> dmp_from_sympy([[S(1)], [S(2)]], 1, ZZ) == [[ZZ(1)], [ZZ(2)]] True """ if not u: return dup_from_sympy(f, K) v = u - 1 return dmp_strip([ dmp_from_sympy(c, v, K) for c in f ], u) def dup_nth(f, n, K): """ Return the ``n``-th coefficient of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_nth >>> f = ZZ.map([1, 2, 3]) >>> dup_nth(f, 0, ZZ) 3 >>> dup_nth(f, 4, ZZ) 0 """ if n < 0: raise IndexError("'n' must be non-negative, got %i" % n) elif n >= len(f): return K.zero else: return f[dup_degree(f) - n] def dmp_nth(f, n, u, K): """ Return the ``n``-th coefficient of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_nth >>> f = ZZ.map([[1], [2], [3]]) >>> dmp_nth(f, 0, 1, ZZ) [3] >>> dmp_nth(f, 4, 1, ZZ) [] """ if n < 0: raise IndexError("'n' must be non-negative, got %i" % n) elif n >= len(f): return dmp_zero(u - 1) else: return f[dmp_degree(f, u) - n] def dmp_ground_nth(f, N, u, K): """ Return the ground ``n``-th coefficient of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_ground_nth >>> f = ZZ.map([[1], [2, 3]]) >>> dmp_ground_nth(f, (0, 1), 1, ZZ) 2 """ v = u for n in N: if n < 0: raise IndexError("`n` must be non-negative, got %i" % n) elif n >= len(f): return K.zero else: d = dmp_degree(f, v) if d == -oo: d = -1 f, v = f[d - n], v - 1 return f def dmp_zero_p(f, u): """ Return ``True`` if ``f`` is zero in ``K[X]``. Examples ======== >>> from sympy.polys.densebasic import dmp_zero_p >>> dmp_zero_p([[[[[]]]]], 4) True >>> dmp_zero_p([[[[[1]]]]], 4) False """ while u: if len(f) != 1: return False f = f[0] u -= 1 return not f def dmp_zero(u): """ Return a multivariate zero. Examples ======== >>> from sympy.polys.densebasic import dmp_zero >>> dmp_zero(4) [[[[[]]]]] """ r = [] for i in range(u): r = [r] return r def dmp_one_p(f, u, K): """ Return ``True`` if ``f`` is one in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_one_p >>> dmp_one_p([[[ZZ(1)]]], 2, ZZ) True """ return dmp_ground_p(f, K.one, u) def dmp_one(u, K): """ Return a multivariate one over ``K``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_one >>> dmp_one(2, ZZ) [[[1]]] """ return dmp_ground(K.one, u) def dmp_ground_p(f, c, u): """ Return True if ``f`` is constant in ``K[X]``. Examples ======== >>> from sympy.polys.densebasic import dmp_ground_p >>> dmp_ground_p([[[3]]], 3, 2) True >>> dmp_ground_p([[[4]]], None, 2) True """ if c is not None and not c: return dmp_zero_p(f, u) while u: if len(f) != 1: return False f = f[0] u -= 1 if c is None: return len(f) <= 1 else: return f == [c] def dmp_ground(c, u): """ Return a multivariate constant. Examples ======== >>> from sympy.polys.densebasic import dmp_ground >>> dmp_ground(3, 5) [[[[[[3]]]]]] >>> dmp_ground(1, -1) 1 """ if not c: return dmp_zero(u) for i in range(u + 1): c = [c] return c def dmp_zeros(n, u, K): """ Return a list of multivariate zeros. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_zeros >>> dmp_zeros(3, 2, ZZ) [[[[]]], [[[]]], [[[]]]] >>> dmp_zeros(3, -1, ZZ) [0, 0, 0] """ if not n: return [] if u < 0: return [K.zero]*n else: return [ dmp_zero(u) for i in range(n) ] def dmp_grounds(c, n, u): """ Return a list of multivariate constants. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_grounds >>> dmp_grounds(ZZ(4), 3, 2) [[[[4]]], [[[4]]], [[[4]]]] >>> dmp_grounds(ZZ(4), 3, -1) [4, 4, 4] """ if not n: return [] if u < 0: return [c]*n else: return [ dmp_ground(c, u) for i in range(n) ] def dmp_negative_p(f, u, K): """ Return ``True`` if ``LC(f)`` is negative. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_negative_p >>> dmp_negative_p([[ZZ(1)], [-ZZ(1)]], 1, ZZ) False >>> dmp_negative_p([[-ZZ(1)], [ZZ(1)]], 1, ZZ) True """ return K.is_negative(dmp_ground_LC(f, u, K)) def dmp_positive_p(f, u, K): """ Return ``True`` if ``LC(f)`` is positive. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_positive_p >>> dmp_positive_p([[ZZ(1)], [-ZZ(1)]], 1, ZZ) True >>> dmp_positive_p([[-ZZ(1)], [ZZ(1)]], 1, ZZ) False """ return K.is_positive(dmp_ground_LC(f, u, K)) def dup_from_dict(f, K): """ Create a ``K[x]`` polynomial from a ``dict``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_from_dict >>> dup_from_dict({(0,): ZZ(7), (2,): ZZ(5), (4,): ZZ(1)}, ZZ) [1, 0, 5, 0, 7] >>> dup_from_dict({}, ZZ) [] """ if not f: return [] n, h = max(f.keys()), [] if type(n) is int: for k in range(n, -1, -1): h.append(f.get(k, K.zero)) else: (n,) = n for k in range(n, -1, -1): h.append(f.get((k,), K.zero)) return dup_strip(h) def dup_from_raw_dict(f, K): """ Create a ``K[x]`` polynomial from a raw ``dict``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_from_raw_dict >>> dup_from_raw_dict({0: ZZ(7), 2: ZZ(5), 4: ZZ(1)}, ZZ) [1, 0, 5, 0, 7] """ if not f: return [] n, h = max(f.keys()), [] for k in range(n, -1, -1): h.append(f.get(k, K.zero)) return dup_strip(h) def dmp_from_dict(f, u, K): """ Create a ``K[X]`` polynomial from a ``dict``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_from_dict >>> dmp_from_dict({(0, 0): ZZ(3), (0, 1): ZZ(2), (2, 1): ZZ(1)}, 1, ZZ) [[1, 0], [], [2, 3]] >>> dmp_from_dict({}, 0, ZZ) [] """ if not u: return dup_from_dict(f, K) if not f: return dmp_zero(u) coeffs = {} for monom, coeff in f.items(): head, tail = monom[0], monom[1:] if head in coeffs: coeffs[head][tail] = coeff else: coeffs[head] = { tail: coeff } n, v, h = max(coeffs.keys()), u - 1, [] for k in range(n, -1, -1): coeff = coeffs.get(k) if coeff is not None: h.append(dmp_from_dict(coeff, v, K)) else: h.append(dmp_zero(v)) return dmp_strip(h, u) def dup_to_dict(f, K=None, zero=False): """ Convert ``K[x]`` polynomial to a ``dict``. Examples ======== >>> from sympy.polys.densebasic import dup_to_dict >>> dup_to_dict([1, 0, 5, 0, 7]) {(0,): 7, (2,): 5, (4,): 1} >>> dup_to_dict([]) {} """ if not f and zero: return {(0,): K.zero} n, result = len(f) - 1, {} for k in range(0, n + 1): if f[n - k]: result[(k,)] = f[n - k] return result def dup_to_raw_dict(f, K=None, zero=False): """ Convert a ``K[x]`` polynomial to a raw ``dict``. Examples ======== >>> from sympy.polys.densebasic import dup_to_raw_dict >>> dup_to_raw_dict([1, 0, 5, 0, 7]) {0: 7, 2: 5, 4: 1} """ if not f and zero: return {0: K.zero} n, result = len(f) - 1, {} for k in range(0, n + 1): if f[n - k]: result[k] = f[n - k] return result def dmp_to_dict(f, u, K=None, zero=False): """ Convert a ``K[X]`` polynomial to a ``dict````. Examples ======== >>> from sympy.polys.densebasic import dmp_to_dict >>> dmp_to_dict([[1, 0], [], [2, 3]], 1) {(0, 0): 3, (0, 1): 2, (2, 1): 1} >>> dmp_to_dict([], 0) {} """ if not u: return dup_to_dict(f, K, zero=zero) if dmp_zero_p(f, u) and zero: return {(0,)*(u + 1): K.zero} n, v, result = dmp_degree(f, u), u - 1, {} if n == -oo: n = -1 for k in range(0, n + 1): h = dmp_to_dict(f[n - k], v) for exp, coeff in h.items(): result[(k,) + exp] = coeff return result def dmp_swap(f, i, j, u, K): """ Transform ``K[..x_i..x_j..]`` to ``K[..x_j..x_i..]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_swap >>> f = ZZ.map([[[2], [1, 0]], []]) >>> dmp_swap(f, 0, 1, 2, ZZ) [[[2], []], [[1, 0], []]] >>> dmp_swap(f, 1, 2, 2, ZZ) [[[1], [2, 0]], [[]]] >>> dmp_swap(f, 0, 2, 2, ZZ) [[[1, 0]], [[2, 0], []]] """ if i < 0 or j < 0 or i > u or j > u: raise IndexError("0 <= i < j <= %s expected" % u) elif i == j: return f F, H = dmp_to_dict(f, u), {} for exp, coeff in F.items(): H[exp[:i] + (exp[j],) + exp[i + 1:j] + (exp[i],) + exp[j + 1:]] = coeff return dmp_from_dict(H, u, K) def dmp_permute(f, P, u, K): """ Return a polynomial in ``K[x_{P(1)},..,x_{P(n)}]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_permute >>> f = ZZ.map([[[2], [1, 0]], []]) >>> dmp_permute(f, [1, 0, 2], 2, ZZ) [[[2], []], [[1, 0], []]] >>> dmp_permute(f, [1, 2, 0], 2, ZZ) [[[1], []], [[2, 0], []]] """ F, H = dmp_to_dict(f, u), {} for exp, coeff in F.items(): new_exp = [0]*len(exp) for e, p in zip(exp, P): new_exp[p] = e H[tuple(new_exp)] = coeff return dmp_from_dict(H, u, K) def dmp_nest(f, l, K): """ Return a multivariate value nested ``l``-levels. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_nest >>> dmp_nest([[ZZ(1)]], 2, ZZ) [[[[1]]]] """ if not isinstance(f, list): return dmp_ground(f, l) for i in range(l): f = [f] return f def dmp_raise(f, l, u, K): """ Return a multivariate polynomial raised ``l``-levels. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_raise >>> f = ZZ.map([[], [1, 2]]) >>> dmp_raise(f, 2, 1, ZZ) [[[[]]], [[[1]], [[2]]]] """ if not l: return f if not u: if not f: return dmp_zero(l) k = l - 1 return [ dmp_ground(c, k) for c in f ] v = u - 1 return [ dmp_raise(c, l, v, K) for c in f ] def dup_deflate(f, K): """ Map ``x**m`` to ``y`` in a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_deflate >>> f = ZZ.map([1, 0, 0, 1, 0, 0, 1]) >>> dup_deflate(f, ZZ) (3, [1, 1, 1]) """ if dup_degree(f) <= 0: return 1, f g = 0 for i in range(len(f)): if not f[-i - 1]: continue g = igcd(g, i) if g == 1: return 1, f return g, f[::g] def dmp_deflate(f, u, K): """ Map ``x_i**m_i`` to ``y_i`` in a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_deflate >>> f = ZZ.map([[1, 0, 0, 2], [], [3, 0, 0, 4]]) >>> dmp_deflate(f, 1, ZZ) ((2, 3), [[1, 2], [3, 4]]) """ if dmp_zero_p(f, u): return (1,)*(u + 1), f F = dmp_to_dict(f, u) B = [0]*(u + 1) for M in F.keys(): for i, m in enumerate(M): B[i] = igcd(B[i], m) for i, b in enumerate(B): if not b: B[i] = 1 B = tuple(B) if all(b == 1 for b in B): return B, f H = {} for A, coeff in F.items(): N = [ a // b for a, b in zip(A, B) ] H[tuple(N)] = coeff return B, dmp_from_dict(H, u, K) def dup_multi_deflate(polys, K): """ Map ``x**m`` to ``y`` in a set of polynomials in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_multi_deflate >>> f = ZZ.map([1, 0, 2, 0, 3]) >>> g = ZZ.map([4, 0, 0]) >>> dup_multi_deflate((f, g), ZZ) (2, ([1, 2, 3], [4, 0])) """ G = 0 for p in polys: if dup_degree(p) <= 0: return 1, polys g = 0 for i in range(len(p)): if not p[-i - 1]: continue g = igcd(g, i) if g == 1: return 1, polys G = igcd(G, g) return G, tuple([ p[::G] for p in polys ]) def dmp_multi_deflate(polys, u, K): """ Map ``x_i**m_i`` to ``y_i`` in a set of polynomials in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_multi_deflate >>> f = ZZ.map([[1, 0, 0, 2], [], [3, 0, 0, 4]]) >>> g = ZZ.map([[1, 0, 2], [], [3, 0, 4]]) >>> dmp_multi_deflate((f, g), 1, ZZ) ((2, 1), ([[1, 0, 0, 2], [3, 0, 0, 4]], [[1, 0, 2], [3, 0, 4]])) """ if not u: M, H = dup_multi_deflate(polys, K) return (M,), H F, B = [], [0]*(u + 1) for p in polys: f = dmp_to_dict(p, u) if not dmp_zero_p(p, u): for M in f.keys(): for i, m in enumerate(M): B[i] = igcd(B[i], m) F.append(f) for i, b in enumerate(B): if not b: B[i] = 1 B = tuple(B) if all(b == 1 for b in B): return B, polys H = [] for f in F: h = {} for A, coeff in f.items(): N = [ a // b for a, b in zip(A, B) ] h[tuple(N)] = coeff H.append(dmp_from_dict(h, u, K)) return B, tuple(H) def dup_inflate(f, m, K): """ Map ``y`` to ``x**m`` in a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_inflate >>> f = ZZ.map([1, 1, 1]) >>> dup_inflate(f, 3, ZZ) [1, 0, 0, 1, 0, 0, 1] """ if m <= 0: raise IndexError("'m' must be positive, got %s" % m) if m == 1 or not f: return f result = [f[0]] for coeff in f[1:]: result.extend([K.zero]*(m - 1)) result.append(coeff) return result def _rec_inflate(g, M, v, i, K): """Recursive helper for :func:`dmp_inflate`.""" if not v: return dup_inflate(g, M[i], K) if M[i] <= 0: raise IndexError("all M[i] must be positive, got %s" % M[i]) w, j = v - 1, i + 1 g = [ _rec_inflate(c, M, w, j, K) for c in g ] result = [g[0]] for coeff in g[1:]: for _ in range(1, M[i]): result.append(dmp_zero(w)) result.append(coeff) return result def dmp_inflate(f, M, u, K): """ Map ``y_i`` to ``x_i**k_i`` in a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_inflate >>> f = ZZ.map([[1, 2], [3, 4]]) >>> dmp_inflate(f, (2, 3), 1, ZZ) [[1, 0, 0, 2], [], [3, 0, 0, 4]] """ if not u: return dup_inflate(f, M[0], K) if all(m == 1 for m in M): return f else: return _rec_inflate(f, M, u, 0, K) def dmp_exclude(f, u, K): """ Exclude useless levels from ``f``. Return the levels excluded, the new excluded ``f``, and the new ``u``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_exclude >>> f = ZZ.map([[[1]], [[1], [2]]]) >>> dmp_exclude(f, 2, ZZ) ([2], [[1], [1, 2]], 1) """ if not u or dmp_ground_p(f, None, u): return [], f, u J, F = [], dmp_to_dict(f, u) for j in range(0, u + 1): for monom in F.keys(): if monom[j]: break else: J.append(j) if not J: return [], f, u f = {} for monom, coeff in F.items(): monom = list(monom) for j in reversed(J): del monom[j] f[tuple(monom)] = coeff u -= len(J) return J, dmp_from_dict(f, u, K), u def dmp_include(f, J, u, K): """ Include useless levels in ``f``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_include >>> f = ZZ.map([[1], [1, 2]]) >>> dmp_include(f, [2], 1, ZZ) [[[1]], [[1], [2]]] """ if not J: return f F, f = dmp_to_dict(f, u), {} for monom, coeff in F.items(): monom = list(monom) for j in J: monom.insert(j, 0) f[tuple(monom)] = coeff u += len(J) return dmp_from_dict(f, u, K) def dmp_inject(f, u, K, front=False): """ Convert ``f`` from ``K[X][Y]`` to ``K[X,Y]``. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_inject >>> R, x,y = ring("x,y", ZZ) >>> dmp_inject([R(1), x + 2], 0, R.to_domain()) ([[[1]], [[1], [2]]], 2) >>> dmp_inject([R(1), x + 2], 0, R.to_domain(), front=True) ([[[1]], [[1, 2]]], 2) """ f, h = dmp_to_dict(f, u), {} v = K.ngens - 1 for f_monom, g in f.items(): g = g.to_dict() for g_monom, c in g.items(): if front: h[g_monom + f_monom] = c else: h[f_monom + g_monom] = c w = u + v + 1 return dmp_from_dict(h, w, K.dom), w def dmp_eject(f, u, K, front=False): """ Convert ``f`` from ``K[X,Y]`` to ``K[X][Y]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_eject >>> dmp_eject([[[1]], [[1], [2]]], 2, ZZ['x', 'y']) [1, x + 2] """ f, h = dmp_to_dict(f, u), {} n = K.ngens v = u - K.ngens + 1 for monom, c in f.items(): if front: g_monom, f_monom = monom[:n], monom[n:] else: g_monom, f_monom = monom[-n:], monom[:-n] if f_monom in h: h[f_monom][g_monom] = c else: h[f_monom] = {g_monom: c} for monom, c in h.items(): h[monom] = K(c) return dmp_from_dict(h, v - 1, K) def dup_terms_gcd(f, K): """ Remove GCD of terms from ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_terms_gcd >>> f = ZZ.map([1, 0, 1, 0, 0]) >>> dup_terms_gcd(f, ZZ) (2, [1, 0, 1]) """ if dup_TC(f, K) or not f: return 0, f i = 0 for c in reversed(f): if not c: i += 1 else: break return i, f[:-i] def dmp_terms_gcd(f, u, K): """ Remove GCD of terms from ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_terms_gcd >>> f = ZZ.map([[1, 0], [1, 0, 0], [], []]) >>> dmp_terms_gcd(f, 1, ZZ) ((2, 1), [[1], [1, 0]]) """ if dmp_ground_TC(f, u, K) or dmp_zero_p(f, u): return (0,)*(u + 1), f F = dmp_to_dict(f, u) G = monomial_min(*list(F.keys())) if all(g == 0 for g in G): return G, f f = {} for monom, coeff in F.items(): f[monomial_div(monom, G)] = coeff return G, dmp_from_dict(f, u, K) def _rec_list_terms(g, v, monom): """Recursive helper for :func:`dmp_list_terms`.""" d, terms = dmp_degree(g, v), [] if not v: for i, c in enumerate(g): if not c: continue terms.append((monom + (d - i,), c)) else: w = v - 1 for i, c in enumerate(g): terms.extend(_rec_list_terms(c, w, monom + (d - i,))) return terms def dmp_list_terms(f, u, K, order=None): """ List all non-zero terms from ``f`` in the given order ``order``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_list_terms >>> f = ZZ.map([[1, 1], [2, 3]]) >>> dmp_list_terms(f, 1, ZZ) [((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)] >>> dmp_list_terms(f, 1, ZZ, order='grevlex') [((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)] """ def sort(terms, O): return sorted(terms, key=lambda term: O(term[0]), reverse=True) terms = _rec_list_terms(f, u, ()) if not terms: return [((0,)*(u + 1), K.zero)] if order is None: return terms else: return sort(terms, monomial_key(order)) def dup_apply_pairs(f, g, h, args, K): """ Apply ``h`` to pairs of coefficients of ``f`` and ``g``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_apply_pairs >>> h = lambda x, y, z: 2*x + y - z >>> dup_apply_pairs([1, 2, 3], [3, 2, 1], h, (1,), ZZ) [4, 5, 6] """ n, m = len(f), len(g) if n != m: if n > m: g = [K.zero]*(n - m) + g else: f = [K.zero]*(m - n) + f result = [] for a, b in zip(f, g): result.append(h(a, b, *args)) return dup_strip(result) def dmp_apply_pairs(f, g, h, args, u, K): """ Apply ``h`` to pairs of coefficients of ``f`` and ``g``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dmp_apply_pairs >>> h = lambda x, y, z: 2*x + y - z >>> dmp_apply_pairs([[1], [2, 3]], [[3], [2, 1]], h, (1,), 1, ZZ) [[4], [5, 6]] """ if not u: return dup_apply_pairs(f, g, h, args, K) n, m, v = len(f), len(g), u - 1 if n != m: if n > m: g = dmp_zeros(n - m, v, K) + g else: f = dmp_zeros(m - n, v, K) + f result = [] for a, b in zip(f, g): result.append(dmp_apply_pairs(a, b, h, args, v, K)) return dmp_strip(result, u) def dup_slice(f, m, n, K): """Take a continuous subsequence of terms of ``f`` in ``K[x]``. """ k = len(f) if k >= m: M = k - m else: M = 0 if k >= n: N = k - n else: N = 0 f = f[N:M] if not f: return [] else: return f + [K.zero]*m def dmp_slice(f, m, n, u, K): """Take a continuous subsequence of terms of ``f`` in ``K[X]``. """ return dmp_slice_in(f, m, n, 0, u, K) def dmp_slice_in(f, m, n, j, u, K): """Take a continuous subsequence of terms of ``f`` in ``x_j`` in ``K[X]``. """ if j < 0 or j > u: raise IndexError("-%s <= j < %s expected, got %s" % (u, u, j)) if not u: return dup_slice(f, m, n, K) f, g = dmp_to_dict(f, u), {} for monom, coeff in f.items(): k = monom[j] if k < m or k >= n: monom = monom[:j] + (0,) + monom[j + 1:] if monom in g: g[monom] += coeff else: g[monom] = coeff return dmp_from_dict(g, u, K) def dup_random(n, a, b, K): """ Return a polynomial of degree ``n`` with coefficients in ``[a, b]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densebasic import dup_random >>> dup_random(3, -10, 10, ZZ) #doctest: +SKIP [-2, -8, 9, -4] """ f = [ K.convert(random.randint(a, b)) for _ in range(0, n + 1) ] while not f[0]: f[0] = K.convert(random.randint(a, b)) return f
eaf72ccfd2170c842e7584950695de763f4a40272a04c60682fc40df0dd33a4a
"""Algorithms for computing symbolic roots of polynomials. """ import math from sympy.core import S, I, pi from sympy.core.compatibility import ordered, reduce from sympy.core.exprtools import factor_terms from sympy.core.function import _mexpand from sympy.core.logic import fuzzy_not from sympy.core.mul import expand_2arg, Mul from sympy.core.numbers import Rational, igcd, comp from sympy.core.power import Pow from sympy.core.relational import Eq from sympy.core.symbol import Dummy, Symbol, symbols from sympy.core.sympify import sympify from sympy.functions import exp, sqrt, im, cos, acos, Piecewise from sympy.functions.elementary.miscellaneous import root from sympy.ntheory import divisors, isprime, nextprime from sympy.polys.domains import EX from sympy.polys.polyerrors import (PolynomialError, GeneratorsNeeded, DomainError) from sympy.polys.polyquinticconst import PolyQuintic from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant from sympy.polys.rationaltools import together from sympy.polys.specialpolys import cyclotomic_poly from sympy.simplify import simplify, powsimp from sympy.utilities import public def roots_linear(f): """Returns a list of roots of a linear polynomial.""" r = -f.nth(0)/f.nth(1) dom = f.get_domain() if not dom.is_Numerical: if dom.is_Composite: r = factor(r) else: r = simplify(r) return [r] def roots_quadratic(f): """Returns a list of roots of a quadratic polynomial. If the domain is ZZ then the roots will be sorted with negatives coming before positives. The ordering will be the same for any numerical coefficients as long as the assumptions tested are correct, otherwise the ordering will not be sorted (but will be canonical). """ a, b, c = f.all_coeffs() dom = f.get_domain() def _sqrt(d): # remove squares from square root since both will be represented # in the results; a similar thing is happening in roots() but # must be duplicated here because not all quadratics are binomials co = [] other = [] for di in Mul.make_args(d): if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0: co.append(Pow(di.base, di.exp//2)) else: other.append(di) if co: d = Mul(*other) co = Mul(*co) return co*sqrt(d) return sqrt(d) def _simplify(expr): if dom.is_Composite: return factor(expr) else: return simplify(expr) if c is S.Zero: r0, r1 = S.Zero, -b/a if not dom.is_Numerical: r1 = _simplify(r1) elif r1.is_negative: r0, r1 = r1, r0 elif b is S.Zero: r = -c/a if not dom.is_Numerical: r = _simplify(r) R = _sqrt(r) r0 = -R r1 = R else: d = b**2 - 4*a*c A = 2*a B = -b/A if not dom.is_Numerical: d = _simplify(d) B = _simplify(B) D = factor_terms(_sqrt(d)/A) r0 = B - D r1 = B + D if a.is_negative: r0, r1 = r1, r0 elif not dom.is_Numerical: r0, r1 = [expand_2arg(i) for i in (r0, r1)] return [r0, r1] def roots_cubic(f, trig=False): """Returns a list of roots of a cubic polynomial. References ========== [1] https://en.wikipedia.org/wiki/Cubic_function, General formula for roots, (accessed November 17, 2014). """ if trig: a, b, c, d = f.all_coeffs() p = (3*a*c - b**2)/3/a**2 q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3) D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2 if (D > 0) == True: rv = [] for k in range(3): rv.append(2*sqrt(-p/3)*cos(acos(q/p*sqrt(-3/p)*Rational(3, 2))/3 - k*pi*Rational(2, 3))) return [i - b/3/a for i in rv] _, a, b, c = f.monic().all_coeffs() if c is S.Zero: x1, x2 = roots([1, a, b], multiple=True) return [x1, S.Zero, x2] p = b - a**2/3 q = c - a*b/3 + 2*a**3/27 pon3 = p/3 aon3 = a/3 u1 = None if p is S.Zero: if q is S.Zero: return [-aon3]*3 if q.is_real: if q.is_positive: u1 = -root(q, 3) elif q.is_negative: u1 = root(-q, 3) elif q is S.Zero: y1, y2 = roots([1, 0, p], multiple=True) return [tmp - aon3 for tmp in [y1, S.Zero, y2]] elif q.is_real and q.is_negative: u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3) coeff = I*sqrt(3)/2 if u1 is None: u1 = S.One u2 = Rational(-1, 2) + coeff u3 = Rational(-1, 2) - coeff a, b, c, d = S(1), a, b, c D0 = b**2 - 3*a*c D1 = 2*b**3 - 9*a*b*c + 27*a**2*d C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3) return [-(b + uk*C + D0/C/uk)/3/a for uk in [u1, u2, u3]] u2 = u1*(Rational(-1, 2) + coeff) u3 = u1*(Rational(-1, 2) - coeff) if p is S.Zero: return [u1 - aon3, u2 - aon3, u3 - aon3] soln = [ -u1 + pon3/u1 - aon3, -u2 + pon3/u2 - aon3, -u3 + pon3/u3 - aon3 ] return soln def _roots_quartic_euler(p, q, r, a): """ Descartes-Euler solution of the quartic equation Parameters ========== p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r`` a: shift of the roots Notes ===== This is a helper function for ``roots_quartic``. Look for solutions of the form :: ``x1 = sqrt(R) - sqrt(A + B*sqrt(R))`` ``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))`` ``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))`` ``x4 = sqrt(R) + sqrt(A + B*sqrt(R))`` To satisfy the quartic equation one must have ``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R`` so that ``R`` must satisfy the Descartes-Euler resolvent equation ``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0`` If the resolvent does not have a rational solution, return None; in that case it is likely that the Ferrari method gives a simpler solution. Examples ======== >>> from sympy import S >>> from sympy.polys.polyroots import _roots_quartic_euler >>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125 >>> _roots_quartic_euler(p, q, r, S(0))[0] -sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5 """ # solve the resolvent equation x = Dummy('x') eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2 xsols = list(roots(Poly(eq, x), cubics=False).keys()) xsols = [sol for sol in xsols if sol.is_rational and sol.is_nonzero] if not xsols: return None R = max(xsols) c1 = sqrt(R) B = -q*c1/(4*R) A = -R - p/2 c2 = sqrt(A + B) c3 = sqrt(A - B) return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a] def roots_quartic(f): r""" Returns a list of roots of a quartic polynomial. There are many references for solving quartic expressions available [1-5]. This reviewer has found that many of them require one to select from among 2 or more possible sets of solutions and that some solutions work when one is searching for real roots but don't work when searching for complex roots (though this is not always stated clearly). The following routine has been tested and found to be correct for 0, 2 or 4 complex roots. The quasisymmetric case solution [6] looks for quartics that have the form `x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`. Although no general solution that is always applicable for all coefficients is known to this reviewer, certain conditions are tested to determine the simplest 4 expressions that can be returned: 1) `f = c + a*(a**2/8 - b/2) == 0` 2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0` 3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then a) `p == 0` b) `p != 0` Examples ======== >>> from sympy import Poly >>> from sympy.polys.polyroots import roots_quartic >>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20')) >>> # 4 complex roots: 1+-I*sqrt(3), 2+-I >>> sorted(str(tmp.evalf(n=2)) for tmp in r) ['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I'] References ========== 1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html 2. https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method 3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html 4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf 5. http://www.albmath.org/files/Math_5713.pdf 6. http://www.statemaster.com/encyclopedia/Quartic-equation 7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf """ _, a, b, c, d = f.monic().all_coeffs() if not d: return [S.Zero] + roots([1, a, b, c], multiple=True) elif (c/a)**2 == d: x, m = f.gen, c/a g = Poly(x**2 + a*x + b - 2*m, x) z1, z2 = roots_quadratic(g) h1 = Poly(x**2 - z1*x + m, x) h2 = Poly(x**2 - z2*x + m, x) r1 = roots_quadratic(h1) r2 = roots_quadratic(h2) return r1 + r2 else: a2 = a**2 e = b - 3*a2/8 f = _mexpand(c + a*(a2/8 - b/2)) g = _mexpand(d - a*(a*(3*a2/256 - b/16) + c/4)) aon4 = a/4 if f is S.Zero: y1, y2 = [sqrt(tmp) for tmp in roots([1, e, g], multiple=True)] return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]] if g is S.Zero: y = [S.Zero] + roots([1, 0, e, f], multiple=True) return [tmp - aon4 for tmp in y] else: # Descartes-Euler method, see [7] sols = _roots_quartic_euler(e, f, g, aon4) if sols: return sols # Ferrari method, see [1, 2] a2 = a**2 e = b - 3*a2/8 f = c + a*(a2/8 - b/2) g = d - a*(a*(3*a2/256 - b/16) + c/4) p = -e**2/12 - g q = -e**3/108 + e*g/3 - f**2/8 TH = Rational(1, 3) def _ans(y): w = sqrt(e + 2*y) arg1 = 3*e + 2*y arg2 = 2*f/w ans = [] for s in [-1, 1]: root = sqrt(-(arg1 + s*arg2)) for t in [-1, 1]: ans.append((s*w - t*root)/2 - aon4) return ans # p == 0 case y1 = e*Rational(-5, 6) - q**TH if p.is_zero: return _ans(y1) # if p != 0 then u below is not 0 root = sqrt(q**2/4 + p**3/27) r = -q/2 + root # or -q/2 - root u = r**TH # primary root of solve(x**3 - r, x) y2 = e*Rational(-5, 6) + u - p/u/3 if fuzzy_not(p.is_zero): return _ans(y2) # sort it out once they know the values of the coefficients return [Piecewise((a1, Eq(p, 0)), (a2, True)) for a1, a2 in zip(_ans(y1), _ans(y2))] def roots_binomial(f): """Returns a list of roots of a binomial polynomial. If the domain is ZZ then the roots will be sorted with negatives coming before positives. The ordering will be the same for any numerical coefficients as long as the assumptions tested are correct, otherwise the ordering will not be sorted (but will be canonical). """ n = f.degree() a, b = f.nth(n), f.nth(0) base = -cancel(b/a) alpha = root(base, n) if alpha.is_number: alpha = alpha.expand(complex=True) # define some parameters that will allow us to order the roots. # If the domain is ZZ this is guaranteed to return roots sorted # with reals before non-real roots and non-real sorted according # to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I neg = base.is_negative even = n % 2 == 0 if neg: if even == True and (base + 1).is_positive: big = True else: big = False # get the indices in the right order so the computed # roots will be sorted when the domain is ZZ ks = [] imax = n//2 if even: ks.append(imax) imax -= 1 if not neg: ks.append(0) for i in range(imax, 0, -1): if neg: ks.extend([i, -i]) else: ks.extend([-i, i]) if neg: ks.append(0) if big: for i in range(0, len(ks), 2): pair = ks[i: i + 2] pair = list(reversed(pair)) # compute the roots roots, d = [], 2*I*pi/n for k in ks: zeta = exp(k*d).expand(complex=True) roots.append((alpha*zeta).expand(power_base=False)) return roots def _inv_totient_estimate(m): """ Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``. Examples ======== >>> from sympy.polys.polyroots import _inv_totient_estimate >>> _inv_totient_estimate(192) (192, 840) >>> _inv_totient_estimate(400) (400, 1750) """ primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ] a, b = 1, 1 for p in primes: a *= p b *= p - 1 L = m U = int(math.ceil(m*(float(a)/b))) P = p = 2 primes = [] while P <= U: p = nextprime(p) primes.append(p) P *= p P //= p b = 1 for p in primes[:-1]: b *= p - 1 U = int(math.ceil(m*(float(P)/b))) return L, U def roots_cyclotomic(f, factor=False): """Compute roots of cyclotomic polynomials. """ L, U = _inv_totient_estimate(f.degree()) for n in range(L, U + 1): g = cyclotomic_poly(n, f.gen, polys=True) if f.expr == g.expr: break else: # pragma: no cover raise RuntimeError("failed to find index of a cyclotomic polynomial") roots = [] if not factor: # get the indices in the right order so the computed # roots will be sorted h = n//2 ks = [i for i in range(1, n + 1) if igcd(i, n) == 1] ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1)) d = 2*I*pi/n for k in reversed(ks): roots.append(exp(k*d).expand(complex=True)) else: g = Poly(f, extension=root(-1, n)) for h, _ in ordered(g.factor_list()[1]): roots.append(-h.TC()) return roots def roots_quintic(f): """ Calculate exact roots of a solvable quintic """ result = [] coeff_5, coeff_4, p, q, r, s = f.all_coeffs() # Eqn must be of the form x^5 + px^3 + qx^2 + rx + s if coeff_4: return result if coeff_5 != 1: l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5] if not all(coeff.is_Rational for coeff in l): return result f = Poly(f/coeff_5) quintic = PolyQuintic(f) # Eqn standardized. Algo for solving starts here if not f.is_irreducible: return result f20 = quintic.f20 # Check if f20 has linear factors over domain Z if f20.is_irreducible: return result # Now, we know that f is solvable for _factor in f20.factor_list()[1]: if _factor[0].is_linear: theta = _factor[0].root(0) break d = discriminant(f) delta = sqrt(d) # zeta = a fifth root of unity zeta1, zeta2, zeta3, zeta4 = quintic.zeta T = quintic.T(theta, d) tol = S(1e-10) alpha = T[1] + T[2]*delta alpha_bar = T[1] - T[2]*delta beta = T[3] + T[4]*delta beta_bar = T[3] - T[4]*delta disc = alpha**2 - 4*beta disc_bar = alpha_bar**2 - 4*beta_bar l0 = quintic.l0(theta) l1 = _quintic_simplify((-alpha + sqrt(disc)) / S(2)) l4 = _quintic_simplify((-alpha - sqrt(disc)) / S(2)) l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / S(2)) l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / S(2)) order = quintic.order(theta, d) test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) ) # Comparing floats if not comp(test, 0, tol): l2, l3 = l3, l2 # Now we have correct order of l's R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4 R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4 R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4 R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4 Res = [None, [None]*5, [None]*5, [None]*5, [None]*5] Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5] sol = Symbol('sol') # Simplifying improves performance a lot for exact expressions R1 = _quintic_simplify(R1) R2 = _quintic_simplify(R2) R3 = _quintic_simplify(R3) R4 = _quintic_simplify(R4) # Solve imported here. Causing problems if imported as 'solve' # and hence the changed name from sympy.solvers.solvers import solve as _solve a, b = symbols('a b', cls=Dummy) _sol = _solve( sol**5 - a - I*b, sol) for i in range(5): _sol[i] = factor(_sol[i]) R1 = R1.as_real_imag() R2 = R2.as_real_imag() R3 = R3.as_real_imag() R4 = R4.as_real_imag() for i, currentroot in enumerate(_sol): Res[1][i] = _quintic_simplify(currentroot.subs({ a: R1[0], b: R1[1] })) Res[2][i] = _quintic_simplify(currentroot.subs({ a: R2[0], b: R2[1] })) Res[3][i] = _quintic_simplify(currentroot.subs({ a: R3[0], b: R3[1] })) Res[4][i] = _quintic_simplify(currentroot.subs({ a: R4[0], b: R4[1] })) for i in range(1, 5): for j in range(5): Res_n[i][j] = Res[i][j].n() Res[i][j] = _quintic_simplify(Res[i][j]) r1 = Res[1][0] r1_n = Res_n[1][0] for i in range(5): if comp(im(r1_n*Res_n[4][i]), 0, tol): r4 = Res[4][i] break # Now we have various Res values. Each will be a list of five # values. We have to pick one r value from those five for each Res u, v = quintic.uv(theta, d) testplus = (u + v*delta*sqrt(5)).n() testminus = (u - v*delta*sqrt(5)).n() # Evaluated numbers suffixed with _n # We will use evaluated numbers for calculation. Much faster. r4_n = r4.n() r2 = r3 = None for i in range(5): r2temp_n = Res_n[2][i] for j in range(5): # Again storing away the exact number and using # evaluated numbers in computations r3temp_n = Res_n[3][j] if (comp((r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus).n(), 0, tol) and comp((r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus).n(), 0, tol)): r2 = Res[2][i] r3 = Res[3][j] break if r2: break else: return [] # fall back to normal solve # Now, we have r's so we can get roots x1 = (r1 + r2 + r3 + r4)/5 x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5 x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5 x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5 x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5 result = [x1, x2, x3, x4, x5] # Now check if solutions are distinct saw = set() for r in result: r = r.n(2) if r in saw: # Roots were identical. Abort, return [] # and fall back to usual solve return [] saw.add(r) return result def _quintic_simplify(expr): expr = powsimp(expr) expr = cancel(expr) return together(expr) def _integer_basis(poly): """Compute coefficient basis for a polynomial over integers. Returns the integer ``div`` such that substituting ``x = div*y`` ``p(x) = m*q(y)`` where the coefficients of ``q`` are smaller than those of ``p``. For example ``x**5 + 512*x + 1024 = 0`` with ``div = 4`` becomes ``y**5 + 2*y + 1 = 0`` Returns the integer ``div`` or ``None`` if there is no possible scaling. Examples ======== >>> from sympy.polys import Poly >>> from sympy.abc import x >>> from sympy.polys.polyroots import _integer_basis >>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ') >>> _integer_basis(p) 4 """ monoms, coeffs = list(zip(*poly.terms())) monoms, = list(zip(*monoms)) coeffs = list(map(abs, coeffs)) if coeffs[0] < coeffs[-1]: coeffs = list(reversed(coeffs)) n = monoms[0] monoms = [n - i for i in reversed(monoms)] else: return None monoms = monoms[:-1] coeffs = coeffs[:-1] divs = reversed(divisors(gcd_list(coeffs))[1:]) try: div = next(divs) except StopIteration: return None while True: for monom, coeff in zip(monoms, coeffs): if coeff % div**monom != 0: try: div = next(divs) except StopIteration: return None else: break else: return div def preprocess_roots(poly): """Try to get rid of symbolic coefficients from ``poly``. """ coeff = S.One poly_func = poly.func try: _, poly = poly.clear_denoms(convert=True) except DomainError: return coeff, poly poly = poly.primitive()[1] poly = poly.retract() # TODO: This is fragile. Figure out how to make this independent of construct_domain(). if poly.get_domain().is_Poly and all(c.is_term for c in poly.rep.coeffs()): poly = poly.inject() strips = list(zip(*poly.monoms())) gens = list(poly.gens[1:]) base, strips = strips[0], strips[1:] for gen, strip in zip(list(gens), strips): reverse = False if strip[0] < strip[-1]: strip = reversed(strip) reverse = True ratio = None for a, b in zip(base, strip): if not a and not b: continue elif not a or not b: break elif b % a != 0: break else: _ratio = b // a if ratio is None: ratio = _ratio elif ratio != _ratio: break else: if reverse: ratio = -ratio poly = poly.eval(gen, 1) coeff *= gen**(-ratio) gens.remove(gen) if gens: poly = poly.eject(*gens) if poly.is_univariate and poly.get_domain().is_ZZ: basis = _integer_basis(poly) if basis is not None: n = poly.degree() def func(k, coeff): return coeff//basis**(n - k[0]) poly = poly.termwise(func) coeff *= basis if not isinstance(poly, poly_func): poly = poly_func(poly) return coeff, poly @public def roots(f, *gens, auto=True, cubics=True, trig=False, quartics=True, quintics=False, multiple=False, filter=None, predicate=None, **flags): """ Computes symbolic roots of a univariate polynomial. Given a univariate polynomial f with symbolic coefficients (or a list of the polynomial's coefficients), returns a dictionary with its roots and their multiplicities. Only roots expressible via radicals will be returned. To get a complete set of roots use RootOf class or numerical methods instead. By default cubic and quartic formulas are used in the algorithm. To disable them because of unreadable output set ``cubics=False`` or ``quartics=False`` respectively. If cubic roots are real but are expressed in terms of complex numbers (casus irreducibilis [1]) the ``trig`` flag can be set to True to have the solutions returned in terms of cosine and inverse cosine functions. To get roots from a specific domain set the ``filter`` flag with one of the following specifiers: Z, Q, R, I, C. By default all roots are returned (this is equivalent to setting ``filter='C'``). By default a dictionary is returned giving a compact result in case of multiple roots. However to get a list containing all those roots set the ``multiple`` flag to True; the list will have identical roots appearing next to each other in the result. (For a given Poly, the all_roots method will give the roots in sorted numerical order.) Examples ======== >>> from sympy import Poly, roots >>> from sympy.abc import x, y >>> roots(x**2 - 1, x) {-1: 1, 1: 1} >>> p = Poly(x**2-1, x) >>> roots(p) {-1: 1, 1: 1} >>> p = Poly(x**2-y, x, y) >>> roots(Poly(p, x)) {-sqrt(y): 1, sqrt(y): 1} >>> roots(x**2 - y, x) {-sqrt(y): 1, sqrt(y): 1} >>> roots([1, 0, -1]) {-1: 1, 1: 1} References ========== .. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method """ from sympy.polys.polytools import to_rational_coeffs flags = dict(flags) if isinstance(f, list): if gens: raise ValueError('redundant generators given') x = Dummy('x') poly, i = {}, len(f) - 1 for coeff in f: poly[i], i = sympify(coeff), i - 1 f = Poly(poly, x, field=True) else: try: F = Poly(f, *gens, **flags) if not isinstance(f, Poly) and not F.gen.is_Symbol: raise PolynomialError("generator must be a Symbol") else: f = F if f.length == 2 and f.degree() != 1: # check for foo**n factors in the constant n = f.degree() npow_bases = [] others = [] expr = f.as_expr() con = expr.as_independent(*gens)[0] for p in Mul.make_args(con): if p.is_Pow and not p.exp % n: npow_bases.append(p.base**(p.exp/n)) else: others.append(p) if npow_bases: b = Mul(*npow_bases) B = Dummy() d = roots(Poly(expr - con + B**n*Mul(*others), *gens, **flags), *gens, **flags) rv = {} for k, v in d.items(): rv[k.subs(B, b)] = v return rv except GeneratorsNeeded: if multiple: return [] else: return {} if f.is_multivariate: raise PolynomialError('multivariate polynomials are not supported') def _update_dict(result, currentroot, k): if currentroot in result: result[currentroot] += k else: result[currentroot] = k def _try_decompose(f): """Find roots using functional decomposition. """ factors, roots = f.decompose(), [] for currentroot in _try_heuristics(factors[0]): roots.append(currentroot) for currentfactor in factors[1:]: previous, roots = list(roots), [] for currentroot in previous: g = currentfactor - Poly(currentroot, f.gen) for currentroot in _try_heuristics(g): roots.append(currentroot) return roots def _try_heuristics(f): """Find roots using formulas and some tricks. """ if f.is_ground: return [] if f.is_monomial: return [S.Zero]*f.degree() if f.length() == 2: if f.degree() == 1: return list(map(cancel, roots_linear(f))) else: return roots_binomial(f) result = [] for i in [-1, 1]: if not f.eval(i): f = f.quo(Poly(f.gen - i, f.gen)) result.append(i) break n = f.degree() if n == 1: result += list(map(cancel, roots_linear(f))) elif n == 2: result += list(map(cancel, roots_quadratic(f))) elif f.is_cyclotomic: result += roots_cyclotomic(f) elif n == 3 and cubics: result += roots_cubic(f, trig=trig) elif n == 4 and quartics: result += roots_quartic(f) elif n == 5 and quintics: result += roots_quintic(f) return result # Convert the generators to symbols dumgens = symbols('x:%d' % len(f.gens), cls=Dummy) f = f.per(f.rep, dumgens) (k,), f = f.terms_gcd() if not k: zeros = {} else: zeros = {S.Zero: k} coeff, f = preprocess_roots(f) if auto and f.get_domain().is_Ring: f = f.to_field() # Use EX instead of ZZ_I or QQ_I if f.get_domain().is_QQ_I: f = f.per(f.rep.convert(EX)) rescale_x = None translate_x = None result = {} if not f.is_ground: dom = f.get_domain() if not dom.is_Exact and dom.is_Numerical: for r in f.nroots(): _update_dict(result, r, 1) elif f.degree() == 1: result[roots_linear(f)[0]] = 1 elif f.length() == 2: roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial for r in roots_fun(f): _update_dict(result, r, 1) else: _, factors = Poly(f.as_expr()).factor_list() if len(factors) == 1 and f.degree() == 2: for r in roots_quadratic(f): _update_dict(result, r, 1) else: if len(factors) == 1 and factors[0][1] == 1: if f.get_domain().is_EX: res = to_rational_coeffs(f) if res: if res[0] is None: translate_x, f = res[2:] else: rescale_x, f = res[1], res[-1] result = roots(f) if not result: for currentroot in _try_decompose(f): _update_dict(result, currentroot, 1) else: for r in _try_heuristics(f): _update_dict(result, r, 1) else: for currentroot in _try_decompose(f): _update_dict(result, currentroot, 1) else: for currentfactor, k in factors: for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)): _update_dict(result, r, k) if coeff is not S.One: _result, result, = result, {} for currentroot, k in _result.items(): result[coeff*currentroot] = k if filter not in [None, 'C']: handlers = { 'Z': lambda r: r.is_Integer, 'Q': lambda r: r.is_Rational, 'R': lambda r: all(a.is_real for a in r.as_numer_denom()), 'I': lambda r: r.is_imaginary, } try: query = handlers[filter] except KeyError: raise ValueError("Invalid filter: %s" % filter) for zero in dict(result).keys(): if not query(zero): del result[zero] if predicate is not None: for zero in dict(result).keys(): if not predicate(zero): del result[zero] if rescale_x: result1 = {} for k, v in result.items(): result1[k*rescale_x] = v result = result1 if translate_x: result1 = {} for k, v in result.items(): result1[k + translate_x] = v result = result1 # adding zero roots after non-trivial roots have been translated result.update(zeros) if not multiple: return result else: zeros = [] for zero in ordered(result): zeros.extend([zero]*result[zero]) return zeros def root_factors(f, *gens, filter=None, **args): """ Returns all factors of a univariate polynomial. Examples ======== >>> from sympy.abc import x, y >>> from sympy.polys.polyroots import root_factors >>> root_factors(x**2 - y, x) [x - sqrt(y), x + sqrt(y)] """ args = dict(args) F = Poly(f, *gens, **args) if not F.is_Poly: return [f] if F.is_multivariate: raise ValueError('multivariate polynomials are not supported') x = F.gens[0] zeros = roots(F, filter=filter) if not zeros: factors = [F] else: factors, N = [], 0 for r, n in ordered(zeros.items()): factors, N = factors + [Poly(x - r, x)]*n, N + n if N < F.degree(): G = reduce(lambda p, q: p*q, factors) factors.append(F.quo(G)) if not isinstance(f, Poly): factors = [ f.as_expr() for f in factors ] return factors
44eba916e598f9302e1370474391f6ed72992d6b66f2c90246e1c8a373464b17
from operator import mul from sympy.core.sympify import _sympify from sympy.matrices.common import (NonInvertibleMatrixError, NonSquareMatrixError, ShapeError) from sympy.polys.constructor import construct_domain class DDMError(Exception): """Base class for errors raised by DDM""" pass class DDMBadInputError(DDMError): """list of lists is inconsistent with shape""" pass class DDMDomainError(DDMError): """domains do not match""" pass class DDMShapeError(DDMError): """shapes are inconsistent""" pass class DDM(list): """Dense matrix based on polys domain elements This is a list subclass and is a wrapper for a list of lists that supports basic matrix arithmetic +, -, *, **. """ def __init__(self, rowslist, shape, domain): super().__init__(rowslist) self.shape = self.rows, self.cols = m, n = shape self.domain = domain if not (len(self) == m and all(len(row) == n for row in self)): raise DDMBadInputError("Inconsistent row-list/shape") def __str__(self): cls = type(self).__name__ rows = list.__str__(self) return '%s(%s, %s, %s)' % (cls, rows, self.shape, self.domain) def __eq__(self, other): if not isinstance(other, DDM): return False return (super().__eq__(other) and self.domain == other.domain) def __ne__(self, other): return not self.__eq__(other) @classmethod def zeros(cls, shape, domain): z = domain.zero m, n = shape rowslist = ([z] * n for _ in range(m)) return DDM(rowslist, shape, domain) @classmethod def eye(cls, size, domain): one = domain.one ddm = cls.zeros((size, size), domain) for i in range(size): ddm[i][i] = one return ddm def copy(self): copyrows = (row[:] for row in self) return DDM(copyrows, self.shape, self.domain) def __add__(a, b): if not isinstance(b, DDM): return NotImplemented return a.add(b) def __sub__(a, b): if not isinstance(b, DDM): return NotImplemented return a.sub(b) def __neg__(a): return a.neg() def __mul__(a, b): if b in a.domain: return a.mul(b) else: return NotImplemented def __matmul__(a, b): if isinstance(b, DDM): return a.matmul(b) else: return NotImplemented @classmethod def _check(cls, a, op, b, ashape, bshape): if a.domain != b.domain: msg = "Domain mismatch: %s %s %s" % (a.domain, op, b.domain) raise DDMDomainError(msg) if ashape != bshape: msg = "Shape mismatch: %s %s %s" % (a.shape, op, b.shape) raise DDMShapeError(msg) def add(a, b): """a + b""" a._check(a, '+', b, a.shape, b.shape) c = a.copy() ddm_iadd(c, b) return c def sub(a, b): """a - b""" a._check(a, '-', b, a.shape, b.shape) c = a.copy() ddm_isub(c, b) return c def neg(a): """-a""" b = a.copy() ddm_ineg(b) return b def matmul(a, b): """a @ b (matrix product)""" m, o = a.shape o2, n = b.shape a._check(a, '*', b, o, o2) c = a.zeros((m, n), a.domain) ddm_imatmul(c, a, b) return c def rref(a): """Reduced-row echelon form of a and list of pivots""" b = a.copy() pivots = ddm_irref(b) return b, pivots def det(a): """Determinant of a""" m, n = a.shape if m != n: raise DDMShapeError("Determinant of non-square matrix") b = a.copy() K = b.domain deta = ddm_idet(b, K) return deta def inv(a): """Inverse of a""" m, n = a.shape if m != n: raise DDMShapeError("Determinant of non-square matrix") ainv = a.copy() K = a.domain ddm_iinv(ainv, a, K) return ainv def lu(a): """L, U decomposition of a""" m, n = a.shape K = a.domain U = a.copy() L = a.eye(m, K) swaps = ddm_ilu_split(L, U, K) return L, U, swaps def lu_solve(a, b): """x where a*x = b""" m, n = a.shape m2, o = b.shape a._check(a, 'lu_solve', b, m, m2) L, U, swaps = a.lu() x = a.zeros((n, o), a.domain) ddm_ilu_solve(x, L, U, swaps, b) return x def charpoly(a): """Coefficients of characteristic polynomial of a""" K = a.domain m, n = a.shape if m != n: raise DDMShapeError("Charpoly of non-square matrix") vec = ddm_berk(a, K) coeffs = [vec[i][0] for i in range(n+1)] return coeffs def ddm_iadd(a, b): """a += b""" for ai, bi in zip(a, b): for j, bij in enumerate(bi): ai[j] += bij def ddm_isub(a, b): """a -= b""" for ai, bi in zip(a, b): for j, bij in enumerate(bi): ai[j] -= bij def ddm_ineg(a): """a <-- -a""" for ai in a: for j, aij in enumerate(ai): ai[j] = -aij def ddm_imatmul(a, b, c): """a += b @ c""" cT = list(zip(*c)) for bi, ai in zip(b, a): for j, cTj in enumerate(cT): ai[j] = sum(map(mul, bi, cTj), ai[j]) def ddm_irref(a): """a <-- rref(a)""" # a is (m x n) m = len(a) if not m: return [] n = len(a[0]) i = 0 pivots = [] for j in range(n): # pivot aij = a[i][j] # zero-pivot if not aij: for ip in range(i+1, m): aij = a[ip][j] # row-swap if aij: a[i], a[ip] = a[ip], a[i] break else: # next column continue # normalise row ai = a[i] for l in range(j, n): ai[l] /= aij # ai[j] = one # eliminate above and below to the right for k, ak in enumerate(a): if k == i or not ak[j]: continue akj = ak[j] ak[j] -= akj # ak[j] = zero for l in range(j+1, n): ak[l] -= akj * ai[l] # next row pivots.append(j) i += 1 # no more rows? if i >= m: break return pivots def ddm_idet(a, K): """a <-- echelon(a); return det""" # Fraction-free Gaussian elimination # https://www.math.usm.edu/perry/Research/Thesis_DRL.pdf # a is (m x n) m = len(a) if not m: return K.one n = len(a[0]) is_field = K.is_Field # uf keeps track of the effect of row swaps and multiplies uf = K.one for j in range(n-1): # if zero on the diagonal need to swap if not a[j][j]: for l in range(j+1, n): if a[l][j]: a[j], a[l] = a[l], a[j] uf = -uf break else: # unable to swap: det = 0 return K.zero for i in range(j+1, n): if a[i][j]: if not is_field: d = K.gcd(a[j][j], a[i][j]) b = a[j][j] // d c = a[i][j] // d else: b = a[j][j] c = a[i][j] # account for multiplying row i by b uf = b * uf for k in range(j+1, n): a[i][k] = b*a[i][k] - c*a[j][k] # triangular det is product of diagonal prod = K.one for i in range(n): prod = prod * a[i][i] # incorporate swaps and multiplies if not is_field: D = prod // uf else: D = prod / uf return D def ddm_iinv(ainv, a, K): if not K.is_Field: raise ValueError('Not a field') # a is (m x n) m = len(a) if not m: return n = len(a[0]) if m != n: raise NonSquareMatrixError eye = [[K.one if i==j else K.zero for j in range(n)] for i in range(n)] Aaug = [row + eyerow for row, eyerow in zip(a, eye)] pivots = ddm_irref(Aaug) if pivots != list(range(n)): raise NonInvertibleMatrixError('Matrix det == 0; not invertible.') ainv[:] = [row[n:] for row in Aaug] def ddm_ilu_split(L, U, K): """L, U <-- LU(U)""" m = len(U) if not m: return [] n = len(U[0]) swaps = ddm_ilu(U) zeros = [K.zero] * min(m, n) for i in range(1, m): j = min(i, n) L[i][:j] = U[i][:j] U[i][:j] = zeros[:j] return swaps def ddm_ilu(a): """a <-- LU(a)""" m = len(a) if not m: return [] n = len(a[0]) swaps = [] for i in range(min(m, n)): if not a[i][i]: for ip in range(i+1, m): if a[ip][i]: swaps.append((i, ip)) a[i], a[ip] = a[ip], a[i] break else: # M = Matrix([[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 2]]) continue for j in range(i+1, m): l_ji = a[j][i] / a[i][i] a[j][i] = l_ji for k in range(i+1, n): a[j][k] -= l_ji * a[i][k] return swaps def ddm_ilu_solve(x, L, U, swaps, b): """x <-- solve(L*U*x = swaps(b))""" m = len(U) if not m: return n = len(U[0]) m2 = len(b) if not m2: raise DDMShapeError("Shape mismtch") o = len(b[0]) if m != m2: raise DDMShapeError("Shape mismtch") if m < n: raise NotImplementedError("Underdetermined") if swaps: b = [row[:] for row in b] for i1, i2 in swaps: b[i1], b[i2] = b[i2], b[i1] # solve Ly = b y = [[None] * o for _ in range(m)] for k in range(o): for i in range(m): rhs = b[i][k] for j in range(i): rhs -= L[i][j] * y[j][k] y[i][k] = rhs if m > n: for i in range(n, m): for j in range(o): if y[i][j]: raise NonInvertibleMatrixError # Solve Ux = y for k in range(o): for i in reversed(range(n)): if not U[i][i]: raise NonInvertibleMatrixError rhs = y[i][k] for j in range(i+1, n): rhs -= U[i][j] * x[j][k] x[i][k] = rhs / U[i][i] def ddm_berk(M, K): m = len(M) if not m: return [[K.one]] n = len(M[0]) if m != n: raise DDMShapeError("Not square") if n == 1: return [[K.one], [-M[0][0]]] a = M[0][0] R = [M[0][1:]] C = [[row[0]] for row in M[1:]] A = [row[1:] for row in M[1:]] q = ddm_berk(A, K) T = [[K.zero] * n for _ in range(n+1)] for i in range(n): T[i][i] = K.one T[i+1][i] = -a for i in range(2, n+1): if i == 2: AnC = C else: C = AnC AnC = [[K.zero] for row in C] ddm_imatmul(AnC, A, C) RAnC = [[K.zero]] ddm_imatmul(RAnC, R, AnC) for j in range(0, n+1-i): T[i+j][j] = -RAnC[0][0] qout = [[K.zero] for _ in range(n+1)] ddm_imatmul(qout, T, q) return qout class DomainMatrix: def __init__(self, rows, shape, domain): self.rep = DDM(rows, shape, domain) self.shape = shape self.domain = domain @classmethod def from_ddm(cls, ddm): return cls(ddm, ddm.shape, ddm.domain) @classmethod def from_list_sympy(cls, nrows, ncols, rows): assert len(rows) == nrows assert all(len(row) == ncols for row in rows) items_sympy = [_sympify(item) for row in rows for item in row] domain, items_domain = cls.get_domain(items_sympy) domain_rows = [[items_domain[ncols*r + c] for c in range(ncols)] for r in range(nrows)] return DomainMatrix(domain_rows, (nrows, ncols), domain) @classmethod def get_domain(cls, items_sympy, **kwargs): K, items_K = construct_domain(items_sympy, **kwargs) return K, items_K def convert_to(self, K): Kold = self.domain new_rows = [[K.convert_from(e, Kold) for e in row] for row in self.rep] return DomainMatrix(new_rows, self.shape, K) def to_field(self): K = self.domain.get_field() return self.convert_to(K) def unify(self, other): K1 = self.domain K2 = other.domain if K1 == K2: return self, other K = K1.unify(K2) if K1 != K: self = self.convert_to(K) if K2 != K: other = other.convert_to(K) return self, other def to_Matrix(self): from sympy.matrices.dense import MutableDenseMatrix rows_sympy = [[self.domain.to_sympy(e) for e in row] for row in self.rep] return MutableDenseMatrix(rows_sympy) def __repr__(self): rows_str = ['[%s]' % (', '.join(map(str, row))) for row in self.rep] rowstr = '[%s]' % ', '.join(rows_str) return 'DomainMatrix(%s, %r, %r)' % (rowstr, self.shape, self.domain) def __add__(A, B): if not isinstance(B, DomainMatrix): return NotImplemented return A.add(B) def __sub__(A, B): if not isinstance(B, DomainMatrix): return NotImplemented return A.sub(B) def __neg__(A): return A.neg() def __mul__(A, B): """A * B""" if not isinstance(B, DomainMatrix): return NotImplemented return A.matmul(B) def __pow__(A, n): """A ** n""" if not isinstance(n, int): return NotImplemented return A.pow(n) def add(A, B): if A.shape != B.shape: raise ShapeError("shape") if A.domain != B.domain: raise ValueError("domain") return A.from_ddm(A.rep.add(B.rep)) def sub(A, B): if A.shape != B.shape: raise ShapeError("shape") if A.domain != B.domain: raise ValueError("domain") return A.from_ddm(A.rep.sub(B.rep)) def neg(A): return A.from_ddm(A.rep.neg()) def matmul(A, B): return A.from_ddm(A.rep.matmul(B.rep)) def pow(A, n): if n < 0: raise NotImplementedError('Negative powers') elif n == 0: m, n = A.shape rows = [[A.domain.zero] * m for _ in range(m)] for i in range(m): rows[i][i] = A.domain.one return type(A)(rows, A.shape, A.domain) elif n == 1: return A elif n % 2 == 1: return A * A**(n - 1) else: sqrtAn = A ** (n // 2) return sqrtAn * sqrtAn def rref(self): if not self.domain.is_Field: raise ValueError('Not a field') rref_ddm, pivots = self.rep.rref() return self.from_ddm(rref_ddm), tuple(pivots) def inv(self): if not self.domain.is_Field: raise ValueError('Not a field') m, n = self.shape if m != n: raise NonSquareMatrixError inv = self.rep.inv() return self.from_ddm(inv) def det(self): m, n = self.shape if m != n: raise NonSquareMatrixError return self.rep.det() def lu(self): if not self.domain.is_Field: raise ValueError('Not a field') L, U, swaps = self.rep.lu() return self.from_ddm(L), self.from_ddm(U), swaps def lu_solve(self, rhs): if self.shape[0] != rhs.shape[0]: raise ShapeError("Shape") if not self.domain.is_Field: raise ValueError('Not a field') sol = self.rep.lu_solve(rhs.rep) return self.from_ddm(sol) def charpoly(self): m, n = self.shape if m != n: raise NonSquareMatrixError("not square") return self.rep.charpoly() def __eq__(A, B): """A == B""" if not isinstance(B, DomainMatrix): return NotImplemented return A.rep == B.rep
d07a82e76e01b4d0a2ae00bb2494b7fc64ed8f8625c9aec94056cedf3c308bc9
"""Arithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``. """ from sympy.polys.densebasic import ( dup_slice, dup_LC, dmp_LC, dup_degree, dmp_degree, dup_strip, dmp_strip, dmp_zero_p, dmp_zero, dmp_one_p, dmp_one, dmp_ground, dmp_zeros) from sympy.polys.polyerrors import (ExactQuotientFailed, PolynomialDivisionFailed) def dup_add_term(f, c, i, K): """ Add ``c*x**i`` to ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_term(x**2 - 1, ZZ(2), 4) 2*x**4 + x**2 - 1 """ if not c: return f n = len(f) m = n - i - 1 if i == n - 1: return dup_strip([f[0] + c] + f[1:]) else: if i >= n: return [c] + [K.zero]*(i - n) + f else: return f[:m] + [f[m] + c] + f[m + 1:] def dmp_add_term(f, c, i, u, K): """ Add ``c(x_2..x_u)*x_0**i`` to ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_term(x*y + 1, 2, 2) 2*x**2 + x*y + 1 """ if not u: return dup_add_term(f, c, i, K) v = u - 1 if dmp_zero_p(c, v): return f n = len(f) m = n - i - 1 if i == n - 1: return dmp_strip([dmp_add(f[0], c, v, K)] + f[1:], u) else: if i >= n: return [c] + dmp_zeros(i - n, v, K) + f else: return f[:m] + [dmp_add(f[m], c, v, K)] + f[m + 1:] def dup_sub_term(f, c, i, K): """ Subtract ``c*x**i`` from ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_term(2*x**4 + x**2 - 1, ZZ(2), 4) x**2 - 1 """ if not c: return f n = len(f) m = n - i - 1 if i == n - 1: return dup_strip([f[0] - c] + f[1:]) else: if i >= n: return [-c] + [K.zero]*(i - n) + f else: return f[:m] + [f[m] - c] + f[m + 1:] def dmp_sub_term(f, c, i, u, K): """ Subtract ``c(x_2..x_u)*x_0**i`` from ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_term(2*x**2 + x*y + 1, 2, 2) x*y + 1 """ if not u: return dup_add_term(f, -c, i, K) v = u - 1 if dmp_zero_p(c, v): return f n = len(f) m = n - i - 1 if i == n - 1: return dmp_strip([dmp_sub(f[0], c, v, K)] + f[1:], u) else: if i >= n: return [dmp_neg(c, v, K)] + dmp_zeros(i - n, v, K) + f else: return f[:m] + [dmp_sub(f[m], c, v, K)] + f[m + 1:] def dup_mul_term(f, c, i, K): """ Multiply ``f`` by ``c*x**i`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul_term(x**2 - 1, ZZ(3), 2) 3*x**4 - 3*x**2 """ if not c or not f: return [] else: return [ cf * c for cf in f ] + [K.zero]*i def dmp_mul_term(f, c, i, u, K): """ Multiply ``f`` by ``c(x_2..x_u)*x_0**i`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul_term(x**2*y + x, 3*y, 2) 3*x**4*y**2 + 3*x**3*y """ if not u: return dup_mul_term(f, c, i, K) v = u - 1 if dmp_zero_p(f, u): return f if dmp_zero_p(c, v): return dmp_zero(u) else: return [ dmp_mul(cf, c, v, K) for cf in f ] + dmp_zeros(i, v, K) def dup_add_ground(f, c, K): """ Add an element of the ground domain to ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x + 8 """ return dup_add_term(f, c, 0, K) def dmp_add_ground(f, c, u, K): """ Add an element of the ground domain to ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x + 8 """ return dmp_add_term(f, dmp_ground(c, u - 1), 0, u, K) def dup_sub_ground(f, c, K): """ Subtract an element of the ground domain from ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x """ return dup_sub_term(f, c, 0, K) def dmp_sub_ground(f, c, u, K): """ Subtract an element of the ground domain from ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x """ return dmp_sub_term(f, dmp_ground(c, u - 1), 0, u, K) def dup_mul_ground(f, c, K): """ Multiply ``f`` by a constant value in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul_ground(x**2 + 2*x - 1, ZZ(3)) 3*x**2 + 6*x - 3 """ if not c or not f: return [] else: return [ cf * c for cf in f ] def dmp_mul_ground(f, c, u, K): """ Multiply ``f`` by a constant value in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul_ground(2*x + 2*y, ZZ(3)) 6*x + 6*y """ if not u: return dup_mul_ground(f, c, K) v = u - 1 return [ dmp_mul_ground(cf, c, v, K) for cf in f ] def dup_quo_ground(f, c, K): """ Quotient by a constant in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_quo_ground(3*x**2 + 2, ZZ(2)) x**2 + 1 >>> R, x = ring("x", QQ) >>> R.dup_quo_ground(3*x**2 + 2, QQ(2)) 3/2*x**2 + 1 """ if not c: raise ZeroDivisionError('polynomial division') if not f: return f if K.is_Field: return [ K.quo(cf, c) for cf in f ] else: return [ cf // c for cf in f ] def dmp_quo_ground(f, c, u, K): """ Quotient by a constant in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_quo_ground(2*x**2*y + 3*x, ZZ(2)) x**2*y + x >>> R, x,y = ring("x,y", QQ) >>> R.dmp_quo_ground(2*x**2*y + 3*x, QQ(2)) x**2*y + 3/2*x """ if not u: return dup_quo_ground(f, c, K) v = u - 1 return [ dmp_quo_ground(cf, c, v, K) for cf in f ] def dup_exquo_ground(f, c, K): """ Exact quotient by a constant in ``K[x]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_exquo_ground(x**2 + 2, QQ(2)) 1/2*x**2 + 1 """ if not c: raise ZeroDivisionError('polynomial division') if not f: return f return [ K.exquo(cf, c) for cf in f ] def dmp_exquo_ground(f, c, u, K): """ Exact quotient by a constant in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_exquo_ground(x**2*y + 2*x, QQ(2)) 1/2*x**2*y + x """ if not u: return dup_exquo_ground(f, c, K) v = u - 1 return [ dmp_exquo_ground(cf, c, v, K) for cf in f ] def dup_lshift(f, n, K): """ Efficiently multiply ``f`` by ``x**n`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_lshift(x**2 + 1, 2) x**4 + x**2 """ if not f: return f else: return f + [K.zero]*n def dup_rshift(f, n, K): """ Efficiently divide ``f`` by ``x**n`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rshift(x**4 + x**2, 2) x**2 + 1 >>> R.dup_rshift(x**4 + x**2 + 2, 2) x**2 + 1 """ return f[:-n] def dup_abs(f, K): """ Make all coefficients positive in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_abs(x**2 - 1) x**2 + 1 """ return [ K.abs(coeff) for coeff in f ] def dmp_abs(f, u, K): """ Make all coefficients positive in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_abs(x**2*y - x) x**2*y + x """ if not u: return dup_abs(f, K) v = u - 1 return [ dmp_abs(cf, v, K) for cf in f ] def dup_neg(f, K): """ Negate a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_neg(x**2 - 1) -x**2 + 1 """ return [ -coeff for coeff in f ] def dmp_neg(f, u, K): """ Negate a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_neg(x**2*y - x) -x**2*y + x """ if not u: return dup_neg(f, K) v = u - 1 return [ dmp_neg(cf, v, K) for cf in f ] def dup_add(f, g, K): """ Add dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add(x**2 - 1, x - 2) x**2 + x - 3 """ if not f: return g if not g: return f df = dup_degree(f) dg = dup_degree(g) if df == dg: return dup_strip([ a + b for a, b in zip(f, g) ]) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = g[:k], g[k:] return h + [ a + b for a, b in zip(f, g) ] def dmp_add(f, g, u, K): """ Add dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add(x**2 + y, x**2*y + x) x**2*y + x**2 + x + y """ if not u: return dup_add(f, g, K) df = dmp_degree(f, u) if df < 0: return g dg = dmp_degree(g, u) if dg < 0: return f v = u - 1 if df == dg: return dmp_strip([ dmp_add(a, b, v, K) for a, b in zip(f, g) ], u) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = g[:k], g[k:] return h + [ dmp_add(a, b, v, K) for a, b in zip(f, g) ] def dup_sub(f, g, K): """ Subtract dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub(x**2 - 1, x - 2) x**2 - x + 1 """ if not f: return dup_neg(g, K) if not g: return f df = dup_degree(f) dg = dup_degree(g) if df == dg: return dup_strip([ a - b for a, b in zip(f, g) ]) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = dup_neg(g[:k], K), g[k:] return h + [ a - b for a, b in zip(f, g) ] def dmp_sub(f, g, u, K): """ Subtract dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub(x**2 + y, x**2*y + x) -x**2*y + x**2 - x + y """ if not u: return dup_sub(f, g, K) df = dmp_degree(f, u) if df < 0: return dmp_neg(g, u, K) dg = dmp_degree(g, u) if dg < 0: return f v = u - 1 if df == dg: return dmp_strip([ dmp_sub(a, b, v, K) for a, b in zip(f, g) ], u) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = dmp_neg(g[:k], u, K), g[k:] return h + [ dmp_sub(a, b, v, K) for a, b in zip(f, g) ] def dup_add_mul(f, g, h, K): """ Returns ``f + g*h`` where ``f, g, h`` are in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_mul(x**2 - 1, x - 2, x + 2) 2*x**2 - 5 """ return dup_add(f, dup_mul(g, h, K), K) def dmp_add_mul(f, g, h, u, K): """ Returns ``f + g*h`` where ``f, g, h`` are in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_mul(x**2 + y, x, x + 2) 2*x**2 + 2*x + y """ return dmp_add(f, dmp_mul(g, h, u, K), u, K) def dup_sub_mul(f, g, h, K): """ Returns ``f - g*h`` where ``f, g, h`` are in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_mul(x**2 - 1, x - 2, x + 2) 3 """ return dup_sub(f, dup_mul(g, h, K), K) def dmp_sub_mul(f, g, h, u, K): """ Returns ``f - g*h`` where ``f, g, h`` are in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_mul(x**2 + y, x, x + 2) -2*x + y """ return dmp_sub(f, dmp_mul(g, h, u, K), u, K) def dup_mul(f, g, K): """ Multiply dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul(x - 2, x + 2) x**2 - 4 """ if f == g: return dup_sqr(f, K) if not (f and g): return [] df = dup_degree(f) dg = dup_degree(g) n = max(df, dg) + 1 if n < 100: h = [] for i in range(0, df + dg + 1): coeff = K.zero for j in range(max(0, i - dg), min(df, i) + 1): coeff += f[j]*g[i - j] h.append(coeff) return dup_strip(h) else: # Use Karatsuba's algorithm (divide and conquer), see e.g.: # Joris van der Hoeven, Relax But Don't Be Too Lazy, # J. Symbolic Computation, 11 (2002), section 3.1.1. n2 = n//2 fl, gl = dup_slice(f, 0, n2, K), dup_slice(g, 0, n2, K) fh = dup_rshift(dup_slice(f, n2, n, K), n2, K) gh = dup_rshift(dup_slice(g, n2, n, K), n2, K) lo, hi = dup_mul(fl, gl, K), dup_mul(fh, gh, K) mid = dup_mul(dup_add(fl, fh, K), dup_add(gl, gh, K), K) mid = dup_sub(mid, dup_add(lo, hi, K), K) return dup_add(dup_add(lo, dup_lshift(mid, n2, K), K), dup_lshift(hi, 2*n2, K), K) def dmp_mul(f, g, u, K): """ Multiply dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul(x*y + 1, x) x**2*y + x """ if not u: return dup_mul(f, g, K) if f == g: return dmp_sqr(f, u, K) df = dmp_degree(f, u) if df < 0: return f dg = dmp_degree(g, u) if dg < 0: return g h, v = [], u - 1 for i in range(0, df + dg + 1): coeff = dmp_zero(v) for j in range(max(0, i - dg), min(df, i) + 1): coeff = dmp_add(coeff, dmp_mul(f[j], g[i - j], v, K), v, K) h.append(coeff) return dmp_strip(h, u) def dup_sqr(f, K): """ Square dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqr(x**2 + 1) x**4 + 2*x**2 + 1 """ df, h = len(f) - 1, [] for i in range(0, 2*df + 1): c = K.zero jmin = max(0, i - df) jmax = min(i, df) n = jmax - jmin + 1 jmax = jmin + n // 2 - 1 for j in range(jmin, jmax + 1): c += f[j]*f[i - j] c += c if n & 1: elem = f[jmax + 1] c += elem**2 h.append(c) return dup_strip(h) def dmp_sqr(f, u, K): """ Square dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqr(x**2 + x*y + y**2) x**4 + 2*x**3*y + 3*x**2*y**2 + 2*x*y**3 + y**4 """ if not u: return dup_sqr(f, K) df = dmp_degree(f, u) if df < 0: return f h, v = [], u - 1 for i in range(0, 2*df + 1): c = dmp_zero(v) jmin = max(0, i - df) jmax = min(i, df) n = jmax - jmin + 1 jmax = jmin + n // 2 - 1 for j in range(jmin, jmax + 1): c = dmp_add(c, dmp_mul(f[j], f[i - j], v, K), v, K) c = dmp_mul_ground(c, K(2), v, K) if n & 1: elem = dmp_sqr(f[jmax + 1], v, K) c = dmp_add(c, elem, v, K) h.append(c) return dmp_strip(h, u) def dup_pow(f, n, K): """ Raise ``f`` to the ``n``-th power in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pow(x - 2, 3) x**3 - 6*x**2 + 12*x - 8 """ if not n: return [K.one] if n < 0: raise ValueError("can't raise polynomial to a negative power") if n == 1 or not f or f == [K.one]: return f g = [K.one] while True: n, m = n//2, n if m % 2: g = dup_mul(g, f, K) if not n: break f = dup_sqr(f, K) return g def dmp_pow(f, n, u, K): """ Raise ``f`` to the ``n``-th power in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_pow(x*y + 1, 3) x**3*y**3 + 3*x**2*y**2 + 3*x*y + 1 """ if not u: return dup_pow(f, n, K) if not n: return dmp_one(u, K) if n < 0: raise ValueError("can't raise polynomial to a negative power") if n == 1 or dmp_zero_p(f, u) or dmp_one_p(f, u, K): return f g = dmp_one(u, K) while True: n, m = n//2, n if m & 1: g = dmp_mul(g, f, u, K) if not n: break f = dmp_sqr(f, u, K) return g def dup_pdiv(f, g, K): """ Polynomial pseudo-division in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pdiv(x**2 + 1, 2*x - 4) (2*x + 4, 20) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r N = df - dg + 1 lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) j, N = dr - dg, N - 1 Q = dup_mul_ground(q, lc_g, K) q = dup_add_term(Q, lc_r, j, K) R = dup_mul_ground(r, lc_g, K) G = dup_mul_term(g, lc_r, j, K) r = dup_sub(R, G, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) c = lc_g**N q = dup_mul_ground(q, c, K) r = dup_mul_ground(r, c, K) return q, r def dup_prem(f, g, K): """ Polynomial pseudo-remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_prem(x**2 + 1, 2*x - 4) 20 """ df = dup_degree(f) dg = dup_degree(g) r, dr = f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return r N = df - dg + 1 lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) j, N = dr - dg, N - 1 R = dup_mul_ground(r, lc_g, K) G = dup_mul_term(g, lc_r, j, K) r = dup_sub(R, G, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return dup_mul_ground(r, lc_g**N, K) def dup_pquo(f, g, K): """ Polynomial exact pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pquo(x**2 - 1, 2*x - 2) 2*x + 2 >>> R.dup_pquo(x**2 + 1, 2*x - 4) 2*x + 4 """ return dup_pdiv(f, g, K)[0] def dup_pexquo(f, g, K): """ Polynomial pseudo-quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pexquo(x**2 - 1, 2*x - 2) 2*x + 2 >>> R.dup_pexquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: [2, -4] does not divide [1, 0, 1] """ q, r = dup_pdiv(f, g, K) if not r: return q else: raise ExactQuotientFailed(f, g) def dmp_pdiv(f, g, u, K): """ Polynomial pseudo-division in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_pdiv(x**2 + x*y, 2*x + 2) (2*x + 2*y - 2, -4*y + 4) """ if not u: return dup_pdiv(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") q, r, dr = dmp_zero(u), f, df if df < dg: return q, r N = df - dg + 1 lc_g = dmp_LC(g, K) while True: lc_r = dmp_LC(r, K) j, N = dr - dg, N - 1 Q = dmp_mul_term(q, lc_g, 0, u, K) q = dmp_add_term(Q, lc_r, j, u, K) R = dmp_mul_term(r, lc_g, 0, u, K) G = dmp_mul_term(g, lc_r, j, u, K) r = dmp_sub(R, G, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) c = dmp_pow(lc_g, N, u - 1, K) q = dmp_mul_term(q, c, 0, u, K) r = dmp_mul_term(r, c, 0, u, K) return q, r def dmp_prem(f, g, u, K): """ Polynomial pseudo-remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_prem(x**2 + x*y, 2*x + 2) -4*y + 4 """ if not u: return dup_prem(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") r, dr = f, df if df < dg: return r N = df - dg + 1 lc_g = dmp_LC(g, K) while True: lc_r = dmp_LC(r, K) j, N = dr - dg, N - 1 R = dmp_mul_term(r, lc_g, 0, u, K) G = dmp_mul_term(g, lc_r, j, u, K) r = dmp_sub(R, G, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) c = dmp_pow(lc_g, N, u - 1, K) return dmp_mul_term(r, c, 0, u, K) def dmp_pquo(f, g, u, K): """ Polynomial exact pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> R.dmp_pquo(f, g) 2*x >>> R.dmp_pquo(f, h) 2*x + 2*y - 2 """ return dmp_pdiv(f, g, u, K)[0] def dmp_pexquo(f, g, u, K): """ Polynomial pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> R.dmp_pexquo(f, g) 2*x >>> R.dmp_pexquo(f, h) Traceback (most recent call last): ... ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []] """ q, r = dmp_pdiv(f, g, u, K) if dmp_zero_p(r, u): return q else: raise ExactQuotientFailed(f, g) def dup_rr_div(f, g, K): """ Univariate division with remainder over a ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rr_div(x**2 + 1, 2*x - 4) (0, x**2 + 1) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) if lc_r % lc_g: break c = K.exquo(lc_r, lc_g) j = dr - dg q = dup_add_term(q, c, j, K) h = dup_mul_term(g, c, j, K) r = dup_sub(r, h, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r def dmp_rr_div(f, g, u, K): """ Multivariate division with remainder over a ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_rr_div(x**2 + x*y, 2*x + 2) (0, x**2 + x*y) """ if not u: return dup_rr_div(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") q, r, dr = dmp_zero(u), f, df if df < dg: return q, r lc_g, v = dmp_LC(g, K), u - 1 while True: lc_r = dmp_LC(r, K) c, R = dmp_rr_div(lc_r, lc_g, v, K) if not dmp_zero_p(R, v): break j = dr - dg q = dmp_add_term(q, c, j, u, K) h = dmp_mul_term(g, c, j, u, K) r = dmp_sub(r, h, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r def dup_ff_div(f, g, K): """ Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_ff_div(x**2 + 1, 2*x - 4) (1/2*x + 1, 5) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) c = K.exquo(lc_r, lc_g) j = dr - dg q = dup_add_term(q, c, j, K) h = dup_mul_term(g, c, j, K) r = dup_sub(r, h, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif dr == _dr and not K.is_Exact: # remove leading term created by rounding error r = dup_strip(r[1:]) dr = dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r def dmp_ff_div(f, g, u, K): """ Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_ff_div(x**2 + x*y, 2*x + 2) (1/2*x + 1/2*y - 1/2, -y + 1) """ if not u: return dup_ff_div(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") q, r, dr = dmp_zero(u), f, df if df < dg: return q, r lc_g, v = dmp_LC(g, K), u - 1 while True: lc_r = dmp_LC(r, K) c, R = dmp_ff_div(lc_r, lc_g, v, K) if not dmp_zero_p(R, v): break j = dr - dg q = dmp_add_term(q, c, j, u, K) h = dmp_mul_term(g, c, j, u, K) r = dmp_sub(r, h, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r def dup_div(f, g, K): """ Polynomial division with remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_div(x**2 + 1, 2*x - 4) (0, x**2 + 1) >>> R, x = ring("x", QQ) >>> R.dup_div(x**2 + 1, 2*x - 4) (1/2*x + 1, 5) """ if K.is_Field: return dup_ff_div(f, g, K) else: return dup_rr_div(f, g, K) def dup_rem(f, g, K): """ Returns polynomial remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_rem(x**2 + 1, 2*x - 4) x**2 + 1 >>> R, x = ring("x", QQ) >>> R.dup_rem(x**2 + 1, 2*x - 4) 5 """ return dup_div(f, g, K)[1] def dup_quo(f, g, K): """ Returns exact polynomial quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_quo(x**2 + 1, 2*x - 4) 0 >>> R, x = ring("x", QQ) >>> R.dup_quo(x**2 + 1, 2*x - 4) 1/2*x + 1 """ return dup_div(f, g, K)[0] def dup_exquo(f, g, K): """ Returns polynomial quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_exquo(x**2 - 1, x - 1) x + 1 >>> R.dup_exquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: [2, -4] does not divide [1, 0, 1] """ q, r = dup_div(f, g, K) if not r: return q else: raise ExactQuotientFailed(f, g) def dmp_div(f, g, u, K): """ Polynomial division with remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_div(x**2 + x*y, 2*x + 2) (0, x**2 + x*y) >>> R, x,y = ring("x,y", QQ) >>> R.dmp_div(x**2 + x*y, 2*x + 2) (1/2*x + 1/2*y - 1/2, -y + 1) """ if K.is_Field: return dmp_ff_div(f, g, u, K) else: return dmp_rr_div(f, g, u, K) def dmp_rem(f, g, u, K): """ Returns polynomial remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_rem(x**2 + x*y, 2*x + 2) x**2 + x*y >>> R, x,y = ring("x,y", QQ) >>> R.dmp_rem(x**2 + x*y, 2*x + 2) -y + 1 """ return dmp_div(f, g, u, K)[1] def dmp_quo(f, g, u, K): """ Returns exact polynomial quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_quo(x**2 + x*y, 2*x + 2) 0 >>> R, x,y = ring("x,y", QQ) >>> R.dmp_quo(x**2 + x*y, 2*x + 2) 1/2*x + 1/2*y - 1/2 """ return dmp_div(f, g, u, K)[0] def dmp_exquo(f, g, u, K): """ Returns polynomial quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = x + y >>> h = 2*x + 2 >>> R.dmp_exquo(f, g) x >>> R.dmp_exquo(f, h) Traceback (most recent call last): ... ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []] """ q, r = dmp_div(f, g, u, K) if dmp_zero_p(r, u): return q else: raise ExactQuotientFailed(f, g) def dup_max_norm(f, K): """ Returns maximum norm of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_max_norm(-x**2 + 2*x - 3) 3 """ if not f: return K.zero else: return max(dup_abs(f, K)) def dmp_max_norm(f, u, K): """ Returns maximum norm of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_max_norm(2*x*y - x - 3) 3 """ if not u: return dup_max_norm(f, K) v = u - 1 return max([ dmp_max_norm(c, v, K) for c in f ]) def dup_l1_norm(f, K): """ Returns l1 norm of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_l1_norm(2*x**3 - 3*x**2 + 1) 6 """ if not f: return K.zero else: return sum(dup_abs(f, K)) def dmp_l1_norm(f, u, K): """ Returns l1 norm of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_l1_norm(2*x*y - x - 3) 6 """ if not u: return dup_l1_norm(f, K) v = u - 1 return sum([ dmp_l1_norm(c, v, K) for c in f ]) def dup_expand(polys, K): """ Multiply together several polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_expand([x**2 - 1, x, 2]) 2*x**3 - 2*x """ if not polys: return [K.one] f = polys[0] for g in polys[1:]: f = dup_mul(f, g, K) return f def dmp_expand(polys, u, K): """ Multiply together several polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_expand([x**2 + y**2, x + 1]) x**3 + x**2 + x*y**2 + y**2 """ if not polys: return dmp_one(u, K) f = polys[0] for g in polys[1:]: f = dmp_mul(f, g, u, K) return f
5e0f9804cbfeaa8664c5d036d7c8d7a09df40a3ea8accba947257d8251dfd9b3
"""Dense univariate polynomials with coefficients in Galois fields. """ from random import uniform from math import ceil as _ceil, sqrt as _sqrt from sympy.core.compatibility import SYMPY_INTS from sympy.core.mul import prod from sympy.ntheory import factorint from sympy.polys.polyconfig import query from sympy.polys.polyerrors import ExactQuotientFailed from sympy.polys.polyutils import _sort_factors def gf_crt(U, M, K=None): """ Chinese Remainder Theorem. Given a set of integer residues ``u_0,...,u_n`` and a set of co-prime integer moduli ``m_0,...,m_n``, returns an integer ``u``, such that ``u = u_i mod m_i`` for ``i = ``0,...,n``. Examples ======== Consider a set of residues ``U = [49, 76, 65]`` and a set of moduli ``M = [99, 97, 95]``. Then we have:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_crt >>> gf_crt([49, 76, 65], [99, 97, 95], ZZ) 639985 This is the correct result because:: >>> [639985 % m for m in [99, 97, 95]] [49, 76, 65] Note: this is a low-level routine with no error checking. See Also ======== sympy.ntheory.modular.crt : a higher level crt routine sympy.ntheory.modular.solve_congruence """ p = prod(M, start=K.one) v = K.zero for u, m in zip(U, M): e = p // m s, _, _ = K.gcdex(e, m) v += e*(u*s % m) return v % p def gf_crt1(M, K): """ First part of the Chinese Remainder Theorem. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_crt1 >>> gf_crt1([99, 97, 95], ZZ) (912285, [9215, 9405, 9603], [62, 24, 12]) """ E, S = [], [] p = prod(M, start=K.one) for m in M: E.append(p // m) S.append(K.gcdex(E[-1], m)[0] % m) return p, E, S def gf_crt2(U, M, p, E, S, K): """ Second part of the Chinese Remainder Theorem. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_crt2 >>> U = [49, 76, 65] >>> M = [99, 97, 95] >>> p = 912285 >>> E = [9215, 9405, 9603] >>> S = [62, 24, 12] >>> gf_crt2(U, M, p, E, S, ZZ) 639985 """ v = K.zero for u, m, e, s in zip(U, M, E, S): v += e*(u*s % m) return v % p def gf_int(a, p): """ Coerce ``a mod p`` to an integer in the range ``[-p/2, p/2]``. Examples ======== >>> from sympy.polys.galoistools import gf_int >>> gf_int(2, 7) 2 >>> gf_int(5, 7) -2 """ if a <= p // 2: return a else: return a - p def gf_degree(f): """ Return the leading degree of ``f``. Examples ======== >>> from sympy.polys.galoistools import gf_degree >>> gf_degree([1, 1, 2, 0]) 3 >>> gf_degree([]) -1 """ return len(f) - 1 def gf_LC(f, K): """ Return the leading coefficient of ``f``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_LC >>> gf_LC([3, 0, 1], ZZ) 3 """ if not f: return K.zero else: return f[0] def gf_TC(f, K): """ Return the trailing coefficient of ``f``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_TC >>> gf_TC([3, 0, 1], ZZ) 1 """ if not f: return K.zero else: return f[-1] def gf_strip(f): """ Remove leading zeros from ``f``. Examples ======== >>> from sympy.polys.galoistools import gf_strip >>> gf_strip([0, 0, 0, 3, 0, 1]) [3, 0, 1] """ if not f or f[0]: return f k = 0 for coeff in f: if coeff: break else: k += 1 return f[k:] def gf_trunc(f, p): """ Reduce all coefficients modulo ``p``. Examples ======== >>> from sympy.polys.galoistools import gf_trunc >>> gf_trunc([7, -2, 3], 5) [2, 3, 3] """ return gf_strip([ a % p for a in f ]) def gf_normal(f, p, K): """ Normalize all coefficients in ``K``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_normal >>> gf_normal([5, 10, 21, -3], 5, ZZ) [1, 2] """ return gf_trunc(list(map(K, f)), p) def gf_from_dict(f, p, K): """ Create a ``GF(p)[x]`` polynomial from a dict. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_from_dict >>> gf_from_dict({10: ZZ(4), 4: ZZ(33), 0: ZZ(-1)}, 5, ZZ) [4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4] """ n, h = max(f.keys()), [] if isinstance(n, SYMPY_INTS): for k in range(n, -1, -1): h.append(f.get(k, K.zero) % p) else: (n,) = n for k in range(n, -1, -1): h.append(f.get((k,), K.zero) % p) return gf_trunc(h, p) def gf_to_dict(f, p, symmetric=True): """ Convert a ``GF(p)[x]`` polynomial to a dict. Examples ======== >>> from sympy.polys.galoistools import gf_to_dict >>> gf_to_dict([4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4], 5) {0: -1, 4: -2, 10: -1} >>> gf_to_dict([4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4], 5, symmetric=False) {0: 4, 4: 3, 10: 4} """ n, result = gf_degree(f), {} for k in range(0, n + 1): if symmetric: a = gf_int(f[n - k], p) else: a = f[n - k] if a: result[k] = a return result def gf_from_int_poly(f, p): """ Create a ``GF(p)[x]`` polynomial from ``Z[x]``. Examples ======== >>> from sympy.polys.galoistools import gf_from_int_poly >>> gf_from_int_poly([7, -2, 3], 5) [2, 3, 3] """ return gf_trunc(f, p) def gf_to_int_poly(f, p, symmetric=True): """ Convert a ``GF(p)[x]`` polynomial to ``Z[x]``. Examples ======== >>> from sympy.polys.galoistools import gf_to_int_poly >>> gf_to_int_poly([2, 3, 3], 5) [2, -2, -2] >>> gf_to_int_poly([2, 3, 3], 5, symmetric=False) [2, 3, 3] """ if symmetric: return [ gf_int(c, p) for c in f ] else: return f def gf_neg(f, p, K): """ Negate a polynomial in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_neg >>> gf_neg([3, 2, 1, 0], 5, ZZ) [2, 3, 4, 0] """ return [ -coeff % p for coeff in f ] def gf_add_ground(f, a, p, K): """ Compute ``f + a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_add_ground >>> gf_add_ground([3, 2, 4], 2, 5, ZZ) [3, 2, 1] """ if not f: a = a % p else: a = (f[-1] + a) % p if len(f) > 1: return f[:-1] + [a] if not a: return [] else: return [a] def gf_sub_ground(f, a, p, K): """ Compute ``f - a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sub_ground >>> gf_sub_ground([3, 2, 4], 2, 5, ZZ) [3, 2, 2] """ if not f: a = -a % p else: a = (f[-1] - a) % p if len(f) > 1: return f[:-1] + [a] if not a: return [] else: return [a] def gf_mul_ground(f, a, p, K): """ Compute ``f * a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_mul_ground >>> gf_mul_ground([3, 2, 4], 2, 5, ZZ) [1, 4, 3] """ if not a: return [] else: return [ (a*b) % p for b in f ] def gf_quo_ground(f, a, p, K): """ Compute ``f/a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_quo_ground >>> gf_quo_ground(ZZ.map([3, 2, 4]), ZZ(2), 5, ZZ) [4, 1, 2] """ return gf_mul_ground(f, K.invert(a, p), p, K) def gf_add(f, g, p, K): """ Add polynomials in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_add >>> gf_add([3, 2, 4], [2, 2, 2], 5, ZZ) [4, 1] """ if not f: return g if not g: return f df = gf_degree(f) dg = gf_degree(g) if df == dg: return gf_strip([ (a + b) % p for a, b in zip(f, g) ]) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = g[:k], g[k:] return h + [ (a + b) % p for a, b in zip(f, g) ] def gf_sub(f, g, p, K): """ Subtract polynomials in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sub >>> gf_sub([3, 2, 4], [2, 2, 2], 5, ZZ) [1, 0, 2] """ if not g: return f if not f: return gf_neg(g, p, K) df = gf_degree(f) dg = gf_degree(g) if df == dg: return gf_strip([ (a - b) % p for a, b in zip(f, g) ]) else: k = abs(df - dg) if df > dg: h, f = f[:k], f[k:] else: h, g = gf_neg(g[:k], p, K), g[k:] return h + [ (a - b) % p for a, b in zip(f, g) ] def gf_mul(f, g, p, K): """ Multiply polynomials in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_mul >>> gf_mul([3, 2, 4], [2, 2, 2], 5, ZZ) [1, 0, 3, 2, 3] """ df = gf_degree(f) dg = gf_degree(g) dh = df + dg h = [0]*(dh + 1) for i in range(0, dh + 1): coeff = K.zero for j in range(max(0, i - dg), min(i, df) + 1): coeff += f[j]*g[i - j] h[i] = coeff % p return gf_strip(h) def gf_sqr(f, p, K): """ Square polynomials in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sqr >>> gf_sqr([3, 2, 4], 5, ZZ) [4, 2, 3, 1, 1] """ df = gf_degree(f) dh = 2*df h = [0]*(dh + 1) for i in range(0, dh + 1): coeff = K.zero jmin = max(0, i - df) jmax = min(i, df) n = jmax - jmin + 1 jmax = jmin + n // 2 - 1 for j in range(jmin, jmax + 1): coeff += f[j]*f[i - j] coeff += coeff if n & 1: elem = f[jmax + 1] coeff += elem**2 h[i] = coeff % p return gf_strip(h) def gf_add_mul(f, g, h, p, K): """ Returns ``f + g*h`` where ``f``, ``g``, ``h`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_add_mul >>> gf_add_mul([3, 2, 4], [2, 2, 2], [1, 4], 5, ZZ) [2, 3, 2, 2] """ return gf_add(f, gf_mul(g, h, p, K), p, K) def gf_sub_mul(f, g, h, p, K): """ Compute ``f - g*h`` where ``f``, ``g``, ``h`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sub_mul >>> gf_sub_mul([3, 2, 4], [2, 2, 2], [1, 4], 5, ZZ) [3, 3, 2, 1] """ return gf_sub(f, gf_mul(g, h, p, K), p, K) def gf_expand(F, p, K): """ Expand results of :func:`~.factor` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_expand >>> gf_expand([([3, 2, 4], 1), ([2, 2], 2), ([3, 1], 3)], 5, ZZ) [4, 3, 0, 3, 0, 1, 4, 1] """ if type(F) is tuple: lc, F = F else: lc = K.one g = [lc] for f, k in F: f = gf_pow(f, k, p, K) g = gf_mul(g, f, p, K) return g def gf_div(f, g, p, K): """ Division with remainder in ``GF(p)[x]``. Given univariate polynomials ``f`` and ``g`` with coefficients in a finite field with ``p`` elements, returns polynomials ``q`` and ``r`` (quotient and remainder) such that ``f = q*g + r``. Consider polynomials ``x**3 + x + 1`` and ``x**2 + x`` in GF(2):: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_div, gf_add_mul >>> gf_div(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ) ([1, 1], [1]) As result we obtained quotient ``x + 1`` and remainder ``1``, thus:: >>> gf_add_mul(ZZ.map([1]), ZZ.map([1, 1]), ZZ.map([1, 1, 0]), 2, ZZ) [1, 0, 1, 1] References ========== .. [1] [Monagan93]_ .. [2] [Gathen99]_ """ df = gf_degree(f) dg = gf_degree(g) if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return [], f inv = K.invert(g[0], p) h, dq, dr = list(f), df - dg, dg - 1 for i in range(0, df + 1): coeff = h[i] for j in range(max(0, dg - i), min(df - i, dr) + 1): coeff -= h[i + j - dg] * g[dg - j] if i <= dq: coeff *= inv h[i] = coeff % p return h[:dq + 1], gf_strip(h[dq + 1:]) def gf_rem(f, g, p, K): """ Compute polynomial remainder in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_rem >>> gf_rem(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ) [1] """ return gf_div(f, g, p, K)[1] def gf_quo(f, g, p, K): """ Compute exact quotient in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_quo >>> gf_quo(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ) [1, 1] >>> gf_quo(ZZ.map([1, 0, 3, 2, 3]), ZZ.map([2, 2, 2]), 5, ZZ) [3, 2, 4] """ df = gf_degree(f) dg = gf_degree(g) if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return [] inv = K.invert(g[0], p) h, dq, dr = f[:], df - dg, dg - 1 for i in range(0, dq + 1): coeff = h[i] for j in range(max(0, dg - i), min(df - i, dr) + 1): coeff -= h[i + j - dg] * g[dg - j] h[i] = (coeff * inv) % p return h[:dq + 1] def gf_exquo(f, g, p, K): """ Compute polynomial quotient in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_exquo >>> gf_exquo(ZZ.map([1, 0, 3, 2, 3]), ZZ.map([2, 2, 2]), 5, ZZ) [3, 2, 4] >>> gf_exquo(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ) Traceback (most recent call last): ... ExactQuotientFailed: [1, 1, 0] does not divide [1, 0, 1, 1] """ q, r = gf_div(f, g, p, K) if not r: return q else: raise ExactQuotientFailed(f, g) def gf_lshift(f, n, K): """ Efficiently multiply ``f`` by ``x**n``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_lshift >>> gf_lshift([3, 2, 4], 4, ZZ) [3, 2, 4, 0, 0, 0, 0] """ if not f: return f else: return f + [K.zero]*n def gf_rshift(f, n, K): """ Efficiently divide ``f`` by ``x**n``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_rshift >>> gf_rshift([1, 2, 3, 4, 0], 3, ZZ) ([1, 2], [3, 4, 0]) """ if not n: return f, [] else: return f[:-n], f[-n:] def gf_pow(f, n, p, K): """ Compute ``f**n`` in ``GF(p)[x]`` using repeated squaring. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_pow >>> gf_pow([3, 2, 4], 3, 5, ZZ) [2, 4, 4, 2, 2, 1, 4] """ if not n: return [K.one] elif n == 1: return f elif n == 2: return gf_sqr(f, p, K) h = [K.one] while True: if n & 1: h = gf_mul(h, f, p, K) n -= 1 n >>= 1 if not n: break f = gf_sqr(f, p, K) return h def gf_frobenius_monomial_base(g, p, K): """ return the list of ``x**(i*p) mod g in Z_p`` for ``i = 0, .., n - 1`` where ``n = gf_degree(g)`` Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_frobenius_monomial_base >>> g = ZZ.map([1, 0, 2, 1]) >>> gf_frobenius_monomial_base(g, 5, ZZ) [[1], [4, 4, 2], [1, 2]] """ n = gf_degree(g) if n == 0: return [] b = [0]*n b[0] = [1] if p < n: for i in range(1, n): mon = gf_lshift(b[i - 1], p, K) b[i] = gf_rem(mon, g, p, K) elif n > 1: b[1] = gf_pow_mod([K.one, K.zero], p, g, p, K) for i in range(2, n): b[i] = gf_mul(b[i - 1], b[1], p, K) b[i] = gf_rem(b[i], g, p, K) return b def gf_frobenius_map(f, g, b, p, K): """ compute gf_pow_mod(f, p, g, p, K) using the Frobenius map Parameters ========== f, g : polynomials in ``GF(p)[x]`` b : frobenius monomial base p : prime number K : domain Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_frobenius_monomial_base, gf_frobenius_map >>> f = ZZ.map([2, 1 , 0, 1]) >>> g = ZZ.map([1, 0, 2, 1]) >>> p = 5 >>> b = gf_frobenius_monomial_base(g, p, ZZ) >>> r = gf_frobenius_map(f, g, b, p, ZZ) >>> gf_frobenius_map(f, g, b, p, ZZ) [4, 0, 3] """ m = gf_degree(g) if gf_degree(f) >= m: f = gf_rem(f, g, p, K) if not f: return [] n = gf_degree(f) sf = [f[-1]] for i in range(1, n + 1): v = gf_mul_ground(b[i], f[n - i], p, K) sf = gf_add(sf, v, p, K) return sf def _gf_pow_pnm1d2(f, n, g, b, p, K): """ utility function for ``gf_edf_zassenhaus`` Compute ``f**((p**n - 1) // 2)`` in ``GF(p)[x]/(g)`` ``f**((p**n - 1) // 2) = (f*f**p*...*f**(p**n - 1))**((p - 1) // 2)`` """ f = gf_rem(f, g, p, K) h = f r = f for i in range(1, n): h = gf_frobenius_map(h, g, b, p, K) r = gf_mul(r, h, p, K) r = gf_rem(r, g, p, K) res = gf_pow_mod(r, (p - 1)//2, g, p, K) return res def gf_pow_mod(f, n, g, p, K): """ Compute ``f**n`` in ``GF(p)[x]/(g)`` using repeated squaring. Given polynomials ``f`` and ``g`` in ``GF(p)[x]`` and a non-negative integer ``n``, efficiently computes ``f**n (mod g)`` i.e. the remainder of ``f**n`` from division by ``g``, using the repeated squaring algorithm. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_pow_mod >>> gf_pow_mod(ZZ.map([3, 2, 4]), 3, ZZ.map([1, 1]), 5, ZZ) [] References ========== .. [1] [Gathen99]_ """ if not n: return [K.one] elif n == 1: return gf_rem(f, g, p, K) elif n == 2: return gf_rem(gf_sqr(f, p, K), g, p, K) h = [K.one] while True: if n & 1: h = gf_mul(h, f, p, K) h = gf_rem(h, g, p, K) n -= 1 n >>= 1 if not n: break f = gf_sqr(f, p, K) f = gf_rem(f, g, p, K) return h def gf_gcd(f, g, p, K): """ Euclidean Algorithm in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_gcd >>> gf_gcd(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ) [1, 3] """ while g: f, g = g, gf_rem(f, g, p, K) return gf_monic(f, p, K)[1] def gf_lcm(f, g, p, K): """ Compute polynomial LCM in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_lcm >>> gf_lcm(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ) [1, 2, 0, 4] """ if not f or not g: return [] h = gf_quo(gf_mul(f, g, p, K), gf_gcd(f, g, p, K), p, K) return gf_monic(h, p, K)[1] def gf_cofactors(f, g, p, K): """ Compute polynomial GCD and cofactors in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_cofactors >>> gf_cofactors(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ) ([1, 3], [3, 3], [2, 1]) """ if not f and not g: return ([], [], []) h = gf_gcd(f, g, p, K) return (h, gf_quo(f, h, p, K), gf_quo(g, h, p, K)) def gf_gcdex(f, g, p, K): """ Extended Euclidean Algorithm in ``GF(p)[x]``. Given polynomials ``f`` and ``g`` in ``GF(p)[x]``, computes polynomials ``s``, ``t`` and ``h``, such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. The typical application of EEA is solving polynomial diophantine equations. Consider polynomials ``f = (x + 7) (x + 1)``, ``g = (x + 7) (x**2 + 1)`` in ``GF(11)[x]``. Application of Extended Euclidean Algorithm gives:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_gcdex, gf_mul, gf_add >>> s, t, g = gf_gcdex(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) >>> s, t, g ([5, 6], [6], [1, 7]) As result we obtained polynomials ``s = 5*x + 6`` and ``t = 6``, and additionally ``gcd(f, g) = x + 7``. This is correct because:: >>> S = gf_mul(s, ZZ.map([1, 8, 7]), 11, ZZ) >>> T = gf_mul(t, ZZ.map([1, 7, 1, 7]), 11, ZZ) >>> gf_add(S, T, 11, ZZ) == [1, 7] True References ========== .. [1] [Gathen99]_ """ if not (f or g): return [K.one], [], [] p0, r0 = gf_monic(f, p, K) p1, r1 = gf_monic(g, p, K) if not f: return [], [K.invert(p1, p)], r1 if not g: return [K.invert(p0, p)], [], r0 s0, s1 = [K.invert(p0, p)], [] t0, t1 = [], [K.invert(p1, p)] while True: Q, R = gf_div(r0, r1, p, K) if not R: break (lc, r1), r0 = gf_monic(R, p, K), r1 inv = K.invert(lc, p) s = gf_sub_mul(s0, s1, Q, p, K) t = gf_sub_mul(t0, t1, Q, p, K) s1, s0 = gf_mul_ground(s, inv, p, K), s1 t1, t0 = gf_mul_ground(t, inv, p, K), t1 return s1, t1, r1 def gf_monic(f, p, K): """ Compute LC and a monic polynomial in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_monic >>> gf_monic(ZZ.map([3, 2, 4]), 5, ZZ) (3, [1, 4, 3]) """ if not f: return K.zero, [] else: lc = f[0] if K.is_one(lc): return lc, list(f) else: return lc, gf_quo_ground(f, lc, p, K) def gf_diff(f, p, K): """ Differentiate polynomial in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_diff >>> gf_diff([3, 2, 4], 5, ZZ) [1, 2] """ df = gf_degree(f) h, n = [K.zero]*df, df for coeff in f[:-1]: coeff *= K(n) coeff %= p if coeff: h[df - n] = coeff n -= 1 return gf_strip(h) def gf_eval(f, a, p, K): """ Evaluate ``f(a)`` in ``GF(p)`` using Horner scheme. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_eval >>> gf_eval([3, 2, 4], 2, 5, ZZ) 0 """ result = K.zero for c in f: result *= a result += c result %= p return result def gf_multi_eval(f, A, p, K): """ Evaluate ``f(a)`` for ``a`` in ``[a_1, ..., a_n]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_multi_eval >>> gf_multi_eval([3, 2, 4], [0, 1, 2, 3, 4], 5, ZZ) [4, 4, 0, 2, 0] """ return [ gf_eval(f, a, p, K) for a in A ] def gf_compose(f, g, p, K): """ Compute polynomial composition ``f(g)`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_compose >>> gf_compose([3, 2, 4], [2, 2, 2], 5, ZZ) [2, 4, 0, 3, 0] """ if len(g) <= 1: return gf_strip([gf_eval(f, gf_LC(g, K), p, K)]) if not f: return [] h = [f[0]] for c in f[1:]: h = gf_mul(h, g, p, K) h = gf_add_ground(h, c, p, K) return h def gf_compose_mod(g, h, f, p, K): """ Compute polynomial composition ``g(h)`` in ``GF(p)[x]/(f)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_compose_mod >>> gf_compose_mod(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 2]), ZZ.map([4, 3]), 5, ZZ) [4] """ if not g: return [] comp = [g[0]] for a in g[1:]: comp = gf_mul(comp, h, p, K) comp = gf_add_ground(comp, a, p, K) comp = gf_rem(comp, f, p, K) return comp def gf_trace_map(a, b, c, n, f, p, K): """ Compute polynomial trace map in ``GF(p)[x]/(f)``. Given a polynomial ``f`` in ``GF(p)[x]``, polynomials ``a``, ``b``, ``c`` in the quotient ring ``GF(p)[x]/(f)`` such that ``b = c**t (mod f)`` for some positive power ``t`` of ``p``, and a positive integer ``n``, returns a mapping:: a -> a**t**n, a + a**t + a**t**2 + ... + a**t**n (mod f) In factorization context, ``b = x**p mod f`` and ``c = x mod f``. This way we can efficiently compute trace polynomials in equal degree factorization routine, much faster than with other methods, like iterated Frobenius algorithm, for large degrees. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_trace_map >>> gf_trace_map([1, 2], [4, 4], [1, 1], 4, [3, 2, 4], 5, ZZ) ([1, 3], [1, 3]) References ========== .. [1] [Gathen92]_ """ u = gf_compose_mod(a, b, f, p, K) v = b if n & 1: U = gf_add(a, u, p, K) V = b else: U = a V = c n >>= 1 while n: u = gf_add(u, gf_compose_mod(u, v, f, p, K), p, K) v = gf_compose_mod(v, v, f, p, K) if n & 1: U = gf_add(U, gf_compose_mod(u, V, f, p, K), p, K) V = gf_compose_mod(v, V, f, p, K) n >>= 1 return gf_compose_mod(a, V, f, p, K), U def _gf_trace_map(f, n, g, b, p, K): """ utility for ``gf_edf_shoup`` """ f = gf_rem(f, g, p, K) h = f r = f for i in range(1, n): h = gf_frobenius_map(h, g, b, p, K) r = gf_add(r, h, p, K) r = gf_rem(r, g, p, K) return r def gf_random(n, p, K): """ Generate a random polynomial in ``GF(p)[x]`` of degree ``n``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_random >>> gf_random(10, 5, ZZ) #doctest: +SKIP [1, 2, 3, 2, 1, 1, 1, 2, 0, 4, 2] """ return [K.one] + [ K(int(uniform(0, p))) for i in range(0, n) ] def gf_irreducible(n, p, K): """ Generate random irreducible polynomial of degree ``n`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_irreducible >>> gf_irreducible(10, 5, ZZ) #doctest: +SKIP [1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4] """ while True: f = gf_random(n, p, K) if gf_irreducible_p(f, p, K): return f def gf_irred_p_ben_or(f, p, K): """ Ben-Or's polynomial irreducibility test over finite fields. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_irred_p_ben_or >>> gf_irred_p_ben_or(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ) True >>> gf_irred_p_ben_or(ZZ.map([3, 2, 4]), 5, ZZ) False """ n = gf_degree(f) if n <= 1: return True _, f = gf_monic(f, p, K) if n < 5: H = h = gf_pow_mod([K.one, K.zero], p, f, p, K) for i in range(0, n//2): g = gf_sub(h, [K.one, K.zero], p, K) if gf_gcd(f, g, p, K) == [K.one]: h = gf_compose_mod(h, H, f, p, K) else: return False else: b = gf_frobenius_monomial_base(f, p, K) H = h = gf_frobenius_map([K.one, K.zero], f, b, p, K) for i in range(0, n//2): g = gf_sub(h, [K.one, K.zero], p, K) if gf_gcd(f, g, p, K) == [K.one]: h = gf_frobenius_map(h, f, b, p, K) else: return False return True def gf_irred_p_rabin(f, p, K): """ Rabin's polynomial irreducibility test over finite fields. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_irred_p_rabin >>> gf_irred_p_rabin(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ) True >>> gf_irred_p_rabin(ZZ.map([3, 2, 4]), 5, ZZ) False """ n = gf_degree(f) if n <= 1: return True _, f = gf_monic(f, p, K) x = [K.one, K.zero] indices = { n//d for d in factorint(n) } b = gf_frobenius_monomial_base(f, p, K) h = b[1] for i in range(1, n): if i in indices: g = gf_sub(h, x, p, K) if gf_gcd(f, g, p, K) != [K.one]: return False h = gf_frobenius_map(h, f, b, p, K) return h == x _irred_methods = { 'ben-or': gf_irred_p_ben_or, 'rabin': gf_irred_p_rabin, } def gf_irreducible_p(f, p, K): """ Test irreducibility of a polynomial ``f`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_irreducible_p >>> gf_irreducible_p(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ) True >>> gf_irreducible_p(ZZ.map([3, 2, 4]), 5, ZZ) False """ method = query('GF_IRRED_METHOD') if method is not None: irred = _irred_methods[method](f, p, K) else: irred = gf_irred_p_rabin(f, p, K) return irred def gf_sqf_p(f, p, K): """ Return ``True`` if ``f`` is square-free in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sqf_p >>> gf_sqf_p(ZZ.map([3, 2, 4]), 5, ZZ) True >>> gf_sqf_p(ZZ.map([2, 4, 4, 2, 2, 1, 4]), 5, ZZ) False """ _, f = gf_monic(f, p, K) if not f: return True else: return gf_gcd(f, gf_diff(f, p, K), p, K) == [K.one] def gf_sqf_part(f, p, K): """ Return square-free part of a ``GF(p)[x]`` polynomial. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_sqf_part >>> gf_sqf_part(ZZ.map([1, 1, 3, 0, 1, 0, 2, 2, 1]), 5, ZZ) [1, 4, 3] """ _, sqf = gf_sqf_list(f, p, K) g = [K.one] for f, _ in sqf: g = gf_mul(g, f, p, K) return g def gf_sqf_list(f, p, K, all=False): """ Return the square-free decomposition of a ``GF(p)[x]`` polynomial. Given a polynomial ``f`` in ``GF(p)[x]``, returns the leading coefficient of ``f`` and a square-free decomposition ``f_1**e_1 f_2**e_2 ... f_k**e_k`` such that all ``f_i`` are monic polynomials and ``(f_i, f_j)`` for ``i != j`` are co-prime and ``e_1 ... e_k`` are given in increasing order. All trivial terms (i.e. ``f_i = 1``) aren't included in the output. Consider polynomial ``f = x**11 + 1`` over ``GF(11)[x]``:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import ( ... gf_from_dict, gf_diff, gf_sqf_list, gf_pow, ... ) ... # doctest: +NORMALIZE_WHITESPACE >>> f = gf_from_dict({11: ZZ(1), 0: ZZ(1)}, 11, ZZ) Note that ``f'(x) = 0``:: >>> gf_diff(f, 11, ZZ) [] This phenomenon doesn't happen in characteristic zero. However we can still compute square-free decomposition of ``f`` using ``gf_sqf()``:: >>> gf_sqf_list(f, 11, ZZ) (1, [([1, 1], 11)]) We obtained factorization ``f = (x + 1)**11``. This is correct because:: >>> gf_pow([1, 1], 11, 11, ZZ) == f True References ========== .. [1] [Geddes92]_ """ n, sqf, factors, r = 1, False, [], int(p) lc, f = gf_monic(f, p, K) if gf_degree(f) < 1: return lc, [] while True: F = gf_diff(f, p, K) if F != []: g = gf_gcd(f, F, p, K) h = gf_quo(f, g, p, K) i = 1 while h != [K.one]: G = gf_gcd(g, h, p, K) H = gf_quo(h, G, p, K) if gf_degree(H) > 0: factors.append((H, i*n)) g, h, i = gf_quo(g, G, p, K), G, i + 1 if g == [K.one]: sqf = True else: f = g if not sqf: d = gf_degree(f) // r for i in range(0, d + 1): f[i] = f[i*r] f, n = f[:d + 1], n*r else: break if all: raise ValueError("'all=True' is not supported yet") return lc, factors def gf_Qmatrix(f, p, K): """ Calculate Berlekamp's ``Q`` matrix. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_Qmatrix >>> gf_Qmatrix([3, 2, 4], 5, ZZ) [[1, 0], [3, 4]] >>> gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ) [[1, 0, 0, 0], [0, 4, 0, 0], [0, 0, 1, 0], [0, 0, 0, 4]] """ n, r = gf_degree(f), int(p) q = [K.one] + [K.zero]*(n - 1) Q = [list(q)] + [[]]*(n - 1) for i in range(1, (n - 1)*r + 1): qq, c = [(-q[-1]*f[-1]) % p], q[-1] for j in range(1, n): qq.append((q[j - 1] - c*f[-j - 1]) % p) if not (i % r): Q[i//r] = list(qq) q = qq return Q def gf_Qbasis(Q, p, K): """ Compute a basis of the kernel of ``Q``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_Qmatrix, gf_Qbasis >>> gf_Qbasis(gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ), 5, ZZ) [[1, 0, 0, 0], [0, 0, 1, 0]] >>> gf_Qbasis(gf_Qmatrix([3, 2, 4], 5, ZZ), 5, ZZ) [[1, 0]] """ Q, n = [ list(q) for q in Q ], len(Q) for k in range(0, n): Q[k][k] = (Q[k][k] - K.one) % p for k in range(0, n): for i in range(k, n): if Q[k][i]: break else: continue inv = K.invert(Q[k][i], p) for j in range(0, n): Q[j][i] = (Q[j][i]*inv) % p for j in range(0, n): t = Q[j][k] Q[j][k] = Q[j][i] Q[j][i] = t for i in range(0, n): if i != k: q = Q[k][i] for j in range(0, n): Q[j][i] = (Q[j][i] - Q[j][k]*q) % p for i in range(0, n): for j in range(0, n): if i == j: Q[i][j] = (K.one - Q[i][j]) % p else: Q[i][j] = (-Q[i][j]) % p basis = [] for q in Q: if any(q): basis.append(q) return basis def gf_berlekamp(f, p, K): """ Factor a square-free ``f`` in ``GF(p)[x]`` for small ``p``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_berlekamp >>> gf_berlekamp([1, 0, 0, 0, 1], 5, ZZ) [[1, 0, 2], [1, 0, 3]] """ Q = gf_Qmatrix(f, p, K) V = gf_Qbasis(Q, p, K) for i, v in enumerate(V): V[i] = gf_strip(list(reversed(v))) factors = [f] for k in range(1, len(V)): for f in list(factors): s = K.zero while s < p: g = gf_sub_ground(V[k], s, p, K) h = gf_gcd(f, g, p, K) if h != [K.one] and h != f: factors.remove(f) f = gf_quo(f, h, p, K) factors.extend([f, h]) if len(factors) == len(V): return _sort_factors(factors, multiple=False) s += K.one return _sort_factors(factors, multiple=False) def gf_ddf_zassenhaus(f, p, K): """ Cantor-Zassenhaus: Deterministic Distinct Degree Factorization Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes partial distinct degree factorization ``f_1 ... f_d`` of ``f`` where ``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a list of pairs ``(f_i, e_i)`` where ``deg(f_i) > 0`` and ``e_i > 0`` is an argument to the equal degree factorization routine. Consider the polynomial ``x**15 - 1`` in ``GF(11)[x]``:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_from_dict >>> f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ) Distinct degree factorization gives:: >>> from sympy.polys.galoistools import gf_ddf_zassenhaus >>> gf_ddf_zassenhaus(f, 11, ZZ) [([1, 0, 0, 0, 0, 10], 1), ([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)] which means ``x**15 - 1 = (x**5 - 1) (x**10 + x**5 + 1)``. To obtain factorization into irreducibles, use equal degree factorization procedure (EDF) with each of the factors. References ========== .. [1] [Gathen99]_ .. [2] [Geddes92]_ """ i, g, factors = 1, [K.one, K.zero], [] b = gf_frobenius_monomial_base(f, p, K) while 2*i <= gf_degree(f): g = gf_frobenius_map(g, f, b, p, K) h = gf_gcd(f, gf_sub(g, [K.one, K.zero], p, K), p, K) if h != [K.one]: factors.append((h, i)) f = gf_quo(f, h, p, K) g = gf_rem(g, f, p, K) b = gf_frobenius_monomial_base(f, p, K) i += 1 if f != [K.one]: return factors + [(f, gf_degree(f))] else: return factors def gf_edf_zassenhaus(f, n, p, K): """ Cantor-Zassenhaus: Probabilistic Equal Degree Factorization Given a monic square-free polynomial ``f`` in ``GF(p)[x]`` and an integer ``n``, such that ``n`` divides ``deg(f)``, returns all irreducible factors ``f_1,...,f_d`` of ``f``, each of degree ``n``. EDF procedure gives complete factorization over Galois fields. Consider the square-free polynomial ``f = x**3 + x**2 + x + 1`` in ``GF(5)[x]``. Let's compute its irreducible factors of degree one:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_edf_zassenhaus >>> gf_edf_zassenhaus([1,1,1,1], 1, 5, ZZ) [[1, 1], [1, 2], [1, 3]] References ========== .. [1] [Gathen99]_ .. [2] [Geddes92]_ """ factors = [f] if gf_degree(f) <= n: return factors N = gf_degree(f) // n if p != 2: b = gf_frobenius_monomial_base(f, p, K) while len(factors) < N: r = gf_random(2*n - 1, p, K) if p == 2: h = r for i in range(0, 2**(n*N - 1)): r = gf_pow_mod(r, 2, f, p, K) h = gf_add(h, r, p, K) g = gf_gcd(f, h, p, K) else: h = _gf_pow_pnm1d2(r, n, f, b, p, K) g = gf_gcd(f, gf_sub_ground(h, K.one, p, K), p, K) if g != [K.one] and g != f: factors = gf_edf_zassenhaus(g, n, p, K) \ + gf_edf_zassenhaus(gf_quo(f, g, p, K), n, p, K) return _sort_factors(factors, multiple=False) def gf_ddf_shoup(f, p, K): """ Kaltofen-Shoup: Deterministic Distinct Degree Factorization Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes partial distinct degree factorization ``f_1,...,f_d`` of ``f`` where ``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a list of pairs ``(f_i, e_i)`` where ``deg(f_i) > 0`` and ``e_i > 0`` is an argument to the equal degree factorization routine. This algorithm is an improved version of Zassenhaus algorithm for large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_ddf_shoup, gf_from_dict >>> f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ) >>> gf_ddf_shoup(f, 3, ZZ) [([1, 1, 0], 1), ([1, 1, 0, 1, 2], 2)] References ========== .. [1] [Kaltofen98]_ .. [2] [Shoup95]_ .. [3] [Gathen92]_ """ n = gf_degree(f) k = int(_ceil(_sqrt(n//2))) b = gf_frobenius_monomial_base(f, p, K) h = gf_frobenius_map([K.one, K.zero], f, b, p, K) # U[i] = x**(p**i) U = [[K.one, K.zero], h] + [K.zero]*(k - 1) for i in range(2, k + 1): U[i] = gf_frobenius_map(U[i-1], f, b, p, K) h, U = U[k], U[:k] # V[i] = x**(p**(k*(i+1))) V = [h] + [K.zero]*(k - 1) for i in range(1, k): V[i] = gf_compose_mod(V[i - 1], h, f, p, K) factors = [] for i, v in enumerate(V): h, j = [K.one], k - 1 for u in U: g = gf_sub(v, u, p, K) h = gf_mul(h, g, p, K) h = gf_rem(h, f, p, K) g = gf_gcd(f, h, p, K) f = gf_quo(f, g, p, K) for u in reversed(U): h = gf_sub(v, u, p, K) F = gf_gcd(g, h, p, K) if F != [K.one]: factors.append((F, k*(i + 1) - j)) g, j = gf_quo(g, F, p, K), j - 1 if f != [K.one]: factors.append((f, gf_degree(f))) return factors def gf_edf_shoup(f, n, p, K): """ Gathen-Shoup: Probabilistic Equal Degree Factorization Given a monic square-free polynomial ``f`` in ``GF(p)[x]`` and integer ``n`` such that ``n`` divides ``deg(f)``, returns all irreducible factors ``f_1,...,f_d`` of ``f``, each of degree ``n``. This is a complete factorization over Galois fields. This algorithm is an improved version of Zassenhaus algorithm for large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``). Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_edf_shoup >>> gf_edf_shoup(ZZ.map([1, 2837, 2277]), 1, 2917, ZZ) [[1, 852], [1, 1985]] References ========== .. [1] [Shoup91]_ .. [2] [Gathen92]_ """ N, q = gf_degree(f), int(p) if not N: return [] if N <= n: return [f] factors, x = [f], [K.one, K.zero] r = gf_random(N - 1, p, K) if p == 2: h = gf_pow_mod(x, q, f, p, K) H = gf_trace_map(r, h, x, n - 1, f, p, K)[1] h1 = gf_gcd(f, H, p, K) h2 = gf_quo(f, h1, p, K) factors = gf_edf_shoup(h1, n, p, K) \ + gf_edf_shoup(h2, n, p, K) else: b = gf_frobenius_monomial_base(f, p, K) H = _gf_trace_map(r, n, f, b, p, K) h = gf_pow_mod(H, (q - 1)//2, f, p, K) h1 = gf_gcd(f, h, p, K) h2 = gf_gcd(f, gf_sub_ground(h, K.one, p, K), p, K) h3 = gf_quo(f, gf_mul(h1, h2, p, K), p, K) factors = gf_edf_shoup(h1, n, p, K) \ + gf_edf_shoup(h2, n, p, K) \ + gf_edf_shoup(h3, n, p, K) return _sort_factors(factors, multiple=False) def gf_zassenhaus(f, p, K): """ Factor a square-free ``f`` in ``GF(p)[x]`` for medium ``p``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_zassenhaus >>> gf_zassenhaus(ZZ.map([1, 4, 3]), 5, ZZ) [[1, 1], [1, 3]] """ factors = [] for factor, n in gf_ddf_zassenhaus(f, p, K): factors += gf_edf_zassenhaus(factor, n, p, K) return _sort_factors(factors, multiple=False) def gf_shoup(f, p, K): """ Factor a square-free ``f`` in ``GF(p)[x]`` for large ``p``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_shoup >>> gf_shoup(ZZ.map([1, 4, 3]), 5, ZZ) [[1, 1], [1, 3]] """ factors = [] for factor, n in gf_ddf_shoup(f, p, K): factors += gf_edf_shoup(factor, n, p, K) return _sort_factors(factors, multiple=False) _factor_methods = { 'berlekamp': gf_berlekamp, # ``p`` : small 'zassenhaus': gf_zassenhaus, # ``p`` : medium 'shoup': gf_shoup, # ``p`` : large } def gf_factor_sqf(f, p, K, method=None): """ Factor a square-free polynomial ``f`` in ``GF(p)[x]``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_factor_sqf >>> gf_factor_sqf(ZZ.map([3, 2, 4]), 5, ZZ) (3, [[1, 1], [1, 3]]) """ lc, f = gf_monic(f, p, K) if gf_degree(f) < 1: return lc, [] method = method or query('GF_FACTOR_METHOD') if method is not None: factors = _factor_methods[method](f, p, K) else: factors = gf_zassenhaus(f, p, K) return lc, factors def gf_factor(f, p, K): """ Factor (non square-free) polynomials in ``GF(p)[x]``. Given a possibly non square-free polynomial ``f`` in ``GF(p)[x]``, returns its complete factorization into irreducibles:: f_1(x)**e_1 f_2(x)**e_2 ... f_d(x)**e_d where each ``f_i`` is a monic polynomial and ``gcd(f_i, f_j) == 1``, for ``i != j``. The result is given as a tuple consisting of the leading coefficient of ``f`` and a list of factors of ``f`` with their multiplicities. The algorithm proceeds by first computing square-free decomposition of ``f`` and then iteratively factoring each of square-free factors. Consider a non square-free polynomial ``f = (7*x + 1) (x + 2)**2`` in ``GF(11)[x]``. We obtain its factorization into irreducibles as follows:: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.galoistools import gf_factor >>> gf_factor(ZZ.map([5, 2, 7, 2]), 11, ZZ) (5, [([1, 2], 1), ([1, 8], 2)]) We arrived with factorization ``f = 5 (x + 2) (x + 8)**2``. We didn't recover the exact form of the input polynomial because we requested to get monic factors of ``f`` and its leading coefficient separately. Square-free factors of ``f`` can be factored into irreducibles over ``GF(p)`` using three very different methods: Berlekamp efficient for very small values of ``p`` (usually ``p < 25``) Cantor-Zassenhaus efficient on average input and with "typical" ``p`` Shoup-Kaltofen-Gathen efficient with very large inputs and modulus If you want to use a specific factorization method, instead of the default one, set ``GF_FACTOR_METHOD`` with one of ``berlekamp``, ``zassenhaus`` or ``shoup`` values. References ========== .. [1] [Gathen99]_ """ lc, f = gf_monic(f, p, K) if gf_degree(f) < 1: return lc, [] factors = [] for g, n in gf_sqf_list(f, p, K)[1]: for h in gf_factor_sqf(g, p, K)[1]: factors.append((h, n)) return lc, _sort_factors(factors) def gf_value(f, a): """ Value of polynomial 'f' at 'a' in field R. Examples ======== >>> from sympy.polys.galoistools import gf_value >>> gf_value([1, 7, 2, 4], 11) 2204 """ result = 0 for c in f: result *= a result += c return result def linear_congruence(a, b, m): """ Returns the values of x satisfying a*x congruent b mod(m) Here m is positive integer and a, b are natural numbers. This function returns only those values of x which are distinct mod(m). Examples ======== >>> from sympy.polys.galoistools import linear_congruence >>> linear_congruence(3, 12, 15) [4, 9, 14] There are 3 solutions distinct mod(15) since gcd(a, m) = gcd(3, 15) = 3. References ========== .. [1] https://en.wikipedia.org/wiki/Linear_congruence_theorem """ from sympy.polys.polytools import gcdex if a % m == 0: if b % m == 0: return list(range(m)) else: return [] r, _, g = gcdex(a, m) if b % g != 0: return [] return [(r * b // g + t * m // g) % m for t in range(g)] def _raise_mod_power(x, s, p, f): """ Used in gf_csolve to generate solutions of f(x) cong 0 mod(p**(s + 1)) from the solutions of f(x) cong 0 mod(p**s). Examples ======== >>> from sympy.polys.galoistools import _raise_mod_power >>> from sympy.polys.galoistools import csolve_prime These is the solutions of f(x) = x**2 + x + 7 cong 0 mod(3) >>> f = [1, 1, 7] >>> csolve_prime(f, 3) [1] >>> [ i for i in range(3) if not (i**2 + i + 7) % 3] [1] The solutions of f(x) cong 0 mod(9) are constructed from the values returned from _raise_mod_power: >>> x, s, p = 1, 1, 3 >>> V = _raise_mod_power(x, s, p, f) >>> [x + v * p**s for v in V] [1, 4, 7] And these are confirmed with the following: >>> [ i for i in range(3**2) if not (i**2 + i + 7) % 3**2] [1, 4, 7] """ from sympy.polys.domains import ZZ f_f = gf_diff(f, p, ZZ) alpha = gf_value(f_f, x) beta = - gf_value(f, x) // p**s return linear_congruence(alpha, beta, p) def csolve_prime(f, p, e=1): """ Solutions of f(x) congruent 0 mod(p**e). Examples ======== >>> from sympy.polys.galoistools import csolve_prime >>> csolve_prime([1, 1, 7], 3, 1) [1] >>> csolve_prime([1, 1, 7], 3, 2) [1, 4, 7] Solutions [7, 4, 1] (mod 3**2) are generated by ``_raise_mod_power()`` from solution [1] (mod 3). """ from sympy.polys.domains import ZZ X1 = [i for i in range(p) if gf_eval(f, i, p, ZZ) == 0] if e == 1: return X1 X = [] S = list(zip(X1, [1]*len(X1))) while S: x, s = S.pop() if s == e: X.append(x) else: s1 = s + 1 ps = p**s S.extend([(x + v*ps, s1) for v in _raise_mod_power(x, s, p, f)]) return sorted(X) def gf_csolve(f, n): """ To solve f(x) congruent 0 mod(n). n is divided into canonical factors and f(x) cong 0 mod(p**e) will be solved for each factor. Applying the Chinese Remainder Theorem to the results returns the final answers. Examples ======== Solve [1, 1, 7] congruent 0 mod(189): >>> from sympy.polys.galoistools import gf_csolve >>> gf_csolve([1, 1, 7], 189) [13, 49, 76, 112, 139, 175] References ========== .. [1] 'An introduction to the Theory of Numbers' 5th Edition by Ivan Niven, Zuckerman and Montgomery. """ from sympy.polys.domains import ZZ P = factorint(n) X = [csolve_prime(f, p, e) for p, e in P.items()] pools = list(map(tuple, X)) perms = [[]] for pool in pools: perms = [x + [y] for x in perms for y in pool] dist_factors = [pow(p, e) for p, e in P.items()] return sorted([gf_crt(per, dist_factors, ZZ) for per in perms])
feee9db132f2d32fe355d15e3463e1645a78dae1327f7e003422d5ca9ac271d2
""" This module contains functions for the computation of Euclidean, (generalized) Sturmian, (modified) subresultant polynomial remainder sequences (prs's) of two polynomials; included are also three functions for the computation of the resultant of two polynomials. Except for the function res_z(), which computes the resultant of two polynomials, the pseudo-remainder function prem() of sympy is _not_ used by any of the functions in the module. Instead of prem() we use the function rem_z(). Included is also the function quo_z(). An explanation of why we avoid prem() can be found in the references stated in the docstring of rem_z(). 1. Theoretical background: ========================== Consider the polynomials f, g in Z[x] of degrees deg(f) = n and deg(g) = m with n >= m. Definition 1: ============= The sign sequence of a polynomial remainder sequence (prs) is the sequence of signs of the leading coefficients of its polynomials. Sign sequences can be computed with the function: sign_seq(poly_seq, x) Definition 2: ============= A polynomial remainder sequence (prs) is called complete if the degree difference between any two consecutive polynomials is 1; otherwise, it called incomplete. It is understood that f, g belong to the sequences mentioned in the two definitions above. 1A. Euclidean and subresultant prs's: ===================================== The subresultant prs of f, g is a sequence of polynomials in Z[x] analogous to the Euclidean prs, the sequence obtained by applying on f, g Euclid's algorithm for polynomial greatest common divisors (gcd) in Q[x]. The subresultant prs differs from the Euclidean prs in that the coefficients of each polynomial in the former sequence are determinants --- also referred to as subresultants --- of appropriately selected sub-matrices of sylvester1(f, g, x), Sylvester's matrix of 1840 of dimensions (n + m) * (n + m). Recall that the determinant of sylvester1(f, g, x) itself is called the resultant of f, g and serves as a criterion of whether the two polynomials have common roots or not. In sympy the resultant is computed with the function resultant(f, g, x). This function does _not_ evaluate the determinant of sylvester(f, g, x, 1); instead, it returns the last member of the subresultant prs of f, g, multiplied (if needed) by an appropriate power of -1; see the caveat below. In this module we use three functions to compute the resultant of f, g: a) res(f, g, x) computes the resultant by evaluating the determinant of sylvester(f, g, x, 1); b) res_q(f, g, x) computes the resultant recursively, by performing polynomial divisions in Q[x] with the function rem(); c) res_z(f, g, x) computes the resultant recursively, by performing polynomial divisions in Z[x] with the function prem(). Caveat: If Df = degree(f, x) and Dg = degree(g, x), then: resultant(f, g, x) = (-1)**(Df*Dg) * resultant(g, f, x). For complete prs's the sign sequence of the Euclidean prs of f, g is identical to the sign sequence of the subresultant prs of f, g and the coefficients of one sequence are easily computed from the coefficients of the other. For incomplete prs's the polynomials in the subresultant prs, generally differ in sign from those of the Euclidean prs, and --- unlike the case of complete prs's --- it is not at all obvious how to compute the coefficients of one sequence from the coefficients of the other. 1B. Sturmian and modified subresultant prs's: ============================================= For the same polynomials f, g in Z[x] mentioned above, their ``modified'' subresultant prs is a sequence of polynomials similar to the Sturmian prs, the sequence obtained by applying in Q[x] Sturm's algorithm on f, g. The two sequences differ in that the coefficients of each polynomial in the modified subresultant prs are the determinants --- also referred to as modified subresultants --- of appropriately selected sub-matrices of sylvester2(f, g, x), Sylvester's matrix of 1853 of dimensions 2n x 2n. The determinant of sylvester2 itself is called the modified resultant of f, g and it also can serve as a criterion of whether the two polynomials have common roots or not. For complete prs's the sign sequence of the Sturmian prs of f, g is identical to the sign sequence of the modified subresultant prs of f, g and the coefficients of one sequence are easily computed from the coefficients of the other. For incomplete prs's the polynomials in the modified subresultant prs, generally differ in sign from those of the Sturmian prs, and --- unlike the case of complete prs's --- it is not at all obvious how to compute the coefficients of one sequence from the coefficients of the other. As Sylvester pointed out, the coefficients of the polynomial remainders obtained as (modified) subresultants are the smallest possible without introducing rationals and without computing (integer) greatest common divisors. 1C. On terminology: =================== Whence the terminology? Well generalized Sturmian prs's are ``modifications'' of Euclidean prs's; the hint came from the title of the Pell-Gordon paper of 1917. In the literature one also encounters the name ``non signed'' and ``signed'' prs for Euclidean and Sturmian prs respectively. Likewise ``non signed'' and ``signed'' subresultant prs for subresultant and modified subresultant prs respectively. 2. Functions in the module: =========================== No function utilizes sympy's function prem(). 2A. Matrices: ============= The functions sylvester(f, g, x, method=1) and sylvester(f, g, x, method=2) compute either Sylvester matrix. They can be used to compute (modified) subresultant prs's by direct determinant evaluation. The function bezout(f, g, x, method='prs') provides a matrix of smaller dimensions than either Sylvester matrix. It is the function of choice for computing (modified) subresultant prs's by direct determinant evaluation. sylvester(f, g, x, method=1) sylvester(f, g, x, method=2) bezout(f, g, x, method='prs') The following identity holds: bezout(f, g, x, method='prs') = backward_eye(deg(f))*bezout(f, g, x, method='bz')*backward_eye(deg(f)) 2B. Subresultant and modified subresultant prs's by =================================================== determinant evaluations: ======================= We use the Sylvester matrices of 1840 and 1853 to compute, respectively, subresultant and modified subresultant polynomial remainder sequences. However, for large matrices this approach takes a lot of time. Instead of utilizing the Sylvester matrices, we can employ the Bezout matrix which is of smaller dimensions. subresultants_sylv(f, g, x) modified_subresultants_sylv(f, g, x) subresultants_bezout(f, g, x) modified_subresultants_bezout(f, g, x) 2C. Subresultant prs's by ONE determinant evaluation: ===================================================== All three functions in this section evaluate one determinant per remainder polynomial; this is the determinant of an appropriately selected sub-matrix of sylvester1(f, g, x), Sylvester's matrix of 1840. To compute the remainder polynomials the function subresultants_rem(f, g, x) employs rem(f, g, x). By contrast, the other two functions implement Van Vleck's ideas of 1900 and compute the remainder polynomials by trinagularizing sylvester2(f, g, x), Sylvester's matrix of 1853. subresultants_rem(f, g, x) subresultants_vv(f, g, x) subresultants_vv_2(f, g, x). 2E. Euclidean, Sturmian prs's in Q[x]: ====================================== euclid_q(f, g, x) sturm_q(f, g, x) 2F. Euclidean, Sturmian and (modified) subresultant prs's P-G: ============================================================== All functions in this section are based on the Pell-Gordon (P-G) theorem of 1917. Computations are done in Q[x], employing the function rem(f, g, x) for the computation of the remainder polynomials. euclid_pg(f, g, x) sturm pg(f, g, x) subresultants_pg(f, g, x) modified_subresultants_pg(f, g, x) 2G. Euclidean, Sturmian and (modified) subresultant prs's A-M-V: ================================================================ All functions in this section are based on the Akritas-Malaschonok- Vigklas (A-M-V) theorem of 2015. Computations are done in Z[x], employing the function rem_z(f, g, x) for the computation of the remainder polynomials. euclid_amv(f, g, x) sturm_amv(f, g, x) subresultants_amv(f, g, x) modified_subresultants_amv(f, g, x) 2Ga. Exception: =============== subresultants_amv_q(f, g, x) This function employs rem(f, g, x) for the computation of the remainder polynomials, despite the fact that it implements the A-M-V Theorem. It is included in our module in order to show that theorems P-G and A-M-V can be implemented utilizing either the function rem(f, g, x) or the function rem_z(f, g, x). For clearly historical reasons --- since the Collins-Brown-Traub coefficients-reduction factor beta_i was not available in 1917 --- we have implemented the Pell-Gordon theorem with the function rem(f, g, x) and the A-M-V Theorem with the function rem_z(f, g, x). 2H. Resultants: =============== res(f, g, x) res_q(f, g, x) res_z(f, g, x) """ from sympy import (Abs, degree, expand, eye, floor, LC, Matrix, nan, Poly, pprint) from sympy import (QQ, pquo, quo, prem, rem, S, sign, simplify, summation, var, zeros) from sympy.polys.polyerrors import PolynomialError def sylvester(f, g, x, method = 1): ''' The input polynomials f, g are in Z[x] or in Q[x]. Let m = degree(f, x), n = degree(g, x) and mx = max( m , n ). a. If method = 1 (default), computes sylvester1, Sylvester's matrix of 1840 of dimension (m + n) x (m + n). The determinants of properly chosen submatrices of this matrix (a.k.a. subresultants) can be used to compute the coefficients of the Euclidean PRS of f, g. b. If method = 2, computes sylvester2, Sylvester's matrix of 1853 of dimension (2*mx) x (2*mx). The determinants of properly chosen submatrices of this matrix (a.k.a. ``modified'' subresultants) can be used to compute the coefficients of the Sturmian PRS of f, g. Applications of these Matrices can be found in the references below. Especially, for applications of sylvester2, see the first reference!! References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem by Van Vleck Regarding Sturm Sequences. Serdica Journal of Computing, Vol. 7, No 4, 101-134, 2013. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. ''' # obtain degrees of polys m, n = degree( Poly(f, x), x), degree( Poly(g, x), x) # Special cases: # A:: case m = n < 0 (i.e. both polys are 0) if m == n and n < 0: return Matrix([]) # B:: case m = n = 0 (i.e. both polys are constants) if m == n and n == 0: return Matrix([]) # C:: m == 0 and n < 0 or m < 0 and n == 0 # (i.e. one poly is constant and the other is 0) if m == 0 and n < 0: return Matrix([]) elif m < 0 and n == 0: return Matrix([]) # D:: m >= 1 and n < 0 or m < 0 and n >=1 # (i.e. one poly is of degree >=1 and the other is 0) if m >= 1 and n < 0: return Matrix([0]) elif m < 0 and n >= 1: return Matrix([0]) fp = Poly(f, x).all_coeffs() gp = Poly(g, x).all_coeffs() # Sylvester's matrix of 1840 (default; a.k.a. sylvester1) if method <= 1: M = zeros(m + n) k = 0 for i in range(n): j = k for coeff in fp: M[i, j] = coeff j = j + 1 k = k + 1 k = 0 for i in range(n, m + n): j = k for coeff in gp: M[i, j] = coeff j = j + 1 k = k + 1 return M # Sylvester's matrix of 1853 (a.k.a sylvester2) if method >= 2: if len(fp) < len(gp): h = [] for i in range(len(gp) - len(fp)): h.append(0) fp[ : 0] = h else: h = [] for i in range(len(fp) - len(gp)): h.append(0) gp[ : 0] = h mx = max(m, n) dim = 2*mx M = zeros( dim ) k = 0 for i in range( mx ): j = k for coeff in fp: M[2*i, j] = coeff j = j + 1 j = k for coeff in gp: M[2*i + 1, j] = coeff j = j + 1 k = k + 1 return M def process_matrix_output(poly_seq, x): """ poly_seq is a polynomial remainder sequence computed either by (modified_)subresultants_bezout or by (modified_)subresultants_sylv. This function removes from poly_seq all zero polynomials as well as all those whose degree is equal to the degree of a preceding polynomial in poly_seq, as we scan it from left to right. """ L = poly_seq[:] # get a copy of the input sequence d = degree(L[1], x) i = 2 while i < len(L): d_i = degree(L[i], x) if d_i < 0: # zero poly L.remove(L[i]) i = i - 1 if d == d_i: # poly degree equals degree of previous poly L.remove(L[i]) i = i - 1 if d_i >= 0: d = d_i i = i + 1 return L def subresultants_sylv(f, g, x): """ The input polynomials f, g are in Z[x] or in Q[x]. It is assumed that deg(f) >= deg(g). Computes the subresultant polynomial remainder sequence (prs) of f, g by evaluating determinants of appropriately selected submatrices of sylvester(f, g, x, 1). The dimensions of the latter are (deg(f) + deg(g)) x (deg(f) + deg(g)). Each coefficient is computed by evaluating the determinant of the corresponding submatrix of sylvester(f, g, x, 1). If the subresultant prs is complete, then the output coincides with the Euclidean sequence of the polynomials f, g. References: =========== 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants and Their Applications. Appl. Algebra in Engin., Communic. and Comp., Vol. 15, 233-266, 2004. """ # make sure neither f nor g is 0 if f == 0 or g == 0: return [f, g] n = degF = degree(f, x) m = degG = degree(g, x) # make sure proper degrees if n == 0 and m == 0: return [f, g] if n < m: n, m, degF, degG, f, g = m, n, degG, degF, g, f if n > 0 and m == 0: return [f, g] SR_L = [f, g] # subresultant list # form matrix sylvester(f, g, x, 1) S = sylvester(f, g, x, 1) # pick appropriate submatrices of S # and form subresultant polys j = m - 1 while j > 0: Sp = S[:, :] # copy of S # delete last j rows of coeffs of g for ind in range(m + n - j, m + n): Sp.row_del(m + n - j) # delete last j rows of coeffs of f for ind in range(m - j, m): Sp.row_del(m - j) # evaluate determinants and form coefficients list coeff_L, k, l = [], Sp.rows, 0 while l <= j: coeff_L.append(Sp[ : , 0 : k].det()) Sp.col_swap(k - 1, k + l) l += 1 # form poly and append to SP_L SR_L.append(Poly(coeff_L, x).as_expr()) j -= 1 # j = 0 SR_L.append(S.det()) return process_matrix_output(SR_L, x) def modified_subresultants_sylv(f, g, x): """ The input polynomials f, g are in Z[x] or in Q[x]. It is assumed that deg(f) >= deg(g). Computes the modified subresultant polynomial remainder sequence (prs) of f, g by evaluating determinants of appropriately selected submatrices of sylvester(f, g, x, 2). The dimensions of the latter are (2*deg(f)) x (2*deg(f)). Each coefficient is computed by evaluating the determinant of the corresponding submatrix of sylvester(f, g, x, 2). If the modified subresultant prs is complete, then the output coincides with the Sturmian sequence of the polynomials f, g. References: =========== 1. A. G. Akritas,G.I. Malaschonok and P.S. Vigklas: Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences. Serdica Journal of Computing, Vol. 8, No 1, 29--46, 2014. """ # make sure neither f nor g is 0 if f == 0 or g == 0: return [f, g] n = degF = degree(f, x) m = degG = degree(g, x) # make sure proper degrees if n == 0 and m == 0: return [f, g] if n < m: n, m, degF, degG, f, g = m, n, degG, degF, g, f if n > 0 and m == 0: return [f, g] SR_L = [f, g] # modified subresultant list # form matrix sylvester(f, g, x, 2) S = sylvester(f, g, x, 2) # pick appropriate submatrices of S # and form modified subresultant polys j = m - 1 while j > 0: # delete last 2*j rows of pairs of coeffs of f, g Sp = S[0:2*n - 2*j, :] # copy of first 2*n - 2*j rows of S # evaluate determinants and form coefficients list coeff_L, k, l = [], Sp.rows, 0 while l <= j: coeff_L.append(Sp[ : , 0 : k].det()) Sp.col_swap(k - 1, k + l) l += 1 # form poly and append to SP_L SR_L.append(Poly(coeff_L, x).as_expr()) j -= 1 # j = 0 SR_L.append(S.det()) return process_matrix_output(SR_L, x) def res(f, g, x): """ The input polynomials f, g are in Z[x] or in Q[x]. The output is the resultant of f, g computed by evaluating the determinant of the matrix sylvester(f, g, x, 1). References: =========== 1. J. S. Cohen: Computer Algebra and Symbolic Computation - Mathematical Methods. A. K. Peters, 2003. """ if f == 0 or g == 0: raise PolynomialError("The resultant of %s and %s is not defined" % (f, g)) else: return sylvester(f, g, x, 1).det() def res_q(f, g, x): """ The input polynomials f, g are in Z[x] or in Q[x]. The output is the resultant of f, g computed recursively by polynomial divisions in Q[x], using the function rem. See Cohen's book p. 281. References: =========== 1. J. S. Cohen: Computer Algebra and Symbolic Computation - Mathematical Methods. A. K. Peters, 2003. """ m = degree(f, x) n = degree(g, x) if m < n: return (-1)**(m*n) * res_q(g, f, x) elif n == 0: # g is a constant return g**m else: r = rem(f, g, x) if r == 0: return 0 else: s = degree(r, x) l = LC(g, x) return (-1)**(m*n) * l**(m-s)*res_q(g, r, x) def res_z(f, g, x): """ The input polynomials f, g are in Z[x] or in Q[x]. The output is the resultant of f, g computed recursively by polynomial divisions in Z[x], using the function prem(). See Cohen's book p. 283. References: =========== 1. J. S. Cohen: Computer Algebra and Symbolic Computation - Mathematical Methods. A. K. Peters, 2003. """ m = degree(f, x) n = degree(g, x) if m < n: return (-1)**(m*n) * res_z(g, f, x) elif n == 0: # g is a constant return g**m else: r = prem(f, g, x) if r == 0: return 0 else: delta = m - n + 1 w = (-1)**(m*n) * res_z(g, r, x) s = degree(r, x) l = LC(g, x) k = delta * n - m + s return quo(w, l**k, x) def sign_seq(poly_seq, x): """ Given a sequence of polynomials poly_seq, it returns the sequence of signs of the leading coefficients of the polynomials in poly_seq. """ return [sign(LC(poly_seq[i], x)) for i in range(len(poly_seq))] def bezout(p, q, x, method='bz'): """ The input polynomials p, q are in Z[x] or in Q[x]. Let mx = max( degree(p, x) , degree(q, x) ). The default option bezout(p, q, x, method='bz') returns Bezout's symmetric matrix of p and q, of dimensions (mx) x (mx). The determinant of this matrix is equal to the determinant of sylvester2, Sylvester's matrix of 1853, whose dimensions are (2*mx) x (2*mx); however the subresultants of these two matrices may differ. The other option, bezout(p, q, x, 'prs'), is of interest to us in this module because it returns a matrix equivalent to sylvester2. In this case all subresultants of the two matrices are identical. Both the subresultant polynomial remainder sequence (prs) and the modified subresultant prs of p and q can be computed by evaluating determinants of appropriately selected submatrices of bezout(p, q, x, 'prs') --- one determinant per coefficient of the remainder polynomials. The matrices bezout(p, q, x, 'bz') and bezout(p, q, x, 'prs') are related by the formula bezout(p, q, x, 'prs') = backward_eye(deg(p)) * bezout(p, q, x, 'bz') * backward_eye(deg(p)), where backward_eye() is the backward identity function. References ========== 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants and Their Applications. Appl. Algebra in Engin., Communic. and Comp., Vol. 15, 233-266, 2004. """ # obtain degrees of polys m, n = degree( Poly(p, x), x), degree( Poly(q, x), x) # Special cases: # A:: case m = n < 0 (i.e. both polys are 0) if m == n and n < 0: return Matrix([]) # B:: case m = n = 0 (i.e. both polys are constants) if m == n and n == 0: return Matrix([]) # C:: m == 0 and n < 0 or m < 0 and n == 0 # (i.e. one poly is constant and the other is 0) if m == 0 and n < 0: return Matrix([]) elif m < 0 and n == 0: return Matrix([]) # D:: m >= 1 and n < 0 or m < 0 and n >=1 # (i.e. one poly is of degree >=1 and the other is 0) if m >= 1 and n < 0: return Matrix([0]) elif m < 0 and n >= 1: return Matrix([0]) y = var('y') # expr is 0 when x = y expr = p * q.subs({x:y}) - p.subs({x:y}) * q # hence expr is exactly divisible by x - y poly = Poly( quo(expr, x-y), x, y) # form Bezout matrix and store them in B as indicated to get # the LC coefficient of each poly either in the first position # of each row (method='prs') or in the last (method='bz'). mx = max(m, n) B = zeros(mx) for i in range(mx): for j in range(mx): if method == 'prs': B[mx - 1 - i, mx - 1 - j] = poly.nth(i, j) else: B[i, j] = poly.nth(i, j) return B def backward_eye(n): ''' Returns the backward identity matrix of dimensions n x n. Needed to "turn" the Bezout matrices so that the leading coefficients are first. See docstring of the function bezout(p, q, x, method='bz'). ''' M = eye(n) # identity matrix of order n for i in range(int(M.rows / 2)): M.row_swap(0 + i, M.rows - 1 - i) return M def subresultants_bezout(p, q, x): """ The input polynomials p, q are in Z[x] or in Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant polynomial remainder sequence of p, q by evaluating determinants of appropriately selected submatrices of bezout(p, q, x, 'prs'). The dimensions of the latter are deg(p) x deg(p). Each coefficient is computed by evaluating the determinant of the corresponding submatrix of bezout(p, q, x, 'prs'). bezout(p, q, x, 'prs) is used instead of sylvester(p, q, x, 1), Sylvester's matrix of 1840, because the dimensions of the latter are (deg(p) + deg(q)) x (deg(p) + deg(q)). If the subresultant prs is complete, then the output coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants and Their Applications. Appl. Algebra in Engin., Communic. and Comp., Vol. 15, 233-266, 2004. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] f, g = p, q n = degF = degree(f, x) m = degG = degree(g, x) # make sure proper degrees if n == 0 and m == 0: return [f, g] if n < m: n, m, degF, degG, f, g = m, n, degG, degF, g, f if n > 0 and m == 0: return [f, g] SR_L = [f, g] # subresultant list F = LC(f, x)**(degF - degG) # form the bezout matrix B = bezout(f, g, x, 'prs') # pick appropriate submatrices of B # and form subresultant polys if degF > degG: j = 2 if degF == degG: j = 1 while j <= degF: M = B[0:j, :] k, coeff_L = j - 1, [] while k <= degF - 1: coeff_L.append(M[: ,0 : j].det()) if k < degF - 1: M.col_swap(j - 1, k + 1) k = k + 1 # apply Theorem 2.1 in the paper by Toca & Vega 2004 # to get correct signs SR_L.append(int((-1)**(j*(j-1)/2)) * (Poly(coeff_L, x) / F).as_expr()) j = j + 1 return process_matrix_output(SR_L, x) def modified_subresultants_bezout(p, q, x): """ The input polynomials p, q are in Z[x] or in Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the modified subresultant polynomial remainder sequence of p, q by evaluating determinants of appropriately selected submatrices of bezout(p, q, x, 'prs'). The dimensions of the latter are deg(p) x deg(p). Each coefficient is computed by evaluating the determinant of the corresponding submatrix of bezout(p, q, x, 'prs'). bezout(p, q, x, 'prs') is used instead of sylvester(p, q, x, 2), Sylvester's matrix of 1853, because the dimensions of the latter are 2*deg(p) x 2*deg(p). If the modified subresultant prs is complete, and LC( p ) > 0, the output coincides with the (generalized) Sturm's sequence of the polynomials p, q. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. 2. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants and Their Applications. Appl. Algebra in Engin., Communic. and Comp., Vol. 15, 233-266, 2004. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] f, g = p, q n = degF = degree(f, x) m = degG = degree(g, x) # make sure proper degrees if n == 0 and m == 0: return [f, g] if n < m: n, m, degF, degG, f, g = m, n, degG, degF, g, f if n > 0 and m == 0: return [f, g] SR_L = [f, g] # subresultant list # form the bezout matrix B = bezout(f, g, x, 'prs') # pick appropriate submatrices of B # and form subresultant polys if degF > degG: j = 2 if degF == degG: j = 1 while j <= degF: M = B[0:j, :] k, coeff_L = j - 1, [] while k <= degF - 1: coeff_L.append(M[: ,0 : j].det()) if k < degF - 1: M.col_swap(j - 1, k + 1) k = k + 1 ## Theorem 2.1 in the paper by Toca & Vega 2004 is _not needed_ ## in this case since ## the bezout matrix is equivalent to sylvester2 SR_L.append(( Poly(coeff_L, x)).as_expr()) j = j + 1 return process_matrix_output(SR_L, x) def sturm_pg(p, q, x, method=0): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the (generalized) Sturm sequence of p and q in Z[x] or Q[x]. If q = diff(p, x, 1) it is the usual Sturm sequence. A. If method == 0, default, the remainder coefficients of the sequence are (in absolute value) ``modified'' subresultants, which for non-monic polynomials are greater than the coefficients of the corresponding subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). B. If method == 1, the remainder coefficients of the sequence are (in absolute value) subresultants, which for non-monic polynomials are smaller than the coefficients of the corresponding ``modified'' subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). If the Sturm sequence is complete, method=0 and LC( p ) > 0, the coefficients of the polynomials in the sequence are ``modified'' subresultants. That is, they are determinants of appropriately selected submatrices of sylvester2, Sylvester's matrix of 1853. In this case the Sturm sequence coincides with the ``modified'' subresultant prs, of the polynomials p, q. If the Sturm sequence is incomplete and method=0 then the signs of the coefficients of the polynomials in the sequence may differ from the signs of the coefficients of the corresponding polynomials in the ``modified'' subresultant prs; however, the absolute values are the same. To compute the coefficients, no determinant evaluation takes place. Instead, polynomial divisions in Q[x] are performed, using the function rem(p, q, x); the coefficients of the remainders computed this way become (``modified'') subresultants with the help of the Pell-Gordon Theorem of 1917. See also the function euclid_pg(p, q, x). References ========== 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding the Highest Common Factor of Two Polynomials. Annals of MatheMatics, Second Series, 18 (1917), No. 4, 188-193. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees d0 = degree(p, x) d1 = degree(q, x) if d0 == 0 and d1 == 0: return [p, q] if d1 > d0: d0, d1 = d1, d0 p, q = q, p if d0 > 0 and d1 == 0: return [p,q] # make sure LC(p) > 0 flag = 0 if LC(p,x) < 0: flag = 1 p = -p q = -q # initialize lcf = LC(p, x)**(d0 - d1) # lcf * subr = modified subr a0, a1 = p, q # the input polys sturm_seq = [a0, a1] # the output list del0 = d0 - d1 # degree difference rho1 = LC(a1, x) # leading coeff of a1 exp_deg = d1 - 1 # expected degree of a2 a2 = - rem(a0, a1, domain=QQ) # first remainder rho2 = LC(a2,x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 deg_diff_new = exp_deg - d2 # expected - actual degree del1 = d1 - d2 # degree difference # mul_fac is the factor by which a2 is multiplied to # get integer coefficients mul_fac_old = rho1**(del0 + del1 - deg_diff_new) # append accordingly if method == 0: sturm_seq.append( simplify(lcf * a2 * Abs(mul_fac_old))) else: sturm_seq.append( simplify( a2 * Abs(mul_fac_old))) # main loop deg_diff_old = deg_diff_new while d2 > 0: a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees del0 = del1 # update degree difference exp_deg = d1 - 1 # new expected degree a2 = - rem(a0, a1, domain=QQ) # new remainder rho3 = LC(a2, x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 deg_diff_new = exp_deg - d2 # expected - actual degree del1 = d1 - d2 # degree difference # take into consideration the power # rho1**deg_diff_old that was "left out" expo_old = deg_diff_old # rho1 raised to this power expo_new = del0 + del1 - deg_diff_new # rho2 raised to this power # update variables and append mul_fac_new = rho2**(expo_new) * rho1**(expo_old) * mul_fac_old deg_diff_old, mul_fac_old = deg_diff_new, mul_fac_new rho1, rho2 = rho2, rho3 if method == 0: sturm_seq.append( simplify(lcf * a2 * Abs(mul_fac_old))) else: sturm_seq.append( simplify( a2 * Abs(mul_fac_old))) if flag: # change the sign of the sequence sturm_seq = [-i for i in sturm_seq] # gcd is of degree > 0 ? m = len(sturm_seq) if sturm_seq[m - 1] == nan or sturm_seq[m - 1] == 0: sturm_seq.pop(m - 1) return sturm_seq def sturm_q(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the (generalized) Sturm sequence of p and q in Q[x]. Polynomial divisions in Q[x] are performed, using the function rem(p, q, x). The coefficients of the polynomials in the Sturm sequence can be uniquely determined from the corresponding coefficients of the polynomials found either in: (a) the ``modified'' subresultant prs, (references 1, 2) or in (b) the subresultant prs (reference 3). References ========== 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding the Highest Common Factor of Two Polynomials. Annals of MatheMatics, Second Series, 18 (1917), No. 4, 188-193. 2 Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees d0 = degree(p, x) d1 = degree(q, x) if d0 == 0 and d1 == 0: return [p, q] if d1 > d0: d0, d1 = d1, d0 p, q = q, p if d0 > 0 and d1 == 0: return [p,q] # make sure LC(p) > 0 flag = 0 if LC(p,x) < 0: flag = 1 p = -p q = -q # initialize a0, a1 = p, q # the input polys sturm_seq = [a0, a1] # the output list a2 = -rem(a0, a1, domain=QQ) # first remainder d2 = degree(a2, x) # degree of a2 sturm_seq.append( a2 ) # main loop while d2 > 0: a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees a2 = -rem(a0, a1, domain=QQ) # new remainder d2 = degree(a2, x) # actual degree of a2 sturm_seq.append( a2 ) if flag: # change the sign of the sequence sturm_seq = [-i for i in sturm_seq] # gcd is of degree > 0 ? m = len(sturm_seq) if sturm_seq[m - 1] == nan or sturm_seq[m - 1] == 0: sturm_seq.pop(m - 1) return sturm_seq def sturm_amv(p, q, x, method=0): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the (generalized) Sturm sequence of p and q in Z[x] or Q[x]. If q = diff(p, x, 1) it is the usual Sturm sequence. A. If method == 0, default, the remainder coefficients of the sequence are (in absolute value) ``modified'' subresultants, which for non-monic polynomials are greater than the coefficients of the corresponding subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). B. If method == 1, the remainder coefficients of the sequence are (in absolute value) subresultants, which for non-monic polynomials are smaller than the coefficients of the corresponding ``modified'' subresultants by the factor Abs( LC(p)**( deg(p)- deg(q)) ). If the Sturm sequence is complete, method=0 and LC( p ) > 0, then the coefficients of the polynomials in the sequence are ``modified'' subresultants. That is, they are determinants of appropriately selected submatrices of sylvester2, Sylvester's matrix of 1853. In this case the Sturm sequence coincides with the ``modified'' subresultant prs, of the polynomials p, q. If the Sturm sequence is incomplete and method=0 then the signs of the coefficients of the polynomials in the sequence may differ from the signs of the coefficients of the corresponding polynomials in the ``modified'' subresultant prs; however, the absolute values are the same. To compute the coefficients, no determinant evaluation takes place. Instead, we first compute the euclidean sequence of p and q using euclid_amv(p, q, x) and then: (a) change the signs of the remainders in the Euclidean sequence according to the pattern "-, -, +, +, -, -, +, +,..." (see Lemma 1 in the 1st reference or Theorem 3 in the 2nd reference) and (b) if method=0, assuming deg(p) > deg(q), we multiply the remainder coefficients of the Euclidean sequence times the factor Abs( LC(p)**( deg(p)- deg(q)) ) to make them modified subresultants. See also the function sturm_pg(p, q, x). References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' Serdica Journal of Computing 9(2) (2015), 123-138. 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' Serdica Journal of Computing 10 (2016), No.3-4, 197-217. """ # compute the euclidean sequence prs = euclid_amv(p, q, x) # defensive if prs == [] or len(prs) == 2: return prs # the coefficients in prs are subresultants and hence are smaller # than the corresponding subresultants by the factor # Abs( LC(prs[0])**( deg(prs[0]) - deg(prs[1])) ); Theorem 2, 2nd reference. lcf = Abs( LC(prs[0])**( degree(prs[0], x) - degree(prs[1], x) ) ) # the signs of the first two polys in the sequence stay the same sturm_seq = [prs[0], prs[1]] # change the signs according to "-, -, +, +, -, -, +, +,..." # and multiply times lcf if needed flag = 0 m = len(prs) i = 2 while i <= m-1: if flag == 0: sturm_seq.append( - prs[i] ) i = i + 1 if i == m: break sturm_seq.append( - prs[i] ) i = i + 1 flag = 1 elif flag == 1: sturm_seq.append( prs[i] ) i = i + 1 if i == m: break sturm_seq.append( prs[i] ) i = i + 1 flag = 0 # subresultants or modified subresultants? if method == 0 and lcf > 1: aux_seq = [sturm_seq[0], sturm_seq[1]] for i in range(2, m): aux_seq.append(simplify(sturm_seq[i] * lcf )) sturm_seq = aux_seq return sturm_seq def euclid_pg(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the Euclidean sequence of p and q in Z[x] or Q[x]. If the Euclidean sequence is complete the coefficients of the polynomials in the sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. In this case the Euclidean sequence coincides with the subresultant prs of the polynomials p, q. If the Euclidean sequence is incomplete the signs of the coefficients of the polynomials in the sequence may differ from the signs of the coefficients of the corresponding polynomials in the subresultant prs; however, the absolute values are the same. To compute the Euclidean sequence, no determinant evaluation takes place. We first compute the (generalized) Sturm sequence of p and q using sturm_pg(p, q, x, 1), in which case the coefficients are (in absolute value) equal to subresultants. Then we change the signs of the remainders in the Sturm sequence according to the pattern "-, -, +, +, -, -, +, +,..." ; see Lemma 1 in the 1st reference or Theorem 3 in the 2nd reference as well as the function sturm_pg(p, q, x). References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' Serdica Journal of Computing 9(2) (2015), 123-138. 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' Serdica Journal of Computing 10 (2016), No.3-4, 197-217. """ # compute the sturmian sequence using the Pell-Gordon (or AMV) theorem # with the coefficients in the prs being (in absolute value) subresultants prs = sturm_pg(p, q, x, 1) ## any other method would do # defensive if prs == [] or len(prs) == 2: return prs # the signs of the first two polys in the sequence stay the same euclid_seq = [prs[0], prs[1]] # change the signs according to "-, -, +, +, -, -, +, +,..." flag = 0 m = len(prs) i = 2 while i <= m-1: if flag == 0: euclid_seq.append(- prs[i] ) i = i + 1 if i == m: break euclid_seq.append(- prs[i] ) i = i + 1 flag = 1 elif flag == 1: euclid_seq.append(prs[i] ) i = i + 1 if i == m: break euclid_seq.append(prs[i] ) i = i + 1 flag = 0 return euclid_seq def euclid_q(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the Euclidean sequence of p and q in Q[x]. Polynomial divisions in Q[x] are performed, using the function rem(p, q, x). The coefficients of the polynomials in the Euclidean sequence can be uniquely determined from the corresponding coefficients of the polynomials found either in: (a) the ``modified'' subresultant polynomial remainder sequence, (references 1, 2) or in (b) the subresultant polynomial remainder sequence (references 3). References ========== 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding the Highest Common Factor of Two Polynomials. Annals of MatheMatics, Second Series, 18 (1917), No. 4, 188-193. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees d0 = degree(p, x) d1 = degree(q, x) if d0 == 0 and d1 == 0: return [p, q] if d1 > d0: d0, d1 = d1, d0 p, q = q, p if d0 > 0 and d1 == 0: return [p,q] # make sure LC(p) > 0 flag = 0 if LC(p,x) < 0: flag = 1 p = -p q = -q # initialize a0, a1 = p, q # the input polys euclid_seq = [a0, a1] # the output list a2 = rem(a0, a1, domain=QQ) # first remainder d2 = degree(a2, x) # degree of a2 euclid_seq.append( a2 ) # main loop while d2 > 0: a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees a2 = rem(a0, a1, domain=QQ) # new remainder d2 = degree(a2, x) # actual degree of a2 euclid_seq.append( a2 ) if flag: # change the sign of the sequence euclid_seq = [-i for i in euclid_seq] # gcd is of degree > 0 ? m = len(euclid_seq) if euclid_seq[m - 1] == nan or euclid_seq[m - 1] == 0: euclid_seq.pop(m - 1) return euclid_seq def euclid_amv(f, g, x): """ f, g are polynomials in Z[x] or Q[x]. It is assumed that degree(f, x) >= degree(g, x). Computes the Euclidean sequence of p and q in Z[x] or Q[x]. If the Euclidean sequence is complete the coefficients of the polynomials in the sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. In this case the Euclidean sequence coincides with the subresultant prs, of the polynomials p, q. If the Euclidean sequence is incomplete the signs of the coefficients of the polynomials in the sequence may differ from the signs of the coefficients of the corresponding polynomials in the subresultant prs; however, the absolute values are the same. To compute the coefficients, no determinant evaluation takes place. Instead, polynomial divisions in Z[x] or Q[x] are performed, using the function rem_z(f, g, x); the coefficients of the remainders computed this way become subresultants with the help of the Collins-Brown-Traub formula for coefficient reduction. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' Serdica Journal of Computing 10 (2016), No.3-4, 197-217. """ # make sure neither f nor g is 0 if f == 0 or g == 0: return [f, g] # make sure proper degrees d0 = degree(f, x) d1 = degree(g, x) if d0 == 0 and d1 == 0: return [f, g] if d1 > d0: d0, d1 = d1, d0 f, g = g, f if d0 > 0 and d1 == 0: return [f, g] # initialize a0 = f a1 = g euclid_seq = [a0, a1] deg_dif_p1, c = degree(a0, x) - degree(a1, x) + 1, -1 # compute the first polynomial of the prs i = 1 a2 = rem_z(a0, a1, x) / Abs( (-1)**deg_dif_p1 ) # first remainder euclid_seq.append( a2 ) d2 = degree(a2, x) # actual degree of a2 # main loop while d2 >= 1: a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees i += 1 sigma0 = -LC(a0) c = (sigma0**(deg_dif_p1 - 1)) / (c**(deg_dif_p1 - 2)) deg_dif_p1 = degree(a0, x) - d2 + 1 a2 = rem_z(a0, a1, x) / Abs( (c**(deg_dif_p1 - 1)) * sigma0 ) euclid_seq.append( a2 ) d2 = degree(a2, x) # actual degree of a2 # gcd is of degree > 0 ? m = len(euclid_seq) if euclid_seq[m - 1] == nan or euclid_seq[m - 1] == 0: euclid_seq.pop(m - 1) return euclid_seq def modified_subresultants_pg(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the ``modified'' subresultant prs of p and q in Z[x] or Q[x]; the coefficients of the polynomials in the sequence are ``modified'' subresultants. That is, they are determinants of appropriately selected submatrices of sylvester2, Sylvester's matrix of 1853. To compute the coefficients, no determinant evaluation takes place. Instead, polynomial divisions in Q[x] are performed, using the function rem(p, q, x); the coefficients of the remainders computed this way become ``modified'' subresultants with the help of the Pell-Gordon Theorem of 1917. If the ``modified'' subresultant prs is complete, and LC( p ) > 0, it coincides with the (generalized) Sturm sequence of the polynomials p, q. References ========== 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding the Highest Common Factor of Two Polynomials. Annals of MatheMatics, Second Series, 18 (1917), No. 4, 188-193. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees d0 = degree(p,x) d1 = degree(q,x) if d0 == 0 and d1 == 0: return [p, q] if d1 > d0: d0, d1 = d1, d0 p, q = q, p if d0 > 0 and d1 == 0: return [p,q] # initialize k = var('k') # index in summation formula u_list = [] # of elements (-1)**u_i subres_l = [p, q] # mod. subr. prs output list a0, a1 = p, q # the input polys del0 = d0 - d1 # degree difference degdif = del0 # save it rho_1 = LC(a0) # lead. coeff (a0) # Initialize Pell-Gordon variables rho_list_minus_1 = sign( LC(a0, x)) # sign of LC(a0) rho1 = LC(a1, x) # leading coeff of a1 rho_list = [ sign(rho1)] # of signs p_list = [del0] # of degree differences u = summation(k, (k, 1, p_list[0])) # value of u u_list.append(u) # of u values v = sum(p_list) # v value # first remainder exp_deg = d1 - 1 # expected degree of a2 a2 = - rem(a0, a1, domain=QQ) # first remainder rho2 = LC(a2, x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 deg_diff_new = exp_deg - d2 # expected - actual degree del1 = d1 - d2 # degree difference # mul_fac is the factor by which a2 is multiplied to # get integer coefficients mul_fac_old = rho1**(del0 + del1 - deg_diff_new) # update Pell-Gordon variables p_list.append(1 + deg_diff_new) # deg_diff_new is 0 for complete seq # apply Pell-Gordon formula (7) in second reference num = 1 # numerator of fraction for k in range(len(u_list)): num *= (-1)**u_list[k] num = num * (-1)**v # denominator depends on complete / incomplete seq if deg_diff_new == 0: # complete seq den = 1 for k in range(len(rho_list)): den *= rho_list[k]**(p_list[k] + p_list[k + 1]) den = den * rho_list_minus_1 else: # incomplete seq den = 1 for k in range(len(rho_list)-1): den *= rho_list[k]**(p_list[k] + p_list[k + 1]) den = den * rho_list_minus_1 expo = (p_list[len(rho_list) - 1] + p_list[len(rho_list)] - deg_diff_new) den = den * rho_list[len(rho_list) - 1]**expo # the sign of the determinant depends on sg(num / den) if sign(num / den) > 0: subres_l.append( simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) else: subres_l.append(- simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) # update Pell-Gordon variables k = var('k') rho_list.append( sign(rho2)) u = summation(k, (k, 1, p_list[len(p_list) - 1])) u_list.append(u) v = sum(p_list) deg_diff_old=deg_diff_new # main loop while d2 > 0: a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees del0 = del1 # update degree difference exp_deg = d1 - 1 # new expected degree a2 = - rem(a0, a1, domain=QQ) # new remainder rho3 = LC(a2, x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 deg_diff_new = exp_deg - d2 # expected - actual degree del1 = d1 - d2 # degree difference # take into consideration the power # rho1**deg_diff_old that was "left out" expo_old = deg_diff_old # rho1 raised to this power expo_new = del0 + del1 - deg_diff_new # rho2 raised to this power mul_fac_new = rho2**(expo_new) * rho1**(expo_old) * mul_fac_old # update variables deg_diff_old, mul_fac_old = deg_diff_new, mul_fac_new rho1, rho2 = rho2, rho3 # update Pell-Gordon variables p_list.append(1 + deg_diff_new) # deg_diff_new is 0 for complete seq # apply Pell-Gordon formula (7) in second reference num = 1 # numerator for k in range(len(u_list)): num *= (-1)**u_list[k] num = num * (-1)**v # denominator depends on complete / incomplete seq if deg_diff_new == 0: # complete seq den = 1 for k in range(len(rho_list)): den *= rho_list[k]**(p_list[k] + p_list[k + 1]) den = den * rho_list_minus_1 else: # incomplete seq den = 1 for k in range(len(rho_list)-1): den *= rho_list[k]**(p_list[k] + p_list[k + 1]) den = den * rho_list_minus_1 expo = (p_list[len(rho_list) - 1] + p_list[len(rho_list)] - deg_diff_new) den = den * rho_list[len(rho_list) - 1]**expo # the sign of the determinant depends on sg(num / den) if sign(num / den) > 0: subres_l.append( simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) else: subres_l.append(- simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) # update Pell-Gordon variables k = var('k') rho_list.append( sign(rho2)) u = summation(k, (k, 1, p_list[len(p_list) - 1])) u_list.append(u) v = sum(p_list) # gcd is of degree > 0 ? m = len(subres_l) if subres_l[m - 1] == nan or subres_l[m - 1] == 0: subres_l.pop(m - 1) # LC( p ) < 0 m = len(subres_l) # list may be shorter now due to deg(gcd ) > 0 if LC( p ) < 0: aux_seq = [subres_l[0], subres_l[1]] for i in range(2, m): aux_seq.append(simplify(subres_l[i] * (-1) )) subres_l = aux_seq return subres_l def subresultants_pg(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant prs of p and q in Z[x] or Q[x], from the modified subresultant prs of p and q. The coefficients of the polynomials in these two sequences differ only in sign and the factor LC(p)**( deg(p)- deg(q)) as stated in Theorem 2 of the reference. The coefficients of the polynomials in the output sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: "On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials." Serdica Journal of Computing 9(2) (2015), 123-138. """ # compute the modified subresultant prs lst = modified_subresultants_pg(p,q,x) ## any other method would do # defensive if lst == [] or len(lst) == 2: return lst # the coefficients in lst are modified subresultants and, hence, are # greater than those of the corresponding subresultants by the factor # LC(lst[0])**( deg(lst[0]) - deg(lst[1])); see Theorem 2 in reference. lcf = LC(lst[0])**( degree(lst[0], x) - degree(lst[1], x) ) # Initialize the subresultant prs list subr_seq = [lst[0], lst[1]] # compute the degree sequences m_i and j_i of Theorem 2 in reference. deg_seq = [degree(Poly(poly, x), x) for poly in lst] deg = deg_seq[0] deg_seq_s = deg_seq[1:-1] m_seq = [m-1 for m in deg_seq_s] j_seq = [deg - m for m in m_seq] # compute the AMV factors of Theorem 2 in reference. fact = [(-1)**( j*(j-1)/S(2) ) for j in j_seq] # shortened list without the first two polys lst_s = lst[2:] # poly lst_s[k] is multiplied times fact[k], divided by lcf # and appended to the subresultant prs list m = len(fact) for k in range(m): if sign(fact[k]) == -1: subr_seq.append(-lst_s[k] / lcf) else: subr_seq.append(lst_s[k] / lcf) return subr_seq def subresultants_amv_q(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant prs of p and q in Q[x]; the coefficients of the polynomials in the sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. To compute the coefficients, no determinant evaluation takes place. Instead, polynomial divisions in Q[x] are performed, using the function rem(p, q, x); the coefficients of the remainders computed this way become subresultants with the help of the Akritas-Malaschonok-Vigklas Theorem of 2015. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' Serdica Journal of Computing 10 (2016), No.3-4, 197-217. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees d0 = degree(p, x) d1 = degree(q, x) if d0 == 0 and d1 == 0: return [p, q] if d1 > d0: d0, d1 = d1, d0 p, q = q, p if d0 > 0 and d1 == 0: return [p, q] # initialize i, s = 0, 0 # counters for remainders & odd elements p_odd_index_sum = 0 # contains the sum of p_1, p_3, etc subres_l = [p, q] # subresultant prs output list a0, a1 = p, q # the input polys sigma1 = LC(a1, x) # leading coeff of a1 p0 = d0 - d1 # degree difference if p0 % 2 == 1: s += 1 phi = floor( (s + 1) / 2 ) mul_fac = 1 d2 = d1 # main loop while d2 > 0: i += 1 a2 = rem(a0, a1, domain= QQ) # new remainder if i == 1: sigma2 = LC(a2, x) else: sigma3 = LC(a2, x) sigma1, sigma2 = sigma2, sigma3 d2 = degree(a2, x) p1 = d1 - d2 psi = i + phi + p_odd_index_sum # new mul_fac mul_fac = sigma1**(p0 + 1) * mul_fac ## compute the sign of the first fraction in formula (9) of the paper # numerator num = (-1)**psi # denominator den = sign(mul_fac) # the sign of the determinant depends on sign( num / den ) != 0 if sign(num / den) > 0: subres_l.append( simplify(expand(a2* Abs(mul_fac)))) else: subres_l.append(- simplify(expand(a2* Abs(mul_fac)))) ## bring into mul_fac the missing power of sigma if there was a degree gap if p1 - 1 > 0: mul_fac = mul_fac * sigma1**(p1 - 1) # update AMV variables a0, a1, d0, d1 = a1, a2, d1, d2 p0 = p1 if p0 % 2 ==1: s += 1 phi = floor( (s + 1) / 2 ) if i%2 == 1: p_odd_index_sum += p0 # p_i has odd index # gcd is of degree > 0 ? m = len(subres_l) if subres_l[m - 1] == nan or subres_l[m - 1] == 0: subres_l.pop(m - 1) return subres_l def compute_sign(base, expo): ''' base != 0 and expo >= 0 are integers; returns the sign of base**expo without evaluating the power itself! ''' sb = sign(base) if sb == 1: return 1 pe = expo % 2 if pe == 0: return -sb else: return sb def rem_z(p, q, x): ''' Intended mainly for p, q polynomials in Z[x] so that, on dividing p by q, the remainder will also be in Z[x]. (However, it also works fine for polynomials in Q[x].) It is assumed that degree(p, x) >= degree(q, x). It premultiplies p by the _absolute_ value of the leading coefficient of q, raised to the power deg(p) - deg(q) + 1 and then performs polynomial division in Q[x], using the function rem(p, q, x). By contrast the function prem(p, q, x) does _not_ use the absolute value of the leading coefficient of q. This results not only in ``messing up the signs'' of the Euclidean and Sturmian prs's as mentioned in the second reference, but also in violation of the main results of the first and third references --- Theorem 4 and Theorem 1 respectively. Theorems 4 and 1 establish a one-to-one correspondence between the Euclidean and the Sturmian prs of p, q, on one hand, and the subresultant prs of p, q, on the other. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' Serdica Journal of Computing, 9(2) (2015), 123-138. 2. http://planetMath.org/sturmstheorem 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. ''' if (p.as_poly().is_univariate and q.as_poly().is_univariate and p.as_poly().gens == q.as_poly().gens): delta = (degree(p, x) - degree(q, x) + 1) return rem(Abs(LC(q, x))**delta * p, q, x) else: return prem(p, q, x) def quo_z(p, q, x): """ Intended mainly for p, q polynomials in Z[x] so that, on dividing p by q, the quotient will also be in Z[x]. (However, it also works fine for polynomials in Q[x].) It is assumed that degree(p, x) >= degree(q, x). It premultiplies p by the _absolute_ value of the leading coefficient of q, raised to the power deg(p) - deg(q) + 1 and then performs polynomial division in Q[x], using the function quo(p, q, x). By contrast the function pquo(p, q, x) does _not_ use the absolute value of the leading coefficient of q. See also function rem_z(p, q, x) for additional comments and references. """ if (p.as_poly().is_univariate and q.as_poly().is_univariate and p.as_poly().gens == q.as_poly().gens): delta = (degree(p, x) - degree(q, x) + 1) return quo(Abs(LC(q, x))**delta * p, q, x) else: return pquo(p, q, x) def subresultants_amv(f, g, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(f, x) >= degree(g, x). Computes the subresultant prs of p and q in Z[x] or Q[x]; the coefficients of the polynomials in the sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. To compute the coefficients, no determinant evaluation takes place. Instead, polynomial divisions in Z[x] or Q[x] are performed, using the function rem_z(p, q, x); the coefficients of the remainders computed this way become subresultants with the help of the Akritas-Malaschonok-Vigklas Theorem of 2015 and the Collins-Brown- Traub formula for coefficient reduction. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' Serdica Journal of Computing 10 (2016), No.3-4, 197-217. """ # make sure neither f nor g is 0 if f == 0 or g == 0: return [f, g] # make sure proper degrees d0 = degree(f, x) d1 = degree(g, x) if d0 == 0 and d1 == 0: return [f, g] if d1 > d0: d0, d1 = d1, d0 f, g = g, f if d0 > 0 and d1 == 0: return [f, g] # initialize a0 = f a1 = g subres_l = [a0, a1] deg_dif_p1, c = degree(a0, x) - degree(a1, x) + 1, -1 # initialize AMV variables sigma1 = LC(a1, x) # leading coeff of a1 i, s = 0, 0 # counters for remainders & odd elements p_odd_index_sum = 0 # contains the sum of p_1, p_3, etc p0 = deg_dif_p1 - 1 if p0 % 2 == 1: s += 1 phi = floor( (s + 1) / 2 ) # compute the first polynomial of the prs i += 1 a2 = rem_z(a0, a1, x) / Abs( (-1)**deg_dif_p1 ) # first remainder sigma2 = LC(a2, x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 p1 = d1 - d2 # degree difference # sgn_den is the factor, the denominator 1st fraction of (9), # by which a2 is multiplied to get integer coefficients sgn_den = compute_sign( sigma1, p0 + 1 ) ## compute sign of the 1st fraction in formula (9) of the paper # numerator psi = i + phi + p_odd_index_sum num = (-1)**psi # denominator den = sgn_den # the sign of the determinant depends on sign(num / den) != 0 if sign(num / den) > 0: subres_l.append( a2 ) else: subres_l.append( -a2 ) # update AMV variable if p1 % 2 == 1: s += 1 # bring in the missing power of sigma if there was gap if p1 - 1 > 0: sgn_den = sgn_den * compute_sign( sigma1, p1 - 1 ) # main loop while d2 >= 1: phi = floor( (s + 1) / 2 ) if i%2 == 1: p_odd_index_sum += p1 # p_i has odd index a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees p0 = p1 # update degree difference i += 1 sigma0 = -LC(a0) c = (sigma0**(deg_dif_p1 - 1)) / (c**(deg_dif_p1 - 2)) deg_dif_p1 = degree(a0, x) - d2 + 1 a2 = rem_z(a0, a1, x) / Abs( (c**(deg_dif_p1 - 1)) * sigma0 ) sigma3 = LC(a2, x) # leading coeff of a2 d2 = degree(a2, x) # actual degree of a2 p1 = d1 - d2 # degree difference psi = i + phi + p_odd_index_sum # update variables sigma1, sigma2 = sigma2, sigma3 # new sgn_den sgn_den = compute_sign( sigma1, p0 + 1 ) * sgn_den # compute the sign of the first fraction in formula (9) of the paper # numerator num = (-1)**psi # denominator den = sgn_den # the sign of the determinant depends on sign( num / den ) != 0 if sign(num / den) > 0: subres_l.append( a2 ) else: subres_l.append( -a2 ) # update AMV variable if p1 % 2 ==1: s += 1 # bring in the missing power of sigma if there was gap if p1 - 1 > 0: sgn_den = sgn_den * compute_sign( sigma1, p1 - 1 ) # gcd is of degree > 0 ? m = len(subres_l) if subres_l[m - 1] == nan or subres_l[m - 1] == 0: subres_l.pop(m - 1) return subres_l def modified_subresultants_amv(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the modified subresultant prs of p and q in Z[x] or Q[x], from the subresultant prs of p and q. The coefficients of the polynomials in the two sequences differ only in sign and the factor LC(p)**( deg(p)- deg(q)) as stated in Theorem 2 of the reference. The coefficients of the polynomials in the output sequence are modified subresultants. That is, they are determinants of appropriately selected submatrices of sylvester2, Sylvester's matrix of 1853. If the modified subresultant prs is complete, and LC( p ) > 0, it coincides with the (generalized) Sturm's sequence of the polynomials p, q. References ========== 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: "On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials." Serdica Journal of Computing, Serdica Journal of Computing, 9(2) (2015), 123-138. """ # compute the subresultant prs lst = subresultants_amv(p,q,x) ## any other method would do # defensive if lst == [] or len(lst) == 2: return lst # the coefficients in lst are subresultants and, hence, smaller than those # of the corresponding modified subresultants by the factor # LC(lst[0])**( deg(lst[0]) - deg(lst[1])); see Theorem 2. lcf = LC(lst[0])**( degree(lst[0], x) - degree(lst[1], x) ) # Initialize the modified subresultant prs list subr_seq = [lst[0], lst[1]] # compute the degree sequences m_i and j_i of Theorem 2 deg_seq = [degree(Poly(poly, x), x) for poly in lst] deg = deg_seq[0] deg_seq_s = deg_seq[1:-1] m_seq = [m-1 for m in deg_seq_s] j_seq = [deg - m for m in m_seq] # compute the AMV factors of Theorem 2 fact = [(-1)**( j*(j-1)/S(2) ) for j in j_seq] # shortened list without the first two polys lst_s = lst[2:] # poly lst_s[k] is multiplied times fact[k] and times lcf # and appended to the subresultant prs list m = len(fact) for k in range(m): if sign(fact[k]) == -1: subr_seq.append( simplify(-lst_s[k] * lcf) ) else: subr_seq.append( simplify(lst_s[k] * lcf) ) return subr_seq def correct_sign(deg_f, deg_g, s1, rdel, cdel): """ Used in various subresultant prs algorithms. Evaluates the determinant, (a.k.a. subresultant) of a properly selected submatrix of s1, Sylvester's matrix of 1840, to get the correct sign and value of the leading coefficient of a given polynomial remainder. deg_f, deg_g are the degrees of the original polynomials p, q for which the matrix s1 = sylvester(p, q, x, 1) was constructed. rdel denotes the expected degree of the remainder; it is the number of rows to be deleted from each group of rows in s1 as described in the reference below. cdel denotes the expected degree minus the actual degree of the remainder; it is the number of columns to be deleted --- starting with the last column forming the square matrix --- from the matrix resulting after the row deletions. References ========== Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences.'' Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. """ M = s1[:, :] # copy of matrix s1 # eliminate rdel rows from the first deg_g rows for i in range(M.rows - deg_f - 1, M.rows - deg_f - rdel - 1, -1): M.row_del(i) # eliminate rdel rows from the last deg_f rows for i in range(M.rows - 1, M.rows - rdel - 1, -1): M.row_del(i) # eliminate cdel columns for i in range(cdel): M.col_del(M.rows - 1) # define submatrix Md = M[:, 0: M.rows] return Md.det() def subresultants_rem(p, q, x): """ p, q are polynomials in Z[x] or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant prs of p and q in Z[x] or Q[x]; the coefficients of the polynomials in the sequence are subresultants. That is, they are determinants of appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. To compute the coefficients polynomial divisions in Q[x] are performed, using the function rem(p, q, x). The coefficients of the remainders computed this way become subresultants by evaluating one subresultant per remainder --- that of the leading coefficient. This way we obtain the correct sign and value of the leading coefficient of the remainder and we easily ``force'' the rest of the coefficients to become subresultants. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. Akritas, A. G.:``Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees f, g = p, q n = deg_f = degree(f, x) m = deg_g = degree(g, x) if n == 0 and m == 0: return [f, g] if n < m: n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f if n > 0 and m == 0: return [f, g] # initialize s1 = sylvester(f, g, x, 1) sr_list = [f, g] # subresultant list # main loop while deg_g > 0: r = rem(p, q, x) d = degree(r, x) if d < 0: return sr_list # make coefficients subresultants evaluating ONE determinant exp_deg = deg_g - 1 # expected degree sign_value = correct_sign(n, m, s1, exp_deg, exp_deg - d) r = simplify((r / LC(r, x)) * sign_value) # append poly with subresultant coeffs sr_list.append(r) # update degrees and polys deg_f, deg_g = deg_g, d p, q = q, r # gcd is of degree > 0 ? m = len(sr_list) if sr_list[m - 1] == nan or sr_list[m - 1] == 0: sr_list.pop(m - 1) return sr_list def pivot(M, i, j): ''' M is a matrix, and M[i, j] specifies the pivot element. All elements below M[i, j], in the j-th column, will be zeroed, if they are not already 0, according to Dodgson-Bareiss' integer preserving transformations. References ========== 1. Akritas, A. G.: ``A new method for computing polynomial greatest common divisors and polynomial remainder sequences.'' Numerische MatheMatik 52, 119-127, 1988. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem by Van Vleck Regarding Sturm Sequences.'' Serdica Journal of Computing, 7, No 4, 101-134, 2013. ''' ma = M[:, :] # copy of matrix M rs = ma.rows # No. of rows cs = ma.cols # No. of cols for r in range(i+1, rs): if ma[r, j] != 0: for c in range(j + 1, cs): ma[r, c] = ma[i, j] * ma[r, c] - ma[i, c] * ma[r, j] ma[r, j] = 0 return ma def rotate_r(L, k): ''' Rotates right by k. L is a row of a matrix or a list. ''' ll = list(L) if ll == []: return [] for i in range(k): el = ll.pop(len(ll) - 1) ll.insert(0, el) return ll if type(L) is list else Matrix([ll]) def rotate_l(L, k): ''' Rotates left by k. L is a row of a matrix or a list. ''' ll = list(L) if ll == []: return [] for i in range(k): el = ll.pop(0) ll.insert(len(ll) - 1, el) return ll if type(L) is list else Matrix([ll]) def row2poly(row, deg, x): ''' Converts the row of a matrix to a poly of degree deg and variable x. Some entries at the beginning and/or at the end of the row may be zero. ''' k = 0 poly = [] leng = len(row) # find the beginning of the poly ; i.e. the first # non-zero element of the row while row[k] == 0: k = k + 1 # append the next deg + 1 elements to poly for j in range( deg + 1): if k + j <= leng: poly.append(row[k + j]) return Poly(poly, x) def create_ma(deg_f, deg_g, row1, row2, col_num): ''' Creates a ``small'' matrix M to be triangularized. deg_f, deg_g are the degrees of the divident and of the divisor polynomials respectively, deg_g > deg_f. The coefficients of the divident poly are the elements in row2 and those of the divisor poly are the elements in row1. col_num defines the number of columns of the matrix M. ''' if deg_g - deg_f >= 1: print('Reverse degrees') return m = zeros(deg_f - deg_g + 2, col_num) for i in range(deg_f - deg_g + 1): m[i, :] = rotate_r(row1, i) m[deg_f - deg_g + 1, :] = row2 return m def find_degree(M, deg_f): ''' Finds the degree of the poly corresponding (after triangularization) to the _last_ row of the ``small'' matrix M, created by create_ma(). deg_f is the degree of the divident poly. If _last_ row is all 0's returns None. ''' j = deg_f for i in range(0, M.cols): if M[M.rows - 1, i] == 0: j = j - 1 else: return j if j >= 0 else 0 def final_touches(s2, r, deg_g): """ s2 is sylvester2, r is the row pointer in s2, deg_g is the degree of the poly last inserted in s2. After a gcd of degree > 0 has been found with Van Vleck's method, and was inserted into s2, if its last term is not in the last column of s2, then it is inserted as many times as needed, rotated right by one each time, until the condition is met. """ R = s2.row(r-1) # find the first non zero term for i in range(s2.cols): if R[0,i] == 0: continue else: break # missing rows until last term is in last column mr = s2.cols - (i + deg_g + 1) # insert them by replacing the existing entries in the row i = 0 while mr != 0 and r + i < s2.rows : s2[r + i, : ] = rotate_r(R, i + 1) i += 1 mr -= 1 return s2 def subresultants_vv(p, q, x, method = 0): """ p, q are polynomials in Z[x] (intended) or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant prs of p, q by triangularizing, in Z[x] or in Q[x], all the smaller matrices encountered in the process of triangularizing sylvester2, Sylvester's matrix of 1853; see references 1 and 2 for Van Vleck's method. With each remainder, sylvester2 gets updated and is prepared to be printed if requested. If sylvester2 has small dimensions and you want to see the final, triangularized matrix use this version with method=1; otherwise, use either this version with method=0 (default) or the faster version, subresultants_vv_2(p, q, x), where sylvester2 is used implicitly. Sylvester's matrix sylvester1 is also used to compute one subresultant per remainder; namely, that of the leading coefficient, in order to obtain the correct sign and to force the remainder coefficients to become subresultants. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. If the final, triangularized matrix s2 is printed, then: (a) if deg(p) - deg(q) > 1 or deg( gcd(p, q) ) > 0, several of the last rows in s2 will remain unprocessed; (b) if deg(p) - deg(q) == 0, p will not appear in the final matrix. References ========== 1. Akritas, A. G.: ``A new method for computing polynomial greatest common divisors and polynomial remainder sequences.'' Numerische MatheMatik 52, 119-127, 1988. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem by Van Vleck Regarding Sturm Sequences.'' Serdica Journal of Computing, 7, No 4, 101-134, 2013. 3. Akritas, A. G.:``Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees f, g = p, q n = deg_f = degree(f, x) m = deg_g = degree(g, x) if n == 0 and m == 0: return [f, g] if n < m: n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f if n > 0 and m == 0: return [f, g] # initialize s1 = sylvester(f, g, x, 1) s2 = sylvester(f, g, x, 2) sr_list = [f, g] col_num = 2 * n # columns in s2 # make two rows (row0, row1) of poly coefficients row0 = Poly(f, x, domain = QQ).all_coeffs() leng0 = len(row0) for i in range(col_num - leng0): row0.append(0) row0 = Matrix([row0]) row1 = Poly(g,x, domain = QQ).all_coeffs() leng1 = len(row1) for i in range(col_num - leng1): row1.append(0) row1 = Matrix([row1]) # row pointer for deg_f - deg_g == 1; may be reset below r = 2 # modify first rows of s2 matrix depending on poly degrees if deg_f - deg_g > 1: r = 1 # replacing the existing entries in the rows of s2, # insert row0 (deg_f - deg_g - 1) times, rotated each time for i in range(deg_f - deg_g - 1): s2[r + i, : ] = rotate_r(row0, i + 1) r = r + deg_f - deg_g - 1 # insert row1 (deg_f - deg_g) times, rotated each time for i in range(deg_f - deg_g): s2[r + i, : ] = rotate_r(row1, r + i) r = r + deg_f - deg_g if deg_f - deg_g == 0: r = 0 # main loop while deg_g > 0: # create a small matrix M, and triangularize it; M = create_ma(deg_f, deg_g, row1, row0, col_num) # will need only the first and last rows of M for i in range(deg_f - deg_g + 1): M1 = pivot(M, i, i) M = M1[:, :] # treat last row of M as poly; find its degree d = find_degree(M, deg_f) if d is None: break exp_deg = deg_g - 1 # evaluate one determinant & make coefficients subresultants sign_value = correct_sign(n, m, s1, exp_deg, exp_deg - d) poly = row2poly(M[M.rows - 1, :], d, x) temp2 = LC(poly, x) poly = simplify((poly / temp2) * sign_value) # update s2 by inserting first row of M as needed row0 = M[0, :] for i in range(deg_g - d): s2[r + i, :] = rotate_r(row0, r + i) r = r + deg_g - d # update s2 by inserting last row of M as needed row1 = rotate_l(M[M.rows - 1, :], deg_f - d) row1 = (row1 / temp2) * sign_value for i in range(deg_g - d): s2[r + i, :] = rotate_r(row1, r + i) r = r + deg_g - d # update degrees deg_f, deg_g = deg_g, d # append poly with subresultant coeffs sr_list.append(poly) # final touches to print the s2 matrix if method != 0 and s2.rows > 2: s2 = final_touches(s2, r, deg_g) pprint(s2) elif method != 0 and s2.rows == 2: s2[1, :] = rotate_r(s2.row(1), 1) pprint(s2) return sr_list def subresultants_vv_2(p, q, x): """ p, q are polynomials in Z[x] (intended) or Q[x]. It is assumed that degree(p, x) >= degree(q, x). Computes the subresultant prs of p, q by triangularizing, in Z[x] or in Q[x], all the smaller matrices encountered in the process of triangularizing sylvester2, Sylvester's matrix of 1853; see references 1 and 2 for Van Vleck's method. If the sylvester2 matrix has big dimensions use this version, where sylvester2 is used implicitly. If you want to see the final, triangularized matrix sylvester2, then use the first version, subresultants_vv(p, q, x, 1). sylvester1, Sylvester's matrix of 1840, is also used to compute one subresultant per remainder; namely, that of the leading coefficient, in order to obtain the correct sign and to ``force'' the remainder coefficients to become subresultants. If the subresultant prs is complete, then it coincides with the Euclidean sequence of the polynomials p, q. References ========== 1. Akritas, A. G.: ``A new method for computing polynomial greatest common divisors and polynomial remainder sequences.'' Numerische MatheMatik 52, 119-127, 1988. 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem by Van Vleck Regarding Sturm Sequences.'' Serdica Journal of Computing, 7, No 4, 101-134, 2013. 3. Akritas, A. G.:``Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. """ # make sure neither p nor q is 0 if p == 0 or q == 0: return [p, q] # make sure proper degrees f, g = p, q n = deg_f = degree(f, x) m = deg_g = degree(g, x) if n == 0 and m == 0: return [f, g] if n < m: n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f if n > 0 and m == 0: return [f, g] # initialize s1 = sylvester(f, g, x, 1) sr_list = [f, g] # subresultant list col_num = 2 * n # columns in sylvester2 # make two rows (row0, row1) of poly coefficients row0 = Poly(f, x, domain = QQ).all_coeffs() leng0 = len(row0) for i in range(col_num - leng0): row0.append(0) row0 = Matrix([row0]) row1 = Poly(g,x, domain = QQ).all_coeffs() leng1 = len(row1) for i in range(col_num - leng1): row1.append(0) row1 = Matrix([row1]) # main loop while deg_g > 0: # create a small matrix M, and triangularize it M = create_ma(deg_f, deg_g, row1, row0, col_num) for i in range(deg_f - deg_g + 1): M1 = pivot(M, i, i) M = M1[:, :] # treat last row of M as poly; find its degree d = find_degree(M, deg_f) if d is None: return sr_list exp_deg = deg_g - 1 # evaluate one determinant & make coefficients subresultants sign_value = correct_sign(n, m, s1, exp_deg, exp_deg - d) poly = row2poly(M[M.rows - 1, :], d, x) poly = simplify((poly / LC(poly, x)) * sign_value) # append poly with subresultant coeffs sr_list.append(poly) # update degrees and rows deg_f, deg_g = deg_g, d row0 = row1 row1 = Poly(poly, x, domain = QQ).all_coeffs() leng1 = len(row1) for i in range(col_num - leng1): row1.append(0) row1 = Matrix([row1]) return sr_list
260822771c45153ac75f3e6544c0f6a3667f766255193925959e977834c4160e
"""Sparse polynomial rings. """ from typing import Any, Dict from operator import add, mul, lt, le, gt, ge from types import GeneratorType from sympy.core.compatibility import is_sequence, reduce from sympy.core.expr import Expr from sympy.core.numbers import igcd, oo from sympy.core.symbol import Symbol, symbols as _symbols from sympy.core.sympify import CantSympify, sympify from sympy.ntheory.multinomial import multinomial_coefficients from sympy.polys.compatibility import IPolys from sympy.polys.constructor import construct_domain from sympy.polys.densebasic import dmp_to_dict, dmp_from_dict from sympy.polys.domains.domainelement import DomainElement from sympy.polys.domains.polynomialring import PolynomialRing from sympy.polys.heuristicgcd import heugcd from sympy.polys.monomials import MonomialOps from sympy.polys.orderings import lex from sympy.polys.polyerrors import ( CoercionFailed, GeneratorsError, ExactQuotientFailed, MultivariatePolynomialError) from sympy.polys.polyoptions import (Domain as DomainOpt, Order as OrderOpt, build_options) from sympy.polys.polyutils import (expr_from_dict, _dict_reorder, _parallel_dict_from_expr) from sympy.printing.defaults import DefaultPrinting from sympy.utilities import public from sympy.utilities.magic import pollute @public def ring(symbols, domain, order=lex): """Construct a polynomial ring returning ``(ring, x_1, ..., x_n)``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, x, y, z = ring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) <class 'sympy.polys.rings.PolyElement'> """ _ring = PolyRing(symbols, domain, order) return (_ring,) + _ring.gens @public def xring(symbols, domain, order=lex): """Construct a polynomial ring returning ``(ring, (x_1, ..., x_n))``. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import xring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> R, (x, y, z) = xring("x,y,z", ZZ, lex) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z x + y + z >>> type(_) <class 'sympy.polys.rings.PolyElement'> """ _ring = PolyRing(symbols, domain, order) return (_ring, _ring.gens) @public def vring(symbols, domain, order=lex): """Construct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace. Parameters ========== symbols : str Symbol/Expr or sequence of str, Symbol/Expr (non-empty) domain : :class:`~.Domain` or coercible order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` Examples ======== >>> from sympy.polys.rings import vring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex >>> vring("x,y,z", ZZ, lex) Polynomial ring in x, y, z over ZZ with lex order >>> x + y + z # noqa: x + y + z >>> type(_) <class 'sympy.polys.rings.PolyElement'> """ _ring = PolyRing(symbols, domain, order) pollute([ sym.name for sym in _ring.symbols ], _ring.gens) return _ring @public def sring(exprs, *symbols, **options): """Construct a ring deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` options : keyword arguments understood by :class:`~.Options` Examples ======== >>> from sympy.core import symbols >>> from sympy.polys.rings import sring >>> x, y, z = symbols("x,y,z") >>> R, f = sring(x + 2*y + 3*z) >>> R Polynomial ring in x, y, z over ZZ with lex order >>> f x + 2*y + 3*z >>> type(_) <class 'sympy.polys.rings.PolyElement'> """ single = False if not is_sequence(exprs): exprs, single = [exprs], True exprs = list(map(sympify, exprs)) opt = build_options(symbols, options) # TODO: rewrite this so that it doesn't use expand() (see poly()). reps, opt = _parallel_dict_from_expr(exprs, opt) if opt.domain is None: # NOTE: this is inefficient because construct_domain() automatically # performs conversion to the target domain. It shouldn't do this. coeffs = sum([ list(rep.values()) for rep in reps ], []) opt.domain, _ = construct_domain(coeffs, opt=opt) _ring = PolyRing(opt.gens, opt.domain, opt.order) polys = list(map(_ring.from_dict, reps)) if single: return (_ring, polys[0]) else: return (_ring, polys) def _parse_symbols(symbols): if isinstance(symbols, str): return _symbols(symbols, seq=True) if symbols else () elif isinstance(symbols, Expr): return (symbols,) elif is_sequence(symbols): if all(isinstance(s, str) for s in symbols): return _symbols(symbols) elif all(isinstance(s, Expr) for s in symbols): return symbols raise GeneratorsError("expected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions") _ring_cache = {} # type: Dict[Any, Any] class PolyRing(DefaultPrinting, IPolys): """Multivariate distributed polynomial ring. """ def __new__(cls, symbols, domain, order=lex): symbols = tuple(_parse_symbols(symbols)) ngens = len(symbols) domain = DomainOpt.preprocess(domain) order = OrderOpt.preprocess(order) _hash_tuple = (cls.__name__, symbols, ngens, domain, order) obj = _ring_cache.get(_hash_tuple) if obj is None: if domain.is_Composite and set(symbols) & set(domain.symbols): raise GeneratorsError("polynomial ring and it's ground domain share generators") obj = object.__new__(cls) obj._hash_tuple = _hash_tuple obj._hash = hash(_hash_tuple) obj.dtype = type("PolyElement", (PolyElement,), {"ring": obj}) obj.symbols = symbols obj.ngens = ngens obj.domain = domain obj.order = order obj.zero_monom = (0,)*ngens obj.gens = obj._gens() obj._gens_set = set(obj.gens) obj._one = [(obj.zero_monom, domain.one)] if ngens: # These expect monomials in at least one variable codegen = MonomialOps(ngens) obj.monomial_mul = codegen.mul() obj.monomial_pow = codegen.pow() obj.monomial_mulpow = codegen.mulpow() obj.monomial_ldiv = codegen.ldiv() obj.monomial_div = codegen.div() obj.monomial_lcm = codegen.lcm() obj.monomial_gcd = codegen.gcd() else: monunit = lambda a, b: () obj.monomial_mul = monunit obj.monomial_pow = monunit obj.monomial_mulpow = lambda a, b, c: () obj.monomial_ldiv = monunit obj.monomial_div = monunit obj.monomial_lcm = monunit obj.monomial_gcd = monunit if order is lex: obj.leading_expv = lambda f: max(f) else: obj.leading_expv = lambda f: max(f, key=order) for symbol, generator in zip(obj.symbols, obj.gens): if isinstance(symbol, Symbol): name = symbol.name if not hasattr(obj, name): setattr(obj, name, generator) _ring_cache[_hash_tuple] = obj return obj def _gens(self): """Return a list of polynomial generators. """ one = self.domain.one _gens = [] for i in range(self.ngens): expv = self.monomial_basis(i) poly = self.zero poly[expv] = one _gens.append(poly) return tuple(_gens) def __getnewargs__(self): return (self.symbols, self.domain, self.order) def __getstate__(self): state = self.__dict__.copy() del state["leading_expv"] for key, value in state.items(): if key.startswith("monomial_"): del state[key] return state def __hash__(self): return self._hash def __eq__(self, other): return isinstance(other, PolyRing) and \ (self.symbols, self.domain, self.ngens, self.order) == \ (other.symbols, other.domain, other.ngens, other.order) def __ne__(self, other): return not self == other def clone(self, symbols=None, domain=None, order=None): return self.__class__(symbols or self.symbols, domain or self.domain, order or self.order) def monomial_basis(self, i): """Return the ith-basis element. """ basis = [0]*self.ngens basis[i] = 1 return tuple(basis) @property def zero(self): return self.dtype() @property def one(self): return self.dtype(self._one) def domain_new(self, element, orig_domain=None): return self.domain.convert(element, orig_domain) def ground_new(self, coeff): return self.term_new(self.zero_monom, coeff) def term_new(self, monom, coeff): coeff = self.domain_new(coeff) poly = self.zero if coeff: poly[monom] = coeff return poly def ring_new(self, element): if isinstance(element, PolyElement): if self == element.ring: return element elif isinstance(self.domain, PolynomialRing) and self.domain.ring == element.ring: return self.ground_new(element) else: raise NotImplementedError("conversion") elif isinstance(element, str): raise NotImplementedError("parsing") elif isinstance(element, dict): return self.from_dict(element) elif isinstance(element, list): try: return self.from_terms(element) except ValueError: return self.from_list(element) elif isinstance(element, Expr): return self.from_expr(element) else: return self.ground_new(element) __call__ = ring_new def from_dict(self, element): domain_new = self.domain_new poly = self.zero for monom, coeff in element.items(): coeff = domain_new(coeff) if coeff: poly[monom] = coeff return poly def from_terms(self, element): return self.from_dict(dict(element)) def from_list(self, element): return self.from_dict(dmp_to_dict(element, self.ngens-1, self.domain)) def _rebuild_expr(self, expr, mapping): domain = self.domain def _rebuild(expr): generator = mapping.get(expr) if generator is not None: return generator elif expr.is_Add: return reduce(add, list(map(_rebuild, expr.args))) elif expr.is_Mul: return reduce(mul, list(map(_rebuild, expr.args))) elif expr.is_Pow and expr.exp.is_Integer and expr.exp >= 0: return _rebuild(expr.base)**int(expr.exp) else: return domain.convert(expr) return _rebuild(sympify(expr)) def from_expr(self, expr): mapping = dict(list(zip(self.symbols, self.gens))) try: poly = self._rebuild_expr(expr, mapping) except CoercionFailed: raise ValueError("expected an expression convertible to a polynomial in %s, got %s" % (self, expr)) else: return self.ring_new(poly) def index(self, gen): """Compute index of ``gen`` in ``self.gens``. """ if gen is None: if self.ngens: i = 0 else: i = -1 # indicate impossible choice elif isinstance(gen, int): i = gen if 0 <= i and i < self.ngens: pass elif -self.ngens <= i and i <= -1: i = -i - 1 else: raise ValueError("invalid generator index: %s" % gen) elif isinstance(gen, self.dtype): try: i = self.gens.index(gen) except ValueError: raise ValueError("invalid generator: %s" % gen) elif isinstance(gen, str): try: i = self.symbols.index(gen) except ValueError: raise ValueError("invalid generator: %s" % gen) else: raise ValueError("expected a polynomial generator, an integer, a string or None, got %s" % gen) return i def drop(self, *gens): """Remove specified generators from this ring. """ indices = set(map(self.index, gens)) symbols = [ s for i, s in enumerate(self.symbols) if i not in indices ] if not symbols: return self.domain else: return self.clone(symbols=symbols) def __getitem__(self, key): symbols = self.symbols[key] if not symbols: return self.domain else: return self.clone(symbols=symbols) def to_ground(self): # TODO: should AlgebraicField be a Composite domain? if self.domain.is_Composite or hasattr(self.domain, 'domain'): return self.clone(domain=self.domain.domain) else: raise ValueError("%s is not a composite domain" % self.domain) def to_domain(self): return PolynomialRing(self) def to_field(self): from sympy.polys.fields import FracField return FracField(self.symbols, self.domain, self.order) @property def is_univariate(self): return len(self.gens) == 1 @property def is_multivariate(self): return len(self.gens) > 1 def add(self, *objs): """ Add a sequence of polynomials or containers of polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> R, x = ring("x", ZZ) >>> R.add([ x**2 + 2*i + 3 for i in range(4) ]) 4*x**2 + 24 >>> _.factor_list() (4, [(x**2 + 6, 1)]) """ p = self.zero for obj in objs: if is_sequence(obj, include=GeneratorType): p += self.add(*obj) else: p += obj return p def mul(self, *objs): """ Multiply a sequence of polynomials or containers of polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> R, x = ring("x", ZZ) >>> R.mul([ x**2 + 2*i + 3 for i in range(4) ]) x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945 >>> _.factor_list() (1, [(x**2 + 3, 1), (x**2 + 5, 1), (x**2 + 7, 1), (x**2 + 9, 1)]) """ p = self.one for obj in objs: if is_sequence(obj, include=GeneratorType): p *= self.mul(*obj) else: p *= obj return p def drop_to_ground(self, *gens): r""" Remove specified generators from the ring and inject them into its domain. """ indices = set(map(self.index, gens)) symbols = [s for i, s in enumerate(self.symbols) if i not in indices] gens = [gen for i, gen in enumerate(self.gens) if i not in indices] if not symbols: return self else: return self.clone(symbols=symbols, domain=self.drop(*gens)) def compose(self, other): """Add the generators of ``other`` to ``self``""" if self != other: syms = set(self.symbols).union(set(other.symbols)) return self.clone(symbols=list(syms)) else: return self def add_gens(self, symbols): """Add the elements of ``symbols`` as generators to ``self``""" syms = set(self.symbols).union(set(symbols)) return self.clone(symbols=list(syms)) class PolyElement(DomainElement, DefaultPrinting, CantSympify, dict): """Element of multivariate distributed polynomial ring. """ def new(self, init): return self.__class__(init) def parent(self): return self.ring.to_domain() def __getnewargs__(self): return (self.ring, list(self.iterterms())) _hash = None def __hash__(self): # XXX: This computes a hash of a dictionary, but currently we don't # protect dictionary from being changed so any use site modifications # will make hashing go wrong. Use this feature with caution until we # figure out how to make a safe API without compromising speed of this # low-level class. _hash = self._hash if _hash is None: self._hash = _hash = hash((self.ring, frozenset(self.items()))) return _hash def copy(self): """Return a copy of polynomial self. Polynomials are mutable; if one is interested in preserving a polynomial, and one plans to use inplace operations, one can copy the polynomial. This method makes a shallow copy. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> R, x, y = ring('x, y', ZZ) >>> p = (x + y)**2 >>> p1 = p.copy() >>> p2 = p >>> p[R.zero_monom] = 3 >>> p x**2 + 2*x*y + y**2 + 3 >>> p1 x**2 + 2*x*y + y**2 >>> p2 x**2 + 2*x*y + y**2 + 3 """ return self.new(self) def set_ring(self, new_ring): if self.ring == new_ring: return self elif self.ring.symbols != new_ring.symbols: terms = list(zip(*_dict_reorder(self, self.ring.symbols, new_ring.symbols))) return new_ring.from_terms(terms) else: return new_ring.from_dict(self) def as_expr(self, *symbols): if symbols and len(symbols) != self.ring.ngens: raise ValueError("not enough symbols, expected %s got %s" % (self.ring.ngens, len(symbols))) else: symbols = self.ring.symbols return expr_from_dict(self.as_expr_dict(), *symbols) def as_expr_dict(self): to_sympy = self.ring.domain.to_sympy return {monom: to_sympy(coeff) for monom, coeff in self.iterterms()} def clear_denoms(self): domain = self.ring.domain if not domain.is_Field or not domain.has_assoc_Ring: return domain.one, self ground_ring = domain.get_ring() common = ground_ring.one lcm = ground_ring.lcm denom = domain.denom for coeff in self.values(): common = lcm(common, denom(coeff)) poly = self.new([ (k, v*common) for k, v in self.items() ]) return common, poly def strip_zero(self): """Eliminate monomials with zero coefficient. """ for k, v in list(self.items()): if not v: del self[k] def __eq__(p1, p2): """Equality test for polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p1 = (x + y)**2 + (x - y)**2 >>> p1 == 4*x*y False >>> p1 == 2*(x**2 + y**2) True """ if not p2: return not p1 elif isinstance(p2, PolyElement) and p2.ring == p1.ring: return dict.__eq__(p1, p2) elif len(p1) > 1: return False else: return p1.get(p1.ring.zero_monom) == p2 def __ne__(p1, p2): return not p1 == p2 def almosteq(p1, p2, tolerance=None): """Approximate equality test for polynomials. """ ring = p1.ring if isinstance(p2, ring.dtype): if set(p1.keys()) != set(p2.keys()): return False almosteq = ring.domain.almosteq for k in p1.keys(): if not almosteq(p1[k], p2[k], tolerance): return False return True elif len(p1) > 1: return False else: try: p2 = ring.domain.convert(p2) except CoercionFailed: return False else: return ring.domain.almosteq(p1.const(), p2, tolerance) def sort_key(self): return (len(self), self.terms()) def _cmp(p1, p2, op): if isinstance(p2, p1.ring.dtype): return op(p1.sort_key(), p2.sort_key()) else: return NotImplemented def __lt__(p1, p2): return p1._cmp(p2, lt) def __le__(p1, p2): return p1._cmp(p2, le) def __gt__(p1, p2): return p1._cmp(p2, gt) def __ge__(p1, p2): return p1._cmp(p2, ge) def _drop(self, gen): ring = self.ring i = ring.index(gen) if ring.ngens == 1: return i, ring.domain else: symbols = list(ring.symbols) del symbols[i] return i, ring.clone(symbols=symbols) def drop(self, gen): i, ring = self._drop(gen) if self.ring.ngens == 1: if self.is_ground: return self.coeff(1) else: raise ValueError("can't drop %s" % gen) else: poly = ring.zero for k, v in self.items(): if k[i] == 0: K = list(k) del K[i] poly[tuple(K)] = v else: raise ValueError("can't drop %s" % gen) return poly def _drop_to_ground(self, gen): ring = self.ring i = ring.index(gen) symbols = list(ring.symbols) del symbols[i] return i, ring.clone(symbols=symbols, domain=ring[i]) def drop_to_ground(self, gen): if self.ring.ngens == 1: raise ValueError("can't drop only generator to ground") i, ring = self._drop_to_ground(gen) poly = ring.zero gen = ring.domain.gens[0] for monom, coeff in self.iterterms(): mon = monom[:i] + monom[i+1:] if not mon in poly: poly[mon] = (gen**monom[i]).mul_ground(coeff) else: poly[mon] += (gen**monom[i]).mul_ground(coeff) return poly def to_dense(self): return dmp_from_dict(self, self.ring.ngens-1, self.ring.domain) def to_dict(self): return dict(self) def str(self, printer, precedence, exp_pattern, mul_symbol): if not self: return printer._print(self.ring.domain.zero) prec_mul = precedence["Mul"] prec_atom = precedence["Atom"] ring = self.ring symbols = ring.symbols ngens = ring.ngens zm = ring.zero_monom sexpvs = [] for expv, coeff in self.terms(): negative = ring.domain.is_negative(coeff) sign = " - " if negative else " + " sexpvs.append(sign) if expv == zm: scoeff = printer._print(coeff) if negative and scoeff.startswith("-"): scoeff = scoeff[1:] else: if negative: coeff = -coeff if coeff != self.ring.one: scoeff = printer.parenthesize(coeff, prec_mul, strict=True) else: scoeff = '' sexpv = [] for i in range(ngens): exp = expv[i] if not exp: continue symbol = printer.parenthesize(symbols[i], prec_atom, strict=True) if exp != 1: if exp != int(exp) or exp < 0: sexp = printer.parenthesize(exp, prec_atom, strict=False) else: sexp = exp sexpv.append(exp_pattern % (symbol, sexp)) else: sexpv.append('%s' % symbol) if scoeff: sexpv = [scoeff] + sexpv sexpvs.append(mul_symbol.join(sexpv)) if sexpvs[0] in [" + ", " - "]: head = sexpvs.pop(0) if head == " - ": sexpvs.insert(0, "-") return "".join(sexpvs) @property def is_generator(self): return self in self.ring._gens_set @property def is_ground(self): return not self or (len(self) == 1 and self.ring.zero_monom in self) @property def is_monomial(self): return not self or (len(self) == 1 and self.LC == 1) @property def is_term(self): return len(self) <= 1 @property def is_negative(self): return self.ring.domain.is_negative(self.LC) @property def is_positive(self): return self.ring.domain.is_positive(self.LC) @property def is_nonnegative(self): return self.ring.domain.is_nonnegative(self.LC) @property def is_nonpositive(self): return self.ring.domain.is_nonpositive(self.LC) @property def is_zero(f): return not f @property def is_one(f): return f == f.ring.one @property def is_monic(f): return f.ring.domain.is_one(f.LC) @property def is_primitive(f): return f.ring.domain.is_one(f.content()) @property def is_linear(f): return all(sum(monom) <= 1 for monom in f.itermonoms()) @property def is_quadratic(f): return all(sum(monom) <= 2 for monom in f.itermonoms()) @property def is_squarefree(f): if not f.ring.ngens: return True return f.ring.dmp_sqf_p(f) @property def is_irreducible(f): if not f.ring.ngens: return True return f.ring.dmp_irreducible_p(f) @property def is_cyclotomic(f): if f.ring.is_univariate: return f.ring.dup_cyclotomic_p(f) else: raise MultivariatePolynomialError("cyclotomic polynomial") def __neg__(self): return self.new([ (monom, -coeff) for monom, coeff in self.iterterms() ]) def __pos__(self): return self def __add__(p1, p2): """Add two polynomials. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> (x + y)**2 + (x - y)**2 2*x**2 + 2*y**2 """ if not p2: return p1.copy() ring = p1.ring if isinstance(p2, ring.dtype): p = p1.copy() get = p.get zero = ring.domain.zero for k, v in p2.items(): v = get(k, zero) + v if v: p[k] = v else: del p[k] return p elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__radd__(p1) else: return NotImplemented try: cp2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: p = p1.copy() if not cp2: return p zm = ring.zero_monom if zm not in p1.keys(): p[zm] = cp2 else: if p2 == -p[zm]: del p[zm] else: p[zm] += cp2 return p def __radd__(p1, n): p = p1.copy() if not n: return p ring = p1.ring try: n = ring.domain_new(n) except CoercionFailed: return NotImplemented else: zm = ring.zero_monom if zm not in p1.keys(): p[zm] = n else: if n == -p[zm]: del p[zm] else: p[zm] += n return p def __sub__(p1, p2): """Subtract polynomial p2 from p1. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p1 = x + y**2 >>> p2 = x*y + y**2 >>> p1 - p2 -x*y + x """ if not p2: return p1.copy() ring = p1.ring if isinstance(p2, ring.dtype): p = p1.copy() get = p.get zero = ring.domain.zero for k, v in p2.items(): v = get(k, zero) - v if v: p[k] = v else: del p[k] return p elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__rsub__(p1) else: return NotImplemented try: p2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: p = p1.copy() zm = ring.zero_monom if zm not in p1.keys(): p[zm] = -p2 else: if p2 == p[zm]: del p[zm] else: p[zm] -= p2 return p def __rsub__(p1, n): """n - p1 with n convertible to the coefficient domain. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y >>> 4 - p -x - y + 4 """ ring = p1.ring try: n = ring.domain_new(n) except CoercionFailed: return NotImplemented else: p = ring.zero for expv in p1: p[expv] = -p1[expv] p += n return p def __mul__(p1, p2): """Multiply two polynomials. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', QQ) >>> p1 = x + y >>> p2 = x - y >>> p1*p2 x**2 - y**2 """ ring = p1.ring p = ring.zero if not p1 or not p2: return p elif isinstance(p2, ring.dtype): get = p.get zero = ring.domain.zero monomial_mul = ring.monomial_mul p2it = list(p2.items()) for exp1, v1 in p1.items(): for exp2, v2 in p2it: exp = monomial_mul(exp1, exp2) p[exp] = get(exp, zero) + v1*v2 p.strip_zero() return p elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__rmul__(p1) else: return NotImplemented try: p2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: for exp1, v1 in p1.items(): v = v1*p2 if v: p[exp1] = v return p def __rmul__(p1, p2): """p2 * p1 with p2 in the coefficient domain of p1. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y >>> 4 * p 4*x + 4*y """ p = p1.ring.zero if not p2: return p try: p2 = p.ring.domain_new(p2) except CoercionFailed: return NotImplemented else: for exp1, v1 in p1.items(): v = p2*v1 if v: p[exp1] = v return p def __pow__(self, n): """raise polynomial to power `n` Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.rings import ring >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p**3 x**3 + 3*x**2*y**2 + 3*x*y**4 + y**6 """ ring = self.ring if not n: if self: return ring.one else: raise ValueError("0**0") elif len(self) == 1: monom, coeff = list(self.items())[0] p = ring.zero if coeff == 1: p[ring.monomial_pow(monom, n)] = coeff else: p[ring.monomial_pow(monom, n)] = coeff**n return p # For ring series, we need negative and rational exponent support only # with monomials. n = int(n) if n < 0: raise ValueError("Negative exponent") elif n == 1: return self.copy() elif n == 2: return self.square() elif n == 3: return self*self.square() elif len(self) <= 5: # TODO: use an actual density measure return self._pow_multinomial(n) else: return self._pow_generic(n) def _pow_generic(self, n): p = self.ring.one c = self while True: if n & 1: p = p*c n -= 1 if not n: break c = c.square() n = n // 2 return p def _pow_multinomial(self, n): multinomials = list(multinomial_coefficients(len(self), n).items()) monomial_mulpow = self.ring.monomial_mulpow zero_monom = self.ring.zero_monom terms = list(self.iterterms()) zero = self.ring.domain.zero poly = self.ring.zero for multinomial, multinomial_coeff in multinomials: product_monom = zero_monom product_coeff = multinomial_coeff for exp, (monom, coeff) in zip(multinomial, terms): if exp: product_monom = monomial_mulpow(product_monom, monom, exp) product_coeff *= coeff**exp monom = tuple(product_monom) coeff = product_coeff coeff = poly.get(monom, zero) + coeff if coeff: poly[monom] = coeff else: del poly[monom] return poly def square(self): """square of a polynomial Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p.square() x**2 + 2*x*y**2 + y**4 """ ring = self.ring p = ring.zero get = p.get keys = list(self.keys()) zero = ring.domain.zero monomial_mul = ring.monomial_mul for i in range(len(keys)): k1 = keys[i] pk = self[k1] for j in range(i): k2 = keys[j] exp = monomial_mul(k1, k2) p[exp] = get(exp, zero) + pk*self[k2] p = p.imul_num(2) get = p.get for k, v in self.items(): k2 = monomial_mul(k, k) p[k2] = get(k2, zero) + v**2 p.strip_zero() return p def __divmod__(p1, p2): ring = p1.ring if not p2: raise ZeroDivisionError("polynomial division") elif isinstance(p2, ring.dtype): return p1.div(p2) elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__rdivmod__(p1) else: return NotImplemented try: p2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: return (p1.quo_ground(p2), p1.rem_ground(p2)) def __rdivmod__(p1, p2): return NotImplemented def __mod__(p1, p2): ring = p1.ring if not p2: raise ZeroDivisionError("polynomial division") elif isinstance(p2, ring.dtype): return p1.rem(p2) elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__rmod__(p1) else: return NotImplemented try: p2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: return p1.rem_ground(p2) def __rmod__(p1, p2): return NotImplemented def __truediv__(p1, p2): ring = p1.ring if not p2: raise ZeroDivisionError("polynomial division") elif isinstance(p2, ring.dtype): if p2.is_monomial: return p1*(p2**(-1)) else: return p1.quo(p2) elif isinstance(p2, PolyElement): if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: pass elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: return p2.__rtruediv__(p1) else: return NotImplemented try: p2 = ring.domain_new(p2) except CoercionFailed: return NotImplemented else: return p1.quo_ground(p2) def __rtruediv__(p1, p2): return NotImplemented __floordiv__ = __truediv__ __rfloordiv__ = __rtruediv__ # TODO: use // (__floordiv__) for exquo()? def _term_div(self): zm = self.ring.zero_monom domain = self.ring.domain domain_quo = domain.quo monomial_div = self.ring.monomial_div if domain.is_Field: def term_div(a_lm_a_lc, b_lm_b_lc): a_lm, a_lc = a_lm_a_lc b_lm, b_lc = b_lm_b_lc if b_lm == zm: # apparently this is a very common case monom = a_lm else: monom = monomial_div(a_lm, b_lm) if monom is not None: return monom, domain_quo(a_lc, b_lc) else: return None else: def term_div(a_lm_a_lc, b_lm_b_lc): a_lm, a_lc = a_lm_a_lc b_lm, b_lc = b_lm_b_lc if b_lm == zm: # apparently this is a very common case monom = a_lm else: monom = monomial_div(a_lm, b_lm) if not (monom is None or a_lc % b_lc): return monom, domain_quo(a_lc, b_lc) else: return None return term_div def div(self, fv): """Division algorithm, see [CLO] p64. fv array of polynomials return qv, r such that self = sum(fv[i]*qv[i]) + r All polynomials are required not to be Laurent polynomials. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> f = x**3 >>> f0 = x - y**2 >>> f1 = x - y >>> qv, r = f.div((f0, f1)) >>> qv[0] x**2 + x*y**2 + y**4 >>> qv[1] 0 >>> r y**6 """ ring = self.ring ret_single = False if isinstance(fv, PolyElement): ret_single = True fv = [fv] if any(not f for f in fv): raise ZeroDivisionError("polynomial division") if not self: if ret_single: return ring.zero, ring.zero else: return [], ring.zero for f in fv: if f.ring != ring: raise ValueError('self and f must have the same ring') s = len(fv) qv = [ring.zero for i in range(s)] p = self.copy() r = ring.zero term_div = self._term_div() expvs = [fx.leading_expv() for fx in fv] while p: i = 0 divoccurred = 0 while i < s and divoccurred == 0: expv = p.leading_expv() term = term_div((expv, p[expv]), (expvs[i], fv[i][expvs[i]])) if term is not None: expv1, c = term qv[i] = qv[i]._iadd_monom((expv1, c)) p = p._iadd_poly_monom(fv[i], (expv1, -c)) divoccurred = 1 else: i += 1 if not divoccurred: expv = p.leading_expv() r = r._iadd_monom((expv, p[expv])) del p[expv] if expv == ring.zero_monom: r += p if ret_single: if not qv: return ring.zero, r else: return qv[0], r else: return qv, r def rem(self, G): f = self if isinstance(G, PolyElement): G = [G] if any(not g for g in G): raise ZeroDivisionError("polynomial division") ring = f.ring domain = ring.domain zero = domain.zero monomial_mul = ring.monomial_mul r = ring.zero term_div = f._term_div() ltf = f.LT f = f.copy() get = f.get while f: for g in G: tq = term_div(ltf, g.LT) if tq is not None: m, c = tq for mg, cg in g.iterterms(): m1 = monomial_mul(mg, m) c1 = get(m1, zero) - c*cg if not c1: del f[m1] else: f[m1] = c1 ltm = f.leading_expv() if ltm is not None: ltf = ltm, f[ltm] break else: ltm, ltc = ltf if ltm in r: r[ltm] += ltc else: r[ltm] = ltc del f[ltm] ltm = f.leading_expv() if ltm is not None: ltf = ltm, f[ltm] return r def quo(f, G): return f.div(G)[0] def exquo(f, G): q, r = f.div(G) if not r: return q else: raise ExactQuotientFailed(f, G) def _iadd_monom(self, mc): """add to self the monomial coeff*x0**i0*x1**i1*... unless self is a generator -- then just return the sum of the two. mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x**4 + 2*y >>> m = (1, 2) >>> p1 = p._iadd_monom((m, 5)) >>> p1 x**4 + 5*x*y**2 + 2*y >>> p1 is p True >>> p = x >>> p1 = p._iadd_monom((m, 5)) >>> p1 5*x*y**2 + x >>> p1 is p False """ if self in self.ring._gens_set: cpself = self.copy() else: cpself = self expv, coeff = mc c = cpself.get(expv) if c is None: cpself[expv] = coeff else: c += coeff if c: cpself[expv] = c else: del cpself[expv] return cpself def _iadd_poly_monom(self, p2, mc): """add to self the product of (p)*(coeff*x0**i0*x1**i1*...) unless self is a generator -- then just return the sum of the two. mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring('x, y, z', ZZ) >>> p1 = x**4 + 2*y >>> p2 = y + z >>> m = (1, 2, 3) >>> p1 = p1._iadd_poly_monom(p2, (m, 3)) >>> p1 x**4 + 3*x*y**3*z**3 + 3*x*y**2*z**4 + 2*y """ p1 = self if p1 in p1.ring._gens_set: p1 = p1.copy() (m, c) = mc get = p1.get zero = p1.ring.domain.zero monomial_mul = p1.ring.monomial_mul for k, v in p2.items(): ka = monomial_mul(k, m) coeff = get(ka, zero) + v*c if coeff: p1[ka] = coeff else: del p1[ka] return p1 def degree(f, x=None): """ The leading degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (the SymPy object -oo). """ i = f.ring.index(x) if not f: return -oo elif i < 0: return 0 else: return max([ monom[i] for monom in f.itermonoms() ]) def degrees(f): """ A tuple containing leading degrees in all variables. Note that the degree of 0 is negative infinity (the SymPy object -oo) """ if not f: return (-oo,)*f.ring.ngens else: return tuple(map(max, list(zip(*f.itermonoms())))) def tail_degree(f, x=None): """ The tail degree in ``x`` or the main variable. Note that the degree of 0 is negative infinity (the SymPy object -oo) """ i = f.ring.index(x) if not f: return -oo elif i < 0: return 0 else: return min([ monom[i] for monom in f.itermonoms() ]) def tail_degrees(f): """ A tuple containing tail degrees in all variables. Note that the degree of 0 is negative infinity (the SymPy object -oo) """ if not f: return (-oo,)*f.ring.ngens else: return tuple(map(min, list(zip(*f.itermonoms())))) def leading_expv(self): """Leading monomial tuple according to the monomial ordering. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring('x, y, z', ZZ) >>> p = x**4 + x**3*y + x**2*z**2 + z**7 >>> p.leading_expv() (4, 0, 0) """ if self: return self.ring.leading_expv(self) else: return None def _get_coeff(self, expv): return self.get(expv, self.ring.domain.zero) def coeff(self, element): """ Returns the coefficient that stands next to the given monomial. Parameters ========== element : PolyElement (with ``is_monomial = True``) or 1 Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y, z = ring("x,y,z", ZZ) >>> f = 3*x**2*y - x*y*z + 7*z**3 + 23 >>> f.coeff(x**2*y) 3 >>> f.coeff(x*y) 0 >>> f.coeff(1) 23 """ if element == 1: return self._get_coeff(self.ring.zero_monom) elif isinstance(element, self.ring.dtype): terms = list(element.iterterms()) if len(terms) == 1: monom, coeff = terms[0] if coeff == self.ring.domain.one: return self._get_coeff(monom) raise ValueError("expected a monomial, got %s" % element) def const(self): """Returns the constant coeffcient. """ return self._get_coeff(self.ring.zero_monom) @property def LC(self): return self._get_coeff(self.leading_expv()) @property def LM(self): expv = self.leading_expv() if expv is None: return self.ring.zero_monom else: return expv def leading_monom(self): """ Leading monomial as a polynomial element. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> (3*x*y + y**2).leading_monom() x*y """ p = self.ring.zero expv = self.leading_expv() if expv: p[expv] = self.ring.domain.one return p @property def LT(self): expv = self.leading_expv() if expv is None: return (self.ring.zero_monom, self.ring.domain.zero) else: return (expv, self._get_coeff(expv)) def leading_term(self): """Leading term as a polynomial element. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> (3*x*y + y**2).leading_term() 3*x*y """ p = self.ring.zero expv = self.leading_expv() if expv is not None: p[expv] = self[expv] return p def _sorted(self, seq, order): if order is None: order = self.ring.order else: order = OrderOpt.preprocess(order) if order is lex: return sorted(seq, key=lambda monom: monom[0], reverse=True) else: return sorted(seq, key=lambda monom: order(monom[0]), reverse=True) def coeffs(self, order=None): """Ordered list of polynomial coefficients. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.coeffs() [2, 1] >>> f.coeffs(grlex) [1, 2] """ return [ coeff for _, coeff in self.terms(order) ] def monoms(self, order=None): """Ordered list of polynomial monomials. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.monoms() [(2, 3), (1, 7)] >>> f.monoms(grlex) [(1, 7), (2, 3)] """ return [ monom for monom, _ in self.terms(order) ] def terms(self, order=None): """Ordered list of polynomial terms. Parameters ========== order : :class:`~.MonomialOrder` or coercible, optional Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> from sympy.polys.orderings import lex, grlex >>> _, x, y = ring("x, y", ZZ, lex) >>> f = x*y**7 + 2*x**2*y**3 >>> f.terms() [((2, 3), 2), ((1, 7), 1)] >>> f.terms(grlex) [((1, 7), 1), ((2, 3), 2)] """ return self._sorted(list(self.items()), order) def itercoeffs(self): """Iterator over coefficients of a polynomial. """ return iter(self.values()) def itermonoms(self): """Iterator over monomials of a polynomial. """ return iter(self.keys()) def iterterms(self): """Iterator over terms of a polynomial. """ return iter(self.items()) def listcoeffs(self): """Unordered list of polynomial coefficients. """ return list(self.values()) def listmonoms(self): """Unordered list of polynomial monomials. """ return list(self.keys()) def listterms(self): """Unordered list of polynomial terms. """ return list(self.items()) def imul_num(p, c): """multiply inplace the polynomial p by an element in the coefficient ring, provided p is not one of the generators; else multiply not inplace Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring('x, y', ZZ) >>> p = x + y**2 >>> p1 = p.imul_num(3) >>> p1 3*x + 3*y**2 >>> p1 is p True >>> p = x >>> p1 = p.imul_num(3) >>> p1 3*x >>> p1 is p False """ if p in p.ring._gens_set: return p*c if not c: p.clear() return for exp in p: p[exp] *= c return p def content(f): """Returns GCD of polynomial's coefficients. """ domain = f.ring.domain cont = domain.zero gcd = domain.gcd for coeff in f.itercoeffs(): cont = gcd(cont, coeff) return cont def primitive(f): """Returns content and a primitive polynomial. """ cont = f.content() return cont, f.quo_ground(cont) def monic(f): """Divides all coefficients by the leading coefficient. """ if not f: return f else: return f.quo_ground(f.LC) def mul_ground(f, x): if not x: return f.ring.zero terms = [ (monom, coeff*x) for monom, coeff in f.iterterms() ] return f.new(terms) def mul_monom(f, monom): monomial_mul = f.ring.monomial_mul terms = [ (monomial_mul(f_monom, monom), f_coeff) for f_monom, f_coeff in f.items() ] return f.new(terms) def mul_term(f, term): monom, coeff = term if not f or not coeff: return f.ring.zero elif monom == f.ring.zero_monom: return f.mul_ground(coeff) monomial_mul = f.ring.monomial_mul terms = [ (monomial_mul(f_monom, monom), f_coeff*coeff) for f_monom, f_coeff in f.items() ] return f.new(terms) def quo_ground(f, x): domain = f.ring.domain if not x: raise ZeroDivisionError('polynomial division') if not f or x == domain.one: return f if domain.is_Field: quo = domain.quo terms = [ (monom, quo(coeff, x)) for monom, coeff in f.iterterms() ] else: terms = [ (monom, coeff // x) for monom, coeff in f.iterterms() if not (coeff % x) ] return f.new(terms) def quo_term(f, term): monom, coeff = term if not coeff: raise ZeroDivisionError("polynomial division") elif not f: return f.ring.zero elif monom == f.ring.zero_monom: return f.quo_ground(coeff) term_div = f._term_div() terms = [ term_div(t, term) for t in f.iterterms() ] return f.new([ t for t in terms if t is not None ]) def trunc_ground(f, p): if f.ring.domain.is_ZZ: terms = [] for monom, coeff in f.iterterms(): coeff = coeff % p if coeff > p // 2: coeff = coeff - p terms.append((monom, coeff)) else: terms = [ (monom, coeff % p) for monom, coeff in f.iterterms() ] poly = f.new(terms) poly.strip_zero() return poly rem_ground = trunc_ground def extract_ground(self, g): f = self fc = f.content() gc = g.content() gcd = f.ring.domain.gcd(fc, gc) f = f.quo_ground(gcd) g = g.quo_ground(gcd) return gcd, f, g def _norm(f, norm_func): if not f: return f.ring.domain.zero else: ground_abs = f.ring.domain.abs return norm_func([ ground_abs(coeff) for coeff in f.itercoeffs() ]) def max_norm(f): return f._norm(max) def l1_norm(f): return f._norm(sum) def deflate(f, *G): ring = f.ring polys = [f] + list(G) J = [0]*ring.ngens for p in polys: for monom in p.itermonoms(): for i, m in enumerate(monom): J[i] = igcd(J[i], m) for i, b in enumerate(J): if not b: J[i] = 1 J = tuple(J) if all(b == 1 for b in J): return J, polys H = [] for p in polys: h = ring.zero for I, coeff in p.iterterms(): N = [ i // j for i, j in zip(I, J) ] h[tuple(N)] = coeff H.append(h) return J, H def inflate(f, J): poly = f.ring.zero for I, coeff in f.iterterms(): N = [ i*j for i, j in zip(I, J) ] poly[tuple(N)] = coeff return poly def lcm(self, g): f = self domain = f.ring.domain if not domain.is_Field: fc, f = f.primitive() gc, g = g.primitive() c = domain.lcm(fc, gc) h = (f*g).quo(f.gcd(g)) if not domain.is_Field: return h.mul_ground(c) else: return h.monic() def gcd(f, g): return f.cofactors(g)[0] def cofactors(f, g): if not f and not g: zero = f.ring.zero return zero, zero, zero elif not f: h, cff, cfg = f._gcd_zero(g) return h, cff, cfg elif not g: h, cfg, cff = g._gcd_zero(f) return h, cff, cfg elif len(f) == 1: h, cff, cfg = f._gcd_monom(g) return h, cff, cfg elif len(g) == 1: h, cfg, cff = g._gcd_monom(f) return h, cff, cfg J, (f, g) = f.deflate(g) h, cff, cfg = f._gcd(g) return (h.inflate(J), cff.inflate(J), cfg.inflate(J)) def _gcd_zero(f, g): one, zero = f.ring.one, f.ring.zero if g.is_nonnegative: return g, zero, one else: return -g, zero, -one def _gcd_monom(f, g): ring = f.ring ground_gcd = ring.domain.gcd ground_quo = ring.domain.quo monomial_gcd = ring.monomial_gcd monomial_ldiv = ring.monomial_ldiv mf, cf = list(f.iterterms())[0] _mgcd, _cgcd = mf, cf for mg, cg in g.iterterms(): _mgcd = monomial_gcd(_mgcd, mg) _cgcd = ground_gcd(_cgcd, cg) h = f.new([(_mgcd, _cgcd)]) cff = f.new([(monomial_ldiv(mf, _mgcd), ground_quo(cf, _cgcd))]) cfg = f.new([(monomial_ldiv(mg, _mgcd), ground_quo(cg, _cgcd)) for mg, cg in g.iterterms()]) return h, cff, cfg def _gcd(f, g): ring = f.ring if ring.domain.is_QQ: return f._gcd_QQ(g) elif ring.domain.is_ZZ: return f._gcd_ZZ(g) else: # TODO: don't use dense representation (port PRS algorithms) return ring.dmp_inner_gcd(f, g) def _gcd_ZZ(f, g): return heugcd(f, g) def _gcd_QQ(self, g): f = self ring = f.ring new_ring = ring.clone(domain=ring.domain.get_ring()) cf, f = f.clear_denoms() cg, g = g.clear_denoms() f = f.set_ring(new_ring) g = g.set_ring(new_ring) h, cff, cfg = f._gcd_ZZ(g) h = h.set_ring(ring) c, h = h.LC, h.monic() cff = cff.set_ring(ring).mul_ground(ring.domain.quo(c, cf)) cfg = cfg.set_ring(ring).mul_ground(ring.domain.quo(c, cg)) return h, cff, cfg def cancel(self, g): """ Cancel common factors in a rational function ``f/g``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1) (2*x + 2, x - 1) """ f = self ring = f.ring if not f: return f, ring.one domain = ring.domain if not (domain.is_Field and domain.has_assoc_Ring): _, p, q = f.cofactors(g) if q.is_negative: p, q = -p, -q else: new_ring = ring.clone(domain=domain.get_ring()) cq, f = f.clear_denoms() cp, g = g.clear_denoms() f = f.set_ring(new_ring) g = g.set_ring(new_ring) _, p, q = f.cofactors(g) _, cp, cq = new_ring.domain.cofactors(cp, cq) p = p.set_ring(ring) q = q.set_ring(ring) p_neg = p.is_negative q_neg = q.is_negative if p_neg and q_neg: p, q = -p, -q elif p_neg: cp, p = -cp, -p elif q_neg: cp, q = -cp, -q p = p.mul_ground(cp) q = q.mul_ground(cq) return p, q def diff(f, x): """Computes partial derivative in ``x``. Examples ======== >>> from sympy.polys.rings import ring >>> from sympy.polys.domains import ZZ >>> _, x, y = ring("x,y", ZZ) >>> p = x + x**2*y**3 >>> p.diff(x) 2*x*y**3 + 1 """ ring = f.ring i = ring.index(x) m = ring.monomial_basis(i) g = ring.zero for expv, coeff in f.iterterms(): if expv[i]: e = ring.monomial_ldiv(expv, m) g[e] = ring.domain_new(coeff*expv[i]) return g def __call__(f, *values): if 0 < len(values) <= f.ring.ngens: return f.evaluate(list(zip(f.ring.gens, values))) else: raise ValueError("expected at least 1 and at most %s values, got %s" % (f.ring.ngens, len(values))) def evaluate(self, x, a=None): f = self if isinstance(x, list) and a is None: (X, a), x = x[0], x[1:] f = f.evaluate(X, a) if not x: return f else: x = [ (Y.drop(X), a) for (Y, a) in x ] return f.evaluate(x) ring = f.ring i = ring.index(x) a = ring.domain.convert(a) if ring.ngens == 1: result = ring.domain.zero for (n,), coeff in f.iterterms(): result += coeff*a**n return result else: poly = ring.drop(x).zero for monom, coeff in f.iterterms(): n, monom = monom[i], monom[:i] + monom[i+1:] coeff = coeff*a**n if monom in poly: coeff = coeff + poly[monom] if coeff: poly[monom] = coeff else: del poly[monom] else: if coeff: poly[monom] = coeff return poly def subs(self, x, a=None): f = self if isinstance(x, list) and a is None: for X, a in x: f = f.subs(X, a) return f ring = f.ring i = ring.index(x) a = ring.domain.convert(a) if ring.ngens == 1: result = ring.domain.zero for (n,), coeff in f.iterterms(): result += coeff*a**n return ring.ground_new(result) else: poly = ring.zero for monom, coeff in f.iterterms(): n, monom = monom[i], monom[:i] + (0,) + monom[i+1:] coeff = coeff*a**n if monom in poly: coeff = coeff + poly[monom] if coeff: poly[monom] = coeff else: del poly[monom] else: if coeff: poly[monom] = coeff return poly def compose(f, x, a=None): ring = f.ring poly = ring.zero gens_map = dict(list(zip(ring.gens, list(range(ring.ngens))))) if a is not None: replacements = [(x, a)] else: if isinstance(x, list): replacements = list(x) elif isinstance(x, dict): replacements = sorted(list(x.items()), key=lambda k: gens_map[k[0]]) else: raise ValueError("expected a generator, value pair a sequence of such pairs") for k, (x, g) in enumerate(replacements): replacements[k] = (gens_map[x], ring.ring_new(g)) for monom, coeff in f.iterterms(): monom = list(monom) subpoly = ring.one for i, g in replacements: n, monom[i] = monom[i], 0 if n: subpoly *= g**n subpoly = subpoly.mul_term((tuple(monom), coeff)) poly += subpoly return poly # TODO: following methods should point to polynomial # representation independent algorithm implementations. def pdiv(f, g): return f.ring.dmp_pdiv(f, g) def prem(f, g): return f.ring.dmp_prem(f, g) def pquo(f, g): return f.ring.dmp_quo(f, g) def pexquo(f, g): return f.ring.dmp_exquo(f, g) def half_gcdex(f, g): return f.ring.dmp_half_gcdex(f, g) def gcdex(f, g): return f.ring.dmp_gcdex(f, g) def subresultants(f, g): return f.ring.dmp_subresultants(f, g) def resultant(f, g): return f.ring.dmp_resultant(f, g) def discriminant(f): return f.ring.dmp_discriminant(f) def decompose(f): if f.ring.is_univariate: return f.ring.dup_decompose(f) else: raise MultivariatePolynomialError("polynomial decomposition") def shift(f, a): if f.ring.is_univariate: return f.ring.dup_shift(f, a) else: raise MultivariatePolynomialError("polynomial shift") def sturm(f): if f.ring.is_univariate: return f.ring.dup_sturm(f) else: raise MultivariatePolynomialError("sturm sequence") def gff_list(f): return f.ring.dmp_gff_list(f) def sqf_norm(f): return f.ring.dmp_sqf_norm(f) def sqf_part(f): return f.ring.dmp_sqf_part(f) def sqf_list(f, all=False): return f.ring.dmp_sqf_list(f, all=all) def factor_list(f): return f.ring.dmp_factor_list(f)
5ea59549abc5f698900f4b29ab9c1e8ca8dd607b5f5e37c6dddd1591db8b7bea
"""Options manager for :class:`~.Poly` and public API functions. """ __all__ = ["Options"] from typing import Dict, Type from typing import List, Optional from sympy.core import Basic, sympify from sympy.polys.polyerrors import GeneratorsError, OptionError, FlagError from sympy.utilities import numbered_symbols, topological_sort, public from sympy.utilities.iterables import has_dups import sympy.polys import re class Option: """Base class for all kinds of options. """ option = None # type: Optional[str] is_Flag = False requires = [] # type: List[str] excludes = [] # type: List[str] after = [] # type: List[str] before = [] # type: List[str] @classmethod def default(cls): return None @classmethod def preprocess(cls, option): return None @classmethod def postprocess(cls, options): pass class Flag(Option): """Base class for all kinds of flags. """ is_Flag = True class BooleanOption(Option): """An option that must have a boolean value or equivalent assigned. """ @classmethod def preprocess(cls, value): if value in [True, False]: return bool(value) else: raise OptionError("'%s' must have a boolean value assigned, got %s" % (cls.option, value)) class OptionType(type): """Base type for all options that does registers options. """ def __init__(cls, *args, **kwargs): @property def getter(self): try: return self[cls.option] except KeyError: return cls.default() setattr(Options, cls.option, getter) Options.__options__[cls.option] = cls @public class Options(dict): """ Options manager for polynomial manipulation module. Examples ======== >>> from sympy.polys.polyoptions import Options >>> from sympy.polys.polyoptions import build_options >>> from sympy.abc import x, y, z >>> Options((x, y, z), {'domain': 'ZZ'}) {'auto': False, 'domain': ZZ, 'gens': (x, y, z)} >>> build_options((x, y, z), {'domain': 'ZZ'}) {'auto': False, 'domain': ZZ, 'gens': (x, y, z)} **Options** * Expand --- boolean option * Gens --- option * Wrt --- option * Sort --- option * Order --- option * Field --- boolean option * Greedy --- boolean option * Domain --- option * Split --- boolean option * Gaussian --- boolean option * Extension --- option * Modulus --- option * Symmetric --- boolean option * Strict --- boolean option **Flags** * Auto --- boolean flag * Frac --- boolean flag * Formal --- boolean flag * Polys --- boolean flag * Include --- boolean flag * All --- boolean flag * Gen --- flag * Series --- boolean flag """ __order__ = None __options__ = {} # type: Dict[str, Type[Option]] def __init__(self, gens, args, flags=None, strict=False): dict.__init__(self) if gens and args.get('gens', ()): raise OptionError( "both '*gens' and keyword argument 'gens' supplied") elif gens: args = dict(args) args['gens'] = gens defaults = args.pop('defaults', {}) def preprocess_options(args): for option, value in args.items(): try: cls = self.__options__[option] except KeyError: raise OptionError("'%s' is not a valid option" % option) if issubclass(cls, Flag): if flags is None or option not in flags: if strict: raise OptionError("'%s' flag is not allowed in this context" % option) if value is not None: self[option] = cls.preprocess(value) preprocess_options(args) for key, value in dict(defaults).items(): if key in self: del defaults[key] else: for option in self.keys(): cls = self.__options__[option] if key in cls.excludes: del defaults[key] break preprocess_options(defaults) for option in self.keys(): cls = self.__options__[option] for require_option in cls.requires: if self.get(require_option) is None: raise OptionError("'%s' option is only allowed together with '%s'" % (option, require_option)) for exclude_option in cls.excludes: if self.get(exclude_option) is not None: raise OptionError("'%s' option is not allowed together with '%s'" % (option, exclude_option)) for option in self.__order__: self.__options__[option].postprocess(self) @classmethod def _init_dependencies_order(cls): """Resolve the order of options' processing. """ if cls.__order__ is None: vertices, edges = [], set() for name, option in cls.__options__.items(): vertices.append(name) for _name in option.after: edges.add((_name, name)) for _name in option.before: edges.add((name, _name)) try: cls.__order__ = topological_sort((vertices, list(edges))) except ValueError: raise RuntimeError( "cycle detected in sympy.polys options framework") def clone(self, updates={}): """Clone ``self`` and update specified options. """ obj = dict.__new__(self.__class__) for option, value in self.items(): obj[option] = value for option, value in updates.items(): obj[option] = value return obj def __setattr__(self, attr, value): if attr in self.__options__: self[attr] = value else: super().__setattr__(attr, value) @property def args(self): args = {} for option, value in self.items(): if value is not None and option != 'gens': cls = self.__options__[option] if not issubclass(cls, Flag): args[option] = value return args @property def options(self): options = {} for option, cls in self.__options__.items(): if not issubclass(cls, Flag): options[option] = getattr(self, option) return options @property def flags(self): flags = {} for option, cls in self.__options__.items(): if issubclass(cls, Flag): flags[option] = getattr(self, option) return flags class Expand(BooleanOption, metaclass=OptionType): """``expand`` option to polynomial manipulation functions. """ option = 'expand' requires = [] # type: List[str] excludes = [] # type: List[str] @classmethod def default(cls): return True class Gens(Option, metaclass=OptionType): """``gens`` option to polynomial manipulation functions. """ option = 'gens' requires = [] # type: List[str] excludes = [] # type: List[str] @classmethod def default(cls): return () @classmethod def preprocess(cls, gens): if isinstance(gens, Basic): gens = (gens,) elif len(gens) == 1 and hasattr(gens[0], '__iter__'): gens = gens[0] if gens == (None,): gens = () elif has_dups(gens): raise GeneratorsError("duplicated generators: %s" % str(gens)) elif any(gen.is_commutative is False for gen in gens): raise GeneratorsError("non-commutative generators: %s" % str(gens)) return tuple(gens) class Wrt(Option, metaclass=OptionType): """``wrt`` option to polynomial manipulation functions. """ option = 'wrt' requires = [] # type: List[str] excludes = [] # type: List[str] _re_split = re.compile(r"\s*,\s*|\s+") @classmethod def preprocess(cls, wrt): if isinstance(wrt, Basic): return [str(wrt)] elif isinstance(wrt, str): wrt = wrt.strip() if wrt.endswith(','): raise OptionError('Bad input: missing parameter.') if not wrt: return [] return [ gen for gen in cls._re_split.split(wrt) ] elif hasattr(wrt, '__getitem__'): return list(map(str, wrt)) else: raise OptionError("invalid argument for 'wrt' option") class Sort(Option, metaclass=OptionType): """``sort`` option to polynomial manipulation functions. """ option = 'sort' requires = [] # type: List[str] excludes = [] # type: List[str] @classmethod def default(cls): return [] @classmethod def preprocess(cls, sort): if isinstance(sort, str): return [ gen.strip() for gen in sort.split('>') ] elif hasattr(sort, '__getitem__'): return list(map(str, sort)) else: raise OptionError("invalid argument for 'sort' option") class Order(Option, metaclass=OptionType): """``order`` option to polynomial manipulation functions. """ option = 'order' requires = [] # type: List[str] excludes = [] # type: List[str] @classmethod def default(cls): return sympy.polys.orderings.lex @classmethod def preprocess(cls, order): return sympy.polys.orderings.monomial_key(order) class Field(BooleanOption, metaclass=OptionType): """``field`` option to polynomial manipulation functions. """ option = 'field' requires = [] # type: List[str] excludes = ['domain', 'split', 'gaussian'] class Greedy(BooleanOption, metaclass=OptionType): """``greedy`` option to polynomial manipulation functions. """ option = 'greedy' requires = [] # type: List[str] excludes = ['domain', 'split', 'gaussian', 'extension', 'modulus', 'symmetric'] class Composite(BooleanOption, metaclass=OptionType): """``composite`` option to polynomial manipulation functions. """ option = 'composite' @classmethod def default(cls): return None requires = [] # type: List[str] excludes = ['domain', 'split', 'gaussian', 'extension', 'modulus', 'symmetric'] class Domain(Option, metaclass=OptionType): """``domain`` option to polynomial manipulation functions. """ option = 'domain' requires = [] # type: List[str] excludes = ['field', 'greedy', 'split', 'gaussian', 'extension'] after = ['gens'] _re_realfield = re.compile(r"^(R|RR)(_(\d+))?$") _re_complexfield = re.compile(r"^(C|CC)(_(\d+))?$") _re_finitefield = re.compile(r"^(FF|GF)\((\d+)\)$") _re_polynomial = re.compile(r"^(Z|ZZ|Q|QQ|ZZ_I|QQ_I|R|RR|C|CC)\[(.+)\]$") _re_fraction = re.compile(r"^(Z|ZZ|Q|QQ)\((.+)\)$") _re_algebraic = re.compile(r"^(Q|QQ)\<(.+)\>$") @classmethod def preprocess(cls, domain): if isinstance(domain, sympy.polys.domains.Domain): return domain elif hasattr(domain, 'to_domain'): return domain.to_domain() elif isinstance(domain, str): if domain in ['Z', 'ZZ']: return sympy.polys.domains.ZZ if domain in ['Q', 'QQ']: return sympy.polys.domains.QQ if domain == 'ZZ_I': return sympy.polys.domains.ZZ_I if domain == 'QQ_I': return sympy.polys.domains.QQ_I if domain == 'EX': return sympy.polys.domains.EX r = cls._re_realfield.match(domain) if r is not None: _, _, prec = r.groups() if prec is None: return sympy.polys.domains.RR else: return sympy.polys.domains.RealField(int(prec)) r = cls._re_complexfield.match(domain) if r is not None: _, _, prec = r.groups() if prec is None: return sympy.polys.domains.CC else: return sympy.polys.domains.ComplexField(int(prec)) r = cls._re_finitefield.match(domain) if r is not None: return sympy.polys.domains.FF(int(r.groups()[1])) r = cls._re_polynomial.match(domain) if r is not None: ground, gens = r.groups() gens = list(map(sympify, gens.split(','))) if ground in ['Z', 'ZZ']: return sympy.polys.domains.ZZ.poly_ring(*gens) elif ground in ['Q', 'QQ']: return sympy.polys.domains.QQ.poly_ring(*gens) elif ground in ['R', 'RR']: return sympy.polys.domains.RR.poly_ring(*gens) elif ground == 'ZZ_I': return sympy.polys.domains.ZZ_I.poly_ring(*gens) elif ground == 'QQ_I': return sympy.polys.domains.QQ_I.poly_ring(*gens) else: return sympy.polys.domains.CC.poly_ring(*gens) r = cls._re_fraction.match(domain) if r is not None: ground, gens = r.groups() gens = list(map(sympify, gens.split(','))) if ground in ['Z', 'ZZ']: return sympy.polys.domains.ZZ.frac_field(*gens) else: return sympy.polys.domains.QQ.frac_field(*gens) r = cls._re_algebraic.match(domain) if r is not None: gens = list(map(sympify, r.groups()[1].split(','))) return sympy.polys.domains.QQ.algebraic_field(*gens) raise OptionError('expected a valid domain specification, got %s' % domain) @classmethod def postprocess(cls, options): if 'gens' in options and 'domain' in options and options['domain'].is_Composite and \ (set(options['domain'].symbols) & set(options['gens'])): raise GeneratorsError( "ground domain and generators interfere together") elif ('gens' not in options or not options['gens']) and \ 'domain' in options and options['domain'] == sympy.polys.domains.EX: raise GeneratorsError("you have to provide generators because EX domain was requested") class Split(BooleanOption, metaclass=OptionType): """``split`` option to polynomial manipulation functions. """ option = 'split' requires = [] # type: List[str] excludes = ['field', 'greedy', 'domain', 'gaussian', 'extension', 'modulus', 'symmetric'] @classmethod def postprocess(cls, options): if 'split' in options: raise NotImplementedError("'split' option is not implemented yet") class Gaussian(BooleanOption, metaclass=OptionType): """``gaussian`` option to polynomial manipulation functions. """ option = 'gaussian' requires = [] # type: List[str] excludes = ['field', 'greedy', 'domain', 'split', 'extension', 'modulus', 'symmetric'] @classmethod def postprocess(cls, options): if 'gaussian' in options and options['gaussian'] is True: options['domain'] = sympy.polys.domains.QQ_I Extension.postprocess(options) class Extension(Option, metaclass=OptionType): """``extension`` option to polynomial manipulation functions. """ option = 'extension' requires = [] # type: List[str] excludes = ['greedy', 'domain', 'split', 'gaussian', 'modulus', 'symmetric'] @classmethod def preprocess(cls, extension): if extension == 1: return bool(extension) elif extension == 0: raise OptionError("'False' is an invalid argument for 'extension'") else: if not hasattr(extension, '__iter__'): extension = {extension} else: if not extension: extension = None else: extension = set(extension) return extension @classmethod def postprocess(cls, options): if 'extension' in options and options['extension'] is not True: options['domain'] = sympy.polys.domains.QQ.algebraic_field( *options['extension']) class Modulus(Option, metaclass=OptionType): """``modulus`` option to polynomial manipulation functions. """ option = 'modulus' requires = [] # type: List[str] excludes = ['greedy', 'split', 'domain', 'gaussian', 'extension'] @classmethod def preprocess(cls, modulus): modulus = sympify(modulus) if modulus.is_Integer and modulus > 0: return int(modulus) else: raise OptionError( "'modulus' must a positive integer, got %s" % modulus) @classmethod def postprocess(cls, options): if 'modulus' in options: modulus = options['modulus'] symmetric = options.get('symmetric', True) options['domain'] = sympy.polys.domains.FF(modulus, symmetric) class Symmetric(BooleanOption, metaclass=OptionType): """``symmetric`` option to polynomial manipulation functions. """ option = 'symmetric' requires = ['modulus'] excludes = ['greedy', 'domain', 'split', 'gaussian', 'extension'] class Strict(BooleanOption, metaclass=OptionType): """``strict`` option to polynomial manipulation functions. """ option = 'strict' @classmethod def default(cls): return True class Auto(BooleanOption, Flag, metaclass=OptionType): """``auto`` flag to polynomial manipulation functions. """ option = 'auto' after = ['field', 'domain', 'extension', 'gaussian'] @classmethod def default(cls): return True @classmethod def postprocess(cls, options): if ('domain' in options or 'field' in options) and 'auto' not in options: options['auto'] = False class Frac(BooleanOption, Flag, metaclass=OptionType): """``auto`` option to polynomial manipulation functions. """ option = 'frac' @classmethod def default(cls): return False class Formal(BooleanOption, Flag, metaclass=OptionType): """``formal`` flag to polynomial manipulation functions. """ option = 'formal' @classmethod def default(cls): return False class Polys(BooleanOption, Flag, metaclass=OptionType): """``polys`` flag to polynomial manipulation functions. """ option = 'polys' class Include(BooleanOption, Flag, metaclass=OptionType): """``include`` flag to polynomial manipulation functions. """ option = 'include' @classmethod def default(cls): return False class All(BooleanOption, Flag, metaclass=OptionType): """``all`` flag to polynomial manipulation functions. """ option = 'all' @classmethod def default(cls): return False class Gen(Flag, metaclass=OptionType): """``gen`` flag to polynomial manipulation functions. """ option = 'gen' @classmethod def default(cls): return 0 @classmethod def preprocess(cls, gen): if isinstance(gen, (Basic, int)): return gen else: raise OptionError("invalid argument for 'gen' option") class Series(BooleanOption, Flag, metaclass=OptionType): """``series`` flag to polynomial manipulation functions. """ option = 'series' @classmethod def default(cls): return False class Symbols(Flag, metaclass=OptionType): """``symbols`` flag to polynomial manipulation functions. """ option = 'symbols' @classmethod def default(cls): return numbered_symbols('s', start=1) @classmethod def preprocess(cls, symbols): if hasattr(symbols, '__iter__'): return iter(symbols) else: raise OptionError("expected an iterator or iterable container, got %s" % symbols) class Method(Flag, metaclass=OptionType): """``method`` flag to polynomial manipulation functions. """ option = 'method' @classmethod def preprocess(cls, method): if isinstance(method, str): return method.lower() else: raise OptionError("expected a string, got %s" % method) def build_options(gens, args=None): """Construct options from keyword arguments or ... options. """ if args is None: gens, args = (), gens if len(args) != 1 or 'opt' not in args or gens: return Options(gens, args) else: return args['opt'] def allowed_flags(args, flags): """ Allow specified flags to be used in the given context. Examples ======== >>> from sympy.polys.polyoptions import allowed_flags >>> from sympy.polys.domains import ZZ >>> allowed_flags({'domain': ZZ}, []) >>> allowed_flags({'domain': ZZ, 'frac': True}, []) Traceback (most recent call last): ... FlagError: 'frac' flag is not allowed in this context >>> allowed_flags({'domain': ZZ, 'frac': True}, ['frac']) """ flags = set(flags) for arg in args.keys(): try: if Options.__options__[arg].is_Flag and not arg in flags: raise FlagError( "'%s' flag is not allowed in this context" % arg) except KeyError: raise OptionError("'%s' is not a valid option" % arg) def set_defaults(options, **defaults): """Update options with default values. """ if 'defaults' not in options: options = dict(options) options['defaults'] = defaults return options Options._init_dependencies_order()
d2183c7be5c0e0c7d30287e562ab99b7b2ff56f0e61d49552f67519340bf4442
"""Groebner bases algorithms. """ from sympy.core.symbol import Dummy from sympy.polys.monomials import monomial_mul, monomial_lcm, monomial_divides, term_div from sympy.polys.orderings import lex from sympy.polys.polyerrors import DomainError from sympy.polys.polyconfig import query def groebner(seq, ring, method=None): """ Computes Groebner basis for a set of polynomials in `K[X]`. Wrapper around the (default) improved Buchberger and the other algorithms for computing Groebner bases. The choice of algorithm can be changed via ``method`` argument or :func:`sympy.polys.polyconfig.setup`, where ``method`` can be either ``buchberger`` or ``f5b``. """ if method is None: method = query('groebner') _groebner_methods = { 'buchberger': _buchberger, 'f5b': _f5b, } try: _groebner = _groebner_methods[method] except KeyError: raise ValueError("'%s' is not a valid Groebner bases algorithm (valid are 'buchberger' and 'f5b')" % method) domain, orig = ring.domain, None if not domain.is_Field or not domain.has_assoc_Field: try: orig, ring = ring, ring.clone(domain=domain.get_field()) except DomainError: raise DomainError("can't compute a Groebner basis over %s" % domain) else: seq = [ s.set_ring(ring) for s in seq ] G = _groebner(seq, ring) if orig is not None: G = [ g.clear_denoms()[1].set_ring(orig) for g in G ] return G def _buchberger(f, ring): """ Computes Groebner basis for a set of polynomials in `K[X]`. Given a set of multivariate polynomials `F`, finds another set `G`, such that Ideal `F = Ideal G` and `G` is a reduced Groebner basis. The resulting basis is unique and has monic generators if the ground domains is a field. Otherwise the result is non-unique but Groebner bases over e.g. integers can be computed (if the input polynomials are monic). Groebner bases can be used to choose specific generators for a polynomial ideal. Because these bases are unique you can check for ideal equality by comparing the Groebner bases. To see if one polynomial lies in an ideal, divide by the elements in the base and see if the remainder vanishes. They can also be used to solve systems of polynomial equations as, by choosing lexicographic ordering, you can eliminate one variable at a time, provided that the ideal is zero-dimensional (finite number of solutions). Notes ===== Algorithm used: an improved version of Buchberger's algorithm as presented in T. Becker, V. Weispfenning, Groebner Bases: A Computational Approach to Commutative Algebra, Springer, 1993, page 232. References ========== .. [1] [Bose03]_ .. [2] [Giovini91]_ .. [3] [Ajwa95]_ .. [4] [Cox97]_ """ order = ring.order monomial_mul = ring.monomial_mul monomial_div = ring.monomial_div monomial_lcm = ring.monomial_lcm def select(P): # normal selection strategy # select the pair with minimum LCM(LM(f), LM(g)) pr = min(P, key=lambda pair: order(monomial_lcm(f[pair[0]].LM, f[pair[1]].LM))) return pr def normal(g, J): h = g.rem([ f[j] for j in J ]) if not h: return None else: h = h.monic() if not h in I: I[h] = len(f) f.append(h) return h.LM, I[h] def update(G, B, ih): # update G using the set of critical pairs B and h # [BW] page 230 h = f[ih] mh = h.LM # filter new pairs (h, g), g in G C = G.copy() D = set() while C: # select a pair (h, g) by popping an element from C ig = C.pop() g = f[ig] mg = g.LM LCMhg = monomial_lcm(mh, mg) def lcm_divides(ip): # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm(mh, f[ip].LM) return monomial_div(LCMhg, m) # HT(h) and HT(g) disjoint: mh*mg == LCMhg if monomial_mul(mh, mg) == LCMhg or ( not any(lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)): D.add((ih, ig)) E = set() while D: # select h, g from D (h the same as above) ih, ig = D.pop() mg = f[ig].LM LCMhg = monomial_lcm(mh, mg) if not monomial_mul(mh, mg) == LCMhg: E.add((ih, ig)) # filter old pairs B_new = set() while B: # select g1, g2 from B (-> CP) ig1, ig2 = B.pop() mg1 = f[ig1].LM mg2 = f[ig2].LM LCM12 = monomial_lcm(mg1, mg2) # if HT(h) does not divide lcm(HT(g1), HT(g2)) if not monomial_div(LCM12, mh) or \ monomial_lcm(mg1, mh) == LCM12 or \ monomial_lcm(mg2, mh) == LCM12: B_new.add((ig1, ig2)) B_new |= E # filter polynomials G_new = set() while G: ig = G.pop() mg = f[ig].LM if not monomial_div(mg, mh): G_new.add(ig) G_new.add(ih) return G_new, B_new # end of update ################################ if not f: return [] # replace f with a reduced list of initial polynomials; see [BW] page 203 f1 = f[:] while True: f = f1[:] f1 = [] for i in range(len(f)): p = f[i] r = p.rem(f[:i]) if r: f1.append(r.monic()) if f == f1: break I = {} # ip = I[p]; p = f[ip] F = set() # set of indices of polynomials G = set() # set of indices of intermediate would-be Groebner basis CP = set() # set of pairs of indices of critical pairs for i, h in enumerate(f): I[h] = i F.add(i) ##################################### # algorithm GROEBNERNEWS2 in [BW] page 232 while F: # select p with minimum monomial according to the monomial ordering h = min([f[x] for x in F], key=lambda f: order(f.LM)) ih = I[h] F.remove(ih) G, CP = update(G, CP, ih) # count the number of critical pairs which reduce to zero reductions_to_zero = 0 while CP: ig1, ig2 = select(CP) CP.remove((ig1, ig2)) h = spoly(f[ig1], f[ig2], ring) # ordering divisors is on average more efficient [Cox] page 111 G1 = sorted(G, key=lambda g: order(f[g].LM)) ht = normal(h, G1) if ht: G, CP = update(G, CP, ht[1]) else: reductions_to_zero += 1 ###################################### # now G is a Groebner basis; reduce it Gr = set() for ig in G: ht = normal(f[ig], G - {ig}) if ht: Gr.add(ht[1]) Gr = [f[ig] for ig in Gr] # order according to the monomial ordering Gr = sorted(Gr, key=lambda f: order(f.LM), reverse=True) return Gr def spoly(p1, p2, ring): """ Compute LCM(LM(p1), LM(p2))/LM(p1)*p1 - LCM(LM(p1), LM(p2))/LM(p2)*p2 This is the S-poly provided p1 and p2 are monic """ LM1 = p1.LM LM2 = p2.LM LCM12 = ring.monomial_lcm(LM1, LM2) m1 = ring.monomial_div(LCM12, LM1) m2 = ring.monomial_div(LCM12, LM2) s1 = p1.mul_monom(m1) s2 = p2.mul_monom(m2) s = s1 - s2 return s # F5B # convenience functions def Sign(f): return f[0] def Polyn(f): return f[1] def Num(f): return f[2] def sig(monomial, index): return (monomial, index) def lbp(signature, polynomial, number): return (signature, polynomial, number) # signature functions def sig_cmp(u, v, order): """ Compare two signatures by extending the term order to K[X]^n. u < v iff - the index of v is greater than the index of u or - the index of v is equal to the index of u and u[0] < v[0] w.r.t. order u > v otherwise """ if u[1] > v[1]: return -1 if u[1] == v[1]: #if u[0] == v[0]: # return 0 if order(u[0]) < order(v[0]): return -1 return 1 def sig_key(s, order): """ Key for comparing two signatures. s = (m, k), t = (n, l) s < t iff [k > l] or [k == l and m < n] s > t otherwise """ return (-s[1], order(s[0])) def sig_mult(s, m): """ Multiply a signature by a monomial. The product of a signature (m, i) and a monomial n is defined as (m * t, i). """ return sig(monomial_mul(s[0], m), s[1]) # labeled polynomial functions def lbp_sub(f, g): """ Subtract labeled polynomial g from f. The signature and number of the difference of f and g are signature and number of the maximum of f and g, w.r.t. lbp_cmp. """ if sig_cmp(Sign(f), Sign(g), Polyn(f).ring.order) < 0: max_poly = g else: max_poly = f ret = Polyn(f) - Polyn(g) return lbp(Sign(max_poly), ret, Num(max_poly)) def lbp_mul_term(f, cx): """ Multiply a labeled polynomial with a term. The product of a labeled polynomial (s, p, k) by a monomial is defined as (m * s, m * p, k). """ return lbp(sig_mult(Sign(f), cx[0]), Polyn(f).mul_term(cx), Num(f)) def lbp_cmp(f, g): """ Compare two labeled polynomials. f < g iff - Sign(f) < Sign(g) or - Sign(f) == Sign(g) and Num(f) > Num(g) f > g otherwise """ if sig_cmp(Sign(f), Sign(g), Polyn(f).ring.order) == -1: return -1 if Sign(f) == Sign(g): if Num(f) > Num(g): return -1 #if Num(f) == Num(g): # return 0 return 1 def lbp_key(f): """ Key for comparing two labeled polynomials. """ return (sig_key(Sign(f), Polyn(f).ring.order), -Num(f)) # algorithm and helper functions def critical_pair(f, g, ring): """ Compute the critical pair corresponding to two labeled polynomials. A critical pair is a tuple (um, f, vm, g), where um and vm are terms such that um * f - vm * g is the S-polynomial of f and g (so, wlog assume um * f > vm * g). For performance sake, a critical pair is represented as a tuple (Sign(um * f), um, f, Sign(vm * g), vm, g), since um * f creates a new, relatively expensive object in memory, whereas Sign(um * f) and um are lightweight and f (in the tuple) is a reference to an already existing object in memory. """ domain = ring.domain ltf = Polyn(f).LT ltg = Polyn(g).LT lt = (monomial_lcm(ltf[0], ltg[0]), domain.one) um = term_div(lt, ltf, domain) vm = term_div(lt, ltg, domain) # The full information is not needed (now), so only the product # with the leading term is considered: fr = lbp_mul_term(lbp(Sign(f), Polyn(f).leading_term(), Num(f)), um) gr = lbp_mul_term(lbp(Sign(g), Polyn(g).leading_term(), Num(g)), vm) # return in proper order, such that the S-polynomial is just # u_first * f_first - u_second * f_second: if lbp_cmp(fr, gr) == -1: return (Sign(gr), vm, g, Sign(fr), um, f) else: return (Sign(fr), um, f, Sign(gr), vm, g) def cp_cmp(c, d): """ Compare two critical pairs c and d. c < d iff - lbp(c[0], _, Num(c[2]) < lbp(d[0], _, Num(d[2])) (this corresponds to um_c * f_c and um_d * f_d) or - lbp(c[0], _, Num(c[2]) >< lbp(d[0], _, Num(d[2])) and lbp(c[3], _, Num(c[5])) < lbp(d[3], _, Num(d[5])) (this corresponds to vm_c * g_c and vm_d * g_d) c > d otherwise """ zero = Polyn(c[2]).ring.zero c0 = lbp(c[0], zero, Num(c[2])) d0 = lbp(d[0], zero, Num(d[2])) r = lbp_cmp(c0, d0) if r == -1: return -1 if r == 0: c1 = lbp(c[3], zero, Num(c[5])) d1 = lbp(d[3], zero, Num(d[5])) r = lbp_cmp(c1, d1) if r == -1: return -1 #if r == 0: # return 0 return 1 def cp_key(c, ring): """ Key for comparing critical pairs. """ return (lbp_key(lbp(c[0], ring.zero, Num(c[2]))), lbp_key(lbp(c[3], ring.zero, Num(c[5])))) def s_poly(cp): """ Compute the S-polynomial of a critical pair. The S-polynomial of a critical pair cp is cp[1] * cp[2] - cp[4] * cp[5]. """ return lbp_sub(lbp_mul_term(cp[2], cp[1]), lbp_mul_term(cp[5], cp[4])) def is_rewritable_or_comparable(sign, num, B): """ Check if a labeled polynomial is redundant by checking if its signature and number imply rewritability or comparability. (sign, num) is comparable if there exists a labeled polynomial h in B, such that sign[1] (the index) is less than Sign(h)[1] and sign[0] is divisible by the leading monomial of h. (sign, num) is rewritable if there exists a labeled polynomial h in B, such thatsign[1] is equal to Sign(h)[1], num < Num(h) and sign[0] is divisible by Sign(h)[0]. """ for h in B: # comparable if sign[1] < Sign(h)[1]: if monomial_divides(Polyn(h).LM, sign[0]): return True # rewritable if sign[1] == Sign(h)[1]: if num < Num(h): if monomial_divides(Sign(h)[0], sign[0]): return True return False def f5_reduce(f, B): """ F5-reduce a labeled polynomial f by B. Continuously searches for non-zero labeled polynomial h in B, such that the leading term lt_h of h divides the leading term lt_f of f and Sign(lt_h * h) < Sign(f). If such a labeled polynomial h is found, f gets replaced by f - lt_f / lt_h * h. If no such h can be found or f is 0, f is no further F5-reducible and f gets returned. A polynomial that is reducible in the usual sense need not be F5-reducible, e.g.: >>> from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn >>> from sympy.polys import ring, QQ, lex >>> R, x,y,z = ring("x,y,z", QQ, lex) >>> f = lbp(sig((1, 1, 1), 4), x, 3) >>> g = lbp(sig((0, 0, 0), 2), x, 2) >>> Polyn(f).rem([Polyn(g)]) 0 >>> f5_reduce(f, [g]) (((1, 1, 1), 4), x, 3) """ order = Polyn(f).ring.order domain = Polyn(f).ring.domain if not Polyn(f): return f while True: g = f for h in B: if Polyn(h): if monomial_divides(Polyn(h).LM, Polyn(f).LM): t = term_div(Polyn(f).LT, Polyn(h).LT, domain) if sig_cmp(sig_mult(Sign(h), t[0]), Sign(f), order) < 0: # The following check need not be done and is in general slower than without. #if not is_rewritable_or_comparable(Sign(gp), Num(gp), B): hp = lbp_mul_term(h, t) f = lbp_sub(f, hp) break if g == f or not Polyn(f): return f def _f5b(F, ring): """ Computes a reduced Groebner basis for the ideal generated by F. f5b is an implementation of the F5B algorithm by Yao Sun and Dingkang Wang. Similarly to Buchberger's algorithm, the algorithm proceeds by computing critical pairs, computing the S-polynomial, reducing it and adjoining the reduced S-polynomial if it is not 0. Unlike Buchberger's algorithm, each polynomial contains additional information, namely a signature and a number. The signature specifies the path of computation (i.e. from which polynomial in the original basis was it derived and how), the number says when the polynomial was added to the basis. With this information it is (often) possible to decide if an S-polynomial will reduce to 0 and can be discarded. Optimizations include: Reducing the generators before computing a Groebner basis, removing redundant critical pairs when a new polynomial enters the basis and sorting the critical pairs and the current basis. Once a Groebner basis has been found, it gets reduced. References ========== .. [1] Yao Sun, Dingkang Wang: "A New Proof for the Correctness of F5 (F5-Like) Algorithm", http://arxiv.org/abs/1004.0084 (specifically v4) .. [2] Thomas Becker, Volker Weispfenning, Groebner bases: A computational approach to commutative algebra, 1993, p. 203, 216 """ order = ring.order # reduce polynomials (like in Mario Pernici's implementation) (Becker, Weispfenning, p. 203) B = F while True: F = B B = [] for i in range(len(F)): p = F[i] r = p.rem(F[:i]) if r: B.append(r) if F == B: break # basis B = [lbp(sig(ring.zero_monom, i + 1), F[i], i + 1) for i in range(len(F))] B.sort(key=lambda f: order(Polyn(f).LM), reverse=True) # critical pairs CP = [critical_pair(B[i], B[j], ring) for i in range(len(B)) for j in range(i + 1, len(B))] CP.sort(key=lambda cp: cp_key(cp, ring), reverse=True) k = len(B) reductions_to_zero = 0 while len(CP): cp = CP.pop() # discard redundant critical pairs: if is_rewritable_or_comparable(cp[0], Num(cp[2]), B): continue if is_rewritable_or_comparable(cp[3], Num(cp[5]), B): continue s = s_poly(cp) p = f5_reduce(s, B) p = lbp(Sign(p), Polyn(p).monic(), k + 1) if Polyn(p): # remove old critical pairs, that become redundant when adding p: indices = [] for i, cp in enumerate(CP): if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p]): indices.append(i) elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p]): indices.append(i) for i in reversed(indices): del CP[i] # only add new critical pairs that are not made redundant by p: for g in B: if Polyn(g): cp = critical_pair(p, g, ring) if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p]): continue elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p]): continue CP.append(cp) # sort (other sorting methods/selection strategies were not as successful) CP.sort(key=lambda cp: cp_key(cp, ring), reverse=True) # insert p into B: m = Polyn(p).LM if order(m) <= order(Polyn(B[-1]).LM): B.append(p) else: for i, q in enumerate(B): if order(m) > order(Polyn(q).LM): B.insert(i, p) break k += 1 #print(len(B), len(CP), "%d critical pairs removed" % len(indices)) else: reductions_to_zero += 1 # reduce Groebner basis: H = [Polyn(g).monic() for g in B] H = red_groebner(H, ring) return sorted(H, key=lambda f: order(f.LM), reverse=True) def red_groebner(G, ring): """ Compute reduced Groebner basis, from BeckerWeispfenning93, p. 216 Selects a subset of generators, that already generate the ideal and computes a reduced Groebner basis for them. """ def reduction(P): """ The actual reduction algorithm. """ Q = [] for i, p in enumerate(P): h = p.rem(P[:i] + P[i + 1:]) if h: Q.append(h) return [p.monic() for p in Q] F = G H = [] while F: f0 = F.pop() if not any(monomial_divides(f.LM, f0.LM) for f in F + H): H.append(f0) # Becker, Weispfenning, p. 217: H is Groebner basis of the ideal generated by G. return reduction(H) def is_groebner(G, ring): """ Check if G is a Groebner basis. """ for i in range(len(G)): for j in range(i + 1, len(G)): s = spoly(G[i], G[j], ring) s = s.rem(G) if s: return False return True def is_minimal(G, ring): """ Checks if G is a minimal Groebner basis. """ order = ring.order domain = ring.domain G.sort(key=lambda g: order(g.LM)) for i, g in enumerate(G): if g.LC != domain.one: return False for h in G[:i] + G[i + 1:]: if monomial_divides(h.LM, g.LM): return False return True def is_reduced(G, ring): """ Checks if G is a reduced Groebner basis. """ order = ring.order domain = ring.domain G.sort(key=lambda g: order(g.LM)) for i, g in enumerate(G): if g.LC != domain.one: return False for term in g.terms(): for h in G[:i] + G[i + 1:]: if monomial_divides(h.LM, term[0]): return False return True def groebner_lcm(f, g): """ Computes LCM of two polynomials using Groebner bases. The LCM is computed as the unique generator of the intersection of the two ideals generated by `f` and `g`. The approach is to compute a Groebner basis with respect to lexicographic ordering of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and then filtering out the solution that doesn't contain `t`. References ========== .. [1] [Cox97]_ """ if f.ring != g.ring: raise ValueError("Values should be equal") ring = f.ring domain = ring.domain if not f or not g: return ring.zero if len(f) <= 1 and len(g) <= 1: monom = monomial_lcm(f.LM, g.LM) coeff = domain.lcm(f.LC, g.LC) return ring.term_new(monom, coeff) fc, f = f.primitive() gc, g = g.primitive() lcm = domain.lcm(fc, gc) f_terms = [ ((1,) + monom, coeff) for monom, coeff in f.terms() ] g_terms = [ ((0,) + monom, coeff) for monom, coeff in g.terms() ] \ + [ ((1,) + monom,-coeff) for monom, coeff in g.terms() ] t = Dummy("t") t_ring = ring.clone(symbols=(t,) + ring.symbols, order=lex) F = t_ring.from_terms(f_terms) G = t_ring.from_terms(g_terms) basis = groebner([F, G], t_ring) def is_independent(h, j): return all(not monom[j] for monom in h.monoms()) H = [ h for h in basis if is_independent(h, 0) ] h_terms = [ (monom[1:], coeff*lcm) for monom, coeff in H[0].terms() ] h = ring.from_terms(h_terms) return h def groebner_gcd(f, g): """Computes GCD of two polynomials using Groebner bases. """ if f.ring != g.ring: raise ValueError("Values should be equal") domain = f.ring.domain if not domain.is_Field: fc, f = f.primitive() gc, g = g.primitive() gcd = domain.gcd(fc, gc) H = (f*g).quo([groebner_lcm(f, g)]) if len(H) != 1: raise ValueError("Length should be 1") h = H[0] if not domain.is_Field: return gcd*h else: return h.monic()
dfdbd8db7055aff76a0f2540da6b68e01d58b5f57ad4f2a1924dac8644518a85
"""Definitions of common exceptions for `polys` module. """ from sympy.utilities import public @public class BasePolynomialError(Exception): """Base class for polynomial related exceptions. """ def new(self, *args): raise NotImplementedError("abstract base class") @public class ExactQuotientFailed(BasePolynomialError): def __init__(self, f, g, dom=None): self.f, self.g, self.dom = f, g, dom def __str__(self): # pragma: no cover from sympy.printing.str import sstr if self.dom is None: return "%s does not divide %s" % (sstr(self.g), sstr(self.f)) else: return "%s does not divide %s in %s" % (sstr(self.g), sstr(self.f), sstr(self.dom)) def new(self, f, g): return self.__class__(f, g, self.dom) @public class PolynomialDivisionFailed(BasePolynomialError): def __init__(self, f, g, domain): self.f = f self.g = g self.domain = domain def __str__(self): if self.domain.is_EX: msg = "You may want to use a different simplification algorithm. Note " \ "that in general it's not possible to guarantee to detect zero " \ "in this domain." elif not self.domain.is_Exact: msg = "Your working precision or tolerance of computations may be set " \ "improperly. Adjust those parameters of the coefficient domain " \ "and try again." else: msg = "Zero detection is guaranteed in this coefficient domain. This " \ "may indicate a bug in SymPy or the domain is user defined and " \ "doesn't implement zero detection properly." return "couldn't reduce degree in a polynomial division algorithm when " \ "dividing %s by %s. This can happen when it's not possible to " \ "detect zero in the coefficient domain. The domain of computation " \ "is %s. %s" % (self.f, self.g, self.domain, msg) @public class OperationNotSupported(BasePolynomialError): def __init__(self, poly, func): self.poly = poly self.func = func def __str__(self): # pragma: no cover return "`%s` operation not supported by %s representation" % (self.func, self.poly.rep.__class__.__name__) @public class HeuristicGCDFailed(BasePolynomialError): pass class ModularGCDFailed(BasePolynomialError): pass @public class HomomorphismFailed(BasePolynomialError): pass @public class IsomorphismFailed(BasePolynomialError): pass @public class ExtraneousFactors(BasePolynomialError): pass @public class EvaluationFailed(BasePolynomialError): pass @public class RefinementFailed(BasePolynomialError): pass @public class CoercionFailed(BasePolynomialError): pass @public class NotInvertible(BasePolynomialError): pass @public class NotReversible(BasePolynomialError): pass @public class NotAlgebraic(BasePolynomialError): pass @public class DomainError(BasePolynomialError): pass @public class PolynomialError(BasePolynomialError): pass @public class UnificationFailed(BasePolynomialError): pass @public class GeneratorsError(BasePolynomialError): pass @public class GeneratorsNeeded(GeneratorsError): pass @public class ComputationFailed(BasePolynomialError): def __init__(self, func, nargs, exc): self.func = func self.nargs = nargs self.exc = exc def __str__(self): return "%s(%s) failed without generators" % (self.func, ', '.join(map(str, self.exc.exprs[:self.nargs]))) @public class UnivariatePolynomialError(PolynomialError): pass @public class MultivariatePolynomialError(PolynomialError): pass @public class PolificationFailed(PolynomialError): def __init__(self, opt, origs, exprs, seq=False): if not seq: self.orig = origs self.expr = exprs self.origs = [origs] self.exprs = [exprs] else: self.origs = origs self.exprs = exprs self.opt = opt self.seq = seq def __str__(self): # pragma: no cover if not self.seq: return "can't construct a polynomial from %s" % str(self.orig) else: return "can't construct polynomials from %s" % ', '.join(map(str, self.origs)) @public class OptionError(BasePolynomialError): pass @public class FlagError(OptionError): pass
32b86a883b044adceb954008aca1f658ec999440e923f25eb2f7776d6e1d6905
"""High-level polynomials manipulation functions. """ from sympy.core import S, Basic, Add, Mul, symbols, Dummy from sympy.polys.polyerrors import ( PolificationFailed, ComputationFailed, MultivariatePolynomialError, OptionError) from sympy.polys.polyoptions import allowed_flags from sympy.polys.polytools import ( poly_from_expr, parallel_poly_from_expr, Poly) from sympy.polys.specialpolys import ( symmetric_poly, interpolating_poly) from sympy.utilities import numbered_symbols, take, public @public def symmetrize(F, *gens, **args): """ Rewrite a polynomial in terms of elementary symmetric polynomials. A symmetric polynomial is a multivariate polynomial that remains invariant under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``, then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where ``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an element of the group ``S_n``). Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that ``f = f1 + f2 + ... + fn``. Examples ======== >>> from sympy.polys.polyfuncs import symmetrize >>> from sympy.abc import x, y >>> symmetrize(x**2 + y**2) (-2*x*y + (x + y)**2, 0) >>> symmetrize(x**2 + y**2, formal=True) (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)]) >>> symmetrize(x**2 - y**2) (-2*x*y + (x + y)**2, -2*y**2) >>> symmetrize(x**2 - y**2, formal=True) (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)]) """ allowed_flags(args, ['formal', 'symbols']) iterable = True if not hasattr(F, '__iter__'): iterable = False F = [F] try: F, opt = parallel_poly_from_expr(F, *gens, **args) except PolificationFailed as exc: result = [] for expr in exc.exprs: if expr.is_Number: result.append((expr, S.Zero)) else: raise ComputationFailed('symmetrize', len(F), exc) if not iterable: result, = result if not exc.opt.formal: return result else: if iterable: return result, [] else: return result + ([],) polys, symbols = [], opt.symbols gens, dom = opt.gens, opt.domain for i in range(len(gens)): poly = symmetric_poly(i + 1, gens, polys=True) polys.append((next(symbols), poly.set_domain(dom))) indices = list(range(len(gens) - 1)) weights = list(range(len(gens), 0, -1)) result = [] for f in F: symmetric = [] if not f.is_homogeneous: symmetric.append(f.TC()) f -= f.TC().as_poly(f.gens) while f: _height, _monom, _coeff = -1, None, None for i, (monom, coeff) in enumerate(f.terms()): if all(monom[i] >= monom[i + 1] for i in indices): height = max([n*m for n, m in zip(weights, monom)]) if height > _height: _height, _monom, _coeff = height, monom, coeff if _height != -1: monom, coeff = _monom, _coeff else: break exponents = [] for m1, m2 in zip(monom, monom[1:] + (0,)): exponents.append(m1 - m2) term = [s**n for (s, _), n in zip(polys, exponents)] poly = [p**n for (_, p), n in zip(polys, exponents)] symmetric.append(Mul(coeff, *term)) product = poly[0].mul(coeff) for p in poly[1:]: product = product.mul(p) f -= product result.append((Add(*symmetric), f.as_expr())) polys = [(s, p.as_expr()) for s, p in polys] if not opt.formal: for i, (sym, non_sym) in enumerate(result): result[i] = (sym.subs(polys), non_sym) if not iterable: result, = result if not opt.formal: return result else: if iterable: return result, polys else: return result + (polys,) @public def horner(f, *gens, **args): """ Rewrite a polynomial in Horner form. Among other applications, evaluation of a polynomial at a point is optimal when it is applied using the Horner scheme ([1]). Examples ======== >>> from sympy.polys.polyfuncs import horner >>> from sympy.abc import x, y, a, b, c, d, e >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) x*(x*(x*(9*x + 8) + 7) + 6) + 5 >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e) e + x*(d + x*(c + x*(a*x + b))) >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y >>> horner(f, wrt=x) x*(x*y*(4*y + 2) + y*(2*y + 1)) >>> horner(f, wrt=y) y*(x*y*(4*x + 2) + x*(2*x + 1)) References ========== [1] - https://en.wikipedia.org/wiki/Horner_scheme """ allowed_flags(args, []) try: F, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: return exc.expr form, gen = S.Zero, F.gen if F.is_univariate: for coeff in F.all_coeffs(): form = form*gen + coeff else: F, gens = Poly(F, gen), gens[1:] for coeff in F.all_coeffs(): form = form*gen + horner(coeff, *gens, **args) return form @public def interpolate(data, x): """ Construct an interpolating polynomial for the data points evaluated at point x (which can be symbolic or numeric). Examples ======== >>> from sympy.polys.polyfuncs import interpolate >>> from sympy.abc import a, b, x A list is interpreted as though it were paired with a range starting from 1: >>> interpolate([1, 4, 9, 16], x) x**2 This can be made explicit by giving a list of coordinates: >>> interpolate([(1, 1), (2, 4), (3, 9)], x) x**2 The (x, y) coordinates can also be given as keys and values of a dictionary (and the points need not be equispaced): >>> interpolate([(-1, 2), (1, 2), (2, 5)], x) x**2 + 1 >>> interpolate({-1: 2, 1: 2, 2: 5}, x) x**2 + 1 If the interpolation is going to be used only once then the value of interest can be passed instead of passing a symbol: >>> interpolate([1, 4, 9], 5) 25 Symbolic coordinates are also supported: >>> [(i,interpolate((a, b), i)) for i in range(1, 4)] [(1, a), (2, b), (3, -a + 2*b)] """ n = len(data) if isinstance(data, dict): if x in data: return S(data[x]) X, Y = list(zip(*data.items())) else: if isinstance(data[0], tuple): X, Y = list(zip(*data)) if x in X: return S(Y[X.index(x)]) else: if x in range(1, n + 1): return S(data[x - 1]) Y = list(data) X = list(range(1, n + 1)) try: return interpolating_poly(n, x, X, Y).expand() except ValueError: d = Dummy() return interpolating_poly(n, d, X, Y).expand().subs(d, x) @public def rational_interpolate(data, degnum, X=symbols('x')): """ Returns a rational interpolation, where the data points are element of any integral domain. The first argument contains the data (as a list of coordinates). The ``degnum`` argument is the degree in the numerator of the rational function. Setting it too high will decrease the maximal degree in the denominator for the same amount of data. Examples ======== >>> from sympy.polys.polyfuncs import rational_interpolate >>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)] >>> rational_interpolate(data, 2) (105*x**2 - 525)/(x + 1) Values do not need to be integers: >>> from sympy import sympify >>> x = [1, 2, 3, 4, 5, 6] >>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]") >>> rational_interpolate(zip(x, y), 2) (3*x**2 - 7*x + 2)/(x + 1) The symbol for the variable can be changed if needed: >>> from sympy import symbols >>> z = symbols('z') >>> rational_interpolate(data, 2, X=z) (105*z**2 - 525)/(z + 1) References ========== .. [1] Algorithm is adapted from: http://axiom-wiki.newsynthesis.org/RationalInterpolation """ from sympy.matrices.dense import ones xdata, ydata = list(zip(*data)) k = len(xdata) - degnum - 1 if k < 0: raise OptionError("Too few values for the required degree.") c = ones(degnum + k + 1, degnum + k + 2) for j in range(max(degnum, k)): for i in range(degnum + k + 1): c[i, j + 1] = c[i, j]*xdata[i] for j in range(k + 1): for i in range(degnum + k + 1): c[i, degnum + k + 1 - j] = -c[i, k - j]*ydata[i] r = c.nullspace()[0] return (sum(r[i] * X**i for i in range(degnum + 1)) / sum(r[i + degnum + 1] * X**i for i in range(k + 1))) @public def viete(f, roots=None, *gens, **args): """ Generate Viete's formulas for ``f``. Examples ======== >>> from sympy.polys.polyfuncs import viete >>> from sympy import symbols >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') >>> viete(a*x**2 + b*x + c, [r1, r2], x) [(r1 + r2, -b/a), (r1*r2, c/a)] """ allowed_flags(args, []) if isinstance(roots, Basic): gens, roots = (roots,) + gens, None try: f, opt = poly_from_expr(f, *gens, **args) except PolificationFailed as exc: raise ComputationFailed('viete', 1, exc) if f.is_multivariate: raise MultivariatePolynomialError( "multivariate polynomials are not allowed") n = f.degree() if n < 1: raise ValueError( "can't derive Viete's formulas for a constant polynomial") if roots is None: roots = numbered_symbols('r', start=1) roots = take(roots, n) if n != len(roots): raise ValueError("required %s roots, got %s" % (n, len(roots))) lc, coeffs = f.LC(), f.all_coeffs() result, sign = [], -1 for i, coeff in enumerate(coeffs[1:]): poly = symmetric_poly(i + 1, roots) coeff = sign*(coeff/lc) result.append((poly, coeff)) sign = -sign return result
ac3de29ed80a0ad000307decaa582a6c92f55d2abd4237de1bf318cf008e7a4d
"""Heuristic polynomial GCD algorithm (HEUGCD). """ from .polyerrors import HeuristicGCDFailed HEU_GCD_MAX = 6 def heugcd(f, g): """ Heuristic polynomial GCD in ``Z[X]``. Given univariate polynomials ``f`` and ``g`` in ``Z[X]``, returns their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg`` such that:: h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h) The algorithm is purely heuristic which means it may fail to compute the GCD. This will be signaled by raising an exception. In this case you will need to switch to another GCD method. The algorithm computes the polynomial GCD by evaluating polynomials ``f`` and ``g`` at certain points and computing (fast) integer GCD of those evaluations. The polynomial GCD is recovered from the integer image by interpolation. The evaluation process reduces f and g variable by variable into a large integer. The final step is to verify if the interpolated polynomial is the correct GCD. This gives cofactors of the input polynomials as a side effect. Examples ======== >>> from sympy.polys.heuristicgcd import heugcd >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> h, cff, cfg = heugcd(f, g) >>> h, cff, cfg (x + y, x + y, x) >>> cff*h == f True >>> cfg*h == g True References ========== .. [1] [Liao95]_ """ assert f.ring == g.ring and f.ring.domain.is_ZZ ring = f.ring x0 = ring.gens[0] domain = ring.domain gcd, f, g = f.extract_ground(g) f_norm = f.max_norm() g_norm = g.max_norm() B = domain(2*min(f_norm, g_norm) + 29) x = max(min(B, 99*domain.sqrt(B)), 2*min(f_norm // abs(f.LC), g_norm // abs(g.LC)) + 4) for i in range(0, HEU_GCD_MAX): ff = f.evaluate(x0, x) gg = g.evaluate(x0, x) if ff and gg: if ring.ngens == 1: h, cff, cfg = domain.cofactors(ff, gg) else: h, cff, cfg = heugcd(ff, gg) h = _gcd_interpolate(h, x, ring) h = h.primitive()[1] cff_, r = f.div(h) if not r: cfg_, r = g.div(h) if not r: h = h.mul_ground(gcd) return h, cff_, cfg_ cff = _gcd_interpolate(cff, x, ring) h, r = f.div(cff) if not r: cfg_, r = g.div(h) if not r: h = h.mul_ground(gcd) return h, cff, cfg_ cfg = _gcd_interpolate(cfg, x, ring) h, r = g.div(cfg) if not r: cff_, r = f.div(h) if not r: h = h.mul_ground(gcd) return h, cff_, cfg x = 73794*x * domain.sqrt(domain.sqrt(x)) // 27011 raise HeuristicGCDFailed('no luck') def _gcd_interpolate(h, x, ring): """Interpolate polynomial GCD from integer GCD. """ f, i = ring.zero, 0 # TODO: don't expose poly repr implementation details if ring.ngens == 1: while h: g = h % x if g > x // 2: g -= x h = (h - g) // x # f += X**i*g if g: f[(i,)] = g i += 1 else: while h: g = h.trunc_ground(x) h = (h - g).quo_ground(x) # f += X**i*g if g: for monom, coeff in g.iterterms(): f[(i,) + monom] = coeff i += 1 if f.LC < 0: return -f else: return f
c0e0e9d87794f85087372030da24f1e9fecef7e33a2a0a91abbe8b7a6e0c2253
"""Tools and arithmetics for monomials of distributed polynomials. """ from itertools import combinations_with_replacement, product from textwrap import dedent from sympy.core import Mul, S, Tuple, sympify from sympy.core.compatibility import exec_, iterable from sympy.polys.polyerrors import ExactQuotientFailed from sympy.polys.polyutils import PicklableWithSlots, dict_from_expr from sympy.utilities import public from sympy.core.compatibility import is_sequence @public def itermonomials(variables, max_degrees, min_degrees=None): r""" `max_degrees` and `min_degrees` are either both integers or both lists. Unless otherwise specified, `min_degrees` is either 0 or [0,...,0]. A generator of all monomials `monom` is returned, such that either min_degree <= total_degree(monom) <= max_degree, or min_degrees[i] <= degree_list(monom)[i] <= max_degrees[i], for all i. Case I:: `max_degrees` and `min_degrees` are both integers. =========================================================== Given a set of variables `V` and a min_degree `N` and a max_degree `M` generate a set of monomials of degree less than or equal to `N` and greater than or equal to `M`. The total number of monomials in commutative variables is huge and is given by the following formula if `M = 0`: .. math:: \frac{(\#V + N)!}{\#V! N!} For example if we would like to generate a dense polynomial of a total degree `N = 50` and `M = 0`, which is the worst case, in 5 variables, assuming that exponents and all of coefficients are 32-bit long and stored in an array we would need almost 80 GiB of memory! Fortunately most polynomials, that we will encounter, are sparse. Examples ======== Consider monomials in commutative variables `x` and `y` and non-commutative variables `a` and `b`:: >>> from sympy import symbols >>> from sympy.polys.monomials import itermonomials >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> sorted(itermonomials([x, y], 2), key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2] >>> sorted(itermonomials([x, y], 3), key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3] >>> a, b = symbols('a, b', commutative=False) >>> set(itermonomials([a, b, x], 2)) {1, a, a**2, b, b**2, x, x**2, a*b, b*a, x*a, x*b} >>> sorted(itermonomials([x, y], 2, 1), key=monomial_key('grlex', [y, x])) [x, y, x**2, x*y, y**2] Case II:: `max_degrees` and `min_degrees` are both lists. ========================================================= If max_degrees = [d_1, ..., d_n] and min_degrees = [e_1, ..., e_n], the number of monomials generated is: (d_1 - e_1 + 1) * ... * (d_n - e_n + 1) Example ======= Let us generate all monomials `monom` in variables `x`, and `y` such that [1, 2][i] <= degree_list(monom)[i] <= [2, 4][i], i = 0, 1 :: >>> from sympy import symbols >>> from sympy.polys.monomials import itermonomials >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> sorted(itermonomials([x, y], [2, 4], [1, 2]), reverse=True, key=monomial_key('lex', [x, y])) [x**2*y**4, x**2*y**3, x**2*y**2, x*y**4, x*y**3, x*y**2] """ n = len(variables) if is_sequence(max_degrees): if len(max_degrees) != n: raise ValueError('Argument sizes do not match') if min_degrees is None: min_degrees = [0]*n elif not is_sequence(min_degrees): raise ValueError('min_degrees is not a list') else: if len(min_degrees) != n: raise ValueError('Argument sizes do not match') if any(i < 0 for i in min_degrees): raise ValueError("min_degrees can't contain negative numbers") total_degree = False else: max_degree = max_degrees if max_degree < 0: raise ValueError("max_degrees can't be negative") if min_degrees is None: min_degree = 0 else: if min_degrees < 0: raise ValueError("min_degrees can't be negative") min_degree = min_degrees total_degree = True if total_degree: if min_degree > max_degree: return if not variables or max_degree == 0: yield S.One return # Force to list in case of passed tuple or other incompatible collection variables = list(variables) + [S.One] if all(variable.is_commutative for variable in variables): monomials_list_comm = [] for item in combinations_with_replacement(variables, max_degree): powers = dict() for variable in variables: powers[variable] = 0 for variable in item: if variable != 1: powers[variable] += 1 if max(powers.values()) >= min_degree: monomials_list_comm.append(Mul(*item)) yield from set(monomials_list_comm) else: monomials_list_non_comm = [] for item in product(variables, repeat=max_degree): powers = dict() for variable in variables: powers[variable] = 0 for variable in item: if variable != 1: powers[variable] += 1 if max(powers.values()) >= min_degree: monomials_list_non_comm.append(Mul(*item)) yield from set(monomials_list_non_comm) else: if any(min_degrees[i] > max_degrees[i] for i in range(n)): raise ValueError('min_degrees[i] must be <= max_degrees[i] for all i') power_lists = [] for var, min_d, max_d in zip(variables, min_degrees, max_degrees): power_lists.append([var**i for i in range(min_d, max_d + 1)]) for powers in product(*power_lists): yield Mul(*powers) def monomial_count(V, N): r""" Computes the number of monomials. The number of monomials is given by the following formula: .. math:: \frac{(\#V + N)!}{\#V! N!} where `N` is a total degree and `V` is a set of variables. Examples ======== >>> from sympy.polys.monomials import itermonomials, monomial_count >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> monomial_count(2, 2) 6 >>> M = list(itermonomials([x, y], 2)) >>> sorted(M, key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2] >>> len(M) 6 """ from sympy import factorial return factorial(V + N) / factorial(V) / factorial(N) def monomial_mul(A, B): """ Multiplication of tuples representing monomials. Examples ======== Lets multiply `x**3*y**4*z` with `x*y**2`:: >>> from sympy.polys.monomials import monomial_mul >>> monomial_mul((3, 4, 1), (1, 2, 0)) (4, 6, 1) which gives `x**4*y**5*z`. """ return tuple([ a + b for a, b in zip(A, B) ]) def monomial_div(A, B): """ Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_div >>> monomial_div((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. However:: >>> monomial_div((3, 4, 1), (1, 2, 2)) is None True `x*y**2*z**2` does not divide `x**3*y**4*z`. """ C = monomial_ldiv(A, B) if all(c >= 0 for c in C): return tuple(C) else: return None def monomial_ldiv(A, B): """ Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_ldiv >>> monomial_ldiv((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. >>> monomial_ldiv((3, 4, 1), (1, 2, 2)) (2, 2, -1) which gives `x**2*y**2*z**-1`. """ return tuple([ a - b for a, b in zip(A, B) ]) def monomial_pow(A, n): """Return the n-th pow of the monomial. """ return tuple([ a*n for a in A ]) def monomial_gcd(A, B): """ Greatest common divisor of tuples representing monomials. Examples ======== Lets compute GCD of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_gcd >>> monomial_gcd((1, 4, 1), (3, 2, 0)) (1, 2, 0) which gives `x*y**2`. """ return tuple([ min(a, b) for a, b in zip(A, B) ]) def monomial_lcm(A, B): """ Least common multiple of tuples representing monomials. Examples ======== Lets compute LCM of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_lcm >>> monomial_lcm((1, 4, 1), (3, 2, 0)) (3, 4, 1) which gives `x**3*y**4*z`. """ return tuple([ max(a, b) for a, b in zip(A, B) ]) def monomial_divides(A, B): """ Does there exist a monomial X such that XA == B? Examples ======== >>> from sympy.polys.monomials import monomial_divides >>> monomial_divides((1, 2), (3, 4)) True >>> monomial_divides((1, 2), (0, 2)) False """ return all(a <= b for a, b in zip(A, B)) def monomial_max(*monoms): """ Returns maximal degree for each variable in a set of monomials. Examples ======== Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`. We wish to find out what is the maximal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_max >>> monomial_max((3,4,5), (0,5,1), (6,3,9)) (6, 5, 9) """ M = list(monoms[0]) for N in monoms[1:]: for i, n in enumerate(N): M[i] = max(M[i], n) return tuple(M) def monomial_min(*monoms): """ Returns minimal degree for each variable in a set of monomials. Examples ======== Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`. We wish to find out what is the minimal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_min >>> monomial_min((3,4,5), (0,5,1), (6,3,9)) (0, 3, 1) """ M = list(monoms[0]) for N in monoms[1:]: for i, n in enumerate(N): M[i] = min(M[i], n) return tuple(M) def monomial_deg(M): """ Returns the total degree of a monomial. Examples ======== The total degree of `xy^2` is 3: >>> from sympy.polys.monomials import monomial_deg >>> monomial_deg((1, 2)) 3 """ return sum(M) def term_div(a, b, domain): """Division of two terms in over a ring/field. """ a_lm, a_lc = a b_lm, b_lc = b monom = monomial_div(a_lm, b_lm) if domain.is_Field: if monom is not None: return monom, domain.quo(a_lc, b_lc) else: return None else: if not (monom is None or a_lc % b_lc): return monom, domain.quo(a_lc, b_lc) else: return None class MonomialOps: """Code generator of fast monomial arithmetic functions. """ def __init__(self, ngens): self.ngens = ngens def _build(self, code, name): ns = {} exec_(code, ns) return ns[name] def _vars(self, name): return [ "%s%s" % (name, i) for i in range(self.ngens) ] def mul(self): name = "monomial_mul" template = dedent("""\ def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) """) A = self._vars("a") B = self._vars("b") AB = [ "%s + %s" % (a, b) for a, b in zip(A, B) ] code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB)) return self._build(code, name) def pow(self): name = "monomial_pow" template = dedent("""\ def %(name)s(A, k): (%(A)s,) = A return (%(Ak)s,) """) A = self._vars("a") Ak = [ "%s*k" % a for a in A ] code = template % dict(name=name, A=", ".join(A), Ak=", ".join(Ak)) return self._build(code, name) def mulpow(self): name = "monomial_mulpow" template = dedent("""\ def %(name)s(A, B, k): (%(A)s,) = A (%(B)s,) = B return (%(ABk)s,) """) A = self._vars("a") B = self._vars("b") ABk = [ "%s + %s*k" % (a, b) for a, b in zip(A, B) ] code = template % dict(name=name, A=", ".join(A), B=", ".join(B), ABk=", ".join(ABk)) return self._build(code, name) def ldiv(self): name = "monomial_ldiv" template = dedent("""\ def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) """) A = self._vars("a") B = self._vars("b") AB = [ "%s - %s" % (a, b) for a, b in zip(A, B) ] code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB)) return self._build(code, name) def div(self): name = "monomial_div" template = dedent("""\ def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B %(RAB)s return (%(R)s,) """) A = self._vars("a") B = self._vars("b") RAB = [ "r%(i)s = a%(i)s - b%(i)s\n if r%(i)s < 0: return None" % dict(i=i) for i in range(self.ngens) ] R = self._vars("r") code = template % dict(name=name, A=", ".join(A), B=", ".join(B), RAB="\n ".join(RAB), R=", ".join(R)) return self._build(code, name) def lcm(self): name = "monomial_lcm" template = dedent("""\ def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) """) A = self._vars("a") B = self._vars("b") AB = [ "%s if %s >= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ] code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB)) return self._build(code, name) def gcd(self): name = "monomial_gcd" template = dedent("""\ def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) """) A = self._vars("a") B = self._vars("b") AB = [ "%s if %s <= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ] code = template % dict(name=name, A=", ".join(A), B=", ".join(B), AB=", ".join(AB)) return self._build(code, name) @public class Monomial(PicklableWithSlots): """Class representing a monomial, i.e. a product of powers. """ __slots__ = ('exponents', 'gens') def __init__(self, monom, gens=None): if not iterable(monom): rep, gens = dict_from_expr(sympify(monom), gens=gens) if len(rep) == 1 and list(rep.values())[0] == 1: monom = list(rep.keys())[0] else: raise ValueError("Expected a monomial got {}".format(monom)) self.exponents = tuple(map(int, monom)) self.gens = gens def rebuild(self, exponents, gens=None): return self.__class__(exponents, gens or self.gens) def __len__(self): return len(self.exponents) def __iter__(self): return iter(self.exponents) def __getitem__(self, item): return self.exponents[item] def __hash__(self): return hash((self.__class__.__name__, self.exponents, self.gens)) def __str__(self): if self.gens: return "*".join([ "%s**%s" % (gen, exp) for gen, exp in zip(self.gens, self.exponents) ]) else: return "%s(%s)" % (self.__class__.__name__, self.exponents) def as_expr(self, *gens): """Convert a monomial instance to a SymPy expression. """ gens = gens or self.gens if not gens: raise ValueError( "can't convert %s to an expression without generators" % self) return Mul(*[ gen**exp for gen, exp in zip(gens, self.exponents) ]) def __eq__(self, other): if isinstance(other, Monomial): exponents = other.exponents elif isinstance(other, (tuple, Tuple)): exponents = other else: return False return self.exponents == exponents def __ne__(self, other): return not self == other def __mul__(self, other): if isinstance(other, Monomial): exponents = other.exponents elif isinstance(other, (tuple, Tuple)): exponents = other else: raise NotImplementedError return self.rebuild(monomial_mul(self.exponents, exponents)) def __truediv__(self, other): if isinstance(other, Monomial): exponents = other.exponents elif isinstance(other, (tuple, Tuple)): exponents = other else: raise NotImplementedError result = monomial_div(self.exponents, exponents) if result is not None: return self.rebuild(result) else: raise ExactQuotientFailed(self, Monomial(other)) __floordiv__ = __truediv__ def __pow__(self, other): n = int(other) if not n: return self.rebuild([0]*len(self)) elif n > 0: exponents = self.exponents for i in range(1, n): exponents = monomial_mul(exponents, self.exponents) return self.rebuild(exponents) else: raise ValueError("a non-negative integer expected, got %s" % other) def gcd(self, other): """Greatest common divisor of monomials. """ if isinstance(other, Monomial): exponents = other.exponents elif isinstance(other, (tuple, Tuple)): exponents = other else: raise TypeError( "an instance of Monomial class expected, got %s" % other) return self.rebuild(monomial_gcd(self.exponents, exponents)) def lcm(self, other): """Least common multiple of monomials. """ if isinstance(other, Monomial): exponents = other.exponents elif isinstance(other, (tuple, Tuple)): exponents = other else: raise TypeError( "an instance of Monomial class expected, got %s" % other) return self.rebuild(monomial_lcm(self.exponents, exponents))
a6c9f256fc34593a2422fe25741cc431ebabc3e35b5db9d4bc9e8a23595c77ee
"""Sparse rational function fields. """ from typing import Any, Dict from operator import add, mul, lt, le, gt, ge from sympy.core.compatibility import is_sequence, reduce from sympy.core.expr import Expr from sympy.core.mod import Mod from sympy.core.numbers import Exp1 from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.core.sympify import CantSympify, sympify from sympy.functions.elementary.exponential import ExpBase from sympy.polys.domains.domainelement import DomainElement from sympy.polys.domains.fractionfield import FractionField from sympy.polys.domains.polynomialring import PolynomialRing from sympy.polys.constructor import construct_domain from sympy.polys.orderings import lex from sympy.polys.polyerrors import CoercionFailed from sympy.polys.polyoptions import build_options from sympy.polys.polyutils import _parallel_dict_from_expr from sympy.polys.rings import PolyElement from sympy.printing.defaults import DefaultPrinting from sympy.utilities import public from sympy.utilities.magic import pollute @public def field(symbols, domain, order=lex): """Construct new rational function field returning (field, x1, ..., xn). """ _field = FracField(symbols, domain, order) return (_field,) + _field.gens @public def xfield(symbols, domain, order=lex): """Construct new rational function field returning (field, (x1, ..., xn)). """ _field = FracField(symbols, domain, order) return (_field, _field.gens) @public def vfield(symbols, domain, order=lex): """Construct new rational function field and inject generators into global namespace. """ _field = FracField(symbols, domain, order) pollute([ sym.name for sym in _field.symbols ], _field.gens) return _field @public def sfield(exprs, *symbols, **options): """Construct a field deriving generators and domain from options and input expressions. Parameters ========== exprs : :class:`Expr` or sequence of :class:`Expr` (sympifiable) symbols : sequence of :class:`Symbol`/:class:`Expr` options : keyword arguments understood by :class:`Options` Examples ======== >>> from sympy.core import symbols >>> from sympy.functions import exp, log >>> from sympy.polys.fields import sfield >>> x = symbols("x") >>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2) >>> K Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order >>> f (4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5) """ single = False if not is_sequence(exprs): exprs, single = [exprs], True exprs = list(map(sympify, exprs)) opt = build_options(symbols, options) numdens = [] for expr in exprs: numdens.extend(expr.as_numer_denom()) reps, opt = _parallel_dict_from_expr(numdens, opt) if opt.domain is None: # NOTE: this is inefficient because construct_domain() automatically # performs conversion to the target domain. It shouldn't do this. coeffs = sum([list(rep.values()) for rep in reps], []) opt.domain, _ = construct_domain(coeffs, opt=opt) _field = FracField(opt.gens, opt.domain, opt.order) fracs = [] for i in range(0, len(reps), 2): fracs.append(_field(tuple(reps[i:i+2]))) if single: return (_field, fracs[0]) else: return (_field, fracs) _field_cache = {} # type: Dict[Any, Any] class FracField(DefaultPrinting): """Multivariate distributed rational function field. """ def __new__(cls, symbols, domain, order=lex): from sympy.polys.rings import PolyRing ring = PolyRing(symbols, domain, order) symbols = ring.symbols ngens = ring.ngens domain = ring.domain order = ring.order _hash_tuple = (cls.__name__, symbols, ngens, domain, order) obj = _field_cache.get(_hash_tuple) if obj is None: obj = object.__new__(cls) obj._hash_tuple = _hash_tuple obj._hash = hash(_hash_tuple) obj.ring = ring obj.dtype = type("FracElement", (FracElement,), {"field": obj}) obj.symbols = symbols obj.ngens = ngens obj.domain = domain obj.order = order obj.zero = obj.dtype(ring.zero) obj.one = obj.dtype(ring.one) obj.gens = obj._gens() for symbol, generator in zip(obj.symbols, obj.gens): if isinstance(symbol, Symbol): name = symbol.name if not hasattr(obj, name): setattr(obj, name, generator) _field_cache[_hash_tuple] = obj return obj def _gens(self): """Return a list of polynomial generators. """ return tuple([ self.dtype(gen) for gen in self.ring.gens ]) def __getnewargs__(self): return (self.symbols, self.domain, self.order) def __hash__(self): return self._hash def __eq__(self, other): return isinstance(other, FracField) and \ (self.symbols, self.ngens, self.domain, self.order) == \ (other.symbols, other.ngens, other.domain, other.order) def __ne__(self, other): return not self == other def raw_new(self, numer, denom=None): return self.dtype(numer, denom) def new(self, numer, denom=None): if denom is None: denom = self.ring.one numer, denom = numer.cancel(denom) return self.raw_new(numer, denom) def domain_new(self, element): return self.domain.convert(element) def ground_new(self, element): try: return self.new(self.ring.ground_new(element)) except CoercionFailed: domain = self.domain if not domain.is_Field and domain.has_assoc_Field: ring = self.ring ground_field = domain.get_field() element = ground_field.convert(element) numer = ring.ground_new(ground_field.numer(element)) denom = ring.ground_new(ground_field.denom(element)) return self.raw_new(numer, denom) else: raise def field_new(self, element): if isinstance(element, FracElement): if self == element.field: return element if isinstance(self.domain, FractionField) and \ self.domain.field == element.field: return self.ground_new(element) elif isinstance(self.domain, PolynomialRing) and \ self.domain.ring.to_field() == element.field: return self.ground_new(element) else: raise NotImplementedError("conversion") elif isinstance(element, PolyElement): denom, numer = element.clear_denoms() if isinstance(self.domain, PolynomialRing) and \ numer.ring == self.domain.ring: numer = self.ring.ground_new(numer) elif isinstance(self.domain, FractionField) and \ numer.ring == self.domain.field.to_ring(): numer = self.ring.ground_new(numer) else: numer = numer.set_ring(self.ring) denom = self.ring.ground_new(denom) return self.raw_new(numer, denom) elif isinstance(element, tuple) and len(element) == 2: numer, denom = list(map(self.ring.ring_new, element)) return self.new(numer, denom) elif isinstance(element, str): raise NotImplementedError("parsing") elif isinstance(element, Expr): return self.from_expr(element) else: return self.ground_new(element) __call__ = field_new def _rebuild_expr(self, expr, mapping): domain = self.domain powers = tuple((gen, gen.as_base_exp()) for gen in mapping.keys() if gen.is_Pow or isinstance(gen, ExpBase)) def _rebuild(expr): generator = mapping.get(expr) if generator is not None: return generator elif expr.is_Add: return reduce(add, list(map(_rebuild, expr.args))) elif expr.is_Mul: return reduce(mul, list(map(_rebuild, expr.args))) elif expr.is_Pow or isinstance(expr, (ExpBase, Exp1)): b, e = expr.as_base_exp() # look for bg**eg whose integer power may be b**e for gen, (bg, eg) in powers: if bg == b and Mod(e, eg) == 0: return mapping.get(gen)**int(e/eg) if e.is_Integer and e is not S.One: return _rebuild(b)**int(e) try: return domain.convert(expr) except CoercionFailed: if not domain.is_Field and domain.has_assoc_Field: return domain.get_field().convert(expr) else: raise return _rebuild(sympify(expr)) def from_expr(self, expr): mapping = dict(list(zip(self.symbols, self.gens))) try: frac = self._rebuild_expr(expr, mapping) except CoercionFailed: raise ValueError("expected an expression convertible to a rational function in %s, got %s" % (self, expr)) else: return self.field_new(frac) def to_domain(self): return FractionField(self) def to_ring(self): from sympy.polys.rings import PolyRing return PolyRing(self.symbols, self.domain, self.order) class FracElement(DomainElement, DefaultPrinting, CantSympify): """Element of multivariate distributed rational function field. """ def __init__(self, numer, denom=None): if denom is None: denom = self.field.ring.one elif not denom: raise ZeroDivisionError("zero denominator") self.numer = numer self.denom = denom def raw_new(f, numer, denom): return f.__class__(numer, denom) def new(f, numer, denom): return f.raw_new(*numer.cancel(denom)) def to_poly(f): if f.denom != 1: raise ValueError("f.denom should be 1") return f.numer def parent(self): return self.field.to_domain() def __getnewargs__(self): return (self.field, self.numer, self.denom) _hash = None def __hash__(self): _hash = self._hash if _hash is None: self._hash = _hash = hash((self.field, self.numer, self.denom)) return _hash def copy(self): return self.raw_new(self.numer.copy(), self.denom.copy()) def set_field(self, new_field): if self.field == new_field: return self else: new_ring = new_field.ring numer = self.numer.set_ring(new_ring) denom = self.denom.set_ring(new_ring) return new_field.new(numer, denom) def as_expr(self, *symbols): return self.numer.as_expr(*symbols)/self.denom.as_expr(*symbols) def __eq__(f, g): if isinstance(g, FracElement) and f.field == g.field: return f.numer == g.numer and f.denom == g.denom else: return f.numer == g and f.denom == f.field.ring.one def __ne__(f, g): return not f == g def __bool__(f): return bool(f.numer) def sort_key(self): return (self.denom.sort_key(), self.numer.sort_key()) def _cmp(f1, f2, op): if isinstance(f2, f1.field.dtype): return op(f1.sort_key(), f2.sort_key()) else: return NotImplemented def __lt__(f1, f2): return f1._cmp(f2, lt) def __le__(f1, f2): return f1._cmp(f2, le) def __gt__(f1, f2): return f1._cmp(f2, gt) def __ge__(f1, f2): return f1._cmp(f2, ge) def __pos__(f): """Negate all coefficients in ``f``. """ return f.raw_new(f.numer, f.denom) def __neg__(f): """Negate all coefficients in ``f``. """ return f.raw_new(-f.numer, f.denom) def _extract_ground(self, element): domain = self.field.domain try: element = domain.convert(element) except CoercionFailed: if not domain.is_Field and domain.has_assoc_Field: ground_field = domain.get_field() try: element = ground_field.convert(element) except CoercionFailed: pass else: return -1, ground_field.numer(element), ground_field.denom(element) return 0, None, None else: return 1, element, None def __add__(f, g): """Add rational functions ``f`` and ``g``. """ field = f.field if not g: return f elif not f: return g elif isinstance(g, field.dtype): if f.denom == g.denom: return f.new(f.numer + g.numer, f.denom) else: return f.new(f.numer*g.denom + f.denom*g.numer, f.denom*g.denom) elif isinstance(g, field.ring.dtype): return f.new(f.numer + f.denom*g, f.denom) else: if isinstance(g, FracElement): if isinstance(field.domain, FractionField) and field.domain.field == g.field: pass elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field: return g.__radd__(f) else: return NotImplemented elif isinstance(g, PolyElement): if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring: pass else: return g.__radd__(f) return f.__radd__(g) def __radd__(f, c): if isinstance(c, f.field.ring.dtype): return f.new(f.numer + f.denom*c, f.denom) op, g_numer, g_denom = f._extract_ground(c) if op == 1: return f.new(f.numer + f.denom*g_numer, f.denom) elif not op: return NotImplemented else: return f.new(f.numer*g_denom + f.denom*g_numer, f.denom*g_denom) def __sub__(f, g): """Subtract rational functions ``f`` and ``g``. """ field = f.field if not g: return f elif not f: return -g elif isinstance(g, field.dtype): if f.denom == g.denom: return f.new(f.numer - g.numer, f.denom) else: return f.new(f.numer*g.denom - f.denom*g.numer, f.denom*g.denom) elif isinstance(g, field.ring.dtype): return f.new(f.numer - f.denom*g, f.denom) else: if isinstance(g, FracElement): if isinstance(field.domain, FractionField) and field.domain.field == g.field: pass elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field: return g.__rsub__(f) else: return NotImplemented elif isinstance(g, PolyElement): if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring: pass else: return g.__rsub__(f) op, g_numer, g_denom = f._extract_ground(g) if op == 1: return f.new(f.numer - f.denom*g_numer, f.denom) elif not op: return NotImplemented else: return f.new(f.numer*g_denom - f.denom*g_numer, f.denom*g_denom) def __rsub__(f, c): if isinstance(c, f.field.ring.dtype): return f.new(-f.numer + f.denom*c, f.denom) op, g_numer, g_denom = f._extract_ground(c) if op == 1: return f.new(-f.numer + f.denom*g_numer, f.denom) elif not op: return NotImplemented else: return f.new(-f.numer*g_denom + f.denom*g_numer, f.denom*g_denom) def __mul__(f, g): """Multiply rational functions ``f`` and ``g``. """ field = f.field if not f or not g: return field.zero elif isinstance(g, field.dtype): return f.new(f.numer*g.numer, f.denom*g.denom) elif isinstance(g, field.ring.dtype): return f.new(f.numer*g, f.denom) else: if isinstance(g, FracElement): if isinstance(field.domain, FractionField) and field.domain.field == g.field: pass elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field: return g.__rmul__(f) else: return NotImplemented elif isinstance(g, PolyElement): if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring: pass else: return g.__rmul__(f) return f.__rmul__(g) def __rmul__(f, c): if isinstance(c, f.field.ring.dtype): return f.new(f.numer*c, f.denom) op, g_numer, g_denom = f._extract_ground(c) if op == 1: return f.new(f.numer*g_numer, f.denom) elif not op: return NotImplemented else: return f.new(f.numer*g_numer, f.denom*g_denom) def __truediv__(f, g): """Computes quotient of fractions ``f`` and ``g``. """ field = f.field if not g: raise ZeroDivisionError elif isinstance(g, field.dtype): return f.new(f.numer*g.denom, f.denom*g.numer) elif isinstance(g, field.ring.dtype): return f.new(f.numer, f.denom*g) else: if isinstance(g, FracElement): if isinstance(field.domain, FractionField) and field.domain.field == g.field: pass elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field: return g.__rtruediv__(f) else: return NotImplemented elif isinstance(g, PolyElement): if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring: pass else: return g.__rtruediv__(f) op, g_numer, g_denom = f._extract_ground(g) if op == 1: return f.new(f.numer, f.denom*g_numer) elif not op: return NotImplemented else: return f.new(f.numer*g_denom, f.denom*g_numer) def __rtruediv__(f, c): if not f: raise ZeroDivisionError elif isinstance(c, f.field.ring.dtype): return f.new(f.denom*c, f.numer) op, g_numer, g_denom = f._extract_ground(c) if op == 1: return f.new(f.denom*g_numer, f.numer) elif not op: return NotImplemented else: return f.new(f.denom*g_numer, f.numer*g_denom) def __pow__(f, n): """Raise ``f`` to a non-negative power ``n``. """ if n >= 0: return f.raw_new(f.numer**n, f.denom**n) elif not f: raise ZeroDivisionError else: return f.raw_new(f.denom**-n, f.numer**-n) def diff(f, x): """Computes partial derivative in ``x``. Examples ======== >>> from sympy.polys.fields import field >>> from sympy.polys.domains import ZZ >>> _, x, y, z = field("x,y,z", ZZ) >>> ((x**2 + y)/(z + 1)).diff(x) 2*x/(z + 1) """ x = x.to_poly() return f.new(f.numer.diff(x)*f.denom - f.numer*f.denom.diff(x), f.denom**2) def __call__(f, *values): if 0 < len(values) <= f.field.ngens: return f.evaluate(list(zip(f.field.gens, values))) else: raise ValueError("expected at least 1 and at most %s values, got %s" % (f.field.ngens, len(values))) def evaluate(f, x, a=None): if isinstance(x, list) and a is None: x = [ (X.to_poly(), a) for X, a in x ] numer, denom = f.numer.evaluate(x), f.denom.evaluate(x) else: x = x.to_poly() numer, denom = f.numer.evaluate(x, a), f.denom.evaluate(x, a) field = numer.ring.to_field() return field.new(numer, denom) def subs(f, x, a=None): if isinstance(x, list) and a is None: x = [ (X.to_poly(), a) for X, a in x ] numer, denom = f.numer.subs(x), f.denom.subs(x) else: x = x.to_poly() numer, denom = f.numer.subs(x, a), f.denom.subs(x, a) return f.new(numer, denom) def compose(f, x, a=None): raise NotImplementedError
b28cbbae4c1a8928e5d37de7d41842f5b0553006461380e450dd5e1fe5260535
"""Low-level linear systems solver. """ from sympy.utilities.iterables import connected_components from sympy.matrices import MutableDenseMatrix from sympy.polys.domains import EX from sympy.polys.rings import sring from sympy.polys.polyerrors import NotInvertible from sympy.polys.domainmatrix import DomainMatrix class PolyNonlinearError(Exception): """Raised by solve_lin_sys for nonlinear equations""" pass class RawMatrix(MutableDenseMatrix): _sympify = staticmethod(lambda x: x) def eqs_to_matrix(eqs_coeffs, eqs_rhs, gens, domain): """Get matrix from linear equations in dict format. Explanation =========== Get the matrix representation of a system of linear equations represented as dicts with low-level DomainElement coefficients. This is an *internal* function that is used by solve_lin_sys. Parameters ========== eqs_coeffs: list[dict[Symbol, DomainElement]] The left hand sides of the equations as dicts mapping from symbols to coefficients where the coefficients are instances of DomainElement. eqs_rhs: list[DomainElements] The right hand sides of the equations as instances of DomainElement. gens: list[Symbol] The unknowns in the system of equations. domain: Domain The domain for coefficients of both lhs and rhs. Returns ======= The augmented matrix representation of the system as a DomainMatrix. Examples ======== >>> from sympy import symbols, ZZ >>> from sympy.polys.solvers import eqs_to_matrix >>> x, y = symbols('x, y') >>> eqs_coeff = [{x:ZZ(1), y:ZZ(1)}, {x:ZZ(1), y:ZZ(-1)}] >>> eqs_rhs = [ZZ(0), ZZ(-1)] >>> eqs_to_matrix(eqs_coeff, eqs_rhs, [x, y], ZZ) DomainMatrix([[1, 1, 0], [1, -1, 1]], (2, 3), ZZ) See also ======== solve_lin_sys: Uses :func:`~eqs_to_matrix` internally """ sym2index = {x: n for n, x in enumerate(gens)} nrows = len(eqs_coeffs) ncols = len(gens) + 1 rows = [[domain.zero] * ncols for _ in range(nrows)] for row, eq_coeff, eq_rhs in zip(rows, eqs_coeffs, eqs_rhs): for sym, coeff in eq_coeff.items(): row[sym2index[sym]] = domain.convert(coeff) row[-1] = -domain.convert(eq_rhs) return DomainMatrix(rows, (nrows, ncols), domain) def sympy_eqs_to_ring(eqs, symbols): """Convert a system of equations from Expr to a PolyRing Explanation =========== High-level functions like ``solve`` expect Expr as inputs but can use ``solve_lin_sys`` internally. This function converts equations from ``Expr`` to the low-level poly types used by the ``solve_lin_sys`` function. Parameters ========== eqs: List of Expr A list of equations as Expr instances symbols: List of Symbol A list of the symbols that are the unknowns in the system of equations. Returns ======= Tuple[List[PolyElement], Ring]: The equations as PolyElement instances and the ring of polynomials within which each equation is represented. Examples ======== >>> from sympy import symbols >>> from sympy.polys.solvers import sympy_eqs_to_ring >>> a, x, y = symbols('a, x, y') >>> eqs = [x-y, x+a*y] >>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y]) >>> eqs_ring [x - y, x + a*y] >>> type(eqs_ring[0]) <class 'sympy.polys.rings.PolyElement'> >>> ring ZZ(a)[x,y] With the equations in this form they can be passed to ``solve_lin_sys``: >>> from sympy.polys.solvers import solve_lin_sys >>> solve_lin_sys(eqs_ring, ring) {y: 0, x: 0} """ try: K, eqs_K = sring(eqs, symbols, field=True, extension=True) except NotInvertible: # https://github.com/sympy/sympy/issues/18874 K, eqs_K = sring(eqs, symbols, domain=EX) return eqs_K, K.to_domain() def solve_lin_sys(eqs, ring, _raw=True): """Solve a system of linear equations from a PolynomialRing Explanation =========== Solves a system of linear equations given as PolyElement instances of a PolynomialRing. The basic arithmetic is carried out using instance of DomainElement which is more efficient than :class:`~sympy.core.expr.Expr` for the most common inputs. While this is a public function it is intended primarily for internal use so its interface is not necessarily convenient. Users are suggested to use the :func:`sympy.solvers.solveset.linsolve` function (which uses this function internally) instead. Parameters ========== eqs: list[PolyElement] The linear equations to be solved as elements of a PolynomialRing (assumed equal to zero). ring: PolynomialRing The polynomial ring from which eqs are drawn. The generators of this ring are the unkowns to be solved for and the domain of the ring is the domain of the coefficients of the system of equations. _raw: bool If *_raw* is False, the keys and values in the returned dictionary will be of type Expr (and the unit of the field will be removed from the keys) otherwise the low-level polys types will be returned, e.g. PolyElement: PythonRational. Returns ======= ``None`` if the system has no solution. dict[Symbol, Expr] if _raw=False dict[Symbol, DomainElement] if _raw=True. Examples ======== >>> from sympy import symbols >>> from sympy.polys.solvers import solve_lin_sys, sympy_eqs_to_ring >>> x, y = symbols('x, y') >>> eqs = [x - y, x + y - 2] >>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y]) >>> solve_lin_sys(eqs_ring, ring) {y: 1, x: 1} Passing ``_raw=False`` returns the same result except that the keys are ``Expr`` rather than low-level poly types. >>> solve_lin_sys(eqs_ring, ring, _raw=False) {x: 1, y: 1} See also ======== sympy_eqs_to_ring: prepares the inputs to ``solve_lin_sys``. linsolve: ``linsolve`` uses ``solve_lin_sys`` internally. sympy.solvers.solvers.solve: ``solve`` uses ``solve_lin_sys`` internally. """ as_expr = not _raw assert ring.domain.is_Field eqs_dict = [dict(eq) for eq in eqs] one_monom = ring.one.monoms()[0] zero = ring.domain.zero eqs_rhs = [] eqs_coeffs = [] for eq_dict in eqs_dict: eq_rhs = eq_dict.pop(one_monom, zero) eq_coeffs = {} for monom, coeff in eq_dict.items(): if sum(monom) != 1: msg = "Nonlinear term encountered in solve_lin_sys" raise PolyNonlinearError(msg) eq_coeffs[ring.gens[monom.index(1)]] = coeff if not eq_coeffs: if not eq_rhs: continue else: return None eqs_rhs.append(eq_rhs) eqs_coeffs.append(eq_coeffs) result = _solve_lin_sys(eqs_coeffs, eqs_rhs, ring) if result is not None and as_expr: def to_sympy(x): as_expr = getattr(x, 'as_expr', None) if as_expr: return as_expr() else: return ring.domain.to_sympy(x) tresult = {to_sympy(sym): to_sympy(val) for sym, val in result.items()} # Remove 1.0x result = {} for k, v in tresult.items(): if k.is_Mul: c, s = k.as_coeff_Mul() result[s] = v/c else: result[k] = v return result def _solve_lin_sys(eqs_coeffs, eqs_rhs, ring): """Solve a linear system from dict of PolynomialRing coefficients Explanation =========== This is an **internal** function used by :func:`solve_lin_sys` after the equations have been preprocessed. The role of this function is to split the system into connected components and pass those to :func:`_solve_lin_sys_component`. Examples ======== Setup a system for $x-y=0$ and $x+y=2$ and solve: >>> from sympy import symbols, sring >>> from sympy.polys.solvers import _solve_lin_sys >>> x, y = symbols('x, y') >>> R, (xr, yr) = sring([x, y], [x, y]) >>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}] >>> eqs_rhs = [R.zero, -2*R.one] >>> _solve_lin_sys(eqs, eqs_rhs, R) {y: 1, x: 1} See also ======== solve_lin_sys: This function is used internally by :func:`solve_lin_sys`. """ V = ring.gens E = [] for eq_coeffs in eqs_coeffs: syms = list(eq_coeffs) E.extend(zip(syms[:-1], syms[1:])) G = V, E components = connected_components(G) sym2comp = {} for n, component in enumerate(components): for sym in component: sym2comp[sym] = n subsystems = [([], []) for _ in range(len(components))] for eq_coeff, eq_rhs in zip(eqs_coeffs, eqs_rhs): sym = next(iter(eq_coeff), None) sub_coeff, sub_rhs = subsystems[sym2comp[sym]] sub_coeff.append(eq_coeff) sub_rhs.append(eq_rhs) sol = {} for subsystem in subsystems: subsol = _solve_lin_sys_component(subsystem[0], subsystem[1], ring) if subsol is None: return None sol.update(subsol) return sol def _solve_lin_sys_component(eqs_coeffs, eqs_rhs, ring): """Solve a linear system from dict of PolynomialRing coefficients Explanation =========== This is an **internal** function used by :func:`solve_lin_sys` after the equations have been preprocessed. After :func:`_solve_lin_sys` splits the system into connected components this function is called for each component. The system of equations is solved using Gauss-Jordan elimination with division followed by back-substitution. Examples ======== Setup a system for $x-y=0$ and $x+y=2$ and solve: >>> from sympy import symbols, sring >>> from sympy.polys.solvers import _solve_lin_sys_component >>> x, y = symbols('x, y') >>> R, (xr, yr) = sring([x, y], [x, y]) >>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}] >>> eqs_rhs = [R.zero, -2*R.one] >>> _solve_lin_sys_component(eqs, eqs_rhs, R) {y: 1, x: 1} See also ======== solve_lin_sys: This function is used internally by :func:`solve_lin_sys`. """ # transform from equations to matrix form matrix = eqs_to_matrix(eqs_coeffs, eqs_rhs, ring.gens, ring.domain) # convert to a field for rref if not matrix.domain.is_Field: matrix = matrix.to_field() # solve by row-reduction echelon, pivots = matrix.rref() # construct the returnable form of the solutions keys = ring.gens if pivots and pivots[-1] == len(keys): return None if len(pivots) == len(keys): sol = [] for s in [row[-1] for row in echelon.rep]: a = s sol.append(a) sols = dict(zip(keys, sol)) else: sols = {} g = ring.gens echelon = echelon.rep for i, p in enumerate(pivots): v = echelon[i][-1] - sum(echelon[i][j]*g[j] for j in range(p+1, len(g))) sols[keys[p]] = v return sols
607fcb93ed4bf7cb3e60d4792d44b66933d96b9ffd52f6a6d01f4fcf2837d5d2
"""Algorithms for partial fraction decomposition of rational functions. """ from sympy.core import S, Add, sympify, Function, Lambda, Dummy from sympy.core.basic import preorder_traversal from sympy.polys import Poly, RootSum, cancel, factor from sympy.polys.polyerrors import PolynomialError from sympy.polys.polyoptions import allowed_flags, set_defaults from sympy.polys.polytools import parallel_poly_from_expr from sympy.utilities import numbered_symbols, take, xthreaded, public @xthreaded @public def apart(f, x=None, full=False, **options): """ Compute partial fraction decomposition of a rational function. Given a rational function ``f``, computes the partial fraction decomposition of ``f``. Two algorithms are available: One is based on the undertermined coefficients method, the other is Bronstein's full partial fraction decomposition algorithm. The undetermined coefficients method (selected by ``full=False``) uses polynomial factorization (and therefore accepts the same options as factor) for the denominator. Per default it works over the rational numbers, therefore decomposition of denominators with non-rational roots (e.g. irrational, complex roots) is not supported by default (see options of factor). Bronstein's algorithm can be selected by using ``full=True`` and allows a decomposition of denominators with non-rational roots. A human-readable result can be obtained via ``doit()`` (see examples below). Examples ======== >>> from sympy.polys.partfrac import apart >>> from sympy.abc import x, y By default, using the undetermined coefficients method: >>> apart(y/(x + 2)/(x + 1), x) -y/(x + 2) + y/(x + 1) The undetermined coefficients method does not provide a result when the denominators roots are not rational: >>> apart(y/(x**2 + x + 1), x) y/(x**2 + x + 1) You can choose Bronstein's algorithm by setting ``full=True``: >>> apart(y/(x**2 + x + 1), x, full=True) RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x))) Calling ``doit()`` yields a human-readable result: >>> apart(y/(x**2 + x + 1), x, full=True).doit() (-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 - 2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2) See Also ======== apart_list, assemble_partfrac_list """ allowed_flags(options, []) f = sympify(f) if f.is_Atom: return f else: P, Q = f.as_numer_denom() _options = options.copy() options = set_defaults(options, extension=True) try: (P, Q), opt = parallel_poly_from_expr((P, Q), x, **options) except PolynomialError as msg: if f.is_commutative: raise PolynomialError(msg) # non-commutative if f.is_Mul: c, nc = f.args_cnc(split_1=False) nc = f.func(*nc) if c: c = apart(f.func._from_args(c), x=x, full=full, **_options) return c*nc else: return nc elif f.is_Add: c = [] nc = [] for i in f.args: if i.is_commutative: c.append(i) else: try: nc.append(apart(i, x=x, full=full, **_options)) except NotImplementedError: nc.append(i) return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc) else: reps = [] pot = preorder_traversal(f) next(pot) for e in pot: try: reps.append((e, apart(e, x=x, full=full, **_options))) pot.skip() # this was handled successfully except NotImplementedError: pass return f.xreplace(dict(reps)) if P.is_multivariate: fc = f.cancel() if fc != f: return apart(fc, x=x, full=full, **_options) raise NotImplementedError( "multivariate partial fraction decomposition") common, P, Q = P.cancel(Q) poly, P = P.div(Q, auto=True) P, Q = P.rat_clear_denoms(Q) if Q.degree() <= 1: partial = P/Q else: if not full: partial = apart_undetermined_coeffs(P, Q) else: partial = apart_full_decomposition(P, Q) terms = S.Zero for term in Add.make_args(partial): if term.has(RootSum): terms += term else: terms += factor(term) return common*(poly.as_expr() + terms) def apart_undetermined_coeffs(P, Q): """Partial fractions via method of undetermined coefficients. """ X = numbered_symbols(cls=Dummy) partial, symbols = [], [] _, factors = Q.factor_list() for f, k in factors: n, q = f.degree(), Q for i in range(1, k + 1): coeffs, q = take(X, n), q.quo(f) partial.append((coeffs, q, f, i)) symbols.extend(coeffs) dom = Q.get_domain().inject(*symbols) F = Poly(0, Q.gen, domain=dom) for i, (coeffs, q, f, k) in enumerate(partial): h = Poly(coeffs, Q.gen, domain=dom) partial[i] = (h, f, k) q = q.set_domain(dom) F += h*q system, result = [], S.Zero for (k,), coeff in F.terms(): system.append(coeff - P.nth(k)) from sympy.solvers import solve solution = solve(system, symbols) for h, f, k in partial: h = h.as_expr().subs(solution) result += h/f.as_expr()**k return result def apart_full_decomposition(P, Q): """ Bronstein's full partial fraction decomposition algorithm. Given a univariate rational function ``f``, performing only GCD operations over the algebraic closure of the initial ground domain of definition, compute full partial fraction decomposition with fractions having linear denominators. Note that no factorization of the initial denominator of ``f`` is performed. The final decomposition is formed in terms of a sum of :class:`RootSum` instances. References ========== .. [1] [Bronstein93]_ """ return assemble_partfrac_list(apart_list(P/Q, P.gens[0])) @public def apart_list(f, x=None, dummies=None, **options): """ Compute partial fraction decomposition of a rational function and return the result in structured form. Given a rational function ``f`` compute the partial fraction decomposition of ``f``. Only Bronstein's full partial fraction decomposition algorithm is supported by this method. The return value is highly structured and perfectly suited for further algorithmic treatment rather than being human-readable. The function returns a tuple holding three elements: * The first item is the common coefficient, free of the variable `x` used for decomposition. (It is an element of the base field `K`.) * The second item is the polynomial part of the decomposition. This can be the zero polynomial. (It is an element of `K[x]`.) * The third part itself is a list of quadruples. Each quadruple has the following elements in this order: - The (not necessarily irreducible) polynomial `D` whose roots `w_i` appear in the linear denominator of a bunch of related fraction terms. (This item can also be a list of explicit roots. However, at the moment ``apart_list`` never returns a result this way, but the related ``assemble_partfrac_list`` function accepts this format as input.) - The numerator of the fraction, written as a function of the root `w` - The linear denominator of the fraction *excluding its power exponent*, written as a function of the root `w`. - The power to which the denominator has to be raised. On can always rebuild a plain expression by using the function ``assemble_partfrac_list``. Examples ======== A first example: >>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list >>> from sympy.abc import x, t >>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1) >>> pfd = apart_list(f) >>> pfd (1, Poly(2*x + 4, x, domain='ZZ'), [(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)]) >>> assemble_partfrac_list(pfd) 2*x + 4 + 4/(x - 1) Second example: >>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x) >>> pfd = apart_list(f) >>> pfd (-1, Poly(2/3, x, domain='QQ'), [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)]) >>> assemble_partfrac_list(pfd) -2/3 - 2/(x - 2) Another example, showing symbolic parameters: >>> pfd = apart_list(t/(x**2 + x + t), x) >>> pfd (1, Poly(0, x, domain='ZZ[t]'), [(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'), Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)), Lambda(_a, -_a + x), 1)]) >>> assemble_partfrac_list(pfd) RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x))) This example is taken from Bronstein's original paper: >>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) >>> pfd = apart_list(f) >>> pfd (1, Poly(0, x, domain='ZZ'), [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1), (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2), (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)]) >>> assemble_partfrac_list(pfd) -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2) See also ======== apart, assemble_partfrac_list References ========== .. [1] [Bronstein93]_ """ allowed_flags(options, []) f = sympify(f) if f.is_Atom: return f else: P, Q = f.as_numer_denom() options = set_defaults(options, extension=True) (P, Q), opt = parallel_poly_from_expr((P, Q), x, **options) if P.is_multivariate: raise NotImplementedError( "multivariate partial fraction decomposition") common, P, Q = P.cancel(Q) poly, P = P.div(Q, auto=True) P, Q = P.rat_clear_denoms(Q) polypart = poly if dummies is None: def dummies(name): d = Dummy(name) while True: yield d dummies = dummies("w") rationalpart = apart_list_full_decomposition(P, Q, dummies) return (common, polypart, rationalpart) def apart_list_full_decomposition(P, Q, dummygen): """ Bronstein's full partial fraction decomposition algorithm. Given a univariate rational function ``f``, performing only GCD operations over the algebraic closure of the initial ground domain of definition, compute full partial fraction decomposition with fractions having linear denominators. Note that no factorization of the initial denominator of ``f`` is performed. The final decomposition is formed in terms of a sum of :class:`RootSum` instances. References ========== .. [1] [Bronstein93]_ """ f, x, U = P/Q, P.gen, [] u = Function('u')(x) a = Dummy('a') partial = [] for d, n in Q.sqf_list_include(all=True): b = d.as_expr() U += [ u.diff(x, n - 1) ] h = cancel(f*b**n) / u**n H, subs = [h], [] for j in range(1, n): H += [ H[-1].diff(x) / j ] for j in range(1, n + 1): subs += [ (U[j - 1], b.diff(x, j) / j) ] for j in range(0, n): P, Q = cancel(H[j]).as_numer_denom() for i in range(0, j + 1): P = P.subs(*subs[j - i]) Q = Q.subs(*subs[0]) P = Poly(P, x) Q = Poly(Q, x) G = P.gcd(d) D = d.quo(G) B, g = Q.half_gcdex(D) b = (P * B.quo(g)).rem(D) Dw = D.subs(x, next(dummygen)) numer = Lambda(a, b.as_expr().subs(x, a)) denom = Lambda(a, (x - a)) exponent = n-j partial.append((Dw, numer, denom, exponent)) return partial @public def assemble_partfrac_list(partial_list): r"""Reassemble a full partial fraction decomposition from a structured result obtained by the function ``apart_list``. Examples ======== This example is taken from Bronstein's original paper: >>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list >>> from sympy.abc import x >>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) >>> pfd = apart_list(f) >>> pfd (1, Poly(0, x, domain='ZZ'), [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1), (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2), (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)]) >>> assemble_partfrac_list(pfd) -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2) If we happen to know some roots we can provide them easily inside the structure: >>> pfd = apart_list(2/(x**2-2)) >>> pfd (1, Poly(0, x, domain='ZZ'), [(Poly(_w**2 - 2, _w, domain='ZZ'), Lambda(_a, _a/2), Lambda(_a, -_a + x), 1)]) >>> pfda = assemble_partfrac_list(pfd) >>> pfda RootSum(_w**2 - 2, Lambda(_a, _a/(-_a + x)))/2 >>> pfda.doit() -sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2))) >>> from sympy import Dummy, Poly, Lambda, sqrt >>> a = Dummy("a") >>> pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)]) >>> assemble_partfrac_list(pfd) -sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2))) See Also ======== apart, apart_list """ # Common factor common = partial_list[0] # Polynomial part polypart = partial_list[1] pfd = polypart.as_expr() # Rational parts for r, nf, df, ex in partial_list[2]: if isinstance(r, Poly): # Assemble in case the roots are given implicitly by a polynomials an, nu = nf.variables, nf.expr ad, de = df.variables, df.expr # Hack to make dummies equal because Lambda created new Dummies de = de.subs(ad[0], an[0]) func = Lambda(tuple(an), nu/de**ex) pfd += RootSum(r, func, auto=False, quadratic=False) else: # Assemble in case the roots are given explicitly by a list of algebraic numbers for root in r: pfd += nf(root)/df(root)**ex return common*pfd
4186d17fd2a96ca2659447ed35a260c7b83e5784da69fab29dd3d1c47307d7b3
"""Real and complex root isolation and refinement algorithms. """ from sympy.polys.densearith import ( dup_neg, dup_rshift, dup_rem) from sympy.polys.densebasic import ( dup_LC, dup_TC, dup_degree, dup_strip, dup_reverse, dup_convert, dup_terms_gcd) from sympy.polys.densetools import ( dup_clear_denoms, dup_mirror, dup_scale, dup_shift, dup_transform, dup_diff, dup_eval, dmp_eval_in, dup_sign_variations, dup_real_imag) from sympy.polys.factortools import ( dup_factor_list) from sympy.polys.polyerrors import ( RefinementFailed, DomainError) from sympy.polys.sqfreetools import ( dup_sqf_part, dup_sqf_list) def dup_sturm(f, K): """ Computes the Sturm sequence of ``f`` in ``F[x]``. Given a univariate, square-free polynomial ``f(x)`` returns the associated Sturm sequence ``f_0(x), ..., f_n(x)`` defined by:: f_0(x), f_1(x) = f(x), f'(x) f_n = -rem(f_{n-2}(x), f_{n-1}(x)) Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_sturm(x**3 - 2*x**2 + x - 3) [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2/9*x + 25/9, -2079/4] References ========== .. [1] [Davenport88]_ """ if not K.is_Field: raise DomainError("can't compute Sturm sequence over %s" % K) f = dup_sqf_part(f, K) sturm = [f, dup_diff(f, 1, K)] while sturm[-1]: s = dup_rem(sturm[-2], sturm[-1], K) sturm.append(dup_neg(s, K)) return sturm[:-1] def dup_root_upper_bound(f, K): """Compute the LMQ upper bound for the positive roots of `f`; LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas. References ========== .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials" Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009. """ n, P = len(f), [] t = n * [K.one] if dup_LC(f, K) < 0: f = dup_neg(f, K) f = list(reversed(f)) for i in range(0, n): if f[i] >= 0: continue a, QL = K.log(-f[i], 2), [] for j in range(i + 1, n): if f[j] <= 0: continue q = t[j] + a - K.log(f[j], 2) QL.append([q // (j - i) , j]) if not QL: continue q = min(QL) t[q[1]] = t[q[1]] + 1 P.append(q[0]) if not P: return None else: return K.get_field()(2)**(max(P) + 1) def dup_root_lower_bound(f, K): """Compute the LMQ lower bound for the positive roots of `f`; LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas. References ========== .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials" Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009. """ bound = dup_root_upper_bound(dup_reverse(f), K) if bound is not None: return 1/bound else: return None def _mobius_from_interval(I, field): """Convert an open interval to a Mobius transform. """ s, t = I a, c = field.numer(s), field.denom(s) b, d = field.numer(t), field.denom(t) return a, b, c, d def _mobius_to_interval(M, field): """Convert a Mobius transform to an open interval. """ a, b, c, d = M s, t = field(a, c), field(b, d) if s <= t: return (s, t) else: return (t, s) def dup_step_refine_real_root(f, M, K, fast=False): """One step of positive real root refinement algorithm. """ a, b, c, d = M if a == b and c == d: return f, (a, b, c, d) A = dup_root_lower_bound(f, K) if A is not None: A = K(int(A)) else: A = K.zero if fast and A > 16: f = dup_scale(f, A, K) a, c, A = A*a, A*c, K.one if A >= K.one: f = dup_shift(f, A, K) b, d = A*a + b, A*c + d if not dup_eval(f, K.zero, K): return f, (b, b, d, d) f, g = dup_shift(f, K.one, K), f a1, b1, c1, d1 = a, a + b, c, c + d if not dup_eval(f, K.zero, K): return f, (b1, b1, d1, d1) k = dup_sign_variations(f, K) if k == 1: a, b, c, d = a1, b1, c1, d1 else: f = dup_shift(dup_reverse(g), K.one, K) if not dup_eval(f, K.zero, K): f = dup_rshift(f, 1, K) a, b, c, d = b, a + b, d, c + d return f, (a, b, c, d) def dup_inner_refine_real_root(f, M, K, eps=None, steps=None, disjoint=None, fast=False, mobius=False): """Refine a positive root of `f` given a Mobius transform or an interval. """ F = K.get_field() if len(M) == 2: a, b, c, d = _mobius_from_interval(M, F) else: a, b, c, d = M while not c: f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) if eps is not None and steps is not None: for i in range(0, steps): if abs(F(a, c) - F(b, d)) >= eps: f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) else: break else: if eps is not None: while abs(F(a, c) - F(b, d)) >= eps: f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) if steps is not None: for i in range(0, steps): f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) if disjoint is not None: while True: u, v = _mobius_to_interval((a, b, c, d), F) if v <= disjoint or disjoint <= u: break else: f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) if not mobius: return _mobius_to_interval((a, b, c, d), F) else: return f, (a, b, c, d) def dup_outer_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False): """Refine a positive root of `f` given an interval `(s, t)`. """ a, b, c, d = _mobius_from_interval((s, t), K.get_field()) f = dup_transform(f, dup_strip([a, b]), dup_strip([c, d]), K) if dup_sign_variations(f, K) != 1: raise RefinementFailed("there should be exactly one root in (%s, %s) interval" % (s, t)) return dup_inner_refine_real_root(f, (a, b, c, d), K, eps=eps, steps=steps, disjoint=disjoint, fast=fast) def dup_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False): """Refine real root's approximating interval to the given precision. """ if K.is_QQ: (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() elif not K.is_ZZ: raise DomainError("real root refinement not supported over %s" % K) if s == t: return (s, t) if s > t: s, t = t, s negative = False if s < 0: if t <= 0: f, s, t, negative = dup_mirror(f, K), -t, -s, True else: raise ValueError("can't refine a real root in (%s, %s)" % (s, t)) if negative and disjoint is not None: if disjoint < 0: disjoint = -disjoint else: disjoint = None s, t = dup_outer_refine_real_root( f, s, t, K, eps=eps, steps=steps, disjoint=disjoint, fast=fast) if negative: return (-t, -s) else: return ( s, t) def dup_inner_isolate_real_roots(f, K, eps=None, fast=False): """Internal function for isolation positive roots up to given precision. References ========== 1. Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. 2. Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. """ a, b, c, d = K.one, K.zero, K.zero, K.one k = dup_sign_variations(f, K) if k == 0: return [] if k == 1: roots = [dup_inner_refine_real_root( f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)] else: roots, stack = [], [(a, b, c, d, f, k)] while stack: a, b, c, d, f, k = stack.pop() A = dup_root_lower_bound(f, K) if A is not None: A = K(int(A)) else: A = K.zero if fast and A > 16: f = dup_scale(f, A, K) a, c, A = A*a, A*c, K.one if A >= K.one: f = dup_shift(f, A, K) b, d = A*a + b, A*c + d if not dup_TC(f, K): roots.append((f, (b, b, d, d))) f = dup_rshift(f, 1, K) k = dup_sign_variations(f, K) if k == 0: continue if k == 1: roots.append(dup_inner_refine_real_root( f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)) continue f1 = dup_shift(f, K.one, K) a1, b1, c1, d1, r = a, a + b, c, c + d, 0 if not dup_TC(f1, K): roots.append((f1, (b1, b1, d1, d1))) f1, r = dup_rshift(f1, 1, K), 1 k1 = dup_sign_variations(f1, K) k2 = k - k1 - r a2, b2, c2, d2 = b, a + b, d, c + d if k2 > 1: f2 = dup_shift(dup_reverse(f), K.one, K) if not dup_TC(f2, K): f2 = dup_rshift(f2, 1, K) k2 = dup_sign_variations(f2, K) else: f2 = None if k1 < k2: a1, a2, b1, b2 = a2, a1, b2, b1 c1, c2, d1, d2 = c2, c1, d2, d1 f1, f2, k1, k2 = f2, f1, k2, k1 if not k1: continue if f1 is None: f1 = dup_shift(dup_reverse(f), K.one, K) if not dup_TC(f1, K): f1 = dup_rshift(f1, 1, K) if k1 == 1: roots.append(dup_inner_refine_real_root( f1, (a1, b1, c1, d1), K, eps=eps, fast=fast, mobius=True)) else: stack.append((a1, b1, c1, d1, f1, k1)) if not k2: continue if f2 is None: f2 = dup_shift(dup_reverse(f), K.one, K) if not dup_TC(f2, K): f2 = dup_rshift(f2, 1, K) if k2 == 1: roots.append(dup_inner_refine_real_root( f2, (a2, b2, c2, d2), K, eps=eps, fast=fast, mobius=True)) else: stack.append((a2, b2, c2, d2, f2, k2)) return roots def _discard_if_outside_interval(f, M, inf, sup, K, negative, fast, mobius): """Discard an isolating interval if outside ``(inf, sup)``. """ F = K.get_field() while True: u, v = _mobius_to_interval(M, F) if negative: u, v = -v, -u if (inf is None or u >= inf) and (sup is None or v <= sup): if not mobius: return u, v else: return f, M elif (sup is not None and u > sup) or (inf is not None and v < inf): return None else: f, M = dup_step_refine_real_root(f, M, K, fast=fast) def dup_inner_isolate_positive_roots(f, K, eps=None, inf=None, sup=None, fast=False, mobius=False): """Iteratively compute disjoint positive root isolation intervals. """ if sup is not None and sup < 0: return [] roots = dup_inner_isolate_real_roots(f, K, eps=eps, fast=fast) F, results = K.get_field(), [] if inf is not None or sup is not None: for f, M in roots: result = _discard_if_outside_interval(f, M, inf, sup, K, False, fast, mobius) if result is not None: results.append(result) elif not mobius: for f, M in roots: u, v = _mobius_to_interval(M, F) results.append((u, v)) else: results = roots return results def dup_inner_isolate_negative_roots(f, K, inf=None, sup=None, eps=None, fast=False, mobius=False): """Iteratively compute disjoint negative root isolation intervals. """ if inf is not None and inf >= 0: return [] roots = dup_inner_isolate_real_roots(dup_mirror(f, K), K, eps=eps, fast=fast) F, results = K.get_field(), [] if inf is not None or sup is not None: for f, M in roots: result = _discard_if_outside_interval(f, M, inf, sup, K, True, fast, mobius) if result is not None: results.append(result) elif not mobius: for f, M in roots: u, v = _mobius_to_interval(M, F) results.append((-v, -u)) else: results = roots return results def _isolate_zero(f, K, inf, sup, basis=False, sqf=False): """Handle special case of CF algorithm when ``f`` is homogeneous. """ j, f = dup_terms_gcd(f, K) if j > 0: F = K.get_field() if (inf is None or inf <= 0) and (sup is None or 0 <= sup): if not sqf: if not basis: return [((F.zero, F.zero), j)], f else: return [((F.zero, F.zero), j, [K.one, K.zero])], f else: return [(F.zero, F.zero)], f return [], f def dup_isolate_real_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False): """Isolate real roots of a square-free polynomial using the Vincent-Akritas-Strzebonski (VAS) CF approach. References ========== .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods. Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. """ if K.is_QQ: (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() elif not K.is_ZZ: raise DomainError("isolation of real roots not supported over %s" % K) if dup_degree(f) <= 0: return [] I_zero, f = _isolate_zero(f, K, inf, sup, basis=False, sqf=True) I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) roots = sorted(I_neg + I_zero + I_pos) if not blackbox: return roots else: return [ RealInterval((a, b), f, K) for (a, b) in roots ] def dup_isolate_real_roots(f, K, eps=None, inf=None, sup=None, basis=False, fast=False): """Isolate real roots using Vincent-Akritas-Strzebonski (VAS) continued fractions approach. References ========== .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods. Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. """ if K.is_QQ: (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() elif not K.is_ZZ: raise DomainError("isolation of real roots not supported over %s" % K) if dup_degree(f) <= 0: return [] I_zero, f = _isolate_zero(f, K, inf, sup, basis=basis, sqf=False) _, factors = dup_sqf_list(f, K) if len(factors) == 1: ((f, k),) = factors I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) I_neg = [ ((u, v), k) for u, v in I_neg ] I_pos = [ ((u, v), k) for u, v in I_pos ] else: I_neg, I_pos = _real_isolate_and_disjoin(factors, K, eps=eps, inf=inf, sup=sup, basis=basis, fast=fast) return sorted(I_neg + I_zero + I_pos) def dup_isolate_real_roots_list(polys, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False): """Isolate real roots of a list of square-free polynomial using Vincent-Akritas-Strzebonski (VAS) CF approach. References ========== .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root Isolation Methods. Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. """ if K.is_QQ: K, F, polys = K.get_ring(), K, polys[:] for i, p in enumerate(polys): polys[i] = dup_clear_denoms(p, F, K, convert=True)[1] elif not K.is_ZZ: raise DomainError("isolation of real roots not supported over %s" % K) zeros, factors_dict = False, {} if (inf is None or inf <= 0) and (sup is None or 0 <= sup): zeros, zero_indices = True, {} for i, p in enumerate(polys): j, p = dup_terms_gcd(p, K) if zeros and j > 0: zero_indices[i] = j for f, k in dup_factor_list(p, K)[1]: f = tuple(f) if f not in factors_dict: factors_dict[f] = {i: k} else: factors_dict[f][i] = k factors_list = [] for f, indices in factors_dict.items(): factors_list.append((list(f), indices)) I_neg, I_pos = _real_isolate_and_disjoin(factors_list, K, eps=eps, inf=inf, sup=sup, strict=strict, basis=basis, fast=fast) F = K.get_field() if not zeros or not zero_indices: I_zero = [] else: if not basis: I_zero = [((F.zero, F.zero), zero_indices)] else: I_zero = [((F.zero, F.zero), zero_indices, [K.one, K.zero])] return sorted(I_neg + I_zero + I_pos) def _disjoint_p(M, N, strict=False): """Check if Mobius transforms define disjoint intervals. """ a1, b1, c1, d1 = M a2, b2, c2, d2 = N a1d1, b1c1 = a1*d1, b1*c1 a2d2, b2c2 = a2*d2, b2*c2 if a1d1 == b1c1 and a2d2 == b2c2: return True if a1d1 > b1c1: a1, c1, b1, d1 = b1, d1, a1, c1 if a2d2 > b2c2: a2, c2, b2, d2 = b2, d2, a2, c2 if not strict: return a2*d1 >= c2*b1 or b2*c1 <= d2*a1 else: return a2*d1 > c2*b1 or b2*c1 < d2*a1 def _real_isolate_and_disjoin(factors, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False): """Isolate real roots of a list of polynomials and disjoin intervals. """ I_pos, I_neg = [], [] for i, (f, k) in enumerate(factors): for F, M in dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True): I_pos.append((F, M, k, f)) for G, N in dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True): I_neg.append((G, N, k, f)) for i, (f, M, k, F) in enumerate(I_pos): for j, (g, N, m, G) in enumerate(I_pos[i + 1:]): while not _disjoint_p(M, N, strict=strict): f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) I_pos[i + j + 1] = (g, N, m, G) I_pos[i] = (f, M, k, F) for i, (f, M, k, F) in enumerate(I_neg): for j, (g, N, m, G) in enumerate(I_neg[i + 1:]): while not _disjoint_p(M, N, strict=strict): f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) I_neg[i + j + 1] = (g, N, m, G) I_neg[i] = (f, M, k, F) if strict: for i, (f, M, k, F) in enumerate(I_neg): if not M[0]: while not M[0]: f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) I_neg[i] = (f, M, k, F) break for j, (g, N, m, G) in enumerate(I_pos): if not N[0]: while not N[0]: g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) I_pos[j] = (g, N, m, G) break field = K.get_field() I_neg = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_neg ] I_pos = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_pos ] if not basis: I_neg = [ ((-v, -u), k) for ((u, v), k, _) in I_neg ] I_pos = [ (( u, v), k) for ((u, v), k, _) in I_pos ] else: I_neg = [ ((-v, -u), k, f) for ((u, v), k, f) in I_neg ] I_pos = [ (( u, v), k, f) for ((u, v), k, f) in I_pos ] return I_neg, I_pos def dup_count_real_roots(f, K, inf=None, sup=None): """Returns the number of distinct real roots of ``f`` in ``[inf, sup]``. """ if dup_degree(f) <= 0: return 0 if not K.is_Field: R, K = K, K.get_field() f = dup_convert(f, R, K) sturm = dup_sturm(f, K) if inf is None: signs_inf = dup_sign_variations([ dup_LC(s, K)*(-1)**dup_degree(s) for s in sturm ], K) else: signs_inf = dup_sign_variations([ dup_eval(s, inf, K) for s in sturm ], K) if sup is None: signs_sup = dup_sign_variations([ dup_LC(s, K) for s in sturm ], K) else: signs_sup = dup_sign_variations([ dup_eval(s, sup, K) for s in sturm ], K) count = abs(signs_inf - signs_sup) if inf is not None and not dup_eval(f, inf, K): count += 1 return count OO = 'OO' # Origin of (re, im) coordinate system Q1 = 'Q1' # Quadrant #1 (++): re > 0 and im > 0 Q2 = 'Q2' # Quadrant #2 (-+): re < 0 and im > 0 Q3 = 'Q3' # Quadrant #3 (--): re < 0 and im < 0 Q4 = 'Q4' # Quadrant #4 (+-): re > 0 and im < 0 A1 = 'A1' # Axis #1 (+0): re > 0 and im = 0 A2 = 'A2' # Axis #2 (0+): re = 0 and im > 0 A3 = 'A3' # Axis #3 (-0): re < 0 and im = 0 A4 = 'A4' # Axis #4 (0-): re = 0 and im < 0 _rules_simple = { # Q --> Q (same) => no change (Q1, Q1): 0, (Q2, Q2): 0, (Q3, Q3): 0, (Q4, Q4): 0, # A -- CCW --> Q => +1/4 (CCW) (A1, Q1): 1, (A2, Q2): 1, (A3, Q3): 1, (A4, Q4): 1, # A -- CW --> Q => -1/4 (CCW) (A1, Q4): 2, (A2, Q1): 2, (A3, Q2): 2, (A4, Q3): 2, # Q -- CCW --> A => +1/4 (CCW) (Q1, A2): 3, (Q2, A3): 3, (Q3, A4): 3, (Q4, A1): 3, # Q -- CW --> A => -1/4 (CCW) (Q1, A1): 4, (Q2, A2): 4, (Q3, A3): 4, (Q4, A4): 4, # Q -- CCW --> Q => +1/2 (CCW) (Q1, Q2): +5, (Q2, Q3): +5, (Q3, Q4): +5, (Q4, Q1): +5, # Q -- CW --> Q => -1/2 (CW) (Q1, Q4): -5, (Q2, Q1): -5, (Q3, Q2): -5, (Q4, Q3): -5, } _rules_ambiguous = { # A -- CCW --> Q => { +1/4 (CCW), -9/4 (CW) } (A1, OO, Q1): -1, (A2, OO, Q2): -1, (A3, OO, Q3): -1, (A4, OO, Q4): -1, # A -- CW --> Q => { -1/4 (CCW), +7/4 (CW) } (A1, OO, Q4): -2, (A2, OO, Q1): -2, (A3, OO, Q2): -2, (A4, OO, Q3): -2, # Q -- CCW --> A => { +1/4 (CCW), -9/4 (CW) } (Q1, OO, A2): -3, (Q2, OO, A3): -3, (Q3, OO, A4): -3, (Q4, OO, A1): -3, # Q -- CW --> A => { -1/4 (CCW), +7/4 (CW) } (Q1, OO, A1): -4, (Q2, OO, A2): -4, (Q3, OO, A3): -4, (Q4, OO, A4): -4, # A -- OO --> A => { +1 (CCW), -1 (CW) } (A1, A3): 7, (A2, A4): 7, (A3, A1): 7, (A4, A2): 7, (A1, OO, A3): 7, (A2, OO, A4): 7, (A3, OO, A1): 7, (A4, OO, A2): 7, # Q -- DIA --> Q => { +1 (CCW), -1 (CW) } (Q1, Q3): 8, (Q2, Q4): 8, (Q3, Q1): 8, (Q4, Q2): 8, (Q1, OO, Q3): 8, (Q2, OO, Q4): 8, (Q3, OO, Q1): 8, (Q4, OO, Q2): 8, # A --- R ---> A => { +1/2 (CCW), -3/2 (CW) } (A1, A2): 9, (A2, A3): 9, (A3, A4): 9, (A4, A1): 9, (A1, OO, A2): 9, (A2, OO, A3): 9, (A3, OO, A4): 9, (A4, OO, A1): 9, # A --- L ---> A => { +3/2 (CCW), -1/2 (CW) } (A1, A4): 10, (A2, A1): 10, (A3, A2): 10, (A4, A3): 10, (A1, OO, A4): 10, (A2, OO, A1): 10, (A3, OO, A2): 10, (A4, OO, A3): 10, # Q --- 1 ---> A => { +3/4 (CCW), -5/4 (CW) } (Q1, A3): 11, (Q2, A4): 11, (Q3, A1): 11, (Q4, A2): 11, (Q1, OO, A3): 11, (Q2, OO, A4): 11, (Q3, OO, A1): 11, (Q4, OO, A2): 11, # Q --- 2 ---> A => { +5/4 (CCW), -3/4 (CW) } (Q1, A4): 12, (Q2, A1): 12, (Q3, A2): 12, (Q4, A3): 12, (Q1, OO, A4): 12, (Q2, OO, A1): 12, (Q3, OO, A2): 12, (Q4, OO, A3): 12, # A --- 1 ---> Q => { +5/4 (CCW), -3/4 (CW) } (A1, Q3): 13, (A2, Q4): 13, (A3, Q1): 13, (A4, Q2): 13, (A1, OO, Q3): 13, (A2, OO, Q4): 13, (A3, OO, Q1): 13, (A4, OO, Q2): 13, # A --- 2 ---> Q => { +3/4 (CCW), -5/4 (CW) } (A1, Q2): 14, (A2, Q3): 14, (A3, Q4): 14, (A4, Q1): 14, (A1, OO, Q2): 14, (A2, OO, Q3): 14, (A3, OO, Q4): 14, (A4, OO, Q1): 14, # Q --> OO --> Q => { +1/2 (CCW), -3/2 (CW) } (Q1, OO, Q2): 15, (Q2, OO, Q3): 15, (Q3, OO, Q4): 15, (Q4, OO, Q1): 15, # Q --> OO --> Q => { +3/2 (CCW), -1/2 (CW) } (Q1, OO, Q4): 16, (Q2, OO, Q1): 16, (Q3, OO, Q2): 16, (Q4, OO, Q3): 16, # A --> OO --> A => { +2 (CCW), 0 (CW) } (A1, OO, A1): 17, (A2, OO, A2): 17, (A3, OO, A3): 17, (A4, OO, A4): 17, # Q --> OO --> Q => { +2 (CCW), 0 (CW) } (Q1, OO, Q1): 18, (Q2, OO, Q2): 18, (Q3, OO, Q3): 18, (Q4, OO, Q4): 18, } _values = { 0: [( 0, 1)], 1: [(+1, 4)], 2: [(-1, 4)], 3: [(+1, 4)], 4: [(-1, 4)], -1: [(+9, 4), (+1, 4)], -2: [(+7, 4), (-1, 4)], -3: [(+9, 4), (+1, 4)], -4: [(+7, 4), (-1, 4)], +5: [(+1, 2)], -5: [(-1, 2)], 7: [(+1, 1), (-1, 1)], 8: [(+1, 1), (-1, 1)], 9: [(+1, 2), (-3, 2)], 10: [(+3, 2), (-1, 2)], 11: [(+3, 4), (-5, 4)], 12: [(+5, 4), (-3, 4)], 13: [(+5, 4), (-3, 4)], 14: [(+3, 4), (-5, 4)], 15: [(+1, 2), (-3, 2)], 16: [(+3, 2), (-1, 2)], 17: [(+2, 1), ( 0, 1)], 18: [(+2, 1), ( 0, 1)], } def _classify_point(re, im): """Return the half-axis (or origin) on which (re, im) point is located. """ if not re and not im: return OO if not re: if im > 0: return A2 else: return A4 elif not im: if re > 0: return A1 else: return A3 def _intervals_to_quadrants(intervals, f1, f2, s, t, F): """Generate a sequence of extended quadrants from a list of critical points. """ if not intervals: return [] Q = [] if not f1: (a, b), _, _ = intervals[0] if a == b == s: if len(intervals) == 1: if dup_eval(f2, t, F) > 0: return [OO, A2] else: return [OO, A4] else: (a, _), _, _ = intervals[1] if dup_eval(f2, (s + a)/2, F) > 0: Q.extend([OO, A2]) f2_sgn = +1 else: Q.extend([OO, A4]) f2_sgn = -1 intervals = intervals[1:] else: if dup_eval(f2, s, F) > 0: Q.append(A2) f2_sgn = +1 else: Q.append(A4) f2_sgn = -1 for (a, _), indices, _ in intervals: Q.append(OO) if indices[1] % 2 == 1: f2_sgn = -f2_sgn if a != t: if f2_sgn > 0: Q.append(A2) else: Q.append(A4) return Q if not f2: (a, b), _, _ = intervals[0] if a == b == s: if len(intervals) == 1: if dup_eval(f1, t, F) > 0: return [OO, A1] else: return [OO, A3] else: (a, _), _, _ = intervals[1] if dup_eval(f1, (s + a)/2, F) > 0: Q.extend([OO, A1]) f1_sgn = +1 else: Q.extend([OO, A3]) f1_sgn = -1 intervals = intervals[1:] else: if dup_eval(f1, s, F) > 0: Q.append(A1) f1_sgn = +1 else: Q.append(A3) f1_sgn = -1 for (a, _), indices, _ in intervals: Q.append(OO) if indices[0] % 2 == 1: f1_sgn = -f1_sgn if a != t: if f1_sgn > 0: Q.append(A1) else: Q.append(A3) return Q re = dup_eval(f1, s, F) im = dup_eval(f2, s, F) if not re or not im: Q.append(_classify_point(re, im)) if len(intervals) == 1: re = dup_eval(f1, t, F) im = dup_eval(f2, t, F) else: (a, _), _, _ = intervals[1] re = dup_eval(f1, (s + a)/2, F) im = dup_eval(f2, (s + a)/2, F) intervals = intervals[1:] if re > 0: f1_sgn = +1 else: f1_sgn = -1 if im > 0: f2_sgn = +1 else: f2_sgn = -1 sgn = { (+1, +1): Q1, (-1, +1): Q2, (-1, -1): Q3, (+1, -1): Q4, } Q.append(sgn[(f1_sgn, f2_sgn)]) for (a, b), indices, _ in intervals: if a == b: re = dup_eval(f1, a, F) im = dup_eval(f2, a, F) cls = _classify_point(re, im) if cls is not None: Q.append(cls) if 0 in indices: if indices[0] % 2 == 1: f1_sgn = -f1_sgn if 1 in indices: if indices[1] % 2 == 1: f2_sgn = -f2_sgn if not (a == b and b == t): Q.append(sgn[(f1_sgn, f2_sgn)]) return Q def _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=None): """Transform sequences of quadrants to a sequence of rules. """ if exclude is True: edges = [1, 1, 0, 0] corners = { (0, 1): 1, (1, 2): 1, (2, 3): 0, (3, 0): 1, } else: edges = [0, 0, 0, 0] corners = { (0, 1): 0, (1, 2): 0, (2, 3): 0, (3, 0): 0, } if exclude is not None and exclude is not True: exclude = set(exclude) for i, edge in enumerate(['S', 'E', 'N', 'W']): if edge in exclude: edges[i] = 1 for i, corner in enumerate(['SW', 'SE', 'NE', 'NW']): if corner in exclude: corners[((i - 1) % 4, i)] = 1 QQ, rules = [Q_L1, Q_L2, Q_L3, Q_L4], [] for i, Q in enumerate(QQ): if not Q: continue if Q[-1] == OO: Q = Q[:-1] if Q[0] == OO: j, Q = (i - 1) % 4, Q[1:] qq = (QQ[j][-2], OO, Q[0]) if qq in _rules_ambiguous: rules.append((_rules_ambiguous[qq], corners[(j, i)])) else: raise NotImplementedError("3 element rule (corner): " + str(qq)) q1, k = Q[0], 1 while k < len(Q): q2, k = Q[k], k + 1 if q2 != OO: qq = (q1, q2) if qq in _rules_simple: rules.append((_rules_simple[qq], 0)) elif qq in _rules_ambiguous: rules.append((_rules_ambiguous[qq], edges[i])) else: raise NotImplementedError("2 element rule (inside): " + str(qq)) else: qq, k = (q1, q2, Q[k]), k + 1 if qq in _rules_ambiguous: rules.append((_rules_ambiguous[qq], edges[i])) else: raise NotImplementedError("3 element rule (edge): " + str(qq)) q1 = qq[-1] return rules def _reverse_intervals(intervals): """Reverse intervals for traversal from right to left and from top to bottom. """ return [ ((b, a), indices, f) for (a, b), indices, f in reversed(intervals) ] def _winding_number(T, field): """Compute the winding number of the input polynomial, i.e. the number of roots. """ return int(sum([ field(*_values[t][i]) for t, i in T ]) / field(2)) def dup_count_complex_roots(f, K, inf=None, sup=None, exclude=None): """Count all roots in [u + v*I, s + t*I] rectangle using Collins-Krandick algorithm. """ if not K.is_ZZ and not K.is_QQ: raise DomainError("complex root counting is not supported over %s" % K) if K.is_ZZ: R, F = K, K.get_field() else: R, F = K.get_ring(), K f = dup_convert(f, K, F) if inf is None or sup is None: _, lc = dup_degree(f), abs(dup_LC(f, F)) B = 2*max([ F.quo(abs(c), lc) for c in f ]) if inf is None: (u, v) = (-B, -B) else: (u, v) = inf if sup is None: (s, t) = (+B, +B) else: (s, t) = sup f1, f2 = dup_real_imag(f, F) f1L1F = dmp_eval_in(f1, v, 1, 1, F) f2L1F = dmp_eval_in(f2, v, 1, 1, F) _, f1L1R = dup_clear_denoms(f1L1F, F, R, convert=True) _, f2L1R = dup_clear_denoms(f2L1F, F, R, convert=True) f1L2F = dmp_eval_in(f1, s, 0, 1, F) f2L2F = dmp_eval_in(f2, s, 0, 1, F) _, f1L2R = dup_clear_denoms(f1L2F, F, R, convert=True) _, f2L2R = dup_clear_denoms(f2L2F, F, R, convert=True) f1L3F = dmp_eval_in(f1, t, 1, 1, F) f2L3F = dmp_eval_in(f2, t, 1, 1, F) _, f1L3R = dup_clear_denoms(f1L3F, F, R, convert=True) _, f2L3R = dup_clear_denoms(f2L3F, F, R, convert=True) f1L4F = dmp_eval_in(f1, u, 0, 1, F) f2L4F = dmp_eval_in(f2, u, 0, 1, F) _, f1L4R = dup_clear_denoms(f1L4F, F, R, convert=True) _, f2L4R = dup_clear_denoms(f2L4F, F, R, convert=True) S_L1 = [f1L1R, f2L1R] S_L2 = [f1L2R, f2L2R] S_L3 = [f1L3R, f2L3R] S_L4 = [f1L4R, f2L4R] I_L1 = dup_isolate_real_roots_list(S_L1, R, inf=u, sup=s, fast=True, basis=True, strict=True) I_L2 = dup_isolate_real_roots_list(S_L2, R, inf=v, sup=t, fast=True, basis=True, strict=True) I_L3 = dup_isolate_real_roots_list(S_L3, R, inf=u, sup=s, fast=True, basis=True, strict=True) I_L4 = dup_isolate_real_roots_list(S_L4, R, inf=v, sup=t, fast=True, basis=True, strict=True) I_L3 = _reverse_intervals(I_L3) I_L4 = _reverse_intervals(I_L4) Q_L1 = _intervals_to_quadrants(I_L1, f1L1F, f2L1F, u, s, F) Q_L2 = _intervals_to_quadrants(I_L2, f1L2F, f2L2F, v, t, F) Q_L3 = _intervals_to_quadrants(I_L3, f1L3F, f2L3F, s, u, F) Q_L4 = _intervals_to_quadrants(I_L4, f1L4F, f2L4F, t, v, F) T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=exclude) return _winding_number(T, F) def _vertical_bisection(N, a, b, I, Q, F1, F2, f1, f2, F): """Vertical bisection step in Collins-Krandick root isolation algorithm. """ (u, v), (s, t) = a, b I_L1, I_L2, I_L3, I_L4 = I Q_L1, Q_L2, Q_L3, Q_L4 = Q f1L1F, f1L2F, f1L3F, f1L4F = F1 f2L1F, f2L2F, f2L3F, f2L4F = F2 x = (u + s) / 2 f1V = dmp_eval_in(f1, x, 0, 1, F) f2V = dmp_eval_in(f2, x, 0, 1, F) I_V = dup_isolate_real_roots_list([f1V, f2V], F, inf=v, sup=t, fast=True, strict=True, basis=True) I_L1_L, I_L1_R = [], [] I_L2_L, I_L2_R = I_V, I_L2 I_L3_L, I_L3_R = [], [] I_L4_L, I_L4_R = I_L4, _reverse_intervals(I_V) for I in I_L1: (a, b), indices, h = I if a == b: if a == x: I_L1_L.append(I) I_L1_R.append(I) elif a < x: I_L1_L.append(I) else: I_L1_R.append(I) else: if b <= x: I_L1_L.append(I) elif a >= x: I_L1_R.append(I) else: a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True) if b <= x: I_L1_L.append(((a, b), indices, h)) if a >= x: I_L1_R.append(((a, b), indices, h)) for I in I_L3: (b, a), indices, h = I if a == b: if a == x: I_L3_L.append(I) I_L3_R.append(I) elif a < x: I_L3_L.append(I) else: I_L3_R.append(I) else: if b <= x: I_L3_L.append(I) elif a >= x: I_L3_R.append(I) else: a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True) if b <= x: I_L3_L.append(((b, a), indices, h)) if a >= x: I_L3_R.append(((b, a), indices, h)) Q_L1_L = _intervals_to_quadrants(I_L1_L, f1L1F, f2L1F, u, x, F) Q_L2_L = _intervals_to_quadrants(I_L2_L, f1V, f2V, v, t, F) Q_L3_L = _intervals_to_quadrants(I_L3_L, f1L3F, f2L3F, x, u, F) Q_L4_L = Q_L4 Q_L1_R = _intervals_to_quadrants(I_L1_R, f1L1F, f2L1F, x, s, F) Q_L2_R = Q_L2 Q_L3_R = _intervals_to_quadrants(I_L3_R, f1L3F, f2L3F, s, x, F) Q_L4_R = _intervals_to_quadrants(I_L4_R, f1V, f2V, t, v, F) T_L = _traverse_quadrants(Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L, exclude=True) T_R = _traverse_quadrants(Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R, exclude=True) N_L = _winding_number(T_L, F) N_R = _winding_number(T_R, F) I_L = (I_L1_L, I_L2_L, I_L3_L, I_L4_L) Q_L = (Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L) I_R = (I_L1_R, I_L2_R, I_L3_R, I_L4_R) Q_R = (Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R) F1_L = (f1L1F, f1V, f1L3F, f1L4F) F2_L = (f2L1F, f2V, f2L3F, f2L4F) F1_R = (f1L1F, f1L2F, f1L3F, f1V) F2_R = (f2L1F, f2L2F, f2L3F, f2V) a, b = (u, v), (x, t) c, d = (x, v), (s, t) D_L = (N_L, a, b, I_L, Q_L, F1_L, F2_L) D_R = (N_R, c, d, I_R, Q_R, F1_R, F2_R) return D_L, D_R def _horizontal_bisection(N, a, b, I, Q, F1, F2, f1, f2, F): """Horizontal bisection step in Collins-Krandick root isolation algorithm. """ (u, v), (s, t) = a, b I_L1, I_L2, I_L3, I_L4 = I Q_L1, Q_L2, Q_L3, Q_L4 = Q f1L1F, f1L2F, f1L3F, f1L4F = F1 f2L1F, f2L2F, f2L3F, f2L4F = F2 y = (v + t) / 2 f1H = dmp_eval_in(f1, y, 1, 1, F) f2H = dmp_eval_in(f2, y, 1, 1, F) I_H = dup_isolate_real_roots_list([f1H, f2H], F, inf=u, sup=s, fast=True, strict=True, basis=True) I_L1_B, I_L1_U = I_L1, I_H I_L2_B, I_L2_U = [], [] I_L3_B, I_L3_U = _reverse_intervals(I_H), I_L3 I_L4_B, I_L4_U = [], [] for I in I_L2: (a, b), indices, h = I if a == b: if a == y: I_L2_B.append(I) I_L2_U.append(I) elif a < y: I_L2_B.append(I) else: I_L2_U.append(I) else: if b <= y: I_L2_B.append(I) elif a >= y: I_L2_U.append(I) else: a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True) if b <= y: I_L2_B.append(((a, b), indices, h)) if a >= y: I_L2_U.append(((a, b), indices, h)) for I in I_L4: (b, a), indices, h = I if a == b: if a == y: I_L4_B.append(I) I_L4_U.append(I) elif a < y: I_L4_B.append(I) else: I_L4_U.append(I) else: if b <= y: I_L4_B.append(I) elif a >= y: I_L4_U.append(I) else: a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True) if b <= y: I_L4_B.append(((b, a), indices, h)) if a >= y: I_L4_U.append(((b, a), indices, h)) Q_L1_B = Q_L1 Q_L2_B = _intervals_to_quadrants(I_L2_B, f1L2F, f2L2F, v, y, F) Q_L3_B = _intervals_to_quadrants(I_L3_B, f1H, f2H, s, u, F) Q_L4_B = _intervals_to_quadrants(I_L4_B, f1L4F, f2L4F, y, v, F) Q_L1_U = _intervals_to_quadrants(I_L1_U, f1H, f2H, u, s, F) Q_L2_U = _intervals_to_quadrants(I_L2_U, f1L2F, f2L2F, y, t, F) Q_L3_U = Q_L3 Q_L4_U = _intervals_to_quadrants(I_L4_U, f1L4F, f2L4F, t, y, F) T_B = _traverse_quadrants(Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B, exclude=True) T_U = _traverse_quadrants(Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U, exclude=True) N_B = _winding_number(T_B, F) N_U = _winding_number(T_U, F) I_B = (I_L1_B, I_L2_B, I_L3_B, I_L4_B) Q_B = (Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B) I_U = (I_L1_U, I_L2_U, I_L3_U, I_L4_U) Q_U = (Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U) F1_B = (f1L1F, f1L2F, f1H, f1L4F) F2_B = (f2L1F, f2L2F, f2H, f2L4F) F1_U = (f1H, f1L2F, f1L3F, f1L4F) F2_U = (f2H, f2L2F, f2L3F, f2L4F) a, b = (u, v), (s, y) c, d = (u, y), (s, t) D_B = (N_B, a, b, I_B, Q_B, F1_B, F2_B) D_U = (N_U, c, d, I_U, Q_U, F1_U, F2_U) return D_B, D_U def _depth_first_select(rectangles): """Find a rectangle of minimum area for bisection. """ min_area, j = None, None for i, (_, (u, v), (s, t), _, _, _, _) in enumerate(rectangles): area = (s - u)*(t - v) if min_area is None or area < min_area: min_area, j = area, i return rectangles.pop(j) def _rectangle_small_p(a, b, eps): """Return ``True`` if the given rectangle is small enough. """ (u, v), (s, t) = a, b if eps is not None: return s - u < eps and t - v < eps else: return True def dup_isolate_complex_roots_sqf(f, K, eps=None, inf=None, sup=None, blackbox=False): """Isolate complex roots of a square-free polynomial using Collins-Krandick algorithm. """ if not K.is_ZZ and not K.is_QQ: raise DomainError("isolation of complex roots is not supported over %s" % K) if dup_degree(f) <= 0: return [] if K.is_ZZ: F = K.get_field() else: F = K f = dup_convert(f, K, F) lc = abs(dup_LC(f, F)) B = 2*max([ F.quo(abs(c), lc) for c in f ]) (u, v), (s, t) = (-B, F.zero), (B, B) if inf is not None: u = inf if sup is not None: s = sup if v < 0 or t <= v or s <= u: raise ValueError("not a valid complex isolation rectangle") f1, f2 = dup_real_imag(f, F) f1L1 = dmp_eval_in(f1, v, 1, 1, F) f2L1 = dmp_eval_in(f2, v, 1, 1, F) f1L2 = dmp_eval_in(f1, s, 0, 1, F) f2L2 = dmp_eval_in(f2, s, 0, 1, F) f1L3 = dmp_eval_in(f1, t, 1, 1, F) f2L3 = dmp_eval_in(f2, t, 1, 1, F) f1L4 = dmp_eval_in(f1, u, 0, 1, F) f2L4 = dmp_eval_in(f2, u, 0, 1, F) S_L1 = [f1L1, f2L1] S_L2 = [f1L2, f2L2] S_L3 = [f1L3, f2L3] S_L4 = [f1L4, f2L4] I_L1 = dup_isolate_real_roots_list(S_L1, F, inf=u, sup=s, fast=True, strict=True, basis=True) I_L2 = dup_isolate_real_roots_list(S_L2, F, inf=v, sup=t, fast=True, strict=True, basis=True) I_L3 = dup_isolate_real_roots_list(S_L3, F, inf=u, sup=s, fast=True, strict=True, basis=True) I_L4 = dup_isolate_real_roots_list(S_L4, F, inf=v, sup=t, fast=True, strict=True, basis=True) I_L3 = _reverse_intervals(I_L3) I_L4 = _reverse_intervals(I_L4) Q_L1 = _intervals_to_quadrants(I_L1, f1L1, f2L1, u, s, F) Q_L2 = _intervals_to_quadrants(I_L2, f1L2, f2L2, v, t, F) Q_L3 = _intervals_to_quadrants(I_L3, f1L3, f2L3, s, u, F) Q_L4 = _intervals_to_quadrants(I_L4, f1L4, f2L4, t, v, F) T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4) N = _winding_number(T, F) if not N: return [] I = (I_L1, I_L2, I_L3, I_L4) Q = (Q_L1, Q_L2, Q_L3, Q_L4) F1 = (f1L1, f1L2, f1L3, f1L4) F2 = (f2L1, f2L2, f2L3, f2L4) rectangles, roots = [(N, (u, v), (s, t), I, Q, F1, F2)], [] while rectangles: N, (u, v), (s, t), I, Q, F1, F2 = _depth_first_select(rectangles) if s - u > t - v: D_L, D_R = _vertical_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F) N_L, a, b, I_L, Q_L, F1_L, F2_L = D_L N_R, c, d, I_R, Q_R, F1_R, F2_R = D_R if N_L >= 1: if N_L == 1 and _rectangle_small_p(a, b, eps): roots.append(ComplexInterval(a, b, I_L, Q_L, F1_L, F2_L, f1, f2, F)) else: rectangles.append(D_L) if N_R >= 1: if N_R == 1 and _rectangle_small_p(c, d, eps): roots.append(ComplexInterval(c, d, I_R, Q_R, F1_R, F2_R, f1, f2, F)) else: rectangles.append(D_R) else: D_B, D_U = _horizontal_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F) N_B, a, b, I_B, Q_B, F1_B, F2_B = D_B N_U, c, d, I_U, Q_U, F1_U, F2_U = D_U if N_B >= 1: if N_B == 1 and _rectangle_small_p(a, b, eps): roots.append(ComplexInterval( a, b, I_B, Q_B, F1_B, F2_B, f1, f2, F)) else: rectangles.append(D_B) if N_U >= 1: if N_U == 1 and _rectangle_small_p(c, d, eps): roots.append(ComplexInterval( c, d, I_U, Q_U, F1_U, F2_U, f1, f2, F)) else: rectangles.append(D_U) _roots, roots = sorted(roots, key=lambda r: (r.ax, r.ay)), [] for root in _roots: roots.extend([root.conjugate(), root]) if blackbox: return roots else: return [ r.as_tuple() for r in roots ] def dup_isolate_all_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False): """Isolate real and complex roots of a square-free polynomial ``f``. """ return ( dup_isolate_real_roots_sqf( f, K, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox), dup_isolate_complex_roots_sqf(f, K, eps=eps, inf=inf, sup=sup, blackbox=blackbox)) def dup_isolate_all_roots(f, K, eps=None, inf=None, sup=None, fast=False): """Isolate real and complex roots of a non-square-free polynomial ``f``. """ if not K.is_ZZ and not K.is_QQ: raise DomainError("isolation of real and complex roots is not supported over %s" % K) _, factors = dup_sqf_list(f, K) if len(factors) == 1: ((f, k),) = factors real_part, complex_part = dup_isolate_all_roots_sqf( f, K, eps=eps, inf=inf, sup=sup, fast=fast) real_part = [ ((a, b), k) for (a, b) in real_part ] complex_part = [ ((a, b), k) for (a, b) in complex_part ] return real_part, complex_part else: raise NotImplementedError( "only trivial square-free polynomials are supported") class RealInterval: """A fully qualified representation of a real isolation interval. """ def __init__(self, data, f, dom): """Initialize new real interval with complete information. """ if len(data) == 2: s, t = data self.neg = False if s < 0: if t <= 0: f, s, t, self.neg = dup_mirror(f, dom), -t, -s, True else: raise ValueError("can't refine a real root in (%s, %s)" % (s, t)) a, b, c, d = _mobius_from_interval((s, t), dom.get_field()) f = dup_transform(f, dup_strip([a, b]), dup_strip([c, d]), dom) self.mobius = a, b, c, d else: self.mobius = data[:-1] self.neg = data[-1] self.f, self.dom = f, dom @property def func(self): return RealInterval @property def args(self): i = self return (i.mobius + (i.neg,), i.f, i.dom) def __eq__(self, other): if type(other) != type(self): return False return self.args == other.args @property def a(self): """Return the position of the left end. """ field = self.dom.get_field() a, b, c, d = self.mobius if not self.neg: if a*d < b*c: return field(a, c) return field(b, d) else: if a*d > b*c: return -field(a, c) return -field(b, d) @property def b(self): """Return the position of the right end. """ was = self.neg self.neg = not was rv = -self.a self.neg = was return rv @property def dx(self): """Return width of the real isolating interval. """ return self.b - self.a @property def center(self): """Return the center of the real isolating interval. """ return (self.a + self.b)/2 def as_tuple(self): """Return tuple representation of real isolating interval. """ return (self.a, self.b) def __repr__(self): return "(%s, %s)" % (self.a, self.b) def is_disjoint(self, other): """Return ``True`` if two isolation intervals are disjoint. """ if isinstance(other, RealInterval): return (self.b <= other.a or other.b <= self.a) assert isinstance(other, ComplexInterval) return (self.b <= other.ax or other.bx <= self.a or other.ay*other.by > 0) def _inner_refine(self): """Internal one step real root refinement procedure. """ if self.mobius is None: return self f, mobius = dup_inner_refine_real_root( self.f, self.mobius, self.dom, steps=1, mobius=True) return RealInterval(mobius + (self.neg,), f, self.dom) def refine_disjoint(self, other): """Refine an isolating interval until it is disjoint with another one. """ expr = self while not expr.is_disjoint(other): expr, other = expr._inner_refine(), other._inner_refine() return expr, other def refine_size(self, dx): """Refine an isolating interval until it is of sufficiently small size. """ expr = self while not (expr.dx < dx): expr = expr._inner_refine() return expr def refine_step(self, steps=1): """Perform several steps of real root refinement algorithm. """ expr = self for _ in range(steps): expr = expr._inner_refine() return expr def refine(self): """Perform one step of real root refinement algorithm. """ return self._inner_refine() class ComplexInterval: """A fully qualified representation of a complex isolation interval. The printed form is shown as (ax, bx) x (ay, by) where (ax, ay) and (bx, by) are the coordinates of the southwest and northeast corners of the interval's rectangle, respectively. Examples ======== >>> from sympy import CRootOf, S >>> from sympy.abc import x >>> CRootOf.clear_cache() # for doctest reproducibility >>> root = CRootOf(x**10 - 2*x + 3, 9) >>> i = root._get_interval(); i (3/64, 3/32) x (9/8, 75/64) The real part of the root lies within the range [0, 3/4] while the imaginary part lies within the range [9/8, 3/2]: >>> root.n(3) 0.0766 + 1.14*I The width of the ranges in the x and y directions on the complex plane are: >>> i.dx, i.dy (3/64, 3/64) The center of the range is >>> i.center (9/128, 147/128) The northeast coordinate of the rectangle bounding the root in the complex plane is given by attribute b and the x and y components are accessed by bx and by: >>> i.b, i.bx, i.by ((3/32, 75/64), 3/32, 75/64) The southwest coordinate is similarly given by i.a >>> i.a, i.ax, i.ay ((3/64, 9/8), 3/64, 9/8) Although the interval prints to show only the real and imaginary range of the root, all the information of the underlying root is contained as properties of the interval. For example, an interval with a nonpositive imaginary range is considered to be the conjugate. Since the y values of y are in the range [0, 1/4] it is not the conjugate: >>> i.conj False The conjugate's interval is >>> ic = i.conjugate(); ic (3/64, 3/32) x (-75/64, -9/8) NOTE: the values printed still represent the x and y range in which the root -- conjugate, in this case -- is located, but the underlying a and b values of a root and its conjugate are the same: >>> assert i.a == ic.a and i.b == ic.b What changes are the reported coordinates of the bounding rectangle: >>> (i.ax, i.ay), (i.bx, i.by) ((3/64, 9/8), (3/32, 75/64)) >>> (ic.ax, ic.ay), (ic.bx, ic.by) ((3/64, -75/64), (3/32, -9/8)) The interval can be refined once: >>> i # for reference, this is the current interval (3/64, 3/32) x (9/8, 75/64) >>> i.refine() (3/64, 3/32) x (9/8, 147/128) Several refinement steps can be taken: >>> i.refine_step(2) # 2 steps (9/128, 3/32) x (9/8, 147/128) It is also possible to refine to a given tolerance: >>> tol = min(i.dx, i.dy)/2 >>> i.refine_size(tol) (9/128, 21/256) x (9/8, 291/256) A disjoint interval is one whose bounding rectangle does not overlap with another. An interval, necessarily, is not disjoint with itself, but any interval is disjoint with a conjugate since the conjugate rectangle will always be in the lower half of the complex plane and the non-conjugate in the upper half: >>> i.is_disjoint(i), i.is_disjoint(i.conjugate()) (False, True) The following interval j is not disjoint from i: >>> close = CRootOf(x**10 - 2*x + 300/S(101), 9) >>> j = close._get_interval(); j (75/1616, 75/808) x (225/202, 1875/1616) >>> i.is_disjoint(j) False The two can be made disjoint, however: >>> newi, newj = i.refine_disjoint(j) >>> newi (39/512, 159/2048) x (2325/2048, 4653/4096) >>> newj (3975/51712, 2025/25856) x (29325/25856, 117375/103424) Even though the real ranges overlap, the imaginary do not, so the roots have been resolved as distinct. Intervals are disjoint when either the real or imaginary component of the intervals is distinct. In the case above, the real components have not been resolved (so we don't know, yet, which root has the smaller real part) but the imaginary part of ``close`` is larger than ``root``: >>> close.n(3) 0.0771 + 1.13*I >>> root.n(3) 0.0766 + 1.14*I """ def __init__(self, a, b, I, Q, F1, F2, f1, f2, dom, conj=False): """Initialize new complex interval with complete information. """ # a and b are the SW and NE corner of the bounding interval, # (ax, ay) and (bx, by), respectively, for the NON-CONJUGATE # root (the one with the positive imaginary part); when working # with the conjugate, the a and b value are still non-negative # but the ay, by are reversed and have oppositite sign self.a, self.b = a, b self.I, self.Q = I, Q self.f1, self.F1 = f1, F1 self.f2, self.F2 = f2, F2 self.dom = dom self.conj = conj @property def func(self): return ComplexInterval @property def args(self): i = self return (i.a, i.b, i.I, i.Q, i.F1, i.F2, i.f1, i.f2, i.dom, i.conj) def __eq__(self, other): if type(other) != type(self): return False return self.args == other.args @property def ax(self): """Return ``x`` coordinate of south-western corner. """ return self.a[0] @property def ay(self): """Return ``y`` coordinate of south-western corner. """ if not self.conj: return self.a[1] else: return -self.b[1] @property def bx(self): """Return ``x`` coordinate of north-eastern corner. """ return self.b[0] @property def by(self): """Return ``y`` coordinate of north-eastern corner. """ if not self.conj: return self.b[1] else: return -self.a[1] @property def dx(self): """Return width of the complex isolating interval. """ return self.b[0] - self.a[0] @property def dy(self): """Return height of the complex isolating interval. """ return self.b[1] - self.a[1] @property def center(self): """Return the center of the complex isolating interval. """ return ((self.ax + self.bx)/2, (self.ay + self.by)/2) def as_tuple(self): """Return tuple representation of the complex isolating interval's SW and NE corners, respectively. """ return ((self.ax, self.ay), (self.bx, self.by)) def __repr__(self): return "(%s, %s) x (%s, %s)" % (self.ax, self.bx, self.ay, self.by) def conjugate(self): """This complex interval really is located in lower half-plane. """ return ComplexInterval(self.a, self.b, self.I, self.Q, self.F1, self.F2, self.f1, self.f2, self.dom, conj=True) def is_disjoint(self, other): """Return ``True`` if two isolation intervals are disjoint. """ if isinstance(other, RealInterval): return other.is_disjoint(self) if self.conj != other.conj: # above and below real axis return True re_distinct = (self.bx <= other.ax or other.bx <= self.ax) if re_distinct: return True im_distinct = (self.by <= other.ay or other.by <= self.ay) return im_distinct def _inner_refine(self): """Internal one step complex root refinement procedure. """ (u, v), (s, t) = self.a, self.b I, Q = self.I, self.Q f1, F1 = self.f1, self.F1 f2, F2 = self.f2, self.F2 dom = self.dom if s - u > t - v: D_L, D_R = _vertical_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom) if D_L[0] == 1: _, a, b, I, Q, F1, F2 = D_L else: _, a, b, I, Q, F1, F2 = D_R else: D_B, D_U = _horizontal_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom) if D_B[0] == 1: _, a, b, I, Q, F1, F2 = D_B else: _, a, b, I, Q, F1, F2 = D_U return ComplexInterval(a, b, I, Q, F1, F2, f1, f2, dom, self.conj) def refine_disjoint(self, other): """Refine an isolating interval until it is disjoint with another one. """ expr = self while not expr.is_disjoint(other): expr, other = expr._inner_refine(), other._inner_refine() return expr, other def refine_size(self, dx, dy=None): """Refine an isolating interval until it is of sufficiently small size. """ if dy is None: dy = dx expr = self while not (expr.dx < dx and expr.dy < dy): expr = expr._inner_refine() return expr def refine_step(self, steps=1): """Perform several steps of complex root refinement algorithm. """ expr = self for _ in range(steps): expr = expr._inner_refine() return expr def refine(self): """Perform one step of complex root refinement algorithm. """ return self._inner_refine()
88c5467946f0a3d6d61056a59460da65523314224d69514beda5ccd796bc73b8
"""Compatibility interface between dense and sparse polys. """ from sympy.polys.densearith import dup_add_term from sympy.polys.densearith import dmp_add_term from sympy.polys.densearith import dup_sub_term from sympy.polys.densearith import dmp_sub_term from sympy.polys.densearith import dup_mul_term from sympy.polys.densearith import dmp_mul_term from sympy.polys.densearith import dup_add_ground from sympy.polys.densearith import dmp_add_ground from sympy.polys.densearith import dup_sub_ground from sympy.polys.densearith import dmp_sub_ground from sympy.polys.densearith import dup_mul_ground from sympy.polys.densearith import dmp_mul_ground from sympy.polys.densearith import dup_quo_ground from sympy.polys.densearith import dmp_quo_ground from sympy.polys.densearith import dup_exquo_ground from sympy.polys.densearith import dmp_exquo_ground from sympy.polys.densearith import dup_lshift from sympy.polys.densearith import dup_rshift from sympy.polys.densearith import dup_abs from sympy.polys.densearith import dmp_abs from sympy.polys.densearith import dup_neg from sympy.polys.densearith import dmp_neg from sympy.polys.densearith import dup_add from sympy.polys.densearith import dmp_add from sympy.polys.densearith import dup_sub from sympy.polys.densearith import dmp_sub from sympy.polys.densearith import dup_add_mul from sympy.polys.densearith import dmp_add_mul from sympy.polys.densearith import dup_sub_mul from sympy.polys.densearith import dmp_sub_mul from sympy.polys.densearith import dup_mul from sympy.polys.densearith import dmp_mul from sympy.polys.densearith import dup_sqr from sympy.polys.densearith import dmp_sqr from sympy.polys.densearith import dup_pow from sympy.polys.densearith import dmp_pow from sympy.polys.densearith import dup_pdiv from sympy.polys.densearith import dup_prem from sympy.polys.densearith import dup_pquo from sympy.polys.densearith import dup_pexquo from sympy.polys.densearith import dmp_pdiv from sympy.polys.densearith import dmp_prem from sympy.polys.densearith import dmp_pquo from sympy.polys.densearith import dmp_pexquo from sympy.polys.densearith import dup_rr_div from sympy.polys.densearith import dmp_rr_div from sympy.polys.densearith import dup_ff_div from sympy.polys.densearith import dmp_ff_div from sympy.polys.densearith import dup_div from sympy.polys.densearith import dup_rem from sympy.polys.densearith import dup_quo from sympy.polys.densearith import dup_exquo from sympy.polys.densearith import dmp_div from sympy.polys.densearith import dmp_rem from sympy.polys.densearith import dmp_quo from sympy.polys.densearith import dmp_exquo from sympy.polys.densearith import dup_max_norm from sympy.polys.densearith import dmp_max_norm from sympy.polys.densearith import dup_l1_norm from sympy.polys.densearith import dmp_l1_norm from sympy.polys.densearith import dup_expand from sympy.polys.densearith import dmp_expand from sympy.polys.densebasic import dup_LC from sympy.polys.densebasic import dmp_LC from sympy.polys.densebasic import dup_TC from sympy.polys.densebasic import dmp_TC from sympy.polys.densebasic import dmp_ground_LC from sympy.polys.densebasic import dmp_ground_TC from sympy.polys.densebasic import dup_degree from sympy.polys.densebasic import dmp_degree from sympy.polys.densebasic import dmp_degree_in from sympy.polys.densebasic import dmp_to_dict from sympy.polys.densetools import dup_integrate from sympy.polys.densetools import dmp_integrate from sympy.polys.densetools import dmp_integrate_in from sympy.polys.densetools import dup_diff from sympy.polys.densetools import dmp_diff from sympy.polys.densetools import dmp_diff_in from sympy.polys.densetools import dup_eval from sympy.polys.densetools import dmp_eval from sympy.polys.densetools import dmp_eval_in from sympy.polys.densetools import dmp_eval_tail from sympy.polys.densetools import dmp_diff_eval_in from sympy.polys.densetools import dup_trunc from sympy.polys.densetools import dmp_trunc from sympy.polys.densetools import dmp_ground_trunc from sympy.polys.densetools import dup_monic from sympy.polys.densetools import dmp_ground_monic from sympy.polys.densetools import dup_content from sympy.polys.densetools import dmp_ground_content from sympy.polys.densetools import dup_primitive from sympy.polys.densetools import dmp_ground_primitive from sympy.polys.densetools import dup_extract from sympy.polys.densetools import dmp_ground_extract from sympy.polys.densetools import dup_real_imag from sympy.polys.densetools import dup_mirror from sympy.polys.densetools import dup_scale from sympy.polys.densetools import dup_shift from sympy.polys.densetools import dup_transform from sympy.polys.densetools import dup_compose from sympy.polys.densetools import dmp_compose from sympy.polys.densetools import dup_decompose from sympy.polys.densetools import dmp_lift from sympy.polys.densetools import dup_sign_variations from sympy.polys.densetools import dup_clear_denoms from sympy.polys.densetools import dmp_clear_denoms from sympy.polys.densetools import dup_revert from sympy.polys.euclidtools import dup_half_gcdex from sympy.polys.euclidtools import dmp_half_gcdex from sympy.polys.euclidtools import dup_gcdex from sympy.polys.euclidtools import dmp_gcdex from sympy.polys.euclidtools import dup_invert from sympy.polys.euclidtools import dmp_invert from sympy.polys.euclidtools import dup_euclidean_prs from sympy.polys.euclidtools import dmp_euclidean_prs from sympy.polys.euclidtools import dup_primitive_prs from sympy.polys.euclidtools import dmp_primitive_prs from sympy.polys.euclidtools import dup_inner_subresultants from sympy.polys.euclidtools import dup_subresultants from sympy.polys.euclidtools import dup_prs_resultant from sympy.polys.euclidtools import dup_resultant from sympy.polys.euclidtools import dmp_inner_subresultants from sympy.polys.euclidtools import dmp_subresultants from sympy.polys.euclidtools import dmp_prs_resultant from sympy.polys.euclidtools import dmp_zz_modular_resultant from sympy.polys.euclidtools import dmp_zz_collins_resultant from sympy.polys.euclidtools import dmp_qq_collins_resultant from sympy.polys.euclidtools import dmp_resultant from sympy.polys.euclidtools import dup_discriminant from sympy.polys.euclidtools import dmp_discriminant from sympy.polys.euclidtools import dup_rr_prs_gcd from sympy.polys.euclidtools import dup_ff_prs_gcd from sympy.polys.euclidtools import dmp_rr_prs_gcd from sympy.polys.euclidtools import dmp_ff_prs_gcd from sympy.polys.euclidtools import dup_zz_heu_gcd from sympy.polys.euclidtools import dmp_zz_heu_gcd from sympy.polys.euclidtools import dup_qq_heu_gcd from sympy.polys.euclidtools import dmp_qq_heu_gcd from sympy.polys.euclidtools import dup_inner_gcd from sympy.polys.euclidtools import dmp_inner_gcd from sympy.polys.euclidtools import dup_gcd from sympy.polys.euclidtools import dmp_gcd from sympy.polys.euclidtools import dup_rr_lcm from sympy.polys.euclidtools import dup_ff_lcm from sympy.polys.euclidtools import dup_lcm from sympy.polys.euclidtools import dmp_rr_lcm from sympy.polys.euclidtools import dmp_ff_lcm from sympy.polys.euclidtools import dmp_lcm from sympy.polys.euclidtools import dmp_content from sympy.polys.euclidtools import dmp_primitive from sympy.polys.euclidtools import dup_cancel from sympy.polys.euclidtools import dmp_cancel from sympy.polys.factortools import dup_trial_division from sympy.polys.factortools import dmp_trial_division from sympy.polys.factortools import dup_zz_mignotte_bound from sympy.polys.factortools import dmp_zz_mignotte_bound from sympy.polys.factortools import dup_zz_hensel_step from sympy.polys.factortools import dup_zz_hensel_lift from sympy.polys.factortools import dup_zz_zassenhaus from sympy.polys.factortools import dup_zz_irreducible_p from sympy.polys.factortools import dup_cyclotomic_p from sympy.polys.factortools import dup_zz_cyclotomic_poly from sympy.polys.factortools import dup_zz_cyclotomic_factor from sympy.polys.factortools import dup_zz_factor_sqf from sympy.polys.factortools import dup_zz_factor from sympy.polys.factortools import dmp_zz_wang_non_divisors from sympy.polys.factortools import dmp_zz_wang_lead_coeffs from sympy.polys.factortools import dup_zz_diophantine from sympy.polys.factortools import dmp_zz_diophantine from sympy.polys.factortools import dmp_zz_wang_hensel_lifting from sympy.polys.factortools import dmp_zz_wang from sympy.polys.factortools import dmp_zz_factor from sympy.polys.factortools import dup_qq_i_factor from sympy.polys.factortools import dup_zz_i_factor from sympy.polys.factortools import dmp_qq_i_factor from sympy.polys.factortools import dmp_zz_i_factor from sympy.polys.factortools import dup_ext_factor from sympy.polys.factortools import dmp_ext_factor from sympy.polys.factortools import dup_gf_factor from sympy.polys.factortools import dmp_gf_factor from sympy.polys.factortools import dup_factor_list from sympy.polys.factortools import dup_factor_list_include from sympy.polys.factortools import dmp_factor_list from sympy.polys.factortools import dmp_factor_list_include from sympy.polys.factortools import dup_irreducible_p from sympy.polys.factortools import dmp_irreducible_p from sympy.polys.rootisolation import dup_sturm from sympy.polys.rootisolation import dup_root_upper_bound from sympy.polys.rootisolation import dup_root_lower_bound from sympy.polys.rootisolation import dup_step_refine_real_root from sympy.polys.rootisolation import dup_inner_refine_real_root from sympy.polys.rootisolation import dup_outer_refine_real_root from sympy.polys.rootisolation import dup_refine_real_root from sympy.polys.rootisolation import dup_inner_isolate_real_roots from sympy.polys.rootisolation import dup_inner_isolate_positive_roots from sympy.polys.rootisolation import dup_inner_isolate_negative_roots from sympy.polys.rootisolation import dup_isolate_real_roots_sqf from sympy.polys.rootisolation import dup_isolate_real_roots from sympy.polys.rootisolation import dup_isolate_real_roots_list from sympy.polys.rootisolation import dup_count_real_roots from sympy.polys.rootisolation import dup_count_complex_roots from sympy.polys.rootisolation import dup_isolate_complex_roots_sqf from sympy.polys.rootisolation import dup_isolate_all_roots_sqf from sympy.polys.rootisolation import dup_isolate_all_roots from sympy.polys.sqfreetools import ( dup_sqf_p, dmp_sqf_p, dup_sqf_norm, dmp_sqf_norm, dup_gf_sqf_part, dmp_gf_sqf_part, dup_sqf_part, dmp_sqf_part, dup_gf_sqf_list, dmp_gf_sqf_list, dup_sqf_list, dup_sqf_list_include, dmp_sqf_list, dmp_sqf_list_include, dup_gff_list, dmp_gff_list) from sympy.polys.galoistools import ( gf_degree, gf_LC, gf_TC, gf_strip, gf_from_dict, gf_to_dict, gf_from_int_poly, gf_to_int_poly, gf_neg, gf_add_ground, gf_sub_ground, gf_mul_ground, gf_quo_ground, gf_add, gf_sub, gf_mul, gf_sqr, gf_add_mul, gf_sub_mul, gf_expand, gf_div, gf_rem, gf_quo, gf_exquo, gf_lshift, gf_rshift, gf_pow, gf_pow_mod, gf_gcd, gf_lcm, gf_cofactors, gf_gcdex, gf_monic, gf_diff, gf_eval, gf_multi_eval, gf_compose, gf_compose_mod, gf_trace_map, gf_random, gf_irreducible, gf_irred_p_ben_or, gf_irred_p_rabin, gf_irreducible_p, gf_sqf_p, gf_sqf_part, gf_Qmatrix, gf_berlekamp, gf_ddf_zassenhaus, gf_edf_zassenhaus, gf_ddf_shoup, gf_edf_shoup, gf_zassenhaus, gf_shoup, gf_factor_sqf, gf_factor) from sympy.utilities import public @public class IPolys: symbols = None ngens = None domain = None order = None gens = None def drop(self, gen): pass def clone(self, symbols=None, domain=None, order=None): pass def to_ground(self): pass def ground_new(self, element): pass def domain_new(self, element): pass def from_dict(self, d): pass def wrap(self, element): from sympy.polys.rings import PolyElement if isinstance(element, PolyElement): if element.ring == self: return element else: raise NotImplementedError("domain conversions") else: return self.ground_new(element) def to_dense(self, element): return self.wrap(element).to_dense() def from_dense(self, element): return self.from_dict(dmp_to_dict(element, self.ngens-1, self.domain)) def dup_add_term(self, f, c, i): return self.from_dense(dup_add_term(self.to_dense(f), c, i, self.domain)) def dmp_add_term(self, f, c, i): return self.from_dense(dmp_add_term(self.to_dense(f), self.wrap(c).drop(0).to_dense(), i, self.ngens-1, self.domain)) def dup_sub_term(self, f, c, i): return self.from_dense(dup_sub_term(self.to_dense(f), c, i, self.domain)) def dmp_sub_term(self, f, c, i): return self.from_dense(dmp_sub_term(self.to_dense(f), self.wrap(c).drop(0).to_dense(), i, self.ngens-1, self.domain)) def dup_mul_term(self, f, c, i): return self.from_dense(dup_mul_term(self.to_dense(f), c, i, self.domain)) def dmp_mul_term(self, f, c, i): return self.from_dense(dmp_mul_term(self.to_dense(f), self.wrap(c).drop(0).to_dense(), i, self.ngens-1, self.domain)) def dup_add_ground(self, f, c): return self.from_dense(dup_add_ground(self.to_dense(f), c, self.domain)) def dmp_add_ground(self, f, c): return self.from_dense(dmp_add_ground(self.to_dense(f), c, self.ngens-1, self.domain)) def dup_sub_ground(self, f, c): return self.from_dense(dup_sub_ground(self.to_dense(f), c, self.domain)) def dmp_sub_ground(self, f, c): return self.from_dense(dmp_sub_ground(self.to_dense(f), c, self.ngens-1, self.domain)) def dup_mul_ground(self, f, c): return self.from_dense(dup_mul_ground(self.to_dense(f), c, self.domain)) def dmp_mul_ground(self, f, c): return self.from_dense(dmp_mul_ground(self.to_dense(f), c, self.ngens-1, self.domain)) def dup_quo_ground(self, f, c): return self.from_dense(dup_quo_ground(self.to_dense(f), c, self.domain)) def dmp_quo_ground(self, f, c): return self.from_dense(dmp_quo_ground(self.to_dense(f), c, self.ngens-1, self.domain)) def dup_exquo_ground(self, f, c): return self.from_dense(dup_exquo_ground(self.to_dense(f), c, self.domain)) def dmp_exquo_ground(self, f, c): return self.from_dense(dmp_exquo_ground(self.to_dense(f), c, self.ngens-1, self.domain)) def dup_lshift(self, f, n): return self.from_dense(dup_lshift(self.to_dense(f), n, self.domain)) def dup_rshift(self, f, n): return self.from_dense(dup_rshift(self.to_dense(f), n, self.domain)) def dup_abs(self, f): return self.from_dense(dup_abs(self.to_dense(f), self.domain)) def dmp_abs(self, f): return self.from_dense(dmp_abs(self.to_dense(f), self.ngens-1, self.domain)) def dup_neg(self, f): return self.from_dense(dup_neg(self.to_dense(f), self.domain)) def dmp_neg(self, f): return self.from_dense(dmp_neg(self.to_dense(f), self.ngens-1, self.domain)) def dup_add(self, f, g): return self.from_dense(dup_add(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_add(self, f, g): return self.from_dense(dmp_add(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_sub(self, f, g): return self.from_dense(dup_sub(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_sub(self, f, g): return self.from_dense(dmp_sub(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_add_mul(self, f, g, h): return self.from_dense(dup_add_mul(self.to_dense(f), self.to_dense(g), self.to_dense(h), self.domain)) def dmp_add_mul(self, f, g, h): return self.from_dense(dmp_add_mul(self.to_dense(f), self.to_dense(g), self.to_dense(h), self.ngens-1, self.domain)) def dup_sub_mul(self, f, g, h): return self.from_dense(dup_sub_mul(self.to_dense(f), self.to_dense(g), self.to_dense(h), self.domain)) def dmp_sub_mul(self, f, g, h): return self.from_dense(dmp_sub_mul(self.to_dense(f), self.to_dense(g), self.to_dense(h), self.ngens-1, self.domain)) def dup_mul(self, f, g): return self.from_dense(dup_mul(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_mul(self, f, g): return self.from_dense(dmp_mul(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_sqr(self, f): return self.from_dense(dup_sqr(self.to_dense(f), self.domain)) def dmp_sqr(self, f): return self.from_dense(dmp_sqr(self.to_dense(f), self.ngens-1, self.domain)) def dup_pow(self, f, n): return self.from_dense(dup_pow(self.to_dense(f), n, self.domain)) def dmp_pow(self, f, n): return self.from_dense(dmp_pow(self.to_dense(f), n, self.ngens-1, self.domain)) def dup_pdiv(self, f, g): q, r = dup_pdiv(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(q), self.from_dense(r)) def dup_prem(self, f, g): return self.from_dense(dup_prem(self.to_dense(f), self.to_dense(g), self.domain)) def dup_pquo(self, f, g): return self.from_dense(dup_pquo(self.to_dense(f), self.to_dense(g), self.domain)) def dup_pexquo(self, f, g): return self.from_dense(dup_pexquo(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_pdiv(self, f, g): q, r = dmp_pdiv(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(q), self.from_dense(r)) def dmp_prem(self, f, g): return self.from_dense(dmp_prem(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dmp_pquo(self, f, g): return self.from_dense(dmp_pquo(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dmp_pexquo(self, f, g): return self.from_dense(dmp_pexquo(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_rr_div(self, f, g): q, r = dup_rr_div(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(q), self.from_dense(r)) def dmp_rr_div(self, f, g): q, r = dmp_rr_div(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(q), self.from_dense(r)) def dup_ff_div(self, f, g): q, r = dup_ff_div(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(q), self.from_dense(r)) def dmp_ff_div(self, f, g): q, r = dmp_ff_div(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(q), self.from_dense(r)) def dup_div(self, f, g): q, r = dup_div(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(q), self.from_dense(r)) def dup_rem(self, f, g): return self.from_dense(dup_rem(self.to_dense(f), self.to_dense(g), self.domain)) def dup_quo(self, f, g): return self.from_dense(dup_quo(self.to_dense(f), self.to_dense(g), self.domain)) def dup_exquo(self, f, g): return self.from_dense(dup_exquo(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_div(self, f, g): q, r = dmp_div(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(q), self.from_dense(r)) def dmp_rem(self, f, g): return self.from_dense(dmp_rem(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dmp_quo(self, f, g): return self.from_dense(dmp_quo(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dmp_exquo(self, f, g): return self.from_dense(dmp_exquo(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_max_norm(self, f): return dup_max_norm(self.to_dense(f), self.domain) def dmp_max_norm(self, f): return dmp_max_norm(self.to_dense(f), self.ngens-1, self.domain) def dup_l1_norm(self, f): return dup_l1_norm(self.to_dense(f), self.domain) def dmp_l1_norm(self, f): return dmp_l1_norm(self.to_dense(f), self.ngens-1, self.domain) def dup_expand(self, polys): return self.from_dense(dup_expand(list(map(self.to_dense, polys)), self.domain)) def dmp_expand(self, polys): return self.from_dense(dmp_expand(list(map(self.to_dense, polys)), self.ngens-1, self.domain)) def dup_LC(self, f): return dup_LC(self.to_dense(f), self.domain) def dmp_LC(self, f): LC = dmp_LC(self.to_dense(f), self.domain) if isinstance(LC, list): return self[1:].from_dense(LC) else: return LC def dup_TC(self, f): return dup_TC(self.to_dense(f), self.domain) def dmp_TC(self, f): TC = dmp_TC(self.to_dense(f), self.domain) if isinstance(TC, list): return self[1:].from_dense(TC) else: return TC def dmp_ground_LC(self, f): return dmp_ground_LC(self.to_dense(f), self.ngens-1, self.domain) def dmp_ground_TC(self, f): return dmp_ground_TC(self.to_dense(f), self.ngens-1, self.domain) def dup_degree(self, f): return dup_degree(self.to_dense(f)) def dmp_degree(self, f): return dmp_degree(self.to_dense(f), self.ngens-1) def dmp_degree_in(self, f, j): return dmp_degree_in(self.to_dense(f), j, self.ngens-1) def dup_integrate(self, f, m): return self.from_dense(dup_integrate(self.to_dense(f), m, self.domain)) def dmp_integrate(self, f, m): return self.from_dense(dmp_integrate(self.to_dense(f), m, self.ngens-1, self.domain)) def dup_diff(self, f, m): return self.from_dense(dup_diff(self.to_dense(f), m, self.domain)) def dmp_diff(self, f, m): return self.from_dense(dmp_diff(self.to_dense(f), m, self.ngens-1, self.domain)) def dmp_diff_in(self, f, m, j): return self.from_dense(dmp_diff_in(self.to_dense(f), m, j, self.ngens-1, self.domain)) def dmp_integrate_in(self, f, m, j): return self.from_dense(dmp_integrate_in(self.to_dense(f), m, j, self.ngens-1, self.domain)) def dup_eval(self, f, a): return dup_eval(self.to_dense(f), a, self.domain) def dmp_eval(self, f, a): result = dmp_eval(self.to_dense(f), a, self.ngens-1, self.domain) return self[1:].from_dense(result) def dmp_eval_in(self, f, a, j): result = dmp_eval_in(self.to_dense(f), a, j, self.ngens-1, self.domain) return self.drop(j).from_dense(result) def dmp_diff_eval_in(self, f, m, a, j): result = dmp_diff_eval_in(self.to_dense(f), m, a, j, self.ngens-1, self.domain) return self.drop(j).from_dense(result) def dmp_eval_tail(self, f, A): result = dmp_eval_tail(self.to_dense(f), A, self.ngens-1, self.domain) if isinstance(result, list): return self[:-len(A)].from_dense(result) else: return result def dup_trunc(self, f, p): return self.from_dense(dup_trunc(self.to_dense(f), p, self.domain)) def dmp_trunc(self, f, g): return self.from_dense(dmp_trunc(self.to_dense(f), self[1:].to_dense(g), self.ngens-1, self.domain)) def dmp_ground_trunc(self, f, p): return self.from_dense(dmp_ground_trunc(self.to_dense(f), p, self.ngens-1, self.domain)) def dup_monic(self, f): return self.from_dense(dup_monic(self.to_dense(f), self.domain)) def dmp_ground_monic(self, f): return self.from_dense(dmp_ground_monic(self.to_dense(f), self.ngens-1, self.domain)) def dup_extract(self, f, g): c, F, G = dup_extract(self.to_dense(f), self.to_dense(g), self.domain) return (c, self.from_dense(F), self.from_dense(G)) def dmp_ground_extract(self, f, g): c, F, G = dmp_ground_extract(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (c, self.from_dense(F), self.from_dense(G)) def dup_real_imag(self, f): p, q = dup_real_imag(self.wrap(f).drop(1).to_dense(), self.domain) return (self.from_dense(p), self.from_dense(q)) def dup_mirror(self, f): return self.from_dense(dup_mirror(self.to_dense(f), self.domain)) def dup_scale(self, f, a): return self.from_dense(dup_scale(self.to_dense(f), a, self.domain)) def dup_shift(self, f, a): return self.from_dense(dup_shift(self.to_dense(f), a, self.domain)) def dup_transform(self, f, p, q): return self.from_dense(dup_transform(self.to_dense(f), self.to_dense(p), self.to_dense(q), self.domain)) def dup_compose(self, f, g): return self.from_dense(dup_compose(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_compose(self, f, g): return self.from_dense(dmp_compose(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_decompose(self, f): components = dup_decompose(self.to_dense(f), self.domain) return list(map(self.from_dense, components)) def dmp_lift(self, f): result = dmp_lift(self.to_dense(f), self.ngens-1, self.domain) return self.to_ground().from_dense(result) def dup_sign_variations(self, f): return dup_sign_variations(self.to_dense(f), self.domain) def dup_clear_denoms(self, f, convert=False): c, F = dup_clear_denoms(self.to_dense(f), self.domain, convert=convert) if convert: ring = self.clone(domain=self.domain.get_ring()) else: ring = self return (c, ring.from_dense(F)) def dmp_clear_denoms(self, f, convert=False): c, F = dmp_clear_denoms(self.to_dense(f), self.ngens-1, self.domain, convert=convert) if convert: ring = self.clone(domain=self.domain.get_ring()) else: ring = self return (c, ring.from_dense(F)) def dup_revert(self, f, n): return self.from_dense(dup_revert(self.to_dense(f), n, self.domain)) def dup_half_gcdex(self, f, g): s, h = dup_half_gcdex(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(s), self.from_dense(h)) def dmp_half_gcdex(self, f, g): s, h = dmp_half_gcdex(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(s), self.from_dense(h)) def dup_gcdex(self, f, g): s, t, h = dup_gcdex(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(s), self.from_dense(t), self.from_dense(h)) def dmp_gcdex(self, f, g): s, t, h = dmp_gcdex(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(s), self.from_dense(t), self.from_dense(h)) def dup_invert(self, f, g): return self.from_dense(dup_invert(self.to_dense(f), self.to_dense(g), self.domain)) def dmp_invert(self, f, g): return self.from_dense(dmp_invert(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain)) def dup_euclidean_prs(self, f, g): prs = dup_euclidean_prs(self.to_dense(f), self.to_dense(g), self.domain) return list(map(self.from_dense, prs)) def dmp_euclidean_prs(self, f, g): prs = dmp_euclidean_prs(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return list(map(self.from_dense, prs)) def dup_primitive_prs(self, f, g): prs = dup_primitive_prs(self.to_dense(f), self.to_dense(g), self.domain) return list(map(self.from_dense, prs)) def dmp_primitive_prs(self, f, g): prs = dmp_primitive_prs(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return list(map(self.from_dense, prs)) def dup_inner_subresultants(self, f, g): prs, sres = dup_inner_subresultants(self.to_dense(f), self.to_dense(g), self.domain) return (list(map(self.from_dense, prs)), sres) def dmp_inner_subresultants(self, f, g): prs, sres = dmp_inner_subresultants(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (list(map(self.from_dense, prs)), sres) def dup_subresultants(self, f, g): prs = dup_subresultants(self.to_dense(f), self.to_dense(g), self.domain) return list(map(self.from_dense, prs)) def dmp_subresultants(self, f, g): prs = dmp_subresultants(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return list(map(self.from_dense, prs)) def dup_prs_resultant(self, f, g): res, prs = dup_prs_resultant(self.to_dense(f), self.to_dense(g), self.domain) return (res, list(map(self.from_dense, prs))) def dmp_prs_resultant(self, f, g): res, prs = dmp_prs_resultant(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self[1:].from_dense(res), list(map(self.from_dense, prs))) def dmp_zz_modular_resultant(self, f, g, p): res = dmp_zz_modular_resultant(self.to_dense(f), self.to_dense(g), self.domain_new(p), self.ngens-1, self.domain) return self[1:].from_dense(res) def dmp_zz_collins_resultant(self, f, g): res = dmp_zz_collins_resultant(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self[1:].from_dense(res) def dmp_qq_collins_resultant(self, f, g): res = dmp_qq_collins_resultant(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self[1:].from_dense(res) def dup_resultant(self, f, g): #, includePRS=False): return dup_resultant(self.to_dense(f), self.to_dense(g), self.domain) #, includePRS=includePRS) def dmp_resultant(self, f, g): #, includePRS=False): res = dmp_resultant(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) #, includePRS=includePRS) if isinstance(res, list): return self[1:].from_dense(res) else: return res def dup_discriminant(self, f): return dup_discriminant(self.to_dense(f), self.domain) def dmp_discriminant(self, f): disc = dmp_discriminant(self.to_dense(f), self.ngens-1, self.domain) if isinstance(disc, list): return self[1:].from_dense(disc) else: return disc def dup_rr_prs_gcd(self, f, g): H, F, G = dup_rr_prs_gcd(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dup_ff_prs_gcd(self, f, g): H, F, G = dup_ff_prs_gcd(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dmp_rr_prs_gcd(self, f, g): H, F, G = dmp_rr_prs_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dmp_ff_prs_gcd(self, f, g): H, F, G = dmp_ff_prs_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dup_zz_heu_gcd(self, f, g): H, F, G = dup_zz_heu_gcd(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dmp_zz_heu_gcd(self, f, g): H, F, G = dmp_zz_heu_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dup_qq_heu_gcd(self, f, g): H, F, G = dup_qq_heu_gcd(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dmp_qq_heu_gcd(self, f, g): H, F, G = dmp_qq_heu_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dup_inner_gcd(self, f, g): H, F, G = dup_inner_gcd(self.to_dense(f), self.to_dense(g), self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dmp_inner_gcd(self, f, g): H, F, G = dmp_inner_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return (self.from_dense(H), self.from_dense(F), self.from_dense(G)) def dup_gcd(self, f, g): H = dup_gcd(self.to_dense(f), self.to_dense(g), self.domain) return self.from_dense(H) def dmp_gcd(self, f, g): H = dmp_gcd(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self.from_dense(H) def dup_rr_lcm(self, f, g): H = dup_rr_lcm(self.to_dense(f), self.to_dense(g), self.domain) return self.from_dense(H) def dup_ff_lcm(self, f, g): H = dup_ff_lcm(self.to_dense(f), self.to_dense(g), self.domain) return self.from_dense(H) def dup_lcm(self, f, g): H = dup_lcm(self.to_dense(f), self.to_dense(g), self.domain) return self.from_dense(H) def dmp_rr_lcm(self, f, g): H = dmp_rr_lcm(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self.from_dense(H) def dmp_ff_lcm(self, f, g): H = dmp_ff_lcm(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self.from_dense(H) def dmp_lcm(self, f, g): H = dmp_lcm(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain) return self.from_dense(H) def dup_content(self, f): cont = dup_content(self.to_dense(f), self.domain) return cont def dup_primitive(self, f): cont, prim = dup_primitive(self.to_dense(f), self.domain) return cont, self.from_dense(prim) def dmp_content(self, f): cont = dmp_content(self.to_dense(f), self.ngens-1, self.domain) if isinstance(cont, list): return self[1:].from_dense(cont) else: return cont def dmp_primitive(self, f): cont, prim = dmp_primitive(self.to_dense(f), self.ngens-1, self.domain) if isinstance(cont, list): return (self[1:].from_dense(cont), self.from_dense(prim)) else: return (cont, self.from_dense(prim)) def dmp_ground_content(self, f): cont = dmp_ground_content(self.to_dense(f), self.ngens-1, self.domain) return cont def dmp_ground_primitive(self, f): cont, prim = dmp_ground_primitive(self.to_dense(f), self.ngens-1, self.domain) return (cont, self.from_dense(prim)) def dup_cancel(self, f, g, include=True): result = dup_cancel(self.to_dense(f), self.to_dense(g), self.domain, include=include) if not include: cf, cg, F, G = result return (cf, cg, self.from_dense(F), self.from_dense(G)) else: F, G = result return (self.from_dense(F), self.from_dense(G)) def dmp_cancel(self, f, g, include=True): result = dmp_cancel(self.to_dense(f), self.to_dense(g), self.ngens-1, self.domain, include=include) if not include: cf, cg, F, G = result return (cf, cg, self.from_dense(F), self.from_dense(G)) else: F, G = result return (self.from_dense(F), self.from_dense(G)) def dup_trial_division(self, f, factors): factors = dup_trial_division(self.to_dense(f), list(map(self.to_dense, factors)), self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dmp_trial_division(self, f, factors): factors = dmp_trial_division(self.to_dense(f), list(map(self.to_dense, factors)), self.ngens-1, self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dup_zz_mignotte_bound(self, f): return dup_zz_mignotte_bound(self.to_dense(f), self.domain) def dmp_zz_mignotte_bound(self, f): return dmp_zz_mignotte_bound(self.to_dense(f), self.ngens-1, self.domain) def dup_zz_hensel_step(self, m, f, g, h, s, t): D = self.to_dense G, H, S, T = dup_zz_hensel_step(m, D(f), D(g), D(h), D(s), D(t), self.domain) return (self.from_dense(G), self.from_dense(H), self.from_dense(S), self.from_dense(T)) def dup_zz_hensel_lift(self, p, f, f_list, l): D = self.to_dense polys = dup_zz_hensel_lift(p, D(f), list(map(D, f_list)), l, self.domain) return list(map(self.from_dense, polys)) def dup_zz_zassenhaus(self, f): factors = dup_zz_zassenhaus(self.to_dense(f), self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dup_zz_irreducible_p(self, f): return dup_zz_irreducible_p(self.to_dense(f), self.domain) def dup_cyclotomic_p(self, f, irreducible=False): return dup_cyclotomic_p(self.to_dense(f), self.domain, irreducible=irreducible) def dup_zz_cyclotomic_poly(self, n): F = dup_zz_cyclotomic_poly(n, self.domain) return self.from_dense(F) def dup_zz_cyclotomic_factor(self, f): result = dup_zz_cyclotomic_factor(self.to_dense(f), self.domain) if result is None: return result else: return list(map(self.from_dense, result)) # E: List[ZZ], cs: ZZ, ct: ZZ def dmp_zz_wang_non_divisors(self, E, cs, ct): return dmp_zz_wang_non_divisors(E, cs, ct, self.domain) # f: Poly, T: List[(Poly, int)], ct: ZZ, A: List[ZZ] #def dmp_zz_wang_test_points(f, T, ct, A): # dmp_zz_wang_test_points(self.to_dense(f), T, ct, A, self.ngens-1, self.domain) # f: Poly, T: List[(Poly, int)], cs: ZZ, E: List[ZZ], H: List[Poly], A: List[ZZ] def dmp_zz_wang_lead_coeffs(self, f, T, cs, E, H, A): mv = self[1:] T = [ (mv.to_dense(t), k) for t, k in T ] uv = self[:1] H = list(map(uv.to_dense, H)) f, HH, CC = dmp_zz_wang_lead_coeffs(self.to_dense(f), T, cs, E, H, A, self.ngens-1, self.domain) return self.from_dense(f), list(map(uv.from_dense, HH)), list(map(mv.from_dense, CC)) # f: List[Poly], m: int, p: ZZ def dup_zz_diophantine(self, F, m, p): result = dup_zz_diophantine(list(map(self.to_dense, F)), m, p, self.domain) return list(map(self.from_dense, result)) # f: List[Poly], c: List[Poly], A: List[ZZ], d: int, p: ZZ def dmp_zz_diophantine(self, F, c, A, d, p): result = dmp_zz_diophantine(list(map(self.to_dense, F)), self.to_dense(c), A, d, p, self.ngens-1, self.domain) return list(map(self.from_dense, result)) # f: Poly, H: List[Poly], LC: List[Poly], A: List[ZZ], p: ZZ def dmp_zz_wang_hensel_lifting(self, f, H, LC, A, p): uv = self[:1] mv = self[1:] H = list(map(uv.to_dense, H)) LC = list(map(mv.to_dense, LC)) result = dmp_zz_wang_hensel_lifting(self.to_dense(f), H, LC, A, p, self.ngens-1, self.domain) return list(map(self.from_dense, result)) def dmp_zz_wang(self, f, mod=None, seed=None): factors = dmp_zz_wang(self.to_dense(f), self.ngens-1, self.domain, mod=mod, seed=seed) return [ self.from_dense(g) for g in factors ] def dup_zz_factor_sqf(self, f): coeff, factors = dup_zz_factor_sqf(self.to_dense(f), self.domain) return (coeff, [ self.from_dense(g) for g in factors ]) def dup_zz_factor(self, f): coeff, factors = dup_zz_factor(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_zz_factor(self, f): coeff, factors = dmp_zz_factor(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_qq_i_factor(self, f): coeff, factors = dup_qq_i_factor(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_qq_i_factor(self, f): coeff, factors = dmp_qq_i_factor(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_zz_i_factor(self, f): coeff, factors = dup_zz_i_factor(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_zz_i_factor(self, f): coeff, factors = dmp_zz_i_factor(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_ext_factor(self, f): coeff, factors = dup_ext_factor(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_ext_factor(self, f): coeff, factors = dmp_ext_factor(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_gf_factor(self, f): coeff, factors = dup_gf_factor(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_gf_factor(self, f): coeff, factors = dmp_gf_factor(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_factor_list(self, f): coeff, factors = dup_factor_list(self.to_dense(f), self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_factor_list_include(self, f): factors = dup_factor_list_include(self.to_dense(f), self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dmp_factor_list(self, f): coeff, factors = dmp_factor_list(self.to_dense(f), self.ngens-1, self.domain) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_factor_list_include(self, f): factors = dmp_factor_list_include(self.to_dense(f), self.ngens-1, self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dup_irreducible_p(self, f): return dup_irreducible_p(self.to_dense(f), self.domain) def dmp_irreducible_p(self, f): return dmp_irreducible_p(self.to_dense(f), self.ngens-1, self.domain) def dup_sturm(self, f): seq = dup_sturm(self.to_dense(f), self.domain) return list(map(self.from_dense, seq)) def dup_sqf_p(self, f): return dup_sqf_p(self.to_dense(f), self.domain) def dmp_sqf_p(self, f): return dmp_sqf_p(self.to_dense(f), self.ngens-1, self.domain) def dup_sqf_norm(self, f): s, F, R = dup_sqf_norm(self.to_dense(f), self.domain) return (s, self.from_dense(F), self.to_ground().from_dense(R)) def dmp_sqf_norm(self, f): s, F, R = dmp_sqf_norm(self.to_dense(f), self.ngens-1, self.domain) return (s, self.from_dense(F), self.to_ground().from_dense(R)) def dup_gf_sqf_part(self, f): return self.from_dense(dup_gf_sqf_part(self.to_dense(f), self.domain)) def dmp_gf_sqf_part(self, f): return self.from_dense(dmp_gf_sqf_part(self.to_dense(f), self.domain)) def dup_sqf_part(self, f): return self.from_dense(dup_sqf_part(self.to_dense(f), self.domain)) def dmp_sqf_part(self, f): return self.from_dense(dmp_sqf_part(self.to_dense(f), self.ngens-1, self.domain)) def dup_gf_sqf_list(self, f, all=False): coeff, factors = dup_gf_sqf_list(self.to_dense(f), self.domain, all=all) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_gf_sqf_list(self, f, all=False): coeff, factors = dmp_gf_sqf_list(self.to_dense(f), self.ngens-1, self.domain, all=all) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_sqf_list(self, f, all=False): coeff, factors = dup_sqf_list(self.to_dense(f), self.domain, all=all) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dup_sqf_list_include(self, f, all=False): factors = dup_sqf_list_include(self.to_dense(f), self.domain, all=all) return [ (self.from_dense(g), k) for g, k in factors ] def dmp_sqf_list(self, f, all=False): coeff, factors = dmp_sqf_list(self.to_dense(f), self.ngens-1, self.domain, all=all) return (coeff, [ (self.from_dense(g), k) for g, k in factors ]) def dmp_sqf_list_include(self, f, all=False): factors = dmp_sqf_list_include(self.to_dense(f), self.ngens-1, self.domain, all=all) return [ (self.from_dense(g), k) for g, k in factors ] def dup_gff_list(self, f): factors = dup_gff_list(self.to_dense(f), self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dmp_gff_list(self, f): factors = dmp_gff_list(self.to_dense(f), self.ngens-1, self.domain) return [ (self.from_dense(g), k) for g, k in factors ] def dup_root_upper_bound(self, f): return dup_root_upper_bound(self.to_dense(f), self.domain) def dup_root_lower_bound(self, f): return dup_root_lower_bound(self.to_dense(f), self.domain) def dup_step_refine_real_root(self, f, M, fast=False): return dup_step_refine_real_root(self.to_dense(f), M, self.domain, fast=fast) def dup_inner_refine_real_root(self, f, M, eps=None, steps=None, disjoint=None, fast=False, mobius=False): return dup_inner_refine_real_root(self.to_dense(f), M, self.domain, eps=eps, steps=steps, disjoint=disjoint, fast=fast, mobius=mobius) def dup_outer_refine_real_root(self, f, s, t, eps=None, steps=None, disjoint=None, fast=False): return dup_outer_refine_real_root(self.to_dense(f), s, t, self.domain, eps=eps, steps=steps, disjoint=disjoint, fast=fast) def dup_refine_real_root(self, f, s, t, eps=None, steps=None, disjoint=None, fast=False): return dup_refine_real_root(self.to_dense(f), s, t, self.domain, eps=eps, steps=steps, disjoint=disjoint, fast=fast) def dup_inner_isolate_real_roots(self, f, eps=None, fast=False): return dup_inner_isolate_real_roots(self.to_dense(f), self.domain, eps=eps, fast=fast) def dup_inner_isolate_positive_roots(self, f, eps=None, inf=None, sup=None, fast=False, mobius=False): return dup_inner_isolate_positive_roots(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, fast=fast, mobius=mobius) def dup_inner_isolate_negative_roots(self, f, inf=None, sup=None, eps=None, fast=False, mobius=False): return dup_inner_isolate_negative_roots(self.to_dense(f), self.domain, inf=inf, sup=sup, eps=eps, fast=fast, mobius=mobius) def dup_isolate_real_roots_sqf(self, f, eps=None, inf=None, sup=None, fast=False, blackbox=False): return dup_isolate_real_roots_sqf(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox) def dup_isolate_real_roots(self, f, eps=None, inf=None, sup=None, basis=False, fast=False): return dup_isolate_real_roots(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, basis=basis, fast=fast) def dup_isolate_real_roots_list(self, polys, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False): return dup_isolate_real_roots_list(list(map(self.to_dense, polys)), self.domain, eps=eps, inf=inf, sup=sup, strict=strict, basis=basis, fast=fast) def dup_count_real_roots(self, f, inf=None, sup=None): return dup_count_real_roots(self.to_dense(f), self.domain, inf=inf, sup=sup) def dup_count_complex_roots(self, f, inf=None, sup=None, exclude=None): return dup_count_complex_roots(self.to_dense(f), self.domain, inf=inf, sup=sup, exclude=exclude) def dup_isolate_complex_roots_sqf(self, f, eps=None, inf=None, sup=None, blackbox=False): return dup_isolate_complex_roots_sqf(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, blackbox=blackbox) def dup_isolate_all_roots_sqf(self, f, eps=None, inf=None, sup=None, fast=False, blackbox=False): return dup_isolate_all_roots_sqf(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox) def dup_isolate_all_roots(self, f, eps=None, inf=None, sup=None, fast=False): return dup_isolate_all_roots(self.to_dense(f), self.domain, eps=eps, inf=inf, sup=sup, fast=fast) def fateman_poly_F_1(self): from sympy.polys.specialpolys import dmp_fateman_poly_F_1 return tuple(map(self.from_dense, dmp_fateman_poly_F_1(self.ngens-1, self.domain))) def fateman_poly_F_2(self): from sympy.polys.specialpolys import dmp_fateman_poly_F_2 return tuple(map(self.from_dense, dmp_fateman_poly_F_2(self.ngens-1, self.domain))) def fateman_poly_F_3(self): from sympy.polys.specialpolys import dmp_fateman_poly_F_3 return tuple(map(self.from_dense, dmp_fateman_poly_F_3(self.ngens-1, self.domain))) def to_gf_dense(self, element): return gf_strip([ self.domain.dom.convert(c, self.domain) for c in self.wrap(element).to_dense() ]) def from_gf_dense(self, element): return self.from_dict(dmp_to_dict(element, self.ngens-1, self.domain.dom)) def gf_degree(self, f): return gf_degree(self.to_gf_dense(f)) def gf_LC(self, f): return gf_LC(self.to_gf_dense(f), self.domain.dom) def gf_TC(self, f): return gf_TC(self.to_gf_dense(f), self.domain.dom) def gf_strip(self, f): return self.from_gf_dense(gf_strip(self.to_gf_dense(f))) def gf_trunc(self, f): return self.from_gf_dense(gf_strip(self.to_gf_dense(f), self.domain.mod)) def gf_normal(self, f): return self.from_gf_dense(gf_strip(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_from_dict(self, f): return self.from_gf_dense(gf_from_dict(f, self.domain.mod, self.domain.dom)) def gf_to_dict(self, f, symmetric=True): return gf_to_dict(self.to_gf_dense(f), self.domain.mod, symmetric=symmetric) def gf_from_int_poly(self, f): return self.from_gf_dense(gf_from_int_poly(f, self.domain.mod)) def gf_to_int_poly(self, f, symmetric=True): return gf_to_int_poly(self.to_gf_dense(f), self.domain.mod, symmetric=symmetric) def gf_neg(self, f): return self.from_gf_dense(gf_neg(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_add_ground(self, f, a): return self.from_gf_dense(gf_add_ground(self.to_gf_dense(f), a, self.domain.mod, self.domain.dom)) def gf_sub_ground(self, f, a): return self.from_gf_dense(gf_sub_ground(self.to_gf_dense(f), a, self.domain.mod, self.domain.dom)) def gf_mul_ground(self, f, a): return self.from_gf_dense(gf_mul_ground(self.to_gf_dense(f), a, self.domain.mod, self.domain.dom)) def gf_quo_ground(self, f, a): return self.from_gf_dense(gf_quo_ground(self.to_gf_dense(f), a, self.domain.mod, self.domain.dom)) def gf_add(self, f, g): return self.from_gf_dense(gf_add(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_sub(self, f, g): return self.from_gf_dense(gf_sub(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_mul(self, f, g): return self.from_gf_dense(gf_mul(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_sqr(self, f): return self.from_gf_dense(gf_sqr(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_add_mul(self, f, g, h): return self.from_gf_dense(gf_add_mul(self.to_gf_dense(f), self.to_gf_dense(g), self.to_gf_dense(h), self.domain.mod, self.domain.dom)) def gf_sub_mul(self, f, g, h): return self.from_gf_dense(gf_sub_mul(self.to_gf_dense(f), self.to_gf_dense(g), self.to_gf_dense(h), self.domain.mod, self.domain.dom)) def gf_expand(self, F): return self.from_gf_dense(gf_expand(list(map(self.to_gf_dense, F)), self.domain.mod, self.domain.dom)) def gf_div(self, f, g): q, r = gf_div(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom) return self.from_gf_dense(q), self.from_gf_dense(r) def gf_rem(self, f, g): return self.from_gf_dense(gf_rem(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_quo(self, f, g): return self.from_gf_dense(gf_quo(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_exquo(self, f, g): return self.from_gf_dense(gf_exquo(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_lshift(self, f, n): return self.from_gf_dense(gf_lshift(self.to_gf_dense(f), n, self.domain.dom)) def gf_rshift(self, f, n): return self.from_gf_dense(gf_rshift(self.to_gf_dense(f), n, self.domain.dom)) def gf_pow(self, f, n): return self.from_gf_dense(gf_pow(self.to_gf_dense(f), n, self.domain.mod, self.domain.dom)) def gf_pow_mod(self, f, n, g): return self.from_gf_dense(gf_pow_mod(self.to_gf_dense(f), n, self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_cofactors(self, f, g): h, cff, cfg = gf_cofactors(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom) return self.from_gf_dense(h), self.from_gf_dense(cff), self.from_gf_dense(cfg) def gf_gcd(self, f, g): return self.from_gf_dense(gf_gcd(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_lcm(self, f, g): return self.from_gf_dense(gf_lcm(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_gcdex(self, f, g): return self.from_gf_dense(gf_gcdex(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_monic(self, f): return self.from_gf_dense(gf_monic(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_diff(self, f): return self.from_gf_dense(gf_diff(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_eval(self, f, a): return gf_eval(self.to_gf_dense(f), a, self.domain.mod, self.domain.dom) def gf_multi_eval(self, f, A): return gf_multi_eval(self.to_gf_dense(f), A, self.domain.mod, self.domain.dom) def gf_compose(self, f, g): return self.from_gf_dense(gf_compose(self.to_gf_dense(f), self.to_gf_dense(g), self.domain.mod, self.domain.dom)) def gf_compose_mod(self, g, h, f): return self.from_gf_dense(gf_compose_mod(self.to_gf_dense(g), self.to_gf_dense(h), self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_trace_map(self, a, b, c, n, f): a = self.to_gf_dense(a) b = self.to_gf_dense(b) c = self.to_gf_dense(c) f = self.to_gf_dense(f) U, V = gf_trace_map(a, b, c, n, f, self.domain.mod, self.domain.dom) return self.from_gf_dense(U), self.from_gf_dense(V) def gf_random(self, n): return self.from_gf_dense(gf_random(n, self.domain.mod, self.domain.dom)) def gf_irreducible(self, n): return self.from_gf_dense(gf_irreducible(n, self.domain.mod, self.domain.dom)) def gf_irred_p_ben_or(self, f): return gf_irred_p_ben_or(self.to_gf_dense(f), self.domain.mod, self.domain.dom) def gf_irred_p_rabin(self, f): return gf_irred_p_rabin(self.to_gf_dense(f), self.domain.mod, self.domain.dom) def gf_irreducible_p(self, f): return gf_irreducible_p(self.to_gf_dense(f), self.domain.mod, self.domain.dom) def gf_sqf_p(self, f): return gf_sqf_p(self.to_gf_dense(f), self.domain.mod, self.domain.dom) def gf_sqf_part(self, f): return self.from_gf_dense(gf_sqf_part(self.to_gf_dense(f), self.domain.mod, self.domain.dom)) def gf_sqf_list(self, f, all=False): coeff, factors = gf_sqf_part(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return coeff, [ (self.from_gf_dense(g), k) for g, k in factors ] def gf_Qmatrix(self, f): return gf_Qmatrix(self.to_gf_dense(f), self.domain.mod, self.domain.dom) def gf_berlekamp(self, f): factors = gf_berlekamp(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ self.from_gf_dense(g) for g in factors ] def gf_ddf_zassenhaus(self, f): factors = gf_ddf_zassenhaus(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ (self.from_gf_dense(g), k) for g, k in factors ] def gf_edf_zassenhaus(self, f, n): factors = gf_edf_zassenhaus(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ self.from_gf_dense(g) for g in factors ] def gf_ddf_shoup(self, f): factors = gf_ddf_shoup(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ (self.from_gf_dense(g), k) for g, k in factors ] def gf_edf_shoup(self, f, n): factors = gf_edf_shoup(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ self.from_gf_dense(g) for g in factors ] def gf_zassenhaus(self, f): factors = gf_zassenhaus(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ self.from_gf_dense(g) for g in factors ] def gf_shoup(self, f): factors = gf_shoup(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return [ self.from_gf_dense(g) for g in factors ] def gf_factor_sqf(self, f, method=None): coeff, factors = gf_factor_sqf(self.to_gf_dense(f), self.domain.mod, self.domain.dom, method=method) return coeff, [ self.from_gf_dense(g) for g in factors ] def gf_factor(self, f): coeff, factors = gf_factor(self.to_gf_dense(f), self.domain.mod, self.domain.dom) return coeff, [ (self.from_gf_dense(g), k) for g, k in factors ]
fc2aa3d37a04d03de6328643cebea47cf3f5eaeb0ea9c60a39fb578e7ba829a2
"""Efficient functions for generating orthogonal polynomials. """ from sympy import Dummy from sympy.polys.constructor import construct_domain from sympy.polys.densearith import ( dup_mul, dup_mul_ground, dup_lshift, dup_sub, dup_add ) from sympy.polys.domains import ZZ, QQ from sympy.polys.polyclasses import DMP from sympy.polys.polytools import Poly, PurePoly from sympy.utilities import public def dup_jacobi(n, a, b, K): """Low-level implementation of Jacobi polynomials. """ seq = [[K.one], [(a + b + K(2))/K(2), (a - b)/K(2)]] for i in range(2, n + 1): den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2)) f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den) f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den) f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den p0 = dup_mul_ground(seq[-1], f0, K) p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K) p2 = dup_mul_ground(seq[-2], f2, K) seq.append(dup_sub(dup_add(p0, p1, K), p2, K)) return seq[n] @public def jacobi_poly(n, a, b, x=None, polys=False): """Generates Jacobi polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError("can't generate Jacobi polynomial of degree %s" % n) K, v = construct_domain([a, b], field=True) poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_gegenbauer(n, a, K): """Low-level implementation of Gegenbauer polynomials. """ seq = [[K.one], [K(2)*a, K.zero]] for i in range(2, n + 1): f1 = K(2) * (i + a - K.one) / i f2 = (i + K(2)*a - K(2)) / i p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K) p2 = dup_mul_ground(seq[-2], f2, K) seq.append(dup_sub(p1, p2, K)) return seq[n] def gegenbauer_poly(n, a, x=None, polys=False): """Generates Gegenbauer polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError( "can't generate Gegenbauer polynomial of degree %s" % n) K, a = construct_domain(a, field=True) poly = DMP(dup_gegenbauer(int(n), a, K), K) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_chebyshevt(n, K): """Low-level implementation of Chebyshev polynomials of the 1st kind. """ seq = [[K.one], [K.one, K.zero]] for i in range(2, n + 1): a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2), K) seq.append(dup_sub(a, seq[-2], K)) return seq[n] @public def chebyshevt_poly(n, x=None, polys=False): """Generates Chebyshev polynomial of the first kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError( "can't generate 1st kind Chebyshev polynomial of degree %s" % n) poly = DMP(dup_chebyshevt(int(n), ZZ), ZZ) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_chebyshevu(n, K): """Low-level implementation of Chebyshev polynomials of the 2nd kind. """ seq = [[K.one], [K(2), K.zero]] for i in range(2, n + 1): a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2), K) seq.append(dup_sub(a, seq[-2], K)) return seq[n] @public def chebyshevu_poly(n, x=None, polys=False): """Generates Chebyshev polynomial of the second kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError( "can't generate 2nd kind Chebyshev polynomial of degree %s" % n) poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_hermite(n, K): """Low-level implementation of Hermite polynomials. """ seq = [[K.one], [K(2), K.zero]] for i in range(2, n + 1): a = dup_lshift(seq[-1], 1, K) b = dup_mul_ground(seq[-2], K(i - 1), K) c = dup_mul_ground(dup_sub(a, b, K), K(2), K) seq.append(c) return seq[n] @public def hermite_poly(n, x=None, polys=False): """Generates Hermite polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError("can't generate Hermite polynomial of degree %s" % n) poly = DMP(dup_hermite(int(n), ZZ), ZZ) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_legendre(n, K): """Low-level implementation of Legendre polynomials. """ seq = [[K.one], [K.one, K.zero]] for i in range(2, n + 1): a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2*i - 1, i), K) b = dup_mul_ground(seq[-2], K(i - 1, i), K) seq.append(dup_sub(a, b, K)) return seq[n] @public def legendre_poly(n, x=None, polys=False): """Generates Legendre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError("can't generate Legendre polynomial of degree %s" % n) poly = DMP(dup_legendre(int(n), QQ), QQ) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_laguerre(n, alpha, K): """Low-level implementation of Laguerre polynomials. """ seq = [[K.zero], [K.one]] for i in range(1, n + 1): a = dup_mul(seq[-1], [-K.one/i, alpha/i + K(2*i - 1)/i], K) b = dup_mul_ground(seq[-2], alpha/i + K(i - 1)/i, K) seq.append(dup_sub(a, b, K)) return seq[-1] @public def laguerre_poly(n, x=None, alpha=None, polys=False): """Generates Laguerre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional alpha Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. """ if n < 0: raise ValueError("can't generate Laguerre polynomial of degree %s" % n) if alpha is not None: K, alpha = construct_domain( alpha, field=True) # XXX: ground_field=True else: K, alpha = QQ, QQ(0) poly = DMP(dup_laguerre(int(n), alpha, K), K) if x is not None: poly = Poly.new(poly, x) else: poly = PurePoly.new(poly, Dummy('x')) return poly if polys else poly.as_expr() def dup_spherical_bessel_fn(n, K): """ Low-level implementation of fn(n, x) """ seq = [[K.one], [K.one, K.zero]] for i in range(2, n + 1): a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2*i - 1), K) seq.append(dup_sub(a, seq[-2], K)) return dup_lshift(seq[n], 1, K) def dup_spherical_bessel_fn_minus(n, K): """ Low-level implementation of fn(-n, x) """ seq = [[K.one, K.zero], [K.zero]] for i in range(2, n + 1): a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(3 - 2*i), K) seq.append(dup_sub(a, seq[-2], K)) return seq[n] def spherical_bessel_fn(n, x=None, polys=False): """ Coefficients for the spherical Bessel functions. Those are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 """ if n < 0: dup = dup_spherical_bessel_fn_minus(-int(n), ZZ) else: dup = dup_spherical_bessel_fn(int(n), ZZ) poly = DMP(dup, ZZ) if x is not None: poly = Poly.new(poly, 1/x) else: poly = PurePoly.new(poly, 1/Dummy('x')) return poly if polys else poly.as_expr()
15eed70c48771548d40c920daa82f9cd8c3f560e661cd5d5526c423f3f21019f
"""Useful utilities for higher level polynomial classes. """ from sympy.core import (S, Add, Mul, Pow, Eq, Expr, expand_mul, expand_multinomial) from sympy.core.exprtools import decompose_power, decompose_power_rat from sympy.polys.polyerrors import PolynomialError, GeneratorsError from sympy.polys.polyoptions import build_options import re _gens_order = { 'a': 301, 'b': 302, 'c': 303, 'd': 304, 'e': 305, 'f': 306, 'g': 307, 'h': 308, 'i': 309, 'j': 310, 'k': 311, 'l': 312, 'm': 313, 'n': 314, 'o': 315, 'p': 216, 'q': 217, 'r': 218, 's': 219, 't': 220, 'u': 221, 'v': 222, 'w': 223, 'x': 124, 'y': 125, 'z': 126, } _max_order = 1000 _re_gen = re.compile(r"^(.+?)(\d*)$") def _nsort(roots, separated=False): """Sort the numerical roots putting the real roots first, then sorting according to real and imaginary parts. If ``separated`` is True, then the real and imaginary roots will be returned in two lists, respectively. This routine tries to avoid issue 6137 by separating the roots into real and imaginary parts before evaluation. In addition, the sorting will raise an error if any computation cannot be done with precision. """ if not all(r.is_number for r in roots): raise NotImplementedError # see issue 6137: # get the real part of the evaluated real and imaginary parts of each root key = [[i.n(2).as_real_imag()[0] for i in r.as_real_imag()] for r in roots] # make sure the parts were computed with precision if len(roots) > 1 and any(i._prec == 1 for k in key for i in k): raise NotImplementedError("could not compute root with precision") # insert a key to indicate if the root has an imaginary part key = [(1 if i else 0, r, i) for r, i in key] key = sorted(zip(key, roots)) # return the real and imaginary roots separately if desired if separated: r = [] i = [] for (im, _, _), v in key: if im: i.append(v) else: r.append(v) return r, i _, roots = zip(*key) return list(roots) def _sort_gens(gens, **args): """Sort generators in a reasonably intelligent way. """ opt = build_options(args) gens_order, wrt = {}, None if opt is not None: gens_order, wrt = {}, opt.wrt for i, gen in enumerate(opt.sort): gens_order[gen] = i + 1 def order_key(gen): gen = str(gen) if wrt is not None: try: return (-len(wrt) + wrt.index(gen), gen, 0) except ValueError: pass name, index = _re_gen.match(gen).groups() if index: index = int(index) else: index = 0 try: return ( gens_order[name], name, index) except KeyError: pass try: return (_gens_order[name], name, index) except KeyError: pass return (_max_order, name, index) try: gens = sorted(gens, key=order_key) except TypeError: # pragma: no cover pass return tuple(gens) def _unify_gens(f_gens, g_gens): """Unify generators in a reasonably intelligent way. """ f_gens = list(f_gens) g_gens = list(g_gens) if f_gens == g_gens: return tuple(f_gens) gens, common, k = [], [], 0 for gen in f_gens: if gen in g_gens: common.append(gen) for i, gen in enumerate(g_gens): if gen in common: g_gens[i], k = common[k], k + 1 for gen in common: i = f_gens.index(gen) gens.extend(f_gens[:i]) f_gens = f_gens[i + 1:] i = g_gens.index(gen) gens.extend(g_gens[:i]) g_gens = g_gens[i + 1:] gens.append(gen) gens.extend(f_gens) gens.extend(g_gens) return tuple(gens) def _analyze_gens(gens): """Support for passing generators as `*gens` and `[gens]`. """ if len(gens) == 1 and hasattr(gens[0], '__iter__'): return tuple(gens[0]) else: return tuple(gens) def _sort_factors(factors, **args): """Sort low-level factors in increasing 'complexity' order. """ def order_if_multiple_key(factor): (f, n) = factor return (len(f), n, f) def order_no_multiple_key(f): return (len(f), f) if args.get('multiple', True): return sorted(factors, key=order_if_multiple_key) else: return sorted(factors, key=order_no_multiple_key) illegal = [S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity] finf = [float(i) for i in illegal[1:3]] def _not_a_coeff(expr): """Do not treat NaN and infinities as valid polynomial coefficients. """ if expr in illegal or expr in finf: return True if type(expr) is float and float(expr) != expr: return True # nan return # could be def _parallel_dict_from_expr_if_gens(exprs, opt): """Transform expressions into a multinomial form given generators. """ k, indices = len(opt.gens), {} for i, g in enumerate(opt.gens): indices[g] = i polys = [] for expr in exprs: poly = {} if expr.is_Equality: expr = expr.lhs - expr.rhs for term in Add.make_args(expr): coeff, monom = [], [0]*k for factor in Mul.make_args(term): if not _not_a_coeff(factor) and factor.is_Number: coeff.append(factor) else: try: if opt.series is False: base, exp = decompose_power(factor) if exp < 0: exp, base = -exp, Pow(base, -S.One) else: base, exp = decompose_power_rat(factor) monom[indices[base]] = exp except KeyError: if not factor.free_symbols.intersection(opt.gens): coeff.append(factor) else: raise PolynomialError("%s contains an element of " "the set of generators." % factor) monom = tuple(monom) if monom in poly: poly[monom] += Mul(*coeff) else: poly[monom] = Mul(*coeff) polys.append(poly) return polys, opt.gens def _parallel_dict_from_expr_no_gens(exprs, opt): """Transform expressions into a multinomial form and figure out generators. """ if opt.domain is not None: def _is_coeff(factor): return factor in opt.domain elif opt.extension is True: def _is_coeff(factor): return factor.is_algebraic elif opt.greedy is not False: def _is_coeff(factor): return factor is S.ImaginaryUnit else: def _is_coeff(factor): return factor.is_number gens, reprs = set(), [] for expr in exprs: terms = [] if expr.is_Equality: expr = expr.lhs - expr.rhs for term in Add.make_args(expr): coeff, elements = [], {} for factor in Mul.make_args(term): if not _not_a_coeff(factor) and (factor.is_Number or _is_coeff(factor)): coeff.append(factor) else: if opt.series is False: base, exp = decompose_power(factor) if exp < 0: exp, base = -exp, Pow(base, -S.One) else: base, exp = decompose_power_rat(factor) elements[base] = elements.setdefault(base, 0) + exp gens.add(base) terms.append((coeff, elements)) reprs.append(terms) gens = _sort_gens(gens, opt=opt) k, indices = len(gens), {} for i, g in enumerate(gens): indices[g] = i polys = [] for terms in reprs: poly = {} for coeff, term in terms: monom = [0]*k for base, exp in term.items(): monom[indices[base]] = exp monom = tuple(monom) if monom in poly: poly[monom] += Mul(*coeff) else: poly[monom] = Mul(*coeff) polys.append(poly) return polys, tuple(gens) def _dict_from_expr_if_gens(expr, opt): """Transform an expression into a multinomial form given generators. """ (poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt) return poly, gens def _dict_from_expr_no_gens(expr, opt): """Transform an expression into a multinomial form and figure out generators. """ (poly,), gens = _parallel_dict_from_expr_no_gens((expr,), opt) return poly, gens def parallel_dict_from_expr(exprs, **args): """Transform expressions into a multinomial form. """ reps, opt = _parallel_dict_from_expr(exprs, build_options(args)) return reps, opt.gens def _parallel_dict_from_expr(exprs, opt): """Transform expressions into a multinomial form. """ if opt.expand is not False: exprs = [ expr.expand() for expr in exprs ] if any(expr.is_commutative is False for expr in exprs): raise PolynomialError('non-commutative expressions are not supported') if opt.gens: reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt) else: reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt) return reps, opt.clone({'gens': gens}) def dict_from_expr(expr, **args): """Transform an expression into a multinomial form. """ rep, opt = _dict_from_expr(expr, build_options(args)) return rep, opt.gens def _dict_from_expr(expr, opt): """Transform an expression into a multinomial form. """ if expr.is_commutative is False: raise PolynomialError('non-commutative expressions are not supported') def _is_expandable_pow(expr): return (expr.is_Pow and expr.exp.is_positive and expr.exp.is_Integer and expr.base.is_Add) if opt.expand is not False: if not isinstance(expr, (Expr, Eq)): raise PolynomialError('expression must be of type Expr') expr = expr.expand() # TODO: Integrate this into expand() itself while any(_is_expandable_pow(i) or i.is_Mul and any(_is_expandable_pow(j) for j in i.args) for i in Add.make_args(expr)): expr = expand_multinomial(expr) while any(i.is_Mul and any(j.is_Add for j in i.args) for i in Add.make_args(expr)): expr = expand_mul(expr) if opt.gens: rep, gens = _dict_from_expr_if_gens(expr, opt) else: rep, gens = _dict_from_expr_no_gens(expr, opt) return rep, opt.clone({'gens': gens}) def expr_from_dict(rep, *gens): """Convert a multinomial form into an expression. """ result = [] for monom, coeff in rep.items(): term = [coeff] for g, m in zip(gens, monom): if m: term.append(Pow(g, m)) result.append(Mul(*term)) return Add(*result) parallel_dict_from_basic = parallel_dict_from_expr dict_from_basic = dict_from_expr basic_from_dict = expr_from_dict def _dict_reorder(rep, gens, new_gens): """Reorder levels using dict representation. """ gens = list(gens) monoms = rep.keys() coeffs = rep.values() new_monoms = [ [] for _ in range(len(rep)) ] used_indices = set() for gen in new_gens: try: j = gens.index(gen) used_indices.add(j) for M, new_M in zip(monoms, new_monoms): new_M.append(M[j]) except ValueError: for new_M in new_monoms: new_M.append(0) for i, _ in enumerate(gens): if i not in used_indices: for monom in monoms: if monom[i]: raise GeneratorsError("unable to drop generators") return map(tuple, new_monoms), coeffs class PicklableWithSlots: """ Mixin class that allows to pickle objects with ``__slots__``. Examples ======== First define a class that mixes :class:`PicklableWithSlots` in:: >>> from sympy.polys.polyutils import PicklableWithSlots >>> class Some(PicklableWithSlots): ... __slots__ = ('foo', 'bar') ... ... def __init__(self, foo, bar): ... self.foo = foo ... self.bar = bar To make :mod:`pickle` happy in doctest we have to use these hacks:: >>> from sympy.core.compatibility import builtins >>> builtins.Some = Some >>> from sympy.polys import polyutils >>> polyutils.Some = Some Next lets see if we can create an instance, pickle it and unpickle:: >>> some = Some('abc', 10) >>> some.foo, some.bar ('abc', 10) >>> from pickle import dumps, loads >>> some2 = loads(dumps(some)) >>> some2.foo, some2.bar ('abc', 10) """ __slots__ = () def __getstate__(self, cls=None): if cls is None: # This is the case for the instance that gets pickled cls = self.__class__ d = {} # Get all data that should be stored from super classes for c in cls.__bases__: if hasattr(c, "__getstate__"): d.update(c.__getstate__(self, c)) # Get all information that should be stored from cls and return the dict for name in cls.__slots__: if hasattr(self, name): d[name] = getattr(self, name) return d def __setstate__(self, d): # All values that were pickled are now assigned to a fresh instance for name, value in d.items(): try: setattr(self, name, value) except AttributeError: # This is needed in cases like Rational :> Half pass
40e7ba4160bec8c488766017613fb2f1faa2d099e9e369b994cd7c1a0520ddac
"""Tools for manipulation of rational expressions. """ from sympy.core import Basic, Add, sympify from sympy.core.compatibility import iterable from sympy.core.exprtools import gcd_terms from sympy.utilities import public @public def together(expr, deep=False, fraction=True): """ Denest and combine rational expressions using symbolic methods. This function takes an expression or a container of expressions and puts it (them) together by denesting and combining rational subexpressions. No heroic measures are taken to minimize degree of the resulting numerator and denominator. To obtain completely reduced expression use :func:`~.cancel`. However, :func:`~.together` can preserve as much as possible of the structure of the input expression in the output (no expansion is performed). A wide variety of objects can be put together including lists, tuples, sets, relational objects, integrals and others. It is also possible to transform interior of function applications, by setting ``deep`` flag to ``True``. By definition, :func:`~.together` is a complement to :func:`~.apart`, so ``apart(together(expr))`` should return expr unchanged. Note however, that :func:`~.together` uses only symbolic methods, so it might be necessary to use :func:`~.cancel` to perform algebraic simplification and minimize degree of the numerator and denominator. Examples ======== >>> from sympy import together, exp >>> from sympy.abc import x, y, z >>> together(1/x + 1/y) (x + y)/(x*y) >>> together(1/x + 1/y + 1/z) (x*y + x*z + y*z)/(x*y*z) >>> together(1/(x*y) + 1/y**2) (x + y)/(x*y**2) >>> together(1/(1 + 1/x) + 1/(1 + 1/y)) (x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1)) >>> together(exp(1/x + 1/y)) exp(1/y + 1/x) >>> together(exp(1/x + 1/y), deep=True) exp((x + y)/(x*y)) >>> together(1/exp(x) + 1/(x*exp(x))) (x + 1)*exp(-x)/x >>> together(1/exp(2*x) + 1/(x*exp(3*x))) (x*exp(x) + 1)*exp(-3*x)/x """ def _together(expr): if isinstance(expr, Basic): if expr.is_Atom or (expr.is_Function and not deep): return expr elif expr.is_Add: return gcd_terms(list(map(_together, Add.make_args(expr))), fraction=fraction) elif expr.is_Pow: base = _together(expr.base) if deep: exp = _together(expr.exp) else: exp = expr.exp return expr.__class__(base, exp) else: return expr.__class__(*[ _together(arg) for arg in expr.args ]) elif iterable(expr): return expr.__class__([ _together(ex) for ex in expr ]) return expr return _together(sympify(expr))
c20581eb09a7295d11c6ed89153b990aa4efa0605e879daa5a25afd519f15acd
"""Computational algebraic field theory. """ from sympy import ( S, Rational, AlgebraicNumber, GoldenRatio, TribonacciConstant, Add, Mul, sympify, Dummy, expand_mul, I, pi ) from sympy.functions import sqrt, cbrt from sympy.core.compatibility import reduce from sympy.core.exprtools import Factors from sympy.core.function import _mexpand from sympy.functions.elementary.exponential import exp from sympy.functions.elementary.trigonometric import cos, sin from sympy.ntheory import sieve from sympy.ntheory.factor_ import divisors from sympy.polys.domains import ZZ, QQ from sympy.polys.orthopolys import dup_chebyshevt from sympy.polys.polyerrors import ( IsomorphismFailed, CoercionFailed, NotAlgebraic, GeneratorsError, ) from sympy.polys.polytools import ( Poly, PurePoly, invert, factor_list, groebner, resultant, degree, poly_from_expr, parallel_poly_from_expr, lcm ) from sympy.polys.polyutils import dict_from_expr, expr_from_dict from sympy.polys.ring_series import rs_compose_add from sympy.polys.rings import ring from sympy.polys.rootoftools import CRootOf from sympy.polys.specialpolys import cyclotomic_poly from sympy.printing.lambdarepr import LambdaPrinter from sympy.printing.pycode import PythonCodePrinter, MpmathPrinter from sympy.simplify.radsimp import _split_gcd from sympy.simplify.simplify import _is_sum_surds from sympy.utilities import ( numbered_symbols, variations, lambdify, public, sift ) from mpmath import pslq, mp def _choose_factor(factors, x, v, dom=QQ, prec=200, bound=5): """ Return a factor having root ``v`` It is assumed that one of the factors has root ``v``. """ if isinstance(factors[0], tuple): factors = [f[0] for f in factors] if len(factors) == 1: return factors[0] points = {x:v} symbols = dom.symbols if hasattr(dom, 'symbols') else [] t = QQ(1, 10) for n in range(bound**len(symbols)): prec1 = 10 n_temp = n for s in symbols: points[s] = n_temp % bound n_temp = n_temp // bound while True: candidates = [] eps = t**(prec1 // 2) for f in factors: if abs(f.as_expr().evalf(prec1, points)) < eps: candidates.append(f) if candidates: factors = candidates if len(factors) == 1: return factors[0] if prec1 > prec: break prec1 *= 2 raise NotImplementedError("multiple candidates for the minimal polynomial of %s" % v) def _separate_sq(p): """ helper function for ``_minimal_polynomial_sq`` It selects a rational ``g`` such that the polynomial ``p`` consists of a sum of terms whose surds squared have gcd equal to ``g`` and a sum of terms with surds squared prime with ``g``; then it takes the field norm to eliminate ``sqrt(g)`` See simplify.simplify.split_surds and polytools.sqf_norm. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x >>> from sympy.polys.numberfields import _separate_sq >>> p= -x + sqrt(2) + sqrt(3) + sqrt(7) >>> p = _separate_sq(p); p -x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8 >>> p = _separate_sq(p); p -x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20 >>> p = _separate_sq(p); p -x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400 """ from sympy.utilities.iterables import sift def is_sqrt(expr): return expr.is_Pow and expr.exp is S.Half # p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)] a = [] for y in p.args: if not y.is_Mul: if is_sqrt(y): a.append((S.One, y**2)) elif y.is_Atom: a.append((y, S.One)) elif y.is_Pow and y.exp.is_integer: a.append((y, S.One)) else: raise NotImplementedError continue T, F = sift(y.args, is_sqrt, binary=True) a.append((Mul(*F), Mul(*T)**2)) a.sort(key=lambda z: z[1]) if a[-1][1] is S.One: # there are no surds return p surds = [z for y, z in a] for i in range(len(surds)): if surds[i] != 1: break g, b1, b2 = _split_gcd(*surds[i:]) a1 = [] a2 = [] for y, z in a: if z in b1: a1.append(y*z**S.Half) else: a2.append(y*z**S.Half) p1 = Add(*a1) p2 = Add(*a2) p = _mexpand(p1**2) - _mexpand(p2**2) return p def _minimal_polynomial_sq(p, n, x): """ Returns the minimal polynomial for the ``nth-root`` of a sum of surds or ``None`` if it fails. Parameters ========== p : sum of surds n : positive integer x : variable of the returned polynomial Examples ======== >>> from sympy.polys.numberfields import _minimal_polynomial_sq >>> from sympy import sqrt >>> from sympy.abc import x >>> q = 1 + sqrt(2) + sqrt(3) >>> _minimal_polynomial_sq(q, 3, x) x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8 """ from sympy.simplify.simplify import _is_sum_surds p = sympify(p) n = sympify(n) if not n.is_Integer or not n > 0 or not _is_sum_surds(p): return None pn = p**Rational(1, n) # eliminate the square roots p -= x while 1: p1 = _separate_sq(p) if p1 is p: p = p1.subs({x:x**n}) break else: p = p1 # _separate_sq eliminates field extensions in a minimal way, so that # if n = 1 then `p = constant*(minimal_polynomial(p))` # if n > 1 it contains the minimal polynomial as a factor. if n == 1: p1 = Poly(p) if p.coeff(x**p1.degree(x)) < 0: p = -p p = p.primitive()[1] return p # by construction `p` has root `pn` # the minimal polynomial is the factor vanishing in x = pn factors = factor_list(p)[1] result = _choose_factor(factors, x, pn) return result def _minpoly_op_algebraic_element(op, ex1, ex2, x, dom, mp1=None, mp2=None): """ return the minimal polynomial for ``op(ex1, ex2)`` Parameters ========== op : operation ``Add`` or ``Mul`` ex1, ex2 : expressions for the algebraic elements x : indeterminate of the polynomials dom: ground domain mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None Examples ======== >>> from sympy import sqrt, Add, Mul, QQ >>> from sympy.polys.numberfields import _minpoly_op_algebraic_element >>> from sympy.abc import x, y >>> p1 = sqrt(sqrt(2) + 1) >>> p2 = sqrt(sqrt(2) - 1) >>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ) x - 1 >>> q1 = sqrt(y) >>> q2 = 1 / y >>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y)) x**2*y**2 - 2*x*y - y**3 + 1 References ========== .. [1] https://en.wikipedia.org/wiki/Resultant .. [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638 "Degrees of sums in a separable field extension". """ y = Dummy(str(x)) if mp1 is None: mp1 = _minpoly_compose(ex1, x, dom) if mp2 is None: mp2 = _minpoly_compose(ex2, y, dom) else: mp2 = mp2.subs({x: y}) if op is Add: # mp1a = mp1.subs({x: x - y}) if dom == QQ: R, X = ring('X', QQ) p1 = R(dict_from_expr(mp1)[0]) p2 = R(dict_from_expr(mp2)[0]) else: (p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y) r = p1.compose(p2) mp1a = r.as_expr() elif op is Mul: mp1a = _muly(mp1, x, y) else: raise NotImplementedError('option not available') if op is Mul or dom != QQ: r = resultant(mp1a, mp2, gens=[y, x]) else: r = rs_compose_add(p1, p2) r = expr_from_dict(r.as_expr_dict(), x) deg1 = degree(mp1, x) deg2 = degree(mp2, y) if op is Mul and deg1 == 1 or deg2 == 1: # if deg1 = 1, then mp1 = x - a; mp1a = x - y - a; # r = mp2(x - a), so that `r` is irreducible return r r = Poly(r, x, domain=dom) _, factors = r.factor_list() res = _choose_factor(factors, x, op(ex1, ex2), dom) return res.as_expr() def _invertx(p, x): """ Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))`` """ p1 = poly_from_expr(p, x)[0] n = degree(p1) a = [c * x**(n - i) for (i,), c in p1.terms()] return Add(*a) def _muly(p, x, y): """ Returns ``_mexpand(y**deg*p.subs({x:x / y}))`` """ p1 = poly_from_expr(p, x)[0] n = degree(p1) a = [c * x**i * y**(n - i) for (i,), c in p1.terms()] return Add(*a) def _minpoly_pow(ex, pw, x, dom, mp=None): """ Returns ``minpoly(ex**pw, x)`` Parameters ========== ex : algebraic element pw : rational number x : indeterminate of the polynomial dom: ground domain mp : minimal polynomial of ``p`` Examples ======== >>> from sympy import sqrt, QQ, Rational >>> from sympy.polys.numberfields import _minpoly_pow, minpoly >>> from sympy.abc import x, y >>> p = sqrt(1 + sqrt(2)) >>> _minpoly_pow(p, 2, x, QQ) x**2 - 2*x - 1 >>> minpoly(p**2, x) x**2 - 2*x - 1 >>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y)) x**3 - y >>> minpoly(y**Rational(1, 3), x) x**3 - y """ pw = sympify(pw) if not mp: mp = _minpoly_compose(ex, x, dom) if not pw.is_rational: raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) if pw < 0: if mp == x: raise ZeroDivisionError('%s is zero' % ex) mp = _invertx(mp, x) if pw == -1: return mp pw = -pw ex = 1/ex y = Dummy(str(x)) mp = mp.subs({x: y}) n, d = pw.as_numer_denom() res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom) _, factors = res.factor_list() res = _choose_factor(factors, x, ex**pw, dom) return res.as_expr() def _minpoly_add(x, dom, *a): """ returns ``minpoly(Add(*a), dom, x)`` """ mp = _minpoly_op_algebraic_element(Add, a[0], a[1], x, dom) p = a[0] + a[1] for px in a[2:]: mp = _minpoly_op_algebraic_element(Add, p, px, x, dom, mp1=mp) p = p + px return mp def _minpoly_mul(x, dom, *a): """ returns ``minpoly(Mul(*a), dom, x)`` """ mp = _minpoly_op_algebraic_element(Mul, a[0], a[1], x, dom) p = a[0] * a[1] for px in a[2:]: mp = _minpoly_op_algebraic_element(Mul, p, px, x, dom, mp1=mp) p = p * px return mp def _minpoly_sin(ex, x): """ Returns the minimal polynomial of ``sin(ex)`` see http://mathworld.wolfram.com/TrigonometryAngles.html """ c, a = ex.args[0].as_coeff_Mul() if a is pi: if c.is_rational: n = c.q q = sympify(n) if q.is_prime: # for a = pi*p/q with q odd prime, using chebyshevt # write sin(q*a) = mp(sin(a))*sin(a); # the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1 a = dup_chebyshevt(n, ZZ) return Add(*[x**(n - i - 1)*a[i] for i in range(n)]) if c.p == 1: if q == 9: return 64*x**6 - 96*x**4 + 36*x**2 - 3 if n % 2 == 1: # for a = pi*p/q with q odd, use # sin(q*a) = 0 to see that the minimal polynomial must be # a factor of dup_chebyshevt(n, ZZ) a = dup_chebyshevt(n, ZZ) a = [x**(n - i)*a[i] for i in range(n + 1)] r = Add(*a) _, factors = factor_list(r) res = _choose_factor(factors, x, ex) return res expr = ((1 - cos(2*c*pi))/2)**S.Half res = _minpoly_compose(expr, x, QQ) return res raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) def _minpoly_cos(ex, x): """ Returns the minimal polynomial of ``cos(ex)`` see http://mathworld.wolfram.com/TrigonometryAngles.html """ from sympy import sqrt c, a = ex.args[0].as_coeff_Mul() if a is pi: if c.is_rational: if c.p == 1: if c.q == 7: return 8*x**3 - 4*x**2 - 4*x + 1 if c.q == 9: return 8*x**3 - 6*x + 1 elif c.p == 2: q = sympify(c.q) if q.is_prime: s = _minpoly_sin(ex, x) return _mexpand(s.subs({x:sqrt((1 - x)/2)})) # for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p n = int(c.q) a = dup_chebyshevt(n, ZZ) a = [x**(n - i)*a[i] for i in range(n + 1)] r = Add(*a) - (-1)**c.p _, factors = factor_list(r) res = _choose_factor(factors, x, ex) return res raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) def _minpoly_exp(ex, x): """ Returns the minimal polynomial of ``exp(ex)`` """ c, a = ex.args[0].as_coeff_Mul() q = sympify(c.q) if a == I*pi: if c.is_rational: if c.p == 1 or c.p == -1: if q == 3: return x**2 - x + 1 if q == 4: return x**4 + 1 if q == 6: return x**4 - x**2 + 1 if q == 8: return x**8 + 1 if q == 9: return x**6 - x**3 + 1 if q == 10: return x**8 - x**6 + x**4 - x**2 + 1 if q.is_prime: s = 0 for i in range(q): s += (-x)**i return s # x**(2*q) = product(factors) factors = [cyclotomic_poly(i, x) for i in divisors(2*q)] mp = _choose_factor(factors, x, ex) return mp else: raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) def _minpoly_rootof(ex, x): """ Returns the minimal polynomial of a ``CRootOf`` object. """ p = ex.expr p = p.subs({ex.poly.gens[0]:x}) _, factors = factor_list(p, x) result = _choose_factor(factors, x, ex) return result def _minpoly_compose(ex, x, dom): """ Computes the minimal polynomial of an algebraic element using operations on minimal polynomials Examples ======== >>> from sympy import minimal_polynomial, sqrt, Rational >>> from sympy.abc import x, y >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=True) x**2 - 2*x - 1 >>> minimal_polynomial(sqrt(y) + 1/y, x, compose=True) x**2*y**2 - 2*x*y - y**3 + 1 """ if ex.is_Rational: return ex.q*x - ex.p if ex is I: _, factors = factor_list(x**2 + 1, x, domain=dom) return x**2 + 1 if len(factors) == 1 else x - I if ex is GoldenRatio: _, factors = factor_list(x**2 - x - 1, x, domain=dom) if len(factors) == 1: return x**2 - x - 1 else: return _choose_factor(factors, x, (1 + sqrt(5))/2, dom=dom) if ex is TribonacciConstant: _, factors = factor_list(x**3 - x**2 - x - 1, x, domain=dom) if len(factors) == 1: return x**3 - x**2 - x - 1 else: fac = (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3 return _choose_factor(factors, x, fac, dom=dom) if hasattr(dom, 'symbols') and ex in dom.symbols: return x - ex if dom.is_QQ and _is_sum_surds(ex): # eliminate the square roots ex -= x while 1: ex1 = _separate_sq(ex) if ex1 is ex: return ex else: ex = ex1 if ex.is_Add: res = _minpoly_add(x, dom, *ex.args) elif ex.is_Mul: f = Factors(ex).factors r = sift(f.items(), lambda itx: itx[0].is_Rational and itx[1].is_Rational) if r[True] and dom == QQ: ex1 = Mul(*[bx**ex for bx, ex in r[False] + r[None]]) r1 = dict(r[True]) dens = [y.q for y in r1.values()] lcmdens = reduce(lcm, dens, 1) neg1 = S.NegativeOne expn1 = r1.pop(neg1, S.Zero) nums = [base**(y.p*lcmdens // y.q) for base, y in r1.items()] ex2 = Mul(*nums) mp1 = minimal_polynomial(ex1, x) # use the fact that in SymPy canonicalization products of integers # raised to rational powers are organized in relatively prime # bases, and that in ``base**(n/d)`` a perfect power is # simplified with the root # Powers of -1 have to be treated separately to preserve sign. mp2 = ex2.q*x**lcmdens - ex2.p*neg1**(expn1*lcmdens) ex2 = neg1**expn1 * ex2**Rational(1, lcmdens) res = _minpoly_op_algebraic_element(Mul, ex1, ex2, x, dom, mp1=mp1, mp2=mp2) else: res = _minpoly_mul(x, dom, *ex.args) elif ex.is_Pow: res = _minpoly_pow(ex.base, ex.exp, x, dom) elif ex.__class__ is sin: res = _minpoly_sin(ex, x) elif ex.__class__ is cos: res = _minpoly_cos(ex, x) elif ex.__class__ is exp: res = _minpoly_exp(ex, x) elif ex.__class__ is CRootOf: res = _minpoly_rootof(ex, x) else: raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) return res @public def minimal_polynomial(ex, x=None, compose=True, polys=False, domain=None): """ Computes the minimal polynomial of an algebraic element. Parameters ========== ex : Expr Element or expression whose minimal polynomial is to be calculated. x : Symbol, optional Independent variable of the minimal polynomial compose : boolean, optional (default=True) Method to use for computing minimal polynomial. If ``compose=True`` (default) then ``_minpoly_compose`` is used, if ``compose=False`` then groebner bases are used. polys : boolean, optional (default=False) If ``True`` returns a ``Poly`` object else an ``Expr`` object. domain : Domain, optional Ground domain Notes ===== By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex`` are computed, then the arithmetic operations on them are performed using the resultant and factorization. If ``compose=False``, a bottom-up algorithm is used with ``groebner``. The default algorithm stalls less frequently. If no ground domain is given, it will be generated automatically from the expression. Examples ======== >>> from sympy import minimal_polynomial, sqrt, solve, QQ >>> from sympy.abc import x, y >>> minimal_polynomial(sqrt(2), x) x**2 - 2 >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2))) x - sqrt(2) >>> minimal_polynomial(sqrt(2) + sqrt(3), x) x**4 - 10*x**2 + 1 >>> minimal_polynomial(solve(x**3 + x + 3)[0], x) x**3 + x + 3 >>> minimal_polynomial(sqrt(y), x) x**2 - y """ from sympy.polys.polytools import degree from sympy.polys.domains import FractionField from sympy.core.basic import preorder_traversal ex = sympify(ex) if ex.is_number: # not sure if it's always needed but try it for numbers (issue 8354) ex = _mexpand(ex, recursive=True) for expr in preorder_traversal(ex): if expr.is_AlgebraicNumber: compose = False break if x is not None: x, cls = sympify(x), Poly else: x, cls = Dummy('x'), PurePoly if not domain: if ex.free_symbols: domain = FractionField(QQ, list(ex.free_symbols)) else: domain = QQ if hasattr(domain, 'symbols') and x in domain.symbols: raise GeneratorsError("the variable %s is an element of the ground " "domain %s" % (x, domain)) if compose: result = _minpoly_compose(ex, x, domain) result = result.primitive()[1] c = result.coeff(x**degree(result, x)) if c.is_negative: result = expand_mul(-result) return cls(result, x, field=True) if polys else result.collect(x) if not domain.is_QQ: raise NotImplementedError("groebner method only works for QQ") result = _minpoly_groebner(ex, x, cls) return cls(result, x, field=True) if polys else result.collect(x) def _minpoly_groebner(ex, x, cls): """ Computes the minimal polynomial of an algebraic number using Groebner bases Examples ======== >>> from sympy import minimal_polynomial, sqrt, Rational >>> from sympy.abc import x >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False) x**2 - 2*x - 1 """ from sympy.polys.polytools import degree from sympy.core.function import expand_multinomial generator = numbered_symbols('a', cls=Dummy) mapping, symbols = {}, {} def update_mapping(ex, exp, base=None): a = next(generator) symbols[ex] = a if base is not None: mapping[ex] = a**exp + base else: mapping[ex] = exp.as_expr(a) return a def bottom_up_scan(ex): if ex.is_Atom: if ex is S.ImaginaryUnit: if ex not in mapping: return update_mapping(ex, 2, 1) else: return symbols[ex] elif ex.is_Rational: return ex elif ex.is_Add: return Add(*[ bottom_up_scan(g) for g in ex.args ]) elif ex.is_Mul: return Mul(*[ bottom_up_scan(g) for g in ex.args ]) elif ex.is_Pow: if ex.exp.is_Rational: if ex.exp < 0: minpoly_base = _minpoly_groebner(ex.base, x, cls) inverse = invert(x, minpoly_base).as_expr() base_inv = inverse.subs(x, ex.base).expand() if ex.exp == -1: return bottom_up_scan(base_inv) else: ex = base_inv**(-ex.exp) if not ex.exp.is_Integer: base, exp = ( ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q) else: base, exp = ex.base, ex.exp base = bottom_up_scan(base) expr = base**exp if expr not in mapping: return update_mapping(expr, 1/exp, -base) else: return symbols[expr] elif ex.is_AlgebraicNumber: if ex.root not in mapping: return update_mapping(ex.root, ex.minpoly) else: return symbols[ex.root] raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex) def simpler_inverse(ex): """ Returns True if it is more likely that the minimal polynomial algorithm works better with the inverse """ if ex.is_Pow: if (1/ex.exp).is_integer and ex.exp < 0: if ex.base.is_Add: return True if ex.is_Mul: hit = True for p in ex.args: if p.is_Add: return False if p.is_Pow: if p.base.is_Add and p.exp > 0: return False if hit: return True return False inverted = False ex = expand_multinomial(ex) if ex.is_AlgebraicNumber: return ex.minpoly.as_expr(x) elif ex.is_Rational: result = ex.q*x - ex.p else: inverted = simpler_inverse(ex) if inverted: ex = ex**-1 res = None if ex.is_Pow and (1/ex.exp).is_Integer: n = 1/ex.exp res = _minimal_polynomial_sq(ex.base, n, x) elif _is_sum_surds(ex): res = _minimal_polynomial_sq(ex, S.One, x) if res is not None: result = res if res is None: bus = bottom_up_scan(ex) F = [x - bus] + list(mapping.values()) G = groebner(F, list(symbols.values()) + [x], order='lex') _, factors = factor_list(G[-1]) # by construction G[-1] has root `ex` result = _choose_factor(factors, x, ex) if inverted: result = _invertx(result, x) if result.coeff(x**degree(result, x)) < 0: result = expand_mul(-result) return result minpoly = minimal_polynomial def _coeffs_generator(n): """Generate coefficients for `primitive_element()`. """ for coeffs in variations([1, -1, 2, -2, 3, -3], n, repetition=True): # Two linear combinations with coeffs of opposite signs are # opposites of each other. Hence it suffices to test only one. if coeffs[0] > 0: yield list(coeffs) @public def primitive_element(extension, x=None, *, ex=False, coeffs=_coeffs_generator, polys=False): """Construct a common number field for all extensions. """ if not extension: raise ValueError("can't compute primitive element for empty extension") if x is not None: x, cls = sympify(x), Poly else: x, cls = Dummy('x'), PurePoly coeffs_generator = coeffs if not ex: gen, coeffs = extension[0], [1] # XXX when minimal_polynomial is extended to work # with AlgebraicNumbers this test can be removed if isinstance(gen, AlgebraicNumber): g = gen.minpoly.replace(x) else: g = minimal_polynomial(gen, x, polys=True) for ext in extension[1:]: _, factors = factor_list(g, extension=ext) g = _choose_factor(factors, x, gen) s, _, g = g.sqf_norm() gen += s*ext coeffs.append(s) if not polys: return g.as_expr(), coeffs else: return cls(g), coeffs generator = numbered_symbols('y', cls=Dummy) F, Y = [], [] for ext in extension: y = next(generator) if ext.is_Poly: if ext.is_univariate: f = ext.as_expr(y) else: raise ValueError("expected minimal polynomial, got %s" % ext) else: f = minpoly(ext, y) F.append(f) Y.append(y) for coeffs in coeffs_generator(len(Y)): f = x - sum([ c*y for c, y in zip(coeffs, Y)]) G = groebner(F + [f], Y + [x], order='lex', field=True) H, g = G[:-1], cls(G[-1], x, domain='QQ') for i, (h, y) in enumerate(zip(H, Y)): try: H[i] = Poly(y - h, x, domain='QQ').all_coeffs() # XXX: composite=False except CoercionFailed: # pragma: no cover break # G is not a triangular set else: break else: # pragma: no cover raise RuntimeError("run out of coefficient configurations") _, g = g.clear_denoms() if not polys: return g.as_expr(), coeffs, H else: return g, coeffs, H def is_isomorphism_possible(a, b): """Returns `True` if there is a chance for isomorphism. """ n = a.minpoly.degree() m = b.minpoly.degree() if m % n != 0: return False if n == m: return True da = a.minpoly.discriminant() db = b.minpoly.discriminant() i, k, half = 1, m//n, db//2 while True: p = sieve[i] P = p**k if P > half: break if ((da % p) % 2) and not (db % P): return False i += 1 return True def field_isomorphism_pslq(a, b): """Construct field isomorphism using PSLQ algorithm. """ if not a.root.is_real or not b.root.is_real: raise NotImplementedError("PSLQ doesn't support complex coefficients") f = a.minpoly g = b.minpoly.replace(f.gen) n, m, prev = 100, b.minpoly.degree(), None for i in range(1, 5): A = a.root.evalf(n) B = b.root.evalf(n) basis = [1, B] + [ B**i for i in range(2, m) ] + [A] dps, mp.dps = mp.dps, n coeffs = pslq(basis, maxcoeff=int(1e10), maxsteps=1000) mp.dps = dps if coeffs is None: break if coeffs != prev: prev = coeffs else: break coeffs = [S(c)/coeffs[-1] for c in coeffs[:-1]] while not coeffs[-1]: coeffs.pop() coeffs = list(reversed(coeffs)) h = Poly(coeffs, f.gen, domain='QQ') if f.compose(h).rem(g).is_zero: d, approx = len(coeffs) - 1, 0 for i, coeff in enumerate(coeffs): approx += coeff*B**(d - i) if A*approx < 0: return [ -c for c in coeffs ] else: return coeffs elif f.compose(-h).rem(g).is_zero: return [ -c for c in coeffs ] else: n *= 2 return None def field_isomorphism_factor(a, b): """Construct field isomorphism via factorization. """ _, factors = factor_list(a.minpoly, extension=b) for f, _ in factors: if f.degree() == 1: coeffs = f.rep.TC().to_sympy_list() d, terms = len(coeffs) - 1, [] for i, coeff in enumerate(coeffs): terms.append(coeff*b.root**(d - i)) root = Add(*terms) if (a.root - root).evalf(chop=True) == 0: return coeffs if (a.root + root).evalf(chop=True) == 0: return [-c for c in coeffs] return None @public def field_isomorphism(a, b, *, fast=True): """Construct an isomorphism between two number fields. """ a, b = sympify(a), sympify(b) if not a.is_AlgebraicNumber: a = AlgebraicNumber(a) if not b.is_AlgebraicNumber: b = AlgebraicNumber(b) if a == b: return a.coeffs() n = a.minpoly.degree() m = b.minpoly.degree() if n == 1: return [a.root] if m % n != 0: return None if fast: try: result = field_isomorphism_pslq(a, b) if result is not None: return result except NotImplementedError: pass return field_isomorphism_factor(a, b) @public def to_number_field(extension, theta=None, *, gen=None): """Express `extension` in the field generated by `theta`. """ if hasattr(extension, '__iter__'): extension = list(extension) else: extension = [extension] if len(extension) == 1 and type(extension[0]) is tuple: return AlgebraicNumber(extension[0]) minpoly, coeffs = primitive_element(extension, gen, polys=True) root = sum([ coeff*ext for coeff, ext in zip(coeffs, extension) ]) if theta is None: return AlgebraicNumber((minpoly, root)) else: theta = sympify(theta) if not theta.is_AlgebraicNumber: theta = AlgebraicNumber(theta, gen=gen) coeffs = field_isomorphism(root, theta) if coeffs is not None: return AlgebraicNumber(theta, coeffs) else: raise IsomorphismFailed( "%s is not in a subfield of %s" % (root, theta.root)) class IntervalPrinter(MpmathPrinter, LambdaPrinter): """Use ``lambda`` printer but print numbers as ``mpi`` intervals. """ def _print_Integer(self, expr): return "mpi('%s')" % super(PythonCodePrinter, self)._print_Integer(expr) def _print_Rational(self, expr): return "mpi('%s')" % super(PythonCodePrinter, self)._print_Rational(expr) def _print_Half(self, expr): return "mpi('%s')" % super(PythonCodePrinter, self)._print_Rational(expr) def _print_Pow(self, expr): return super(MpmathPrinter, self)._print_Pow(expr, rational=True) @public def isolate(alg, eps=None, fast=False): """Give a rational isolating interval for an algebraic number. """ alg = sympify(alg) if alg.is_Rational: return (alg, alg) elif not alg.is_real: raise NotImplementedError( "complex algebraic numbers are not supported") func = lambdify((), alg, modules="mpmath", printer=IntervalPrinter()) poly = minpoly(alg, polys=True) intervals = poly.intervals(sqf=True) dps, done = mp.dps, False try: while not done: alg = func() for a, b in intervals: if a <= alg.a and alg.b <= b: done = True break else: mp.dps *= 2 finally: mp.dps = dps if eps is not None: a, b = poly.refine_root(a, b, eps=eps, fast=fast) return (a, b)
b3a188627180e94f638463fb6174104bc1b76f21b8fa43dbaf9e4147aa7e8b3e
from sympy.core import S from sympy.polys import Poly def dispersionset(p, q=None, *gens, **args): r"""Compute the *dispersion set* of two polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as: .. math:: \operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\} For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersion References ========== .. [1] [ManWright94]_ .. [2] [Koepf98]_ .. [3] [Abramov71]_ .. [4] [Man93]_ """ # Check for valid input same = False if q is not None else True if same: q = p p = Poly(p, *gens, **args) q = Poly(q, *gens, **args) if not p.is_univariate or not q.is_univariate: raise ValueError("Polynomials need to be univariate") # The generator if not p.gen == q.gen: raise ValueError("Polynomials must have the same generator") gen = p.gen # We define the dispersion of constant polynomials to be zero if p.degree() < 1 or q.degree() < 1: return {0} # Factor p and q over the rationals fp = p.factor_list() fq = q.factor_list() if not same else fp # Iterate over all pairs of factors J = set() for s, unused in fp[1]: for t, unused in fq[1]: m = s.degree() n = t.degree() if n != m: continue an = s.LC() bn = t.LC() if not (an - bn).is_zero: continue # Note that the roles of `s` and `t` below are switched # w.r.t. the original paper. This is for consistency # with the description in the book of W. Koepf. anm1 = s.coeff_monomial(gen**(m-1)) bnm1 = t.coeff_monomial(gen**(n-1)) alpha = (anm1 - bnm1) / S(n*bn) if not alpha.is_integer: continue if alpha < 0 or alpha in J: continue if n > 1 and not (s - t.shift(alpha)).is_zero: continue J.add(alpha) return J def dispersion(p, q=None, *gens, **args): r"""Compute the *dispersion* of polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as: .. math:: \operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \} and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`. Note that we make the definition `\max\{\} := -\infty`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo The maximum of an empty set is defined to be `-\infty` as seen in this example. Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersionset References ========== .. [1] [ManWright94]_ .. [2] [Koepf98]_ .. [3] [Abramov71]_ .. [4] [Man93]_ """ J = dispersionset(p, q, *gens, **args) if not J: # Definition for maximum of empty set j = S.NegativeInfinity else: j = max(J) return j
81686d2425ad63ec5fae03db7a64dff15283eac9a3821d8e73d234f71fa6699a
"""Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """ from sympy.ntheory import nextprime from sympy.polys.densearith import ( dup_sub_mul, dup_neg, dmp_neg, dmp_add, dmp_sub, dup_mul, dmp_mul, dmp_pow, dup_div, dmp_div, dup_rem, dup_quo, dmp_quo, dup_prem, dmp_prem, dup_mul_ground, dmp_mul_ground, dmp_mul_term, dup_quo_ground, dmp_quo_ground, dup_max_norm, dmp_max_norm) from sympy.polys.densebasic import ( dup_strip, dmp_raise, dmp_zero, dmp_one, dmp_ground, dmp_one_p, dmp_zero_p, dmp_zeros, dup_degree, dmp_degree, dmp_degree_in, dup_LC, dmp_LC, dmp_ground_LC, dmp_multi_deflate, dmp_inflate, dup_convert, dmp_convert, dmp_apply_pairs) from sympy.polys.densetools import ( dup_clear_denoms, dmp_clear_denoms, dup_diff, dmp_diff, dup_eval, dmp_eval, dmp_eval_in, dup_trunc, dmp_ground_trunc, dup_monic, dmp_ground_monic, dup_primitive, dmp_ground_primitive, dup_extract, dmp_ground_extract) from sympy.polys.galoistools import ( gf_int, gf_crt) from sympy.polys.polyconfig import query from sympy.polys.polyerrors import ( MultivariatePolynomialError, HeuristicGCDFailed, HomomorphismFailed, NotInvertible, DomainError) def dup_half_gcdex(f, g, K): """ Half extended Euclidean algorithm in `F[x]`. Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 >>> g = x**3 + x**2 - 4*x - 4 >>> R.dup_half_gcdex(f, g) (-1/5*x + 3/5, x + 1) """ if not K.is_Field: raise DomainError("can't compute half extended GCD over %s" % K) a, b = [K.one], [] while g: q, r = dup_div(f, g, K) f, g = g, r a, b = b, dup_sub_mul(a, q, b, K) a = dup_quo_ground(a, dup_LC(f, K), K) f = dup_monic(f, K) return a, f def dmp_half_gcdex(f, g, u, K): """ Half extended Euclidean algorithm in `F[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) """ if not u: return dup_half_gcdex(f, g, K) else: raise MultivariatePolynomialError(f, g) def dup_gcdex(f, g, K): """ Extended Euclidean algorithm in `F[x]`. Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 >>> g = x**3 + x**2 - 4*x - 4 >>> R.dup_gcdex(f, g) (-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1) """ s, h = dup_half_gcdex(f, g, K) F = dup_sub_mul(h, s, f, K) t = dup_quo(F, g, K) return s, t, h def dmp_gcdex(f, g, u, K): """ Extended Euclidean algorithm in `F[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) """ if not u: return dup_gcdex(f, g, K) else: raise MultivariatePolynomialError(f, g) def dup_invert(f, g, K): """ Compute multiplicative inverse of `f` modulo `g` in `F[x]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = x**2 - 1 >>> g = 2*x - 1 >>> h = x - 1 >>> R.dup_invert(f, g) -4/3 >>> R.dup_invert(f, h) Traceback (most recent call last): ... NotInvertible: zero divisor """ s, h = dup_half_gcdex(f, g, K) if h == [K.one]: return dup_rem(s, g, K) else: raise NotInvertible("zero divisor") def dmp_invert(f, g, u, K): """ Compute multiplicative inverse of `f` modulo `g` in `F[X]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) """ if not u: return dup_invert(f, g, K) else: raise MultivariatePolynomialError(f, g) def dup_euclidean_prs(f, g, K): """ Euclidean polynomial remainder sequence (PRS) in `K[x]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 >>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 >>> prs = R.dup_euclidean_prs(f, g) >>> prs[0] x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 >>> prs[1] 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 >>> prs[2] -5/9*x**4 + 1/9*x**2 - 1/3 >>> prs[3] -117/25*x**2 - 9*x + 441/25 >>> prs[4] 233150/19773*x - 102500/6591 >>> prs[5] -1288744821/543589225 """ prs = [f, g] h = dup_rem(f, g, K) while h: prs.append(h) f, g = g, h h = dup_rem(f, g, K) return prs def dmp_euclidean_prs(f, g, u, K): """ Euclidean polynomial remainder sequence (PRS) in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) """ if not u: return dup_euclidean_prs(f, g, K) else: raise MultivariatePolynomialError(f, g) def dup_primitive_prs(f, g, K): """ Primitive polynomial remainder sequence (PRS) in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 >>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 >>> prs = R.dup_primitive_prs(f, g) >>> prs[0] x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 >>> prs[1] 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 >>> prs[2] -5*x**4 + x**2 - 3 >>> prs[3] 13*x**2 + 25*x - 49 >>> prs[4] 4663*x - 6150 >>> prs[5] 1 """ prs = [f, g] _, h = dup_primitive(dup_prem(f, g, K), K) while h: prs.append(h) f, g = g, h _, h = dup_primitive(dup_prem(f, g, K), K) return prs def dmp_primitive_prs(f, g, u, K): """ Primitive polynomial remainder sequence (PRS) in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) """ if not u: return dup_primitive_prs(f, g, K) else: raise MultivariatePolynomialError(f, g) def dup_inner_subresultants(f, g, K): """ Subresultant PRS algorithm in `K[x]`. Computes the subresultant polynomial remainder sequence (PRS) and the non-zero scalar subresultants of `f` and `g`. By [1] Thm. 3, these are the constants '-c' (- to optimize computation of sign). The first subdeterminant is set to 1 by convention to match the polynomial and the scalar subdeterminants. If 'deg(f) < deg(g)', the subresultants of '(g,f)' are computed. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1) ([x**2 + 1, x**2 - 1, -2], [1, 1, 4]) References ========== .. [1] W.S. Brown, The Subresultant PRS Algorithm. ACM Transaction of Mathematical Software 4 (1978) 237-249 """ n = dup_degree(f) m = dup_degree(g) if n < m: f, g = g, f n, m = m, n if not f: return [], [] if not g: return [f], [K.one] R = [f, g] d = n - m b = (-K.one)**(d + 1) h = dup_prem(f, g, K) h = dup_mul_ground(h, b, K) lc = dup_LC(g, K) c = lc**d # Conventional first scalar subdeterminant is 1 S = [K.one, c] c = -c while h: k = dup_degree(h) R.append(h) f, g, m, d = g, h, k, m - k b = -lc * c**d h = dup_prem(f, g, K) h = dup_quo_ground(h, b, K) lc = dup_LC(g, K) if d > 1: # abnormal case q = c**(d - 1) c = K.quo((-lc)**d, q) else: c = -lc S.append(-c) return R, S def dup_subresultants(f, g, K): """ Computes subresultant PRS of two polynomials in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_subresultants(x**2 + 1, x**2 - 1) [x**2 + 1, x**2 - 1, -2] """ return dup_inner_subresultants(f, g, K)[0] def dup_prs_resultant(f, g, K): """ Resultant algorithm in `K[x]` using subresultant PRS. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_prs_resultant(x**2 + 1, x**2 - 1) (4, [x**2 + 1, x**2 - 1, -2]) """ if not f or not g: return (K.zero, []) R, S = dup_inner_subresultants(f, g, K) if dup_degree(R[-1]) > 0: return (K.zero, R) return S[-1], R def dup_resultant(f, g, K, includePRS=False): """ Computes resultant of two polynomials in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_resultant(x**2 + 1, x**2 - 1) 4 """ if includePRS: return dup_prs_resultant(f, g, K) return dup_prs_resultant(f, g, K)[0] def dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y - y**3 - 4 >>> g = x**2 + x*y**3 - 9 >>> a = 3*x*y**4 + y**3 - 27*y + 4 >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 >>> prs = [f, g, a, b] >>> sres = [[1], [1], [3, 0, 0, 0, 0], [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]] >>> R.dmp_inner_subresultants(f, g) == (prs, sres) True """ if not u: return dup_inner_subresultants(f, g, K) n = dmp_degree(f, u) m = dmp_degree(g, u) if n < m: f, g = g, f n, m = m, n if dmp_zero_p(f, u): return [], [] v = u - 1 if dmp_zero_p(g, u): return [f], [dmp_ground(K.one, v)] R = [f, g] d = n - m b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K) h = dmp_prem(f, g, u, K) h = dmp_mul_term(h, b, 0, u, K) lc = dmp_LC(g, K) c = dmp_pow(lc, d, v, K) S = [dmp_ground(K.one, v), c] c = dmp_neg(c, v, K) while not dmp_zero_p(h, u): k = dmp_degree(h, u) R.append(h) f, g, m, d = g, h, k, m - k b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, d, v, K), v, K) h = dmp_prem(f, g, u, K) h = [ dmp_quo(ch, b, v, K) for ch in h ] lc = dmp_LC(g, K) if d > 1: p = dmp_pow(dmp_neg(lc, v, K), d, v, K) q = dmp_pow(c, d - 1, v, K) c = dmp_quo(p, q, v, K) else: c = dmp_neg(lc, v, K) S.append(dmp_neg(c, v, K)) return R, S def dmp_subresultants(f, g, u, K): """ Computes subresultant PRS of two polynomials in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y - y**3 - 4 >>> g = x**2 + x*y**3 - 9 >>> a = 3*x*y**4 + y**3 - 27*y + 4 >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 >>> R.dmp_subresultants(f, g) == [f, g, a, b] True """ return dmp_inner_subresultants(f, g, u, K)[0] def dmp_prs_resultant(f, g, u, K): """ Resultant algorithm in `K[X]` using subresultant PRS. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y - y**3 - 4 >>> g = x**2 + x*y**3 - 9 >>> a = 3*x*y**4 + y**3 - 27*y + 4 >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 >>> res, prs = R.dmp_prs_resultant(f, g) >>> res == b # resultant has n-1 variables False >>> res == b.drop(x) True >>> prs == [f, g, a, b] True """ if not u: return dup_prs_resultant(f, g, K) if dmp_zero_p(f, u) or dmp_zero_p(g, u): return (dmp_zero(u - 1), []) R, S = dmp_inner_subresultants(f, g, u, K) if dmp_degree(R[-1], u) > 0: return (dmp_zero(u - 1), R) return S[-1], R def dmp_zz_modular_resultant(f, g, p, u, K): """ Compute resultant of `f` and `g` modulo a prime `p`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x + y + 2 >>> g = 2*x*y + x + 3 >>> R.dmp_zz_modular_resultant(f, g, 5) -2*y**2 + 1 """ if not u: return gf_int(dup_prs_resultant(f, g, K)[0] % p, p) v = u - 1 n = dmp_degree(f, u) m = dmp_degree(g, u) N = dmp_degree_in(f, 1, u) M = dmp_degree_in(g, 1, u) B = n*M + m*N D, a = [K.one], -K.one r = dmp_zero(v) while dup_degree(D) <= B: while True: a += K.one if a == p: raise HomomorphismFailed('no luck') F = dmp_eval_in(f, gf_int(a, p), 1, u, K) if dmp_degree(F, v) == n: G = dmp_eval_in(g, gf_int(a, p), 1, u, K) if dmp_degree(G, v) == m: break R = dmp_zz_modular_resultant(F, G, p, v, K) e = dmp_eval(r, a, v, K) if not v: R = dup_strip([R]) e = dup_strip([e]) else: R = [R] e = [e] d = K.invert(dup_eval(D, a, K), p) d = dup_mul_ground(D, d, K) d = dmp_raise(d, v, 0, K) c = dmp_mul(d, dmp_sub(R, e, v, K), v, K) r = dmp_add(r, c, v, K) r = dmp_ground_trunc(r, p, v, K) D = dup_mul(D, [K.one, -a], K) D = dup_trunc(D, p, K) return r def _collins_crt(r, R, P, p, K): """Wrapper of CRT for Collins's resultant algorithm. """ return gf_int(gf_crt([r, R], [P, p], K), P*p) def dmp_zz_collins_resultant(f, g, u, K): """ Collins's modular resultant algorithm in `Z[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x + y + 2 >>> g = 2*x*y + x + 3 >>> R.dmp_zz_collins_resultant(f, g) -2*y**2 - 5*y + 1 """ n = dmp_degree(f, u) m = dmp_degree(g, u) if n < 0 or m < 0: return dmp_zero(u - 1) A = dmp_max_norm(f, u, K) B = dmp_max_norm(g, u, K) a = dmp_ground_LC(f, u, K) b = dmp_ground_LC(g, u, K) v = u - 1 B = K(2)*K.factorial(K(n + m))*A**m*B**n r, p, P = dmp_zero(v), K.one, K.one while P <= B: p = K(nextprime(p)) while not (a % p) or not (b % p): p = K(nextprime(p)) F = dmp_ground_trunc(f, p, u, K) G = dmp_ground_trunc(g, p, u, K) try: R = dmp_zz_modular_resultant(F, G, p, u, K) except HomomorphismFailed: continue if K.is_one(P): r = R else: r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K) P *= p return r def dmp_qq_collins_resultant(f, g, u, K0): """ Collins's modular resultant algorithm in `Q[X]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> f = QQ(1,2)*x + y + QQ(2,3) >>> g = 2*x*y + x + 3 >>> R.dmp_qq_collins_resultant(f, g) -2*y**2 - 7/3*y + 5/6 """ n = dmp_degree(f, u) m = dmp_degree(g, u) if n < 0 or m < 0: return dmp_zero(u - 1) K1 = K0.get_ring() cf, f = dmp_clear_denoms(f, u, K0, K1) cg, g = dmp_clear_denoms(g, u, K0, K1) f = dmp_convert(f, u, K0, K1) g = dmp_convert(g, u, K0, K1) r = dmp_zz_collins_resultant(f, g, u, K1) r = dmp_convert(r, u - 1, K1, K0) c = K0.convert(cf**m * cg**n, K1) return dmp_quo_ground(r, c, u - 1, K0) def dmp_resultant(f, g, u, K, includePRS=False): """ Computes resultant of two polynomials in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y - y**3 - 4 >>> g = x**2 + x*y**3 - 9 >>> R.dmp_resultant(f, g) -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 """ if not u: return dup_resultant(f, g, K, includePRS=includePRS) if includePRS: return dmp_prs_resultant(f, g, u, K) if K.is_Field: if K.is_QQ and query('USE_COLLINS_RESULTANT'): return dmp_qq_collins_resultant(f, g, u, K) else: if K.is_ZZ and query('USE_COLLINS_RESULTANT'): return dmp_zz_collins_resultant(f, g, u, K) return dmp_prs_resultant(f, g, u, K)[0] def dup_discriminant(f, K): """ Computes discriminant of a polynomial in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_discriminant(x**2 + 2*x + 3) -8 """ d = dup_degree(f) if d <= 0: return K.zero else: s = (-1)**((d*(d - 1)) // 2) c = dup_LC(f, K) r = dup_resultant(f, dup_diff(f, 1, K), K) return K.quo(r, c*K(s)) def dmp_discriminant(f, u, K): """ Computes discriminant of a polynomial in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y,z,t = ring("x,y,z,t", ZZ) >>> R.dmp_discriminant(x**2*y + x*z + t) -4*y*t + z**2 """ if not u: return dup_discriminant(f, K) d, v = dmp_degree(f, u), u - 1 if d <= 0: return dmp_zero(v) else: s = (-1)**((d*(d - 1)) // 2) c = dmp_LC(f, K) r = dmp_resultant(f, dmp_diff(f, 1, u, K), u, K) c = dmp_mul_ground(c, K(s), v, K) return dmp_quo(r, c, v, K) def _dup_rr_trivial_gcd(f, g, K): """Handle trivial cases in GCD algorithm over a ring. """ if not (f or g): return [], [], [] elif not f: if K.is_nonnegative(dup_LC(g, K)): return g, [], [K.one] else: return dup_neg(g, K), [], [-K.one] elif not g: if K.is_nonnegative(dup_LC(f, K)): return f, [K.one], [] else: return dup_neg(f, K), [-K.one], [] return None def _dup_ff_trivial_gcd(f, g, K): """Handle trivial cases in GCD algorithm over a field. """ if not (f or g): return [], [], [] elif not f: return dup_monic(g, K), [], [dup_LC(g, K)] elif not g: return dup_monic(f, K), [dup_LC(f, K)], [] else: return None def _dmp_rr_trivial_gcd(f, g, u, K): """Handle trivial cases in GCD algorithm over a ring. """ zero_f = dmp_zero_p(f, u) zero_g = dmp_zero_p(g, u) if_contain_one = dmp_one_p(f, u, K) or dmp_one_p(g, u, K) if zero_f and zero_g: return tuple(dmp_zeros(3, u, K)) elif zero_f: if K.is_nonnegative(dmp_ground_LC(g, u, K)): return g, dmp_zero(u), dmp_one(u, K) else: return dmp_neg(g, u, K), dmp_zero(u), dmp_ground(-K.one, u) elif zero_g: if K.is_nonnegative(dmp_ground_LC(f, u, K)): return f, dmp_one(u, K), dmp_zero(u) else: return dmp_neg(f, u, K), dmp_ground(-K.one, u), dmp_zero(u) elif if_contain_one: return dmp_one(u, K), f, g elif query('USE_SIMPLIFY_GCD'): return _dmp_simplify_gcd(f, g, u, K) else: return None def _dmp_ff_trivial_gcd(f, g, u, K): """Handle trivial cases in GCD algorithm over a field. """ zero_f = dmp_zero_p(f, u) zero_g = dmp_zero_p(g, u) if zero_f and zero_g: return tuple(dmp_zeros(3, u, K)) elif zero_f: return (dmp_ground_monic(g, u, K), dmp_zero(u), dmp_ground(dmp_ground_LC(g, u, K), u)) elif zero_g: return (dmp_ground_monic(f, u, K), dmp_ground(dmp_ground_LC(f, u, K), u), dmp_zero(u)) elif query('USE_SIMPLIFY_GCD'): return _dmp_simplify_gcd(f, g, u, K) else: return None def _dmp_simplify_gcd(f, g, u, K): """Try to eliminate `x_0` from GCD computation in `K[X]`. """ df = dmp_degree(f, u) dg = dmp_degree(g, u) if df > 0 and dg > 0: return None if not (df or dg): F = dmp_LC(f, K) G = dmp_LC(g, K) else: if not df: F = dmp_LC(f, K) G = dmp_content(g, u, K) else: F = dmp_content(f, u, K) G = dmp_LC(g, K) v = u - 1 h = dmp_gcd(F, G, v, K) cff = [ dmp_quo(cf, h, v, K) for cf in f ] cfg = [ dmp_quo(cg, h, v, K) for cg in g ] return [h], cff, cfg def dup_rr_prs_gcd(f, g, K): """ Computes polynomial GCD using subresultants over a ring. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rr_prs_gcd(x**2 - 1, x**2 - 3*x + 2) (x - 1, x + 1, x - 2) """ result = _dup_rr_trivial_gcd(f, g, K) if result is not None: return result fc, F = dup_primitive(f, K) gc, G = dup_primitive(g, K) c = K.gcd(fc, gc) h = dup_subresultants(F, G, K)[-1] _, h = dup_primitive(h, K) if K.is_negative(dup_LC(h, K)): c = -c h = dup_mul_ground(h, c, K) cff = dup_quo(f, h, K) cfg = dup_quo(g, h, K) return h, cff, cfg def dup_ff_prs_gcd(f, g, K): """ Computes polynomial GCD using subresultants over a field. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_ff_prs_gcd(x**2 - 1, x**2 - 3*x + 2) (x - 1, x + 1, x - 2) """ result = _dup_ff_trivial_gcd(f, g, K) if result is not None: return result h = dup_subresultants(f, g, K)[-1] h = dup_monic(h, K) cff = dup_quo(f, h, K) cfg = dup_quo(g, h, K) return h, cff, cfg def dmp_rr_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a ring. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_rr_prs_gcd(f, g) (x + y, x + y, x) """ if not u: return dup_rr_prs_gcd(f, g, K) result = _dmp_rr_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_rr_prs_gcd(fc, gc, u - 1, K) if K.is_negative(dmp_ground_LC(h, u, K)): h = dmp_neg(h, u, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg def dmp_ff_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a field. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y, = ring("x,y", QQ) >>> f = QQ(1,2)*x**2 + x*y + QQ(1,2)*y**2 >>> g = x**2 + x*y >>> R.dmp_ff_prs_gcd(f, g) (x + y, 1/2*x + 1/2*y, x) """ if not u: return dup_ff_prs_gcd(f, g, K) result = _dmp_ff_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_ff_prs_gcd(fc, gc, u - 1, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) h = dmp_ground_monic(h, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg HEU_GCD_MAX = 6 def _dup_zz_gcd_interpolate(h, x, K): """Interpolate polynomial GCD from integer GCD. """ f = [] while h: g = h % x if g > x // 2: g -= x f.insert(0, g) h = (h - g) // x return f def dup_zz_heu_gcd(f, g, K): """ Heuristic polynomial GCD in `Z[x]`. Given univariate polynomials `f` and `g` in `Z[x]`, returns their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg`` such that:: h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h) The algorithm is purely heuristic which means it may fail to compute the GCD. This will be signaled by raising an exception. In this case you will need to switch to another GCD method. The algorithm computes the polynomial GCD by evaluating polynomials f and g at certain points and computing (fast) integer GCD of those evaluations. The polynomial GCD is recovered from the integer image by interpolation. The final step is to verify if the result is the correct GCD. This gives cofactors as a side effect. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_zz_heu_gcd(x**2 - 1, x**2 - 3*x + 2) (x - 1, x + 1, x - 2) References ========== .. [1] [Liao95]_ """ result = _dup_rr_trivial_gcd(f, g, K) if result is not None: return result df = dup_degree(f) dg = dup_degree(g) gcd, f, g = dup_extract(f, g, K) if df == 0 or dg == 0: return [gcd], f, g f_norm = dup_max_norm(f, K) g_norm = dup_max_norm(g, K) B = K(2*min(f_norm, g_norm) + 29) x = max(min(B, 99*K.sqrt(B)), 2*min(f_norm // abs(dup_LC(f, K)), g_norm // abs(dup_LC(g, K))) + 2) for i in range(0, HEU_GCD_MAX): ff = dup_eval(f, x, K) gg = dup_eval(g, x, K) if ff and gg: h = K.gcd(ff, gg) cff = ff // h cfg = gg // h h = _dup_zz_gcd_interpolate(h, x, K) h = dup_primitive(h, K)[1] cff_, r = dup_div(f, h, K) if not r: cfg_, r = dup_div(g, h, K) if not r: h = dup_mul_ground(h, gcd, K) return h, cff_, cfg_ cff = _dup_zz_gcd_interpolate(cff, x, K) h, r = dup_div(f, cff, K) if not r: cfg_, r = dup_div(g, h, K) if not r: h = dup_mul_ground(h, gcd, K) return h, cff, cfg_ cfg = _dup_zz_gcd_interpolate(cfg, x, K) h, r = dup_div(g, cfg, K) if not r: cff_, r = dup_div(f, h, K) if not r: h = dup_mul_ground(h, gcd, K) return h, cff_, cfg x = 73794*x * K.sqrt(K.sqrt(x)) // 27011 raise HeuristicGCDFailed('no luck') def _dmp_zz_gcd_interpolate(h, x, v, K): """Interpolate polynomial GCD from integer GCD. """ f = [] while not dmp_zero_p(h, v): g = dmp_ground_trunc(h, x, v, K) f.insert(0, g) h = dmp_sub(h, g, v, K) h = dmp_quo_ground(h, x, v, K) if K.is_negative(dmp_ground_LC(f, v + 1, K)): return dmp_neg(f, v + 1, K) else: return f def dmp_zz_heu_gcd(f, g, u, K): """ Heuristic polynomial GCD in `Z[X]`. Given univariate polynomials `f` and `g` in `Z[X]`, returns their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg`` such that:: h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h) The algorithm is purely heuristic which means it may fail to compute the GCD. This will be signaled by raising an exception. In this case you will need to switch to another GCD method. The algorithm computes the polynomial GCD by evaluating polynomials f and g at certain points and computing (fast) integer GCD of those evaluations. The polynomial GCD is recovered from the integer image by interpolation. The evaluation process reduces f and g variable by variable into a large integer. The final step is to verify if the interpolated polynomial is the correct GCD. This gives cofactors of the input polynomials as a side effect. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_zz_heu_gcd(f, g) (x + y, x + y, x) References ========== .. [1] [Liao95]_ """ if not u: return dup_zz_heu_gcd(f, g, K) result = _dmp_rr_trivial_gcd(f, g, u, K) if result is not None: return result gcd, f, g = dmp_ground_extract(f, g, u, K) f_norm = dmp_max_norm(f, u, K) g_norm = dmp_max_norm(g, u, K) B = K(2*min(f_norm, g_norm) + 29) x = max(min(B, 99*K.sqrt(B)), 2*min(f_norm // abs(dmp_ground_LC(f, u, K)), g_norm // abs(dmp_ground_LC(g, u, K))) + 2) for i in range(0, HEU_GCD_MAX): ff = dmp_eval(f, x, u, K) gg = dmp_eval(g, x, u, K) v = u - 1 if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)): h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K) h = _dmp_zz_gcd_interpolate(h, x, v, K) h = dmp_ground_primitive(h, u, K)[1] cff_, r = dmp_div(f, h, u, K) if dmp_zero_p(r, u): cfg_, r = dmp_div(g, h, u, K) if dmp_zero_p(r, u): h = dmp_mul_ground(h, gcd, u, K) return h, cff_, cfg_ cff = _dmp_zz_gcd_interpolate(cff, x, v, K) h, r = dmp_div(f, cff, u, K) if dmp_zero_p(r, u): cfg_, r = dmp_div(g, h, u, K) if dmp_zero_p(r, u): h = dmp_mul_ground(h, gcd, u, K) return h, cff, cfg_ cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K) h, r = dmp_div(g, cfg, u, K) if dmp_zero_p(r, u): cff_, r = dmp_div(f, h, u, K) if dmp_zero_p(r, u): h = dmp_mul_ground(h, gcd, u, K) return h, cff_, cfg x = 73794*x * K.sqrt(K.sqrt(x)) // 27011 raise HeuristicGCDFailed('no luck') def dup_qq_heu_gcd(f, g, K0): """ Heuristic polynomial GCD in `Q[x]`. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2) >>> g = QQ(1,2)*x**2 + x >>> R.dup_qq_heu_gcd(f, g) (x + 2, 1/2*x + 3/4, 1/2*x) """ result = _dup_ff_trivial_gcd(f, g, K0) if result is not None: return result K1 = K0.get_ring() cf, f = dup_clear_denoms(f, K0, K1) cg, g = dup_clear_denoms(g, K0, K1) f = dup_convert(f, K0, K1) g = dup_convert(g, K0, K1) h, cff, cfg = dup_zz_heu_gcd(f, g, K1) h = dup_convert(h, K1, K0) c = dup_LC(h, K0) h = dup_monic(h, K0) cff = dup_convert(cff, K1, K0) cfg = dup_convert(cfg, K1, K0) cff = dup_mul_ground(cff, K0.quo(c, cf), K0) cfg = dup_mul_ground(cfg, K0.quo(c, cg), K0) return h, cff, cfg def dmp_qq_heu_gcd(f, g, u, K0): """ Heuristic polynomial GCD in `Q[X]`. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y, = ring("x,y", QQ) >>> f = QQ(1,4)*x**2 + x*y + y**2 >>> g = QQ(1,2)*x**2 + x*y >>> R.dmp_qq_heu_gcd(f, g) (x + 2*y, 1/4*x + 1/2*y, 1/2*x) """ result = _dmp_ff_trivial_gcd(f, g, u, K0) if result is not None: return result K1 = K0.get_ring() cf, f = dmp_clear_denoms(f, u, K0, K1) cg, g = dmp_clear_denoms(g, u, K0, K1) f = dmp_convert(f, u, K0, K1) g = dmp_convert(g, u, K0, K1) h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1) h = dmp_convert(h, u, K1, K0) c = dmp_ground_LC(h, u, K0) h = dmp_ground_monic(h, u, K0) cff = dmp_convert(cff, u, K1, K0) cfg = dmp_convert(cfg, u, K1, K0) cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0) cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0) return h, cff, cfg def dup_inner_gcd(f, g, K): """ Computes polynomial GCD and cofactors of `f` and `g` in `K[x]`. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_inner_gcd(x**2 - 1, x**2 - 3*x + 2) (x - 1, x + 1, x - 2) """ if not K.is_Exact: try: exact = K.get_exact() except DomainError: return [K.one], f, g f = dup_convert(f, K, exact) g = dup_convert(g, K, exact) h, cff, cfg = dup_inner_gcd(f, g, exact) h = dup_convert(h, exact, K) cff = dup_convert(cff, exact, K) cfg = dup_convert(cfg, exact, K) return h, cff, cfg elif K.is_Field: if K.is_QQ and query('USE_HEU_GCD'): try: return dup_qq_heu_gcd(f, g, K) except HeuristicGCDFailed: pass return dup_ff_prs_gcd(f, g, K) else: if K.is_ZZ and query('USE_HEU_GCD'): try: return dup_zz_heu_gcd(f, g, K) except HeuristicGCDFailed: pass return dup_rr_prs_gcd(f, g, K) def _dmp_inner_gcd(f, g, u, K): """Helper function for `dmp_inner_gcd()`. """ if not K.is_Exact: try: exact = K.get_exact() except DomainError: return dmp_one(u, K), f, g f = dmp_convert(f, u, K, exact) g = dmp_convert(g, u, K, exact) h, cff, cfg = _dmp_inner_gcd(f, g, u, exact) h = dmp_convert(h, u, exact, K) cff = dmp_convert(cff, u, exact, K) cfg = dmp_convert(cfg, u, exact, K) return h, cff, cfg elif K.is_Field: if K.is_QQ and query('USE_HEU_GCD'): try: return dmp_qq_heu_gcd(f, g, u, K) except HeuristicGCDFailed: pass return dmp_ff_prs_gcd(f, g, u, K) else: if K.is_ZZ and query('USE_HEU_GCD'): try: return dmp_zz_heu_gcd(f, g, u, K) except HeuristicGCDFailed: pass return dmp_rr_prs_gcd(f, g, u, K) def dmp_inner_gcd(f, g, u, K): """ Computes polynomial GCD and cofactors of `f` and `g` in `K[X]`. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_inner_gcd(f, g) (x + y, x + y, x) """ if not u: return dup_inner_gcd(f, g, K) J, (f, g) = dmp_multi_deflate((f, g), u, K) h, cff, cfg = _dmp_inner_gcd(f, g, u, K) return (dmp_inflate(h, J, u, K), dmp_inflate(cff, J, u, K), dmp_inflate(cfg, J, u, K)) def dup_gcd(f, g, K): """ Computes polynomial GCD of `f` and `g` in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_gcd(x**2 - 1, x**2 - 3*x + 2) x - 1 """ return dup_inner_gcd(f, g, K)[0] def dmp_gcd(f, g, u, K): """ Computes polynomial GCD of `f` and `g` in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_gcd(f, g) x + y """ return dmp_inner_gcd(f, g, u, K)[0] def dup_rr_lcm(f, g, K): """ Computes polynomial LCM over a ring in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rr_lcm(x**2 - 1, x**2 - 3*x + 2) x**3 - 2*x**2 - x + 2 """ fc, f = dup_primitive(f, K) gc, g = dup_primitive(g, K) c = K.lcm(fc, gc) h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K) return dup_mul_ground(h, c, K) def dup_ff_lcm(f, g, K): """ Computes polynomial LCM over a field in `K[x]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2) >>> g = QQ(1,2)*x**2 + x >>> R.dup_ff_lcm(f, g) x**3 + 7/2*x**2 + 3*x """ h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K) return dup_monic(h, K) def dup_lcm(f, g, K): """ Computes polynomial LCM of `f` and `g` in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_lcm(x**2 - 1, x**2 - 3*x + 2) x**3 - 2*x**2 - x + 2 """ if K.is_Field: return dup_ff_lcm(f, g, K) else: return dup_rr_lcm(f, g, K) def dmp_rr_lcm(f, g, u, K): """ Computes polynomial LCM over a ring in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_rr_lcm(f, g) x**3 + 2*x**2*y + x*y**2 """ fc, f = dmp_ground_primitive(f, u, K) gc, g = dmp_ground_primitive(g, u, K) c = K.lcm(fc, gc) h = dmp_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K) return dmp_mul_ground(h, c, u, K) def dmp_ff_lcm(f, g, u, K): """ Computes polynomial LCM over a field in `K[X]`. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y, = ring("x,y", QQ) >>> f = QQ(1,4)*x**2 + x*y + y**2 >>> g = QQ(1,2)*x**2 + x*y >>> R.dmp_ff_lcm(f, g) x**3 + 4*x**2*y + 4*x*y**2 """ h = dmp_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K) return dmp_ground_monic(h, u, K) def dmp_lcm(f, g, u, K): """ Computes polynomial LCM of `f` and `g` in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_lcm(f, g) x**3 + 2*x**2*y + x*y**2 """ if not u: return dup_lcm(f, g, K) if K.is_Field: return dmp_ff_lcm(f, g, u, K) else: return dmp_rr_lcm(f, g, u, K) def dmp_content(f, u, K): """ Returns GCD of multivariate coefficients. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> R.dmp_content(2*x*y + 6*x + 4*y + 12) 2*y + 6 """ cont, v = dmp_LC(f, K), u - 1 if dmp_zero_p(f, u): return cont for c in f[1:]: cont = dmp_gcd(cont, c, v, K) if dmp_one_p(cont, v, K): break if K.is_negative(dmp_ground_LC(cont, v, K)): return dmp_neg(cont, v, K) else: return cont def dmp_primitive(f, u, K): """ Returns multivariate content and a primitive polynomial. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> R.dmp_primitive(2*x*y + 6*x + 4*y + 12) (2*y + 6, x + 2) """ cont, v = dmp_content(f, u, K), u - 1 if dmp_zero_p(f, u) or dmp_one_p(cont, v, K): return cont, f else: return cont, [ dmp_quo(c, cont, v, K) for c in f ] def dup_cancel(f, g, K, include=True): """ Cancel common factors in a rational function `f/g`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_cancel(2*x**2 - 2, x**2 - 2*x + 1) (2*x + 2, x - 1) """ return dmp_cancel(f, g, 0, K, include=include) def dmp_cancel(f, g, u, K, include=True): """ Cancel common factors in a rational function `f/g`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_cancel(2*x**2 - 2, x**2 - 2*x + 1) (2*x + 2, x - 1) """ K0 = None if K.is_Field and K.has_assoc_Ring: K0, K = K, K.get_ring() cq, f = dmp_clear_denoms(f, u, K0, K, convert=True) cp, g = dmp_clear_denoms(g, u, K0, K, convert=True) else: cp, cq = K.one, K.one _, p, q = dmp_inner_gcd(f, g, u, K) if K0 is not None: _, cp, cq = K.cofactors(cp, cq) p = dmp_convert(p, u, K, K0) q = dmp_convert(q, u, K, K0) K = K0 p_neg = K.is_negative(dmp_ground_LC(p, u, K)) q_neg = K.is_negative(dmp_ground_LC(q, u, K)) if p_neg and q_neg: p, q = dmp_neg(p, u, K), dmp_neg(q, u, K) elif p_neg: cp, p = -cp, dmp_neg(p, u, K) elif q_neg: cp, q = -cp, dmp_neg(q, u, K) if not include: return cp, cq, p, q p = dmp_mul_ground(p, cp, u, K) q = dmp_mul_ground(q, cq, u, K) return p, q
90a79c179eef6e23f6dbe8413076f991914f311898d264808fef9d4b5539e447
"""Tools for constructing domains for expressions. """ from sympy.core import sympify from sympy.core.evalf import pure_complex from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, EX from sympy.polys.domains.complexfield import ComplexField from sympy.polys.domains.realfield import RealField from sympy.polys.polyoptions import build_options from sympy.polys.polyutils import parallel_dict_from_basic from sympy.utilities import public def _construct_simple(coeffs, opt): """Handle simple domains, e.g.: ZZ, QQ, RR and algebraic domains. """ rationals = floats = complexes = algebraics = False float_numbers = [] if opt.extension is True: is_algebraic = lambda coeff: coeff.is_number and coeff.is_algebraic else: is_algebraic = lambda coeff: False for coeff in coeffs: if coeff.is_Rational: if not coeff.is_Integer: rationals = True elif coeff.is_Float: if algebraics: # there are both reals and algebraics -> EX return False else: floats = True float_numbers.append(coeff) else: is_complex = pure_complex(coeff) if is_complex: complexes = True x, y = is_complex if x.is_Rational and y.is_Rational: if not (x.is_Integer and y.is_Integer): rationals = True continue else: floats = True if x.is_Float: float_numbers.append(x) if y.is_Float: float_numbers.append(y) if is_algebraic(coeff): if floats: # there are both algebraics and reals -> EX return False algebraics = True else: # this is a composite domain, e.g. ZZ[X], EX return None # Use the maximum precision of all coefficients for the RR or CC # precision max_prec = max(c._prec for c in float_numbers) if float_numbers else 53 if algebraics: domain, result = _construct_algebraic(coeffs, opt) else: if floats and complexes: domain = ComplexField(prec=max_prec) elif floats: domain = RealField(prec=max_prec) elif rationals or opt.field: domain = QQ_I if complexes else QQ else: domain = ZZ_I if complexes else ZZ result = [domain.from_sympy(coeff) for coeff in coeffs] return domain, result def _construct_algebraic(coeffs, opt): """We know that coefficients are algebraic so construct the extension. """ from sympy.polys.numberfields import primitive_element result, exts = [], set() for coeff in coeffs: if coeff.is_Rational: coeff = (None, 0, QQ.from_sympy(coeff)) else: a = coeff.as_coeff_add()[0] coeff -= a b = coeff.as_coeff_mul()[0] coeff /= b exts.add(coeff) a = QQ.from_sympy(a) b = QQ.from_sympy(b) coeff = (coeff, b, a) result.append(coeff) exts = list(exts) g, span, H = primitive_element(exts, ex=True, polys=True) root = sum([ s*ext for s, ext in zip(span, exts) ]) domain, g = QQ.algebraic_field((g, root)), g.rep.rep for i, (coeff, a, b) in enumerate(result): if coeff is not None: coeff = a*domain.dtype.from_list(H[exts.index(coeff)], g, QQ) + b else: coeff = domain.dtype.from_list([b], g, QQ) result[i] = coeff return domain, result def _construct_composite(coeffs, opt): """Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). """ numers, denoms = [], [] for coeff in coeffs: numer, denom = coeff.as_numer_denom() numers.append(numer) denoms.append(denom) polys, gens = parallel_dict_from_basic(numers + denoms) # XXX: sorting if not gens: return None if opt.composite is None: if any(gen.is_number and gen.is_algebraic for gen in gens): return None # generators are number-like so lets better use EX all_symbols = set() for gen in gens: symbols = gen.free_symbols if all_symbols & symbols: return None # there could be algebraic relations between generators else: all_symbols |= symbols n = len(gens) k = len(polys)//2 numers = polys[:k] denoms = polys[k:] if opt.field: fractions = True else: fractions, zeros = False, (0,)*n for denom in denoms: if len(denom) > 1 or zeros not in denom: fractions = True break coeffs = set() if not fractions: for numer, denom in zip(numers, denoms): denom = denom[zeros] for monom, coeff in numer.items(): coeff /= denom coeffs.add(coeff) numer[monom] = coeff else: for numer, denom in zip(numers, denoms): coeffs.update(list(numer.values())) coeffs.update(list(denom.values())) rationals = floats = complexes = False float_numbers = [] for coeff in coeffs: if coeff.is_Rational: if not coeff.is_Integer: rationals = True elif coeff.is_Float: floats = True float_numbers.append(coeff) else: is_complex = pure_complex(coeff) if is_complex is not None: complexes = True x, y = is_complex if x.is_Rational and y.is_Rational: if not (x.is_Integer and y.is_Integer): rationals = True else: floats = True if x.is_Float: float_numbers.append(x) if y.is_Float: float_numbers.append(y) max_prec = max(c._prec for c in float_numbers) if float_numbers else 53 if floats and complexes: ground = ComplexField(prec=max_prec) elif floats: ground = RealField(prec=max_prec) elif complexes: if rationals: ground = QQ_I else: ground = ZZ_I elif rationals: ground = QQ else: ground = ZZ result = [] if not fractions: domain = ground.poly_ring(*gens) for numer in numers: for monom, coeff in numer.items(): numer[monom] = ground.from_sympy(coeff) result.append(domain(numer)) else: domain = ground.frac_field(*gens) for numer, denom in zip(numers, denoms): for monom, coeff in numer.items(): numer[monom] = ground.from_sympy(coeff) for monom, coeff in denom.items(): denom[monom] = ground.from_sympy(coeff) result.append(domain((numer, denom))) return domain, result def _construct_expression(coeffs, opt): """The last resort case, i.e. use the expression domain. """ domain, result = EX, [] for coeff in coeffs: result.append(domain.from_sympy(coeff)) return domain, result @public def construct_domain(obj, **args): """Construct a minimal domain for the list of coefficients. """ opt = build_options(args) if hasattr(obj, '__iter__'): if isinstance(obj, dict): if not obj: monoms, coeffs = [], [] else: monoms, coeffs = list(zip(*list(obj.items()))) else: coeffs = obj else: coeffs = [obj] coeffs = list(map(sympify, coeffs)) result = _construct_simple(coeffs, opt) if result is not None: if result is not False: domain, coeffs = result else: domain, coeffs = _construct_expression(coeffs, opt) else: if opt.composite is False: result = None else: result = _construct_composite(coeffs, opt) if result is not None: domain, coeffs = result else: domain, coeffs = _construct_expression(coeffs, opt) if hasattr(obj, '__iter__'): if isinstance(obj, dict): return domain, dict(list(zip(monoms, coeffs))) else: return domain, coeffs else: return domain, coeffs[0]
b6c81cba0e60620a9270de6c01a71e7a9329f1ea9f9fe9dfd80cd4eeefcc7634
"""Polynomial factorization routines in characteristic zero. """ from sympy.polys.galoistools import ( gf_from_int_poly, gf_to_int_poly, gf_lshift, gf_add_mul, gf_mul, gf_div, gf_rem, gf_gcdex, gf_sqf_p, gf_factor_sqf, gf_factor) from sympy.polys.densebasic import ( dup_LC, dmp_LC, dmp_ground_LC, dup_TC, dup_convert, dmp_convert, dup_degree, dmp_degree, dmp_degree_in, dmp_degree_list, dmp_from_dict, dmp_zero_p, dmp_one, dmp_nest, dmp_raise, dup_strip, dmp_ground, dup_inflate, dmp_exclude, dmp_include, dmp_inject, dmp_eject, dup_terms_gcd, dmp_terms_gcd) from sympy.polys.densearith import ( dup_neg, dmp_neg, dup_add, dmp_add, dup_sub, dmp_sub, dup_mul, dmp_mul, dup_sqr, dmp_pow, dup_div, dmp_div, dup_quo, dmp_quo, dmp_expand, dmp_add_mul, dup_sub_mul, dmp_sub_mul, dup_lshift, dup_max_norm, dmp_max_norm, dup_l1_norm, dup_mul_ground, dmp_mul_ground, dup_quo_ground, dmp_quo_ground) from sympy.polys.densetools import ( dup_clear_denoms, dmp_clear_denoms, dup_trunc, dmp_ground_trunc, dup_content, dup_monic, dmp_ground_monic, dup_primitive, dmp_ground_primitive, dmp_eval_tail, dmp_eval_in, dmp_diff_eval_in, dmp_compose, dup_shift, dup_mirror) from sympy.polys.euclidtools import ( dmp_primitive, dup_inner_gcd, dmp_inner_gcd) from sympy.polys.sqfreetools import ( dup_sqf_p, dup_sqf_norm, dmp_sqf_norm, dup_sqf_part, dmp_sqf_part) from sympy.polys.polyutils import _sort_factors from sympy.polys.polyconfig import query from sympy.polys.polyerrors import ( ExtraneousFactors, DomainError, CoercionFailed, EvaluationFailed) from sympy.ntheory import nextprime, isprime, factorint from sympy.utilities import subsets from math import ceil as _ceil, log as _log def dup_trial_division(f, factors, K): """ Determine multiplicities of factors for a univariate polynomial using trial division. """ result = [] for factor in factors: k = 0 while True: q, r = dup_div(f, factor, K) if not r: f, k = q, k + 1 else: break result.append((factor, k)) return _sort_factors(result) def dmp_trial_division(f, factors, u, K): """ Determine multiplicities of factors for a multivariate polynomial using trial division. """ result = [] for factor in factors: k = 0 while True: q, r = dmp_div(f, factor, u, K) if dmp_zero_p(r, u): f, k = q, k + 1 else: break result.append((factor, k)) return _sort_factors(result) def dup_zz_mignotte_bound(f, K): """ The Knuth-Cohen variant of Mignotte bound for univariate polynomials in `K[x]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**3 + 14*x**2 + 56*x + 64 >>> R.dup_zz_mignotte_bound(f) 152 By checking `factor(f)` we can see that max coeff is 8 Also consider a case that `f` is irreducible for example `f = 2*x**2 + 3*x + 4` To avoid a bug for these cases, we return the bound plus the max coefficient of `f` >>> f = 2*x**2 + 3*x + 4 >>> R.dup_zz_mignotte_bound(f) 6 Lastly,To see the difference between the new and the old Mignotte bound consider the irreducible polynomial:: >>> f = 87*x**7 + 4*x**6 + 80*x**5 + 17*x**4 + 9*x**3 + 12*x**2 + 49*x + 26 >>> R.dup_zz_mignotte_bound(f) 744 The new Mignotte bound is 744 whereas the old one (SymPy 1.5.1) is 1937664. References ========== ..[1] [Abbott2013]_ """ from sympy import binomial d = dup_degree(f) delta = _ceil(d / 2) delta2 = _ceil(delta / 2) # euclidean-norm eucl_norm = K.sqrt( sum( [cf**2 for cf in f] ) ) # biggest values of binomial coefficients (p. 538 of reference) t1 = binomial(delta - 1, delta2) t2 = binomial(delta - 1, delta2 - 1) lc = K.abs(dup_LC(f, K)) # leading coefficient bound = t1 * eucl_norm + t2 * lc # (p. 538 of reference) bound += dup_max_norm(f, K) # add max coeff for irreducible polys bound = _ceil(bound / 2) * 2 # round up to even integer return bound def dmp_zz_mignotte_bound(f, u, K): """Mignotte bound for multivariate polynomials in `K[X]`. """ a = dmp_max_norm(f, u, K) b = abs(dmp_ground_LC(f, u, K)) n = sum(dmp_degree_list(f, u)) return K.sqrt(K(n + 1))*2**n*a*b def dup_zz_hensel_step(m, f, g, h, s, t, K): """ One step in Hensel lifting in `Z[x]`. Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s` and `t` such that:: f = g*h (mod m) s*g + t*h = 1 (mod m) lc(f) is not a zero divisor (mod m) lc(h) = 1 deg(f) = deg(g) + deg(h) deg(s) < deg(h) deg(t) < deg(g) returns polynomials `G`, `H`, `S` and `T`, such that:: f = G*H (mod m**2) S*G + T*H = 1 (mod m**2) References ========== .. [1] [Gathen99]_ """ M = m**2 e = dup_sub_mul(f, g, h, K) e = dup_trunc(e, M, K) q, r = dup_div(dup_mul(s, e, K), h, K) q = dup_trunc(q, M, K) r = dup_trunc(r, M, K) u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K) G = dup_trunc(dup_add(g, u, K), M, K) H = dup_trunc(dup_add(h, r, K), M, K) u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K) b = dup_trunc(dup_sub(u, [K.one], K), M, K) c, d = dup_div(dup_mul(s, b, K), H, K) c = dup_trunc(c, M, K) d = dup_trunc(d, M, K) u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K) S = dup_trunc(dup_sub(s, d, K), M, K) T = dup_trunc(dup_sub(t, u, K), M, K) return G, H, S, T def dup_zz_hensel_lift(p, f, f_list, l, K): """ Multifactor Hensel lifting in `Z[x]`. Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)` is a unit modulo `p`, monic pair-wise coprime polynomials `f_i` over `Z[x]` satisfying:: f = lc(f) f_1 ... f_r (mod p) and a positive integer `l`, returns a list of monic polynomials `F_1`, `F_2`, ..., `F_r` satisfying:: f = lc(f) F_1 ... F_r (mod p**l) F_i = f_i (mod p), i = 1..r References ========== .. [1] [Gathen99]_ """ r = len(f_list) lc = dup_LC(f, K) if r == 1: F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K) return [ dup_trunc(F, p**l, K) ] m = p k = r // 2 d = int(_ceil(_log(l, 2))) g = gf_from_int_poly([lc], p) for f_i in f_list[:k]: g = gf_mul(g, gf_from_int_poly(f_i, p), p, K) h = gf_from_int_poly(f_list[k], p) for f_i in f_list[k + 1:]: h = gf_mul(h, gf_from_int_poly(f_i, p), p, K) s, t, _ = gf_gcdex(g, h, p, K) g = gf_to_int_poly(g, p) h = gf_to_int_poly(h, p) s = gf_to_int_poly(s, p) t = gf_to_int_poly(t, p) for _ in range(1, d + 1): (g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2 return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \ + dup_zz_hensel_lift(p, h, f_list[k:], l, K) def _test_pl(fc, q, pl): if q > pl // 2: q = q - pl if not q: return True return fc % q == 0 def dup_zz_zassenhaus(f, K): """Factor primitive square-free polynomials in `Z[x]`. """ n = dup_degree(f) if n == 1: return [f] fc = f[-1] A = dup_max_norm(f, K) b = dup_LC(f, K) B = int(abs(K.sqrt(K(n + 1))*2**n*A*b)) C = int((n + 1)**(2*n)*A**(2*n - 1)) gamma = int(_ceil(2*_log(C, 2))) bound = int(2*gamma*_log(gamma)) a = [] # choose a prime number `p` such that `f` be square free in Z_p # if there are many factors in Z_p, choose among a few different `p` # the one with fewer factors for px in range(3, bound + 1): if not isprime(px) or b % px == 0: continue px = K.convert(px) F = gf_from_int_poly(f, px) if not gf_sqf_p(F, px, K): continue fsqfx = gf_factor_sqf(F, px, K)[1] a.append((px, fsqfx)) if len(fsqfx) < 15 or len(a) > 4: break p, fsqf = min(a, key=lambda x: len(x[1])) l = int(_ceil(_log(2*B + 1, p))) modular = [gf_to_int_poly(ff, p) for ff in fsqf] g = dup_zz_hensel_lift(p, f, modular, l, K) sorted_T = range(len(g)) T = set(sorted_T) factors, s = [], 1 pl = p**l while 2*s <= len(T): for S in subsets(sorted_T, s): # lift the constant coefficient of the product `G` of the factors # in the subset `S`; if it is does not divide `fc`, `G` does # not divide the input polynomial if b == 1: q = 1 for i in S: q = q*g[i][-1] q = q % pl if not _test_pl(fc, q, pl): continue else: G = [b] for i in S: G = dup_mul(G, g[i], K) G = dup_trunc(G, pl, K) G = dup_primitive(G, K)[1] q = G[-1] if q and fc % q != 0: continue H = [b] S = set(S) T_S = T - S if b == 1: G = [b] for i in S: G = dup_mul(G, g[i], K) G = dup_trunc(G, pl, K) for i in T_S: H = dup_mul(H, g[i], K) H = dup_trunc(H, pl, K) G_norm = dup_l1_norm(G, K) H_norm = dup_l1_norm(H, K) if G_norm*H_norm <= B: T = T_S sorted_T = [i for i in sorted_T if i not in S] G = dup_primitive(G, K)[1] f = dup_primitive(H, K)[1] factors.append(G) b = dup_LC(f, K) break else: s += 1 return factors + [f] def dup_zz_irreducible_p(f, K): """Test irreducibility using Eisenstein's criterion. """ lc = dup_LC(f, K) tc = dup_TC(f, K) e_fc = dup_content(f[1:], K) if e_fc: e_ff = factorint(int(e_fc)) for p in e_ff.keys(): if (lc % p) and (tc % p**2): return True def dup_cyclotomic_p(f, K, irreducible=False): """ Efficiently test if ``f`` is a cyclotomic polynomial. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(f) False >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(g) True """ if K.is_QQ: try: K0, K = K, K.get_ring() f = dup_convert(f, K0, K) except CoercionFailed: return False elif not K.is_ZZ: return False lc = dup_LC(f, K) tc = dup_TC(f, K) if lc != 1 or (tc != -1 and tc != 1): return False if not irreducible: coeff, factors = dup_factor_list(f, K) if coeff != K.one or factors != [(f, 1)]: return False n = dup_degree(f) g, h = [], [] for i in range(n, -1, -2): g.insert(0, f[i]) for i in range(n - 1, -1, -2): h.insert(0, f[i]) g = dup_sqr(dup_strip(g), K) h = dup_sqr(dup_strip(h), K) F = dup_sub(g, dup_lshift(h, 1, K), K) if K.is_negative(dup_LC(F, K)): F = dup_neg(F, K) if F == f: return True g = dup_mirror(f, K) if K.is_negative(dup_LC(g, K)): g = dup_neg(g, K) if F == g and dup_cyclotomic_p(g, K): return True G = dup_sqf_part(F, K) if dup_sqr(G, K) == F and dup_cyclotomic_p(G, K): return True return False def dup_zz_cyclotomic_poly(n, K): """Efficiently generate n-th cyclotomic polynomial. """ h = [K.one, -K.one] for p, k in factorint(n).items(): h = dup_quo(dup_inflate(h, p, K), h, K) h = dup_inflate(h, p**(k - 1), K) return h def _dup_cyclotomic_decompose(n, K): H = [[K.one, -K.one]] for p, k in factorint(n).items(): Q = [ dup_quo(dup_inflate(h, p, K), h, K) for h in H ] H.extend(Q) for i in range(1, k): Q = [ dup_inflate(q, p, K) for q in Q ] H.extend(Q) return H def dup_zz_cyclotomic_factor(f, K): """ Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` returns a list of factors of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for `n >= 1`. Otherwise returns None. Factorization is performed using cyclotomic decomposition of `f`, which makes this method much faster that any other direct factorization approach (e.g. Zassenhaus's). References ========== .. [1] [Weisstein09]_ """ lc_f, tc_f = dup_LC(f, K), dup_TC(f, K) if dup_degree(f) <= 0: return None if lc_f != 1 or tc_f not in [-1, 1]: return None if any(bool(cf) for cf in f[1:-1]): return None n = dup_degree(f) F = _dup_cyclotomic_decompose(n, K) if not K.is_one(tc_f): return F else: H = [] for h in _dup_cyclotomic_decompose(2*n, K): if h not in F: H.append(h) return H def dup_zz_factor_sqf(f, K): """Factor square-free (non-primitive) polynomials in `Z[x]`. """ cont, g = dup_primitive(f, K) n = dup_degree(g) if dup_LC(g, K) < 0: cont, g = -cont, dup_neg(g, K) if n <= 0: return cont, [] elif n == 1: return cont, [g] if query('USE_IRREDUCIBLE_IN_FACTOR'): if dup_zz_irreducible_p(g, K): return cont, [g] factors = None if query('USE_CYCLOTOMIC_FACTOR'): factors = dup_zz_cyclotomic_factor(g, K) if factors is None: factors = dup_zz_zassenhaus(g, K) return cont, _sort_factors(factors, multiple=False) def dup_zz_factor(f, K): """ Factor (non square-free) polynomials in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Zassenhaus algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Examples ======== Consider the polynomial `f = 2*x**4 - 2`:: >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_zz_factor(2*x**4 - 2) (2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)]) In result we got the following factorization:: f = 2 (x - 1) (x + 1) (x**2 + 1) Note that this is a complete factorization over integers, however over Gaussian integers we can factor the last term. By default, polynomials `x**n - 1` and `x**n + 1` are factored using cyclotomic decomposition to speedup computations. To disable this behaviour set cyclotomic=False. References ========== .. [1] [Gathen99]_ """ cont, g = dup_primitive(f, K) n = dup_degree(g) if dup_LC(g, K) < 0: cont, g = -cont, dup_neg(g, K) if n <= 0: return cont, [] elif n == 1: return cont, [(g, 1)] if query('USE_IRREDUCIBLE_IN_FACTOR'): if dup_zz_irreducible_p(g, K): return cont, [(g, 1)] g = dup_sqf_part(g, K) H = None if query('USE_CYCLOTOMIC_FACTOR'): H = dup_zz_cyclotomic_factor(g, K) if H is None: H = dup_zz_zassenhaus(g, K) factors = dup_trial_division(f, H, K) return cont, factors def dmp_zz_wang_non_divisors(E, cs, ct, K): """Wang/EEZ: Compute a set of valid divisors. """ result = [ cs*ct ] for q in E: q = abs(q) for r in reversed(result): while r != 1: r = K.gcd(r, q) q = q // r if K.is_one(q): return None result.append(q) return result[1:] def dmp_zz_wang_test_points(f, T, ct, A, u, K): """Wang/EEZ: Test evaluation points for suitability. """ if not dmp_eval_tail(dmp_LC(f, K), A, u - 1, K): raise EvaluationFailed('no luck') g = dmp_eval_tail(f, A, u, K) if not dup_sqf_p(g, K): raise EvaluationFailed('no luck') c, h = dup_primitive(g, K) if K.is_negative(dup_LC(h, K)): c, h = -c, dup_neg(h, K) v = u - 1 E = [ dmp_eval_tail(t, A, v, K) for t, _ in T ] D = dmp_zz_wang_non_divisors(E, c, ct, K) if D is not None: return c, h, E else: raise EvaluationFailed('no luck') def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K): """Wang/EEZ: Compute correct leading coefficients. """ C, J, v = [], [0]*len(E), u - 1 for h in H: c = dmp_one(v, K) d = dup_LC(h, K)*cs for i in reversed(range(len(E))): k, e, (t, _) = 0, E[i], T[i] while not (d % e): d, k = d//e, k + 1 if k != 0: c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1 C.append(c) if any(not j for j in J): raise ExtraneousFactors # pragma: no cover CC, HH = [], [] for c, h in zip(C, H): d = dmp_eval_tail(c, A, v, K) lc = dup_LC(h, K) if K.is_one(cs): cc = lc//d else: g = K.gcd(lc, d) d, cc = d//g, lc//g h, cs = dup_mul_ground(h, d, K), cs//d c = dmp_mul_ground(c, cc, v, K) CC.append(c) HH.append(h) if K.is_one(cs): return f, HH, CC CCC, HHH = [], [] for c, h in zip(CC, HH): CCC.append(dmp_mul_ground(c, cs, v, K)) HHH.append(dmp_mul_ground(h, cs, 0, K)) f = dmp_mul_ground(f, cs**(len(H) - 1), u, K) return f, HHH, CCC def dup_zz_diophantine(F, m, p, K): """Wang/EEZ: Solve univariate Diophantine equations. """ if len(F) == 2: a, b = F f = gf_from_int_poly(a, p) g = gf_from_int_poly(b, p) s, t, G = gf_gcdex(g, f, p, K) s = gf_lshift(s, m, K) t = gf_lshift(t, m, K) q, s = gf_div(s, f, p, K) t = gf_add_mul(t, q, g, p, K) s = gf_to_int_poly(s, p) t = gf_to_int_poly(t, p) result = [s, t] else: G = [F[-1]] for f in reversed(F[1:-1]): G.insert(0, dup_mul(f, G[0], K)) S, T = [], [[1]] for f, g in zip(F, G): t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K) T.append(t) S.append(s) result, S = [], S + [T[-1]] for s, f in zip(S, F): s = gf_from_int_poly(s, p) f = gf_from_int_poly(f, p) r = gf_rem(gf_lshift(s, m, K), f, p, K) s = gf_to_int_poly(r, p) result.append(s) return result def dmp_zz_diophantine(F, c, A, d, p, u, K): """Wang/EEZ: Solve multivariate Diophantine equations. """ if not A: S = [ [] for _ in F ] n = dup_degree(c) for i, coeff in enumerate(c): if not coeff: continue T = dup_zz_diophantine(F, n - i, p, K) for j, (s, t) in enumerate(zip(S, T)): t = dup_mul_ground(t, coeff, K) S[j] = dup_trunc(dup_add(s, t, K), p, K) else: n = len(A) e = dmp_expand(F, u, K) a, A = A[-1], A[:-1] B, G = [], [] for f in F: B.append(dmp_quo(e, f, u, K)) G.append(dmp_eval_in(f, a, n, u, K)) C = dmp_eval_in(c, a, n, u, K) v = u - 1 S = dmp_zz_diophantine(G, C, A, d, p, v, K) S = [ dmp_raise(s, 1, v, K) for s in S ] for s, b in zip(S, B): c = dmp_sub_mul(c, s, b, u, K) c = dmp_ground_trunc(c, p, u, K) m = dmp_nest([K.one, -a], n, K) M = dmp_one(n, K) for k in K.map(range(0, d)): if dmp_zero_p(c, u): break M = dmp_mul(M, m, u, K) C = dmp_diff_eval_in(c, k + 1, a, n, u, K) if not dmp_zero_p(C, v): C = dmp_quo_ground(C, K.factorial(k + 1), v, K) T = dmp_zz_diophantine(G, C, A, d, p, v, K) for i, t in enumerate(T): T[i] = dmp_mul(dmp_raise(t, 1, v, K), M, u, K) for i, (s, t) in enumerate(zip(S, T)): S[i] = dmp_add(s, t, u, K) for t, b in zip(T, B): c = dmp_sub_mul(c, t, b, u, K) c = dmp_ground_trunc(c, p, u, K) S = [ dmp_ground_trunc(s, p, u, K) for s in S ] return S def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K): """Wang/EEZ: Parallel Hensel lifting algorithm. """ S, n, v = [f], len(A), u - 1 H = list(H) for i, a in enumerate(reversed(A[1:])): s = dmp_eval_in(S[0], a, n - i, u - i, K) S.insert(0, dmp_ground_trunc(s, p, v - i, K)) d = max(dmp_degree_list(f, u)[1:]) for j, s, a in zip(range(2, n + 2), S, A): G, w = list(H), j - 1 I, J = A[:j - 2], A[j - 1:] for i, (h, lc) in enumerate(zip(H, LC)): lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w - 1, K) H[i] = [lc] + dmp_raise(h[1:], 1, w - 1, K) m = dmp_nest([K.one, -a], w, K) M = dmp_one(w, K) c = dmp_sub(s, dmp_expand(H, w, K), w, K) dj = dmp_degree_in(s, w, w) for k in K.map(range(0, dj)): if dmp_zero_p(c, w): break M = dmp_mul(M, m, w, K) C = dmp_diff_eval_in(c, k + 1, a, w, w, K) if not dmp_zero_p(C, w - 1): C = dmp_quo_ground(C, K.factorial(k + 1), w - 1, K) T = dmp_zz_diophantine(G, C, I, d, p, w - 1, K) for i, (h, t) in enumerate(zip(H, T)): h = dmp_add_mul(h, dmp_raise(t, 1, w - 1, K), M, w, K) H[i] = dmp_ground_trunc(h, p, w, K) h = dmp_sub(s, dmp_expand(H, w, K), w, K) c = dmp_ground_trunc(h, p, w, K) if dmp_expand(H, u, K) != f: raise ExtraneousFactors # pragma: no cover else: return H def dmp_zz_wang(f, u, K, mod=None, seed=None): """ Factor primitive square-free polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is primitive and square-free in `x_1`, computes factorization of `f` into irreducibles over integers. The procedure is based on Wang's Enhanced Extended Zassenhaus algorithm. The algorithm works by viewing `f` as a univariate polynomial in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed:: x_2 -> a_2, ..., x_n -> a_n where `a_i`, for `i = 2, ..., n`, are carefully chosen integers. The mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`, which can be factored efficiently using Zassenhaus algorithm. The last step is to lift univariate factors to obtain true multivariate factors. For this purpose a parallel Hensel lifting procedure is used. The parameter ``seed`` is passed to _randint and can be used to seed randint (when an integer) or (for testing purposes) can be a sequence of numbers. References ========== .. [1] [Wang78]_ .. [2] [Geddes92]_ """ from sympy.testing.randtest import _randint randint = _randint(seed) ct, T = dmp_zz_factor(dmp_LC(f, K), u - 1, K) b = dmp_zz_mignotte_bound(f, u, K) p = K(nextprime(b)) if mod is None: if u == 1: mod = 2 else: mod = 1 history, configs, A, r = set(), [], [K.zero]*u, None try: cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K) _, H = dup_zz_factor_sqf(s, K) r = len(H) if r == 1: return [f] configs = [(s, cs, E, H, A)] except EvaluationFailed: pass eez_num_configs = query('EEZ_NUMBER_OF_CONFIGS') eez_num_tries = query('EEZ_NUMBER_OF_TRIES') eez_mod_step = query('EEZ_MODULUS_STEP') while len(configs) < eez_num_configs: for _ in range(eez_num_tries): A = [ K(randint(-mod, mod)) for _ in range(u) ] if tuple(A) not in history: history.add(tuple(A)) else: continue try: cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K) except EvaluationFailed: continue _, H = dup_zz_factor_sqf(s, K) rr = len(H) if r is not None: if rr != r: # pragma: no cover if rr < r: configs, r = [], rr else: continue else: r = rr if r == 1: return [f] configs.append((s, cs, E, H, A)) if len(configs) == eez_num_configs: break else: mod += eez_mod_step s_norm, s_arg, i = None, 0, 0 for s, _, _, _, _ in configs: _s_norm = dup_max_norm(s, K) if s_norm is not None: if _s_norm < s_norm: s_norm = _s_norm s_arg = i else: s_norm = _s_norm i += 1 _, cs, E, H, A = configs[s_arg] orig_f = f try: f, H, LC = dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K) factors = dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K) except ExtraneousFactors: # pragma: no cover if query('EEZ_RESTART_IF_NEEDED'): return dmp_zz_wang(orig_f, u, K, mod + 1) else: raise ExtraneousFactors( "we need to restart algorithm with better parameters") result = [] for f in factors: _, f = dmp_ground_primitive(f, u, K) if K.is_negative(dmp_ground_LC(f, u, K)): f = dmp_neg(f, u, K) result.append(f) return result def dmp_zz_factor(f, u, K): """ Factor (non square-free) polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Consider polynomial `f = 2*(x**2 - y**2)`:: >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_zz_factor(2*x**2 - 2*y**2) (2, [(x - y, 1), (x + y, 1)]) In result we got the following factorization:: f = 2 (x - y) (x + y) References ========== .. [1] [Gathen99]_ """ if not u: return dup_zz_factor(f, K) if dmp_zero_p(f, u): return K.zero, [] cont, g = dmp_ground_primitive(f, u, K) if dmp_ground_LC(g, u, K) < 0: cont, g = -cont, dmp_neg(g, u, K) if all(d <= 0 for d in dmp_degree_list(g, u)): return cont, [] G, g = dmp_primitive(g, u, K) factors = [] if dmp_degree(g, u) > 0: g = dmp_sqf_part(g, u, K) H = dmp_zz_wang(g, u, K) factors = dmp_trial_division(f, H, u, K) for g, k in dmp_zz_factor(G, u - 1, K)[1]: factors.insert(0, ([g], k)) return cont, _sort_factors(factors) def dup_qq_i_factor(f, K0): """Factor univariate polynomials into irreducibles in `QQ_I[x]`. """ # Factor in QQ<I> K1 = K0.as_AlgebraicField() f = dup_convert(f, K0, K1) coeff, factors = dup_factor_list(f, K1) factors = [(dup_convert(fac, K1, K0), i) for fac, i in factors] coeff = K0.convert(coeff, K1) return coeff, factors def dup_zz_i_factor(f, K0): """Factor univariate polynomials into irreducibles in `ZZ_I[x]`. """ # First factor in QQ_I K1 = K0.get_field() f = dup_convert(f, K0, K1) coeff, factors = dup_qq_i_factor(f, K1) new_factors = [] for fac, i in factors: # Extract content fac_denom, fac_num = dup_clear_denoms(fac, K1) fac_num_ZZ_I = dup_convert(fac_num, K1, K0) content, fac_prim = dmp_ground_primitive(fac_num_ZZ_I, 0, K1) coeff = (coeff * content ** i) // fac_denom ** i new_factors.append((fac_prim, i)) factors = new_factors coeff = K0.convert(coeff, K1) return coeff, factors def dmp_qq_i_factor(f, u, K0): """Factor multivariate polynomials into irreducibles in `QQ_I[X]`. """ # Factor in QQ<I> K1 = K0.as_AlgebraicField() f = dmp_convert(f, u, K0, K1) coeff, factors = dmp_factor_list(f, u, K1) factors = [(dmp_convert(fac, u, K1, K0), i) for fac, i in factors] coeff = K0.convert(coeff, K1) return coeff, factors def dmp_zz_i_factor(f, u, K0): """Factor multivariate polynomials into irreducibles in `ZZ_I[X]`. """ # First factor in QQ_I K1 = K0.get_field() f = dmp_convert(f, u, K0, K1) coeff, factors = dmp_qq_i_factor(f, u, K1) new_factors = [] for fac, i in factors: # Extract content fac_denom, fac_num = dmp_clear_denoms(fac, u, K1) fac_num_ZZ_I = dmp_convert(fac_num, u, K1, K0) content, fac_prim = dmp_ground_primitive(fac_num_ZZ_I, u, K1) coeff = (coeff * content ** i) // fac_denom ** i new_factors.append((fac_prim, i)) factors = new_factors coeff = K0.convert(coeff, K1) return coeff, factors def dup_ext_factor(f, K): """Factor univariate polynomials over algebraic number fields. """ n, lc = dup_degree(f), dup_LC(f, K) f = dup_monic(f, K) if n <= 0: return lc, [] if n == 1: return lc, [(f, 1)] f, F = dup_sqf_part(f, K), f s, g, r = dup_sqf_norm(f, K) factors = dup_factor_list_include(r, K.dom) if len(factors) == 1: return lc, [(f, n//dup_degree(f))] H = s*K.unit for i, (factor, _) in enumerate(factors): h = dup_convert(factor, K.dom, K) h, _, g = dup_inner_gcd(h, g, K) h = dup_shift(h, H, K) factors[i] = h factors = dup_trial_division(F, factors, K) return lc, factors def dmp_ext_factor(f, u, K): """Factor multivariate polynomials over algebraic number fields. """ if not u: return dup_ext_factor(f, K) lc = dmp_ground_LC(f, u, K) f = dmp_ground_monic(f, u, K) if all(d <= 0 for d in dmp_degree_list(f, u)): return lc, [] f, F = dmp_sqf_part(f, u, K), f s, g, r = dmp_sqf_norm(f, u, K) factors = dmp_factor_list_include(r, u, K.dom) if len(factors) == 1: factors = [f] else: H = dmp_raise([K.one, s*K.unit], u, 0, K) for i, (factor, _) in enumerate(factors): h = dmp_convert(factor, u, K.dom, K) h, _, g = dmp_inner_gcd(h, g, u, K) h = dmp_compose(h, H, u, K) factors[i] = h return lc, dmp_trial_division(F, factors, u, K) def dup_gf_factor(f, K): """Factor univariate polynomials over finite fields. """ f = dup_convert(f, K, K.dom) coeff, factors = gf_factor(f, K.mod, K.dom) for i, (f, k) in enumerate(factors): factors[i] = (dup_convert(f, K.dom, K), k) return K.convert(coeff, K.dom), factors def dmp_gf_factor(f, u, K): """Factor multivariate polynomials over finite fields. """ raise NotImplementedError('multivariate polynomials over finite fields') def dup_factor_list(f, K0): """Factor univariate polynomials into irreducibles in `K[x]`. """ j, f = dup_terms_gcd(f, K0) cont, f = dup_primitive(f, K0) if K0.is_FiniteField: coeff, factors = dup_gf_factor(f, K0) elif K0.is_Algebraic: coeff, factors = dup_ext_factor(f, K0) elif K0.is_GaussianRing: coeff, factors = dup_zz_i_factor(f, K0) elif K0.is_GaussianField: coeff, factors = dup_qq_i_factor(f, K0) else: if not K0.is_Exact: K0_inexact, K0 = K0, K0.get_exact() f = dup_convert(f, K0_inexact, K0) else: K0_inexact = None if K0.is_Field: K = K0.get_ring() denom, f = dup_clear_denoms(f, K0, K) f = dup_convert(f, K0, K) else: K = K0 if K.is_ZZ: coeff, factors = dup_zz_factor(f, K) elif K.is_Poly: f, u = dmp_inject(f, 0, K) coeff, factors = dmp_factor_list(f, u, K.dom) for i, (f, k) in enumerate(factors): factors[i] = (dmp_eject(f, u, K), k) coeff = K.convert(coeff, K.dom) else: # pragma: no cover raise DomainError('factorization not supported over %s' % K0) if K0.is_Field: for i, (f, k) in enumerate(factors): factors[i] = (dup_convert(f, K, K0), k) coeff = K0.convert(coeff, K) coeff = K0.quo(coeff, denom) if K0_inexact: for i, (f, k) in enumerate(factors): max_norm = dup_max_norm(f, K0) f = dup_quo_ground(f, max_norm, K0) f = dup_convert(f, K0, K0_inexact) factors[i] = (f, k) coeff = K0.mul(coeff, K0.pow(max_norm, k)) coeff = K0_inexact.convert(coeff, K0) K0 = K0_inexact if j: factors.insert(0, ([K0.one, K0.zero], j)) return coeff*cont, _sort_factors(factors) def dup_factor_list_include(f, K): """Factor univariate polynomials into irreducibles in `K[x]`. """ coeff, factors = dup_factor_list(f, K) if not factors: return [(dup_strip([coeff]), 1)] else: g = dup_mul_ground(factors[0][0], coeff, K) return [(g, factors[0][1])] + factors[1:] def dmp_factor_list(f, u, K0): """Factor multivariate polynomials into irreducibles in `K[X]`. """ if not u: return dup_factor_list(f, K0) J, f = dmp_terms_gcd(f, u, K0) cont, f = dmp_ground_primitive(f, u, K0) if K0.is_FiniteField: # pragma: no cover coeff, factors = dmp_gf_factor(f, u, K0) elif K0.is_Algebraic: coeff, factors = dmp_ext_factor(f, u, K0) elif K0.is_GaussianRing: coeff, factors = dmp_zz_i_factor(f, u, K0) elif K0.is_GaussianField: coeff, factors = dmp_qq_i_factor(f, u, K0) else: if not K0.is_Exact: K0_inexact, K0 = K0, K0.get_exact() f = dmp_convert(f, u, K0_inexact, K0) else: K0_inexact = None if K0.is_Field: K = K0.get_ring() denom, f = dmp_clear_denoms(f, u, K0, K) f = dmp_convert(f, u, K0, K) else: K = K0 if K.is_ZZ: levels, f, v = dmp_exclude(f, u, K) coeff, factors = dmp_zz_factor(f, v, K) for i, (f, k) in enumerate(factors): factors[i] = (dmp_include(f, levels, v, K), k) elif K.is_Poly: f, v = dmp_inject(f, u, K) coeff, factors = dmp_factor_list(f, v, K.dom) for i, (f, k) in enumerate(factors): factors[i] = (dmp_eject(f, v, K), k) coeff = K.convert(coeff, K.dom) else: # pragma: no cover raise DomainError('factorization not supported over %s' % K0) if K0.is_Field: for i, (f, k) in enumerate(factors): factors[i] = (dmp_convert(f, u, K, K0), k) coeff = K0.convert(coeff, K) coeff = K0.quo(coeff, denom) if K0_inexact: for i, (f, k) in enumerate(factors): max_norm = dmp_max_norm(f, u, K0) f = dmp_quo_ground(f, max_norm, u, K0) f = dmp_convert(f, u, K0, K0_inexact) factors[i] = (f, k) coeff = K0.mul(coeff, K0.pow(max_norm, k)) coeff = K0_inexact.convert(coeff, K0) K0 = K0_inexact for i, j in enumerate(reversed(J)): if not j: continue term = {(0,)*(u - i) + (1,) + (0,)*i: K0.one} factors.insert(0, (dmp_from_dict(term, u, K0), j)) return coeff*cont, _sort_factors(factors) def dmp_factor_list_include(f, u, K): """Factor multivariate polynomials into irreducibles in `K[X]`. """ if not u: return dup_factor_list_include(f, K) coeff, factors = dmp_factor_list(f, u, K) if not factors: return [(dmp_ground(coeff, u), 1)] else: g = dmp_mul_ground(factors[0][0], coeff, u, K) return [(g, factors[0][1])] + factors[1:] def dup_irreducible_p(f, K): """ Returns ``True`` if a univariate polynomial ``f`` has no factors over its domain. """ return dmp_irreducible_p(f, 0, K) def dmp_irreducible_p(f, u, K): """ Returns ``True`` if a multivariate polynomial ``f`` has no factors over its domain. """ _, factors = dmp_factor_list(f, u, K) if not factors: return True elif len(factors) > 1: return False else: _, k = factors[0] return k == 1
0619409f35c1ec98572770f7e9802ad7b1336984753025bf1e9feb4953f6cd3a
r""" Sparse distributed elements of free modules over multivariate (generalized) polynomial rings. This code and its data structures are very much like the distributed polynomials, except that the first "exponent" of the monomial is a module generator index. That is, the multi-exponent ``(i, e_1, ..., e_n)`` represents the "monomial" `x_1^{e_1} \cdots x_n^{e_n} f_i` of the free module `F` generated by `f_1, \ldots, f_r` over (a localization of) the ring `K[x_1, \ldots, x_n]`. A module element is simply stored as a list of terms ordered by the monomial order. Here a term is a pair of a multi-exponent and a coefficient. In general, this coefficient should never be zero (since it can then be omitted). The zero module element is stored as an empty list. The main routines are ``sdm_nf_mora`` and ``sdm_groebner`` which can be used to compute, respectively, weak normal forms and standard bases. They work with arbitrary (not necessarily global) monomial orders. In general, product orders have to be used to construct valid monomial orders for modules. However, ``lex`` can be used as-is. Note that the "level" (number of variables, i.e. parameter u+1 in distributedpolys.py) is never needed in this code. The main reference for this file is [SCA], "A Singular Introduction to Commutative Algebra". """ from itertools import permutations from sympy.polys.monomials import ( monomial_mul, monomial_lcm, monomial_div, monomial_deg ) from sympy.polys.polytools import Poly from sympy.polys.polyutils import parallel_dict_from_expr from sympy import S, sympify # Additional monomial tools. def sdm_monomial_mul(M, X): """ Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple ``M`` representing a monomial of `F`. Examples ======== Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`: >>> from sympy.polys.distributedmodules import sdm_monomial_mul >>> sdm_monomial_mul((1, 1, 0), (1, 3)) (1, 2, 3) """ return (M[0],) + monomial_mul(X, M[1:]) def sdm_monomial_deg(M): """ Return the total degree of ``M``. Examples ======== For example, the total degree of `x^2 y f_5` is 3: >>> from sympy.polys.distributedmodules import sdm_monomial_deg >>> sdm_monomial_deg((5, 2, 1)) 3 """ return monomial_deg(M[1:]) def sdm_monomial_lcm(A, B): r""" Return the "least common multiple" of ``A`` and ``B``. IF `A = M e_j` and `B = N e_j`, where `M` and `N` are polynomial monomials, this returns `\lcm(M, N) e_j`. Note that ``A`` and ``B`` involve distinct monomials. Otherwise the result is undefined. Examples ======== >>> from sympy.polys.distributedmodules import sdm_monomial_lcm >>> sdm_monomial_lcm((1, 2, 3), (1, 0, 5)) (1, 2, 5) """ return (A[0],) + monomial_lcm(A[1:], B[1:]) def sdm_monomial_divides(A, B): """ Does there exist a (polynomial) monomial X such that XA = B? Examples ======== Positive examples: In the following examples, the monomial is given in terms of x, y and the generator(s), f_1, f_2 etc. The tuple form of that monomial is used in the call to sdm_monomial_divides. Note: the generator appears last in the expression but first in the tuple and other factors appear in the same order that they appear in the monomial expression. `A = f_1` divides `B = f_1` >>> from sympy.polys.distributedmodules import sdm_monomial_divides >>> sdm_monomial_divides((1, 0, 0), (1, 0, 0)) True `A = f_1` divides `B = x^2 y f_1` >>> sdm_monomial_divides((1, 0, 0), (1, 2, 1)) True `A = xy f_5` divides `B = x^2 y f_5` >>> sdm_monomial_divides((5, 1, 1), (5, 2, 1)) True Negative examples: `A = f_1` does not divide `B = f_2` >>> sdm_monomial_divides((1, 0, 0), (2, 0, 0)) False `A = x f_1` does not divide `B = f_1` >>> sdm_monomial_divides((1, 1, 0), (1, 0, 0)) False `A = xy^2 f_5` does not divide `B = y f_5` >>> sdm_monomial_divides((5, 1, 2), (5, 0, 1)) False """ return A[0] == B[0] and all(a <= b for a, b in zip(A[1:], B[1:])) # The actual distributed modules code. def sdm_LC(f, K): """Returns the leading coeffcient of ``f``. """ if not f: return K.zero else: return f[0][1] def sdm_to_dict(f): """Make a dictionary from a distributed polynomial. """ return dict(f) def sdm_from_dict(d, O): """ Create an sdm from a dictionary. Here ``O`` is the monomial order to use. Examples ======== >>> from sympy.polys.distributedmodules import sdm_from_dict >>> from sympy.polys import QQ, lex >>> dic = {(1, 1, 0): QQ(1), (1, 0, 0): QQ(2), (0, 1, 0): QQ(0)} >>> sdm_from_dict(dic, lex) [((1, 1, 0), 1), ((1, 0, 0), 2)] """ return sdm_strip(sdm_sort(list(d.items()), O)) def sdm_sort(f, O): """Sort terms in ``f`` using the given monomial order ``O``. """ return sorted(f, key=lambda term: O(term[0]), reverse=True) def sdm_strip(f): """Remove terms with zero coefficients from ``f`` in ``K[X]``. """ return [ (monom, coeff) for monom, coeff in f if coeff ] def sdm_add(f, g, O, K): """ Add two module elements ``f``, ``g``. Addition is done over the ground field ``K``, monomials are ordered according to ``O``. Examples ======== All examples use lexicographic order. `(xy f_1) + (f_2) = f_2 + xy f_1` >>> from sympy.polys.distributedmodules import sdm_add >>> from sympy.polys import lex, QQ >>> sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ) [((2, 0, 0), 1), ((1, 1, 1), 1)] `(xy f_1) + (-xy f_1)` = 0` >>> sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ) [] `(f_1) + (2f_1) = 3f_1` >>> sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ) [((1, 0, 0), 3)] `(yf_1) + (xf_1) = xf_1 + yf_1` >>> sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ) [((1, 1, 0), 1), ((1, 0, 1), 1)] """ h = dict(f) for monom, c in g: if monom in h: coeff = h[monom] + c if not coeff: del h[monom] else: h[monom] = coeff else: h[monom] = c return sdm_from_dict(h, O) def sdm_LM(f): r""" Returns the leading monomial of ``f``. Only valid if `f \ne 0`. Examples ======== >>> from sympy.polys.distributedmodules import sdm_LM, sdm_from_dict >>> from sympy.polys import QQ, lex >>> dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)} >>> sdm_LM(sdm_from_dict(dic, lex)) (4, 0, 1) """ return f[0][0] def sdm_LT(f): r""" Returns the leading term of ``f``. Only valid if `f \ne 0`. Examples ======== >>> from sympy.polys.distributedmodules import sdm_LT, sdm_from_dict >>> from sympy.polys import QQ, lex >>> dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)} >>> sdm_LT(sdm_from_dict(dic, lex)) ((4, 0, 1), 3) """ return f[0] def sdm_mul_term(f, term, O, K): """ Multiply a distributed module element ``f`` by a (polynomial) term ``term``. Multiplication of coefficients is done over the ground field ``K``, and monomials are ordered according to ``O``. Examples ======== `0 f_1 = 0` >>> from sympy.polys.distributedmodules import sdm_mul_term >>> from sympy.polys import lex, QQ >>> sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ) [] `x 0 = 0` >>> sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ) [] `(x) (f_1) = xf_1` >>> sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ) [((1, 1, 0), 1)] `(2xy) (3x f_1 + 4y f_2) = 8xy^2 f_2 + 6x^2y f_1` >>> f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))] >>> sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ) [((2, 1, 2), 8), ((1, 2, 1), 6)] """ X, c = term if not f or not c: return [] else: if K.is_one(c): return [ (sdm_monomial_mul(f_M, X), f_c) for f_M, f_c in f ] else: return [ (sdm_monomial_mul(f_M, X), f_c * c) for f_M, f_c in f ] def sdm_zero(): """Return the zero module element.""" return [] def sdm_deg(f): """ Degree of ``f``. This is the maximum of the degrees of all its monomials. Invalid if ``f`` is zero. Examples ======== >>> from sympy.polys.distributedmodules import sdm_deg >>> sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)]) 7 """ return max(sdm_monomial_deg(M[0]) for M in f) # Conversion def sdm_from_vector(vec, O, K, **opts): """ Create an sdm from an iterable of expressions. Coefficients are created in the ground field ``K``, and terms are ordered according to monomial order ``O``. Named arguments are passed on to the polys conversion code and can be used to specify for example generators. Examples ======== >>> from sympy.polys.distributedmodules import sdm_from_vector >>> from sympy.abc import x, y, z >>> from sympy.polys import QQ, lex >>> sdm_from_vector([x**2+y**2, 2*z], lex, QQ) [((1, 0, 0, 1), 2), ((0, 2, 0, 0), 1), ((0, 0, 2, 0), 1)] """ dics, gens = parallel_dict_from_expr(sympify(vec), **opts) dic = {} for i, d in enumerate(dics): for k, v in d.items(): dic[(i,) + k] = K.convert(v) return sdm_from_dict(dic, O) def sdm_to_vector(f, gens, K, n=None): """ Convert sdm ``f`` into a list of polynomial expressions. The generators for the polynomial ring are specified via ``gens``. The rank of the module is guessed, or passed via ``n``. The ground field is assumed to be ``K``. Examples ======== >>> from sympy.polys.distributedmodules import sdm_to_vector >>> from sympy.abc import x, y, z >>> from sympy.polys import QQ >>> f = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))] >>> sdm_to_vector(f, [x, y, z], QQ) [x**2 + y**2, 2*z] """ dic = sdm_to_dict(f) dics = {} for k, v in dic.items(): dics.setdefault(k[0], []).append((k[1:], v)) n = n or len(dics) res = [] for k in range(n): if k in dics: res.append(Poly(dict(dics[k]), gens=gens, domain=K).as_expr()) else: res.append(S.Zero) return res # Algorithms. def sdm_spoly(f, g, O, K, phantom=None): """ Compute the generalized s-polynomial of ``f`` and ``g``. The ground field is assumed to be ``K``, and monomials ordered according to ``O``. This is invalid if either of ``f`` or ``g`` is zero. If the leading terms of `f` and `g` involve different basis elements of `F`, their s-poly is defined to be zero. Otherwise it is a certain linear combination of `f` and `g` in which the leading terms cancel. See [SCA, defn 2.3.6] for details. If ``phantom`` is not ``None``, it should be a pair of module elements on which to perform the same operation(s) as on ``f`` and ``g``. The in this case both results are returned. Examples ======== >>> from sympy.polys.distributedmodules import sdm_spoly >>> from sympy.polys import QQ, lex >>> f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))] >>> g = [((2, 3, 0), QQ(1))] >>> h = [((1, 2, 3), QQ(1))] >>> sdm_spoly(f, h, lex, QQ) [] >>> sdm_spoly(f, g, lex, QQ) [((1, 2, 1), 1)] """ if not f or not g: return sdm_zero() LM1 = sdm_LM(f) LM2 = sdm_LM(g) if LM1[0] != LM2[0]: return sdm_zero() LM1 = LM1[1:] LM2 = LM2[1:] lcm = monomial_lcm(LM1, LM2) m1 = monomial_div(lcm, LM1) m2 = monomial_div(lcm, LM2) c = K.quo(-sdm_LC(f, K), sdm_LC(g, K)) r1 = sdm_add(sdm_mul_term(f, (m1, K.one), O, K), sdm_mul_term(g, (m2, c), O, K), O, K) if phantom is None: return r1 r2 = sdm_add(sdm_mul_term(phantom[0], (m1, K.one), O, K), sdm_mul_term(phantom[1], (m2, c), O, K), O, K) return r1, r2 def sdm_ecart(f): """ Compute the ecart of ``f``. This is defined to be the difference of the total degree of `f` and the total degree of the leading monomial of `f` [SCA, defn 2.3.7]. Invalid if f is zero. Examples ======== >>> from sympy.polys.distributedmodules import sdm_ecart >>> sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)]) 0 >>> sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)]) 3 """ return sdm_deg(f) - sdm_monomial_deg(sdm_LM(f)) def sdm_nf_mora(f, G, O, K, phantom=None): r""" Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``. The ground field is assumed to be ``K``, and monomials ordered according to ``O``. Weak normal forms are defined in [SCA, defn 2.3.3]. They are not unique. This function deterministically computes a weak normal form, depending on the order of `G`. The most important property of a weak normal form is the following: if `R` is the ring associated with the monomial ordering (if the ordering is global, we just have `R = K[x_1, \ldots, x_n]`, otherwise it is a certain localization thereof), `I` any ideal of `R` and `G` a standard basis for `I`, then for any `f \in R`, we have `f \in I` if and only if `NF(f | G) = 0`. This is the generalized Mora algorithm for computing weak normal forms with respect to arbitrary monomial orders [SCA, algorithm 2.3.9]. If ``phantom`` is not ``None``, it should be a pair of "phantom" arguments on which to perform the same computations as on ``f``, ``G``, both results are then returned. """ from itertools import repeat h = f T = list(G) if phantom is not None: # "phantom" variables with suffix p hp = phantom[0] Tp = list(phantom[1]) phantom = True else: Tp = repeat([]) phantom = False while h: # TODO better data structure!!! Th = [(g, sdm_ecart(g), gp) for g, gp in zip(T, Tp) if sdm_monomial_divides(sdm_LM(g), sdm_LM(h))] if not Th: break g, _, gp = min(Th, key=lambda x: x[1]) if sdm_ecart(g) > sdm_ecart(h): T.append(h) if phantom: Tp.append(hp) if phantom: h, hp = sdm_spoly(h, g, O, K, phantom=(hp, gp)) else: h = sdm_spoly(h, g, O, K) if phantom: return h, hp return h def sdm_nf_buchberger(f, G, O, K, phantom=None): r""" Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``. The ground field is assumed to be ``K``, and monomials ordered according to ``O``. This is the standard Buchberger algorithm for computing weak normal forms with respect to *global* monomial orders [SCA, algorithm 1.6.10]. If ``phantom`` is not ``None``, it should be a pair of "phantom" arguments on which to perform the same computations as on ``f``, ``G``, both results are then returned. """ from itertools import repeat h = f T = list(G) if phantom is not None: # "phantom" variables with suffix p hp = phantom[0] Tp = list(phantom[1]) phantom = True else: Tp = repeat([]) phantom = False while h: try: g, gp = next((g, gp) for g, gp in zip(T, Tp) if sdm_monomial_divides(sdm_LM(g), sdm_LM(h))) except StopIteration: break if phantom: h, hp = sdm_spoly(h, g, O, K, phantom=(hp, gp)) else: h = sdm_spoly(h, g, O, K) if phantom: return h, hp return h def sdm_nf_buchberger_reduced(f, G, O, K): r""" Compute a reduced normal form of ``f`` with respect to ``G`` and order ``O``. The ground field is assumed to be ``K``, and monomials ordered according to ``O``. In contrast to weak normal forms, reduced normal forms *are* unique, but their computation is more expensive. This is the standard Buchberger algorithm for computing reduced normal forms with respect to *global* monomial orders [SCA, algorithm 1.6.11]. The ``pantom`` option is not supported, so this normal form cannot be used as a normal form for the "extended" groebner algorithm. """ h = sdm_zero() g = f while g: g = sdm_nf_buchberger(g, G, O, K) if g: h = sdm_add(h, [sdm_LT(g)], O, K) g = g[1:] return h def sdm_groebner(G, NF, O, K, extended=False): """ Compute a minimal standard basis of ``G`` with respect to order ``O``. The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``. The ground field is assumed to be ``K``, and monomials ordered according to ``O``. Let `N` denote the submodule generated by elements of `G`. A standard basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for any subset `X` of `F`, `in(X)` denotes the submodule generated by the initial forms of elements of `X`. [SCA, defn 2.3.2] A standard basis is called minimal if no subset of it is a standard basis. One may show that standard bases are always generating sets. Minimal standard bases are not unique. This algorithm computes a deterministic result, depending on the particular order of `G`. If ``extended=True``, also compute the transition matrix from the initial generators to the groebner basis. That is, return a list of coefficient vectors, expressing the elements of the groebner basis in terms of the elements of ``G``. This functions implements the "sugar" strategy, see Giovini et al: "One sugar cube, please" OR Selection strategies in Buchberger algorithm. """ # The critical pair set. # A critical pair is stored as (i, j, s, t) where (i, j) defines the pair # (by indexing S), s is the sugar of the pair, and t is the lcm of their # leading monomials. P = [] # The eventual standard basis. S = [] Sugars = [] def Ssugar(i, j): """Compute the sugar of the S-poly corresponding to (i, j).""" LMi = sdm_LM(S[i]) LMj = sdm_LM(S[j]) return max(Sugars[i] - sdm_monomial_deg(LMi), Sugars[j] - sdm_monomial_deg(LMj)) \ + sdm_monomial_deg(sdm_monomial_lcm(LMi, LMj)) ourkey = lambda p: (p[2], O(p[3]), p[1]) def update(f, sugar, P): """Add f with sugar ``sugar`` to S, update P.""" if not f: return P k = len(S) S.append(f) Sugars.append(sugar) LMf = sdm_LM(f) def removethis(pair): i, j, s, t = pair if LMf[0] != t[0]: return False tik = sdm_monomial_lcm(LMf, sdm_LM(S[i])) tjk = sdm_monomial_lcm(LMf, sdm_LM(S[j])) return tik != t and tjk != t and sdm_monomial_divides(tik, t) and \ sdm_monomial_divides(tjk, t) # apply the chain criterion P = [p for p in P if not removethis(p)] # new-pair set N = [(i, k, Ssugar(i, k), sdm_monomial_lcm(LMf, sdm_LM(S[i]))) for i in range(k) if LMf[0] == sdm_LM(S[i])[0]] # TODO apply the product criterion? N.sort(key=ourkey) remove = set() for i, p in enumerate(N): for j in range(i + 1, len(N)): if sdm_monomial_divides(p[3], N[j][3]): remove.add(j) # TODO mergesort? P.extend(reversed([p for i, p in enumerate(N) if not i in remove])) P.sort(key=ourkey, reverse=True) # NOTE reverse-sort, because we want to pop from the end return P # Figure out the number of generators in the ground ring. try: # NOTE: we look for the first non-zero vector, take its first monomial # the number of generators in the ring is one less than the length # (since the zeroth entry is for the module generators) numgens = len(next(x[0] for x in G if x)[0]) - 1 except StopIteration: # No non-zero elements in G ... if extended: return [], [] return [] # This list will store expressions of the elements of S in terms of the # initial generators coefficients = [] # First add all the elements of G to S for i, f in enumerate(G): P = update(f, sdm_deg(f), P) if extended and f: coefficients.append(sdm_from_dict({(i,) + (0,)*numgens: K(1)}, O)) # Now carry out the buchberger algorithm. while P: i, j, s, t = P.pop() f, g = S[i], S[j] if extended: sp, coeff = sdm_spoly(f, g, O, K, phantom=(coefficients[i], coefficients[j])) h, hcoeff = NF(sp, S, O, K, phantom=(coeff, coefficients)) if h: coefficients.append(hcoeff) else: h = NF(sdm_spoly(f, g, O, K), S, O, K) P = update(h, Ssugar(i, j), P) # Finally interreduce the standard basis. # (TODO again, better data structures) S = {(tuple(f), i) for i, f in enumerate(S)} for (a, ai), (b, bi) in permutations(S, 2): A = sdm_LM(a) B = sdm_LM(b) if sdm_monomial_divides(A, B) and (b, bi) in S and (a, ai) in S: S.remove((b, bi)) L = sorted(((list(f), i) for f, i in S), key=lambda p: O(sdm_LM(p[0])), reverse=True) res = [x[0] for x in L] if extended: return res, [coefficients[i] for _, i in L] return res
8ab15e366d70fc7bc9d6db3bfb7e6fbdcb876d6ecc2a19efcd970400f4c837cd
""" Generic Unification algorithm for expression trees with lists of children This implementation is a direct translation of Artificial Intelligence: A Modern Approach by Stuart Russel and Peter Norvig Second edition, section 9.2, page 276 It is modified in the following ways: 1. We allow associative and commutative Compound expressions. This results in combinatorial blowup. 2. We explore the tree lazily. 3. We provide generic interfaces to symbolic algebra libraries in Python. A more traditional version can be found here http://aima.cs.berkeley.edu/python/logic.html """ from sympy.utilities.iterables import kbins class Compound: """ A little class to represent an interior node in the tree This is analogous to SymPy.Basic for non-Atoms """ def __init__(self, op, args): self.op = op self.args = args def __eq__(self, other): return (type(self) == type(other) and self.op == other.op and self.args == other.args) def __hash__(self): return hash((type(self), self.op, self.args)) def __str__(self): return "%s[%s]" % (str(self.op), ', '.join(map(str, self.args))) class Variable: """ A Wild token """ def __init__(self, arg): self.arg = arg def __eq__(self, other): return type(self) == type(other) and self.arg == other.arg def __hash__(self): return hash((type(self), self.arg)) def __str__(self): return "Variable(%s)" % str(self.arg) class CondVariable: """ A wild token that matches conditionally arg - a wild token valid - an additional constraining function on a match """ def __init__(self, arg, valid): self.arg = arg self.valid = valid def __eq__(self, other): return (type(self) == type(other) and self.arg == other.arg and self.valid == other.valid) def __hash__(self): return hash((type(self), self.arg, self.valid)) def __str__(self): return "CondVariable(%s)" % str(self.arg) def unify(x, y, s=None, **fns): """ Unify two expressions Parameters ========== x, y - expression trees containing leaves, Compounds and Variables s - a mapping of variables to subtrees Returns ======= lazy sequence of mappings {Variable: subtree} Examples ======== >>> from sympy.unify.core import unify, Compound, Variable >>> expr = Compound("Add", ("x", "y")) >>> pattern = Compound("Add", ("x", Variable("a"))) >>> next(unify(expr, pattern, {})) {Variable(a): 'y'} """ s = s or {} if x == y: yield s elif isinstance(x, (Variable, CondVariable)): yield from unify_var(x, y, s, **fns) elif isinstance(y, (Variable, CondVariable)): yield from unify_var(y, x, s, **fns) elif isinstance(x, Compound) and isinstance(y, Compound): is_commutative = fns.get('is_commutative', lambda x: False) is_associative = fns.get('is_associative', lambda x: False) for sop in unify(x.op, y.op, s, **fns): if is_associative(x) and is_associative(y): a, b = (x, y) if len(x.args) < len(y.args) else (y, x) if is_commutative(x) and is_commutative(y): combs = allcombinations(a.args, b.args, 'commutative') else: combs = allcombinations(a.args, b.args, 'associative') for aaargs, bbargs in combs: aa = [unpack(Compound(a.op, arg)) for arg in aaargs] bb = [unpack(Compound(b.op, arg)) for arg in bbargs] yield from unify(aa, bb, sop, **fns) elif len(x.args) == len(y.args): yield from unify(x.args, y.args, sop, **fns) elif is_args(x) and is_args(y) and len(x) == len(y): if len(x) == 0: yield s else: for shead in unify(x[0], y[0], s, **fns): yield from unify(x[1:], y[1:], shead, **fns) def unify_var(var, x, s, **fns): if var in s: yield from unify(s[var], x, s, **fns) elif occur_check(var, x): pass elif isinstance(var, CondVariable) and var.valid(x): yield assoc(s, var, x) elif isinstance(var, Variable): yield assoc(s, var, x) def occur_check(var, x): """ var occurs in subtree owned by x? """ if var == x: return True elif isinstance(x, Compound): return occur_check(var, x.args) elif is_args(x): if any(occur_check(var, xi) for xi in x): return True return False def assoc(d, key, val): """ Return copy of d with key associated to val """ d = d.copy() d[key] = val return d def is_args(x): """ Is x a traditional iterable? """ return type(x) in (tuple, list, set) def unpack(x): if isinstance(x, Compound) and len(x.args) == 1: return x.args[0] else: return x def allcombinations(A, B, ordered): """ Restructure A and B to have the same number of elements ordered must be either 'commutative' or 'associative' A and B can be rearranged so that the larger of the two lists is reorganized into smaller sublists. Examples ======== >>> from sympy.unify.core import allcombinations >>> for x in allcombinations((1, 2, 3), (5, 6), 'associative'): print(x) (((1,), (2, 3)), ((5,), (6,))) (((1, 2), (3,)), ((5,), (6,))) >>> for x in allcombinations((1, 2, 3), (5, 6), 'commutative'): print(x) (((1,), (2, 3)), ((5,), (6,))) (((1, 2), (3,)), ((5,), (6,))) (((1,), (3, 2)), ((5,), (6,))) (((1, 3), (2,)), ((5,), (6,))) (((2,), (1, 3)), ((5,), (6,))) (((2, 1), (3,)), ((5,), (6,))) (((2,), (3, 1)), ((5,), (6,))) (((2, 3), (1,)), ((5,), (6,))) (((3,), (1, 2)), ((5,), (6,))) (((3, 1), (2,)), ((5,), (6,))) (((3,), (2, 1)), ((5,), (6,))) (((3, 2), (1,)), ((5,), (6,))) """ if ordered == "commutative": ordered = 11 if ordered == "associative": ordered = None sm, bg = (A, B) if len(A) < len(B) else (B, A) for part in kbins(list(range(len(bg))), len(sm), ordered=ordered): if bg == B: yield tuple((a,) for a in A), partition(B, part) else: yield partition(A, part), tuple((b,) for b in B) def partition(it, part): """ Partition a tuple/list into pieces defined by indices Examples ======== >>> from sympy.unify.core import partition >>> partition((10, 20, 30, 40), [[0, 1, 2], [3]]) ((10, 20, 30), (40,)) """ return type(it)([index(it, ind) for ind in part]) def index(it, ind): """ Fancy indexing into an indexable iterable (tuple, list) Examples ======== >>> from sympy.unify.core import index >>> index([10, 20, 30], (1, 2, 0)) [20, 30, 10] """ return type(it)([it[i] for i in ind])
ccef755d57617b77ed0d394755fd5c2e1e1340c1ef13f2059ea2b125eea3ac42
""" SymPy interface to Unification engine See sympy.unify for module level docstring See sympy.unify.core for algorithmic docstring """ from sympy.core import Basic, Add, Mul, Pow from sympy.core.operations import AssocOp, LatticeOp from sympy.matrices import MatAdd, MatMul, MatrixExpr from sympy.sets.sets import Union, Intersection, FiniteSet from sympy.unify.core import Compound, Variable, CondVariable from sympy.unify import core basic_new_legal = [MatrixExpr] eval_false_legal = [AssocOp, Pow, FiniteSet] illegal = [LatticeOp] def sympy_associative(op): assoc_ops = (AssocOp, MatAdd, MatMul, Union, Intersection, FiniteSet) return any(issubclass(op, aop) for aop in assoc_ops) def sympy_commutative(op): comm_ops = (Add, MatAdd, Union, Intersection, FiniteSet) return any(issubclass(op, cop) for cop in comm_ops) def is_associative(x): return isinstance(x, Compound) and sympy_associative(x.op) def is_commutative(x): if not isinstance(x, Compound): return False if sympy_commutative(x.op): return True if issubclass(x.op, Mul): return all(construct(arg).is_commutative for arg in x.args) def mk_matchtype(typ): def matchtype(x): return (isinstance(x, typ) or isinstance(x, Compound) and issubclass(x.op, typ)) return matchtype def deconstruct(s, variables=()): """ Turn a SymPy object into a Compound """ if s in variables: return Variable(s) if isinstance(s, (Variable, CondVariable)): return s if not isinstance(s, Basic) or s.is_Atom: return s return Compound(s.__class__, tuple(deconstruct(arg, variables) for arg in s.args)) def construct(t): """ Turn a Compound into a SymPy object """ if isinstance(t, (Variable, CondVariable)): return t.arg if not isinstance(t, Compound): return t if any(issubclass(t.op, cls) for cls in eval_false_legal): return t.op(*map(construct, t.args), evaluate=False) elif any(issubclass(t.op, cls) for cls in basic_new_legal): return Basic.__new__(t.op, *map(construct, t.args)) else: return t.op(*map(construct, t.args)) def rebuild(s): """ Rebuild a SymPy expression This removes harm caused by Expr-Rules interactions """ return construct(deconstruct(s)) def unify(x, y, s=None, variables=(), **kwargs): """ Structural unification of two expressions/patterns Examples ======== >>> from sympy.unify.usympy import unify >>> from sympy import Basic >>> from sympy.abc import x, y, z, p, q >>> next(unify(Basic(1, 2), Basic(1, x), variables=[x])) {x: 2} >>> expr = 2*x + y + z >>> pattern = 2*p + q >>> next(unify(expr, pattern, {}, variables=(p, q))) {p: x, q: y + z} Unification supports commutative and associative matching >>> expr = x + y + z >>> pattern = p + q >>> len(list(unify(expr, pattern, {}, variables=(p, q)))) 12 Symbols not indicated to be variables are treated as literal, else they are wild-like and match anything in a sub-expression. >>> expr = x*y*z + 3 >>> pattern = x*y + 3 >>> next(unify(expr, pattern, {}, variables=[x, y])) {x: y, y: x*z} The x and y of the pattern above were in a Mul and matched factors in the Mul of expr. Here, a single symbol matches an entire term: >>> expr = x*y + 3 >>> pattern = p + 3 >>> next(unify(expr, pattern, {}, variables=[p])) {p: x*y} """ decons = lambda x: deconstruct(x, variables) s = s or {} s = {decons(k): decons(v) for k, v in s.items()} ds = core.unify(decons(x), decons(y), s, is_associative=is_associative, is_commutative=is_commutative, **kwargs) for d in ds: yield {construct(k): construct(v) for k, v in d.items()}
9fd129ceeadbd60fb3ef32048498ea4ead18378acc2d94b50ea703020d3edbc2
""" Functions to support rewriting of SymPy expressions """ from sympy import Expr from sympy.assumptions import ask from sympy.strategies.tools import subs from sympy.unify.usympy import rebuild, unify def rewriterule(source, target, variables=(), condition=None, assume=None): """ Rewrite rule Transform expressions that match source into expressions that match target treating all `variables` as wilds. Examples ======== >>> from sympy.abc import w, x, y, z >>> from sympy.unify.rewrite import rewriterule >>> from sympy.utilities import default_sort_key >>> rl = rewriterule(x + y, x**y, [x, y]) >>> sorted(rl(z + 3), key=default_sort_key) [3**z, z**3] Use ``condition`` to specify additional requirements. Inputs are taken in the same order as is found in variables. >>> rl = rewriterule(x + y, x**y, [x, y], lambda x, y: x.is_integer) >>> list(rl(z + 3)) [3**z] Use ``assume`` to specify additional requirements using new assumptions. >>> from sympy.assumptions import Q >>> rl = rewriterule(x + y, x**y, [x, y], assume=Q.integer(x)) >>> list(rl(z + 3)) [3**z] Assumptions for the local context are provided at rule runtime >>> list(rl(w + z, Q.integer(z))) [z**w] """ def rewrite_rl(expr, assumptions=True): for match in unify(source, expr, {}, variables=variables): if (condition and not condition(*[match.get(var, var) for var in variables])): continue if (assume and not ask(assume.xreplace(match), assumptions)): continue expr2 = subs(match)(target) if isinstance(expr2, Expr): expr2 = rebuild(expr2) yield expr2 return rewrite_rl
5b70f0308b54213de082bfb422b967b61357167769bed26e5383ce5737cc8d51
"""py.test hacks to support XFAIL/XPASS""" import sys import functools import os import contextlib import warnings from sympy.core.compatibility import get_function_name from sympy.utilities.exceptions import SymPyDeprecationWarning ON_TRAVIS = os.getenv('TRAVIS_BUILD_NUMBER', None) try: import pytest USE_PYTEST = getattr(sys, '_running_pytest', False) except ImportError: USE_PYTEST = False if USE_PYTEST: raises = pytest.raises warns = pytest.warns skip = pytest.skip XFAIL = pytest.mark.xfail SKIP = pytest.mark.skip slow = pytest.mark.slow nocache_fail = pytest.mark.nocache_fail from _pytest.outcomes import Failed else: # Not using pytest so define the things that would have been imported from # there. # _pytest._code.code.ExceptionInfo class ExceptionInfo: def __init__(self, value): self.value = value def __repr__(self): return "<ExceptionInfo {!r}>".format(self.value) def raises(expectedException, code=None): """ Tests that ``code`` raises the exception ``expectedException``. ``code`` may be a callable, such as a lambda expression or function name. If ``code`` is not given or None, ``raises`` will return a context manager for use in ``with`` statements; the code to execute then comes from the scope of the ``with``. ``raises()`` does nothing if the callable raises the expected exception, otherwise it raises an AssertionError. Examples ======== >>> from sympy.testing.pytest import raises >>> raises(ZeroDivisionError, lambda: 1/0) <ExceptionInfo ZeroDivisionError(...)> >>> raises(ZeroDivisionError, lambda: 1/2) Traceback (most recent call last): ... Failed: DID NOT RAISE >>> with raises(ZeroDivisionError): ... n = 1/0 >>> with raises(ZeroDivisionError): ... n = 1/2 Traceback (most recent call last): ... Failed: DID NOT RAISE Note that you cannot test multiple statements via ``with raises``: >>> with raises(ZeroDivisionError): ... n = 1/0 # will execute and raise, aborting the ``with`` ... n = 9999/0 # never executed This is just what ``with`` is supposed to do: abort the contained statement sequence at the first exception and let the context manager deal with the exception. To test multiple statements, you'll need a separate ``with`` for each: >>> with raises(ZeroDivisionError): ... n = 1/0 # will execute and raise >>> with raises(ZeroDivisionError): ... n = 9999/0 # will also execute and raise """ if code is None: return RaisesContext(expectedException) elif callable(code): try: code() except expectedException as e: return ExceptionInfo(e) raise Failed("DID NOT RAISE") elif isinstance(code, str): raise TypeError( '\'raises(xxx, "code")\' has been phased out; ' 'change \'raises(xxx, "expression")\' ' 'to \'raises(xxx, lambda: expression)\', ' '\'raises(xxx, "statement")\' ' 'to \'with raises(xxx): statement\'') else: raise TypeError( 'raises() expects a callable for the 2nd argument.') class RaisesContext: def __init__(self, expectedException): self.expectedException = expectedException def __enter__(self): return None def __exit__(self, exc_type, exc_value, traceback): if exc_type is None: raise Failed("DID NOT RAISE") return issubclass(exc_type, self.expectedException) class XFail(Exception): pass class XPass(Exception): pass class Skipped(Exception): pass class Failed(Exception): # type: ignore pass def XFAIL(func): def wrapper(): try: func() except Exception as e: message = str(e) if message != "Timeout": raise XFail(get_function_name(func)) else: raise Skipped("Timeout") raise XPass(get_function_name(func)) wrapper = functools.update_wrapper(wrapper, func) return wrapper def skip(str): raise Skipped(str) def SKIP(reason): """Similar to ``skip()``, but this is a decorator. """ def wrapper(func): def func_wrapper(): raise Skipped(reason) func_wrapper = functools.update_wrapper(func_wrapper, func) return func_wrapper return wrapper def slow(func): func._slow = True def func_wrapper(): func() func_wrapper = functools.update_wrapper(func_wrapper, func) func_wrapper.__wrapped__ = func return func_wrapper def nocache_fail(func): "Dummy decorator for marking tests that fail when cache is disabled" return func @contextlib.contextmanager def warns(warningcls, *, match=''): '''Like raises but tests that warnings are emitted. >>> from sympy.testing.pytest import warns >>> import warnings >>> with warns(UserWarning): ... warnings.warn('deprecated', UserWarning) >>> with warns(UserWarning): ... pass Traceback (most recent call last): ... Failed: DID NOT WARN. No warnings of type UserWarning\ was emitted. The list of emitted warnings is: []. ''' # Absorbs all warnings in warnrec with warnings.catch_warnings(record=True) as warnrec: # Hide all warnings but make sure that our warning is emitted warnings.simplefilter("ignore") warnings.filterwarnings("always", match, warningcls) # Now run the test yield # Raise if expected warning not found if not any(issubclass(w.category, warningcls) for w in warnrec): msg = ('Failed: DID NOT WARN.' ' No warnings of type %s was emitted.' ' The list of emitted warnings is: %s.' ) % (warningcls, [w.message for w in warnrec]) raise Failed(msg) @contextlib.contextmanager def warns_deprecated_sympy(): '''Shorthand for ``warns(SymPyDeprecationWarning)`` This is the recommended way to test that ``SymPyDeprecationWarning`` is emitted for deprecated features in SymPy. To test for other warnings use ``warns``. To suppress warnings without asserting that they are emitted use ``ignore_warnings``. >>> from sympy.testing.pytest import warns_deprecated_sympy >>> from sympy.utilities.exceptions import SymPyDeprecationWarning >>> with warns_deprecated_sympy(): ... SymPyDeprecationWarning("Don't use", feature="old thing", ... deprecated_since_version="1.0", issue=123).warn() >>> with warns_deprecated_sympy(): ... pass Traceback (most recent call last): ... Failed: DID NOT WARN. No warnings of type \ SymPyDeprecationWarning was emitted. The list of emitted warnings is: []. ''' with warns(SymPyDeprecationWarning): yield @contextlib.contextmanager def ignore_warnings(warningcls): '''Context manager to suppress warnings during tests. This function is useful for suppressing warnings during tests. The warns function should be used to assert that a warning is raised. The ignore_warnings function is useful in situation when the warning is not guaranteed to be raised (e.g. on importing a module) or if the warning comes from third-party code. When the warning is coming (reliably) from SymPy the warns function should be preferred to ignore_warnings. >>> from sympy.testing.pytest import ignore_warnings >>> import warnings Here's a warning: >>> with warnings.catch_warnings(): # reset warnings in doctest ... warnings.simplefilter('error') ... warnings.warn('deprecated', UserWarning) Traceback (most recent call last): ... UserWarning: deprecated Let's suppress it with ignore_warnings: >>> with warnings.catch_warnings(): # reset warnings in doctest ... warnings.simplefilter('error') ... with ignore_warnings(UserWarning): ... warnings.warn('deprecated', UserWarning) (No warning emitted) ''' # Absorbs all warnings in warnrec with warnings.catch_warnings(record=True) as warnrec: # Make sure our warning doesn't get filtered warnings.simplefilter("always", warningcls) # Now run the test yield # Reissue any warnings that we aren't testing for for w in warnrec: if not issubclass(w.category, warningcls): warnings.warn_explicit(w.message, w.category, w.filename, w.lineno)
b6001dd9f5529863bebaa13052639d4faf5fceabc4846e3699ba5dbdf6734528
"""benchmarking through py.test""" import py from py.__.test.item import Item from py.__.test.terminal.terminal import TerminalSession from math import ceil as _ceil, floor as _floor, log10 import timeit from inspect import getsource from sympy.core.compatibility import exec_ # from IPython.Magic.magic_timeit units = ["s", "ms", "us", "ns"] scaling = [1, 1e3, 1e6, 1e9] unitn = {s: i for i, s in enumerate(units)} precision = 3 # like py.test Directory but scan for 'bench_<smth>.py' class Directory(py.test.collect.Directory): def filefilter(self, path): b = path.purebasename ext = path.ext return b.startswith('bench_') and ext == '.py' # like py.test Module but scane for 'bench_<smth>' and 'timeit_<smth>' class Module(py.test.collect.Module): def funcnamefilter(self, name): return name.startswith('bench_') or name.startswith('timeit_') # Function level benchmarking driver class Timer(timeit.Timer): def __init__(self, stmt, setup='pass', timer=timeit.default_timer, globals=globals()): # copy of timeit.Timer.__init__ # similarity index 95% self.timer = timer stmt = timeit.reindent(stmt, 8) setup = timeit.reindent(setup, 4) src = timeit.template % {'stmt': stmt, 'setup': setup} self.src = src # Save for traceback display code = compile(src, timeit.dummy_src_name, "exec") ns = {} #exec code in globals(), ns -- original timeit code exec_(code, globals, ns) # -- we use caller-provided globals instead self.inner = ns["inner"] class Function(py.__.test.item.Function): def __init__(self, *args, **kw): super().__init__(*args, **kw) self.benchtime = None self.benchtitle = None def execute(self, target, *args): # get func source without first 'def func(...):' line src = getsource(target) src = '\n'.join( src.splitlines()[1:] ) # extract benchmark title if target.func_doc is not None: self.benchtitle = target.func_doc else: self.benchtitle = src.splitlines()[0].strip() # XXX we ignore args timer = Timer(src, globals=target.func_globals) if self.name.startswith('timeit_'): # from IPython.Magic.magic_timeit repeat = 3 number = 1 for i in range(1, 10): t = timer.timeit(number) if t >= 0.2: number *= (0.2 / t) number = int(_ceil(number)) break if t <= 0.02: # we are not close enough to that 0.2s number *= 10 else: # since we are very close to be > 0.2s we'd better adjust number # so that timing time is not too high number *= (0.2 / t) number = int(_ceil(number)) break self.benchtime = min(timer.repeat(repeat, number)) / number # 'bench_<smth>' else: self.benchtime = timer.timeit(1) class BenchSession(TerminalSession): def header(self, colitems): super().header(colitems) def footer(self, colitems): super().footer(colitems) self.out.write('\n') self.print_bench_results() def print_bench_results(self): self.out.write('==============================\n') self.out.write(' *** BENCHMARKING RESULTS *** \n') self.out.write('==============================\n') self.out.write('\n') # benchname, time, benchtitle results = [] for item, outcome in self._memo: if isinstance(item, Item): best = item.benchtime if best is None: # skipped or failed benchmarks tstr = '---' else: # from IPython.Magic.magic_timeit if best > 0.0: order = min(-int(_floor(log10(best)) // 3), 3) else: order = 3 tstr = "%.*g %s" % ( precision, best * scaling[order], units[order]) results.append( [item.name, tstr, item.benchtitle] ) # dot/unit align second column # FIXME simpler? this is crappy -- shame on me... wm = [0]*len(units) we = [0]*len(units) for s in results: tstr = s[1] n, u = tstr.split() # unit n un = unitn[u] try: m, e = n.split('.') except ValueError: m, e = n, '' wm[un] = max(len(m), wm[un]) we[un] = max(len(e), we[un]) for s in results: tstr = s[1] n, u = tstr.split() un = unitn[u] try: m, e = n.split('.') except ValueError: m, e = n, '' m = m.rjust(wm[un]) e = e.ljust(we[un]) if e.strip(): n = '.'.join((m, e)) else: n = ' '.join((m, e)) # let's put the number into the right place txt = '' for i in range(len(units)): if i == un: txt += n else: txt += ' '*(wm[i] + we[i] + 1) s[1] = '%s %s' % (txt, u) # align all columns besides the last one for i in range(2): w = max(len(s[i]) for s in results) for s in results: s[i] = s[i].ljust(w) # show results for s in results: self.out.write('%s | %s | %s\n' % tuple(s)) def main(args=None): # hook our Directory/Module/Function as defaults from py.__.test import defaultconftest defaultconftest.Directory = Directory defaultconftest.Module = Module defaultconftest.Function = Function # hook BenchSession as py.test session config = py.test.config config._getsessionclass = lambda: BenchSession py.test.cmdline.main(args)
b15e02c0ecae9d9e101dc0aa63ef8fa7503d9201082100037f7fb392d41440e6
""" This is our testing framework. Goals: * it should be compatible with py.test and operate very similarly (or identically) * doesn't require any external dependencies * preferably all the functionality should be in this file only * no magic, just import the test file and execute the test functions, that's it * portable """ from __future__ import print_function, division import os import sys import platform import inspect import traceback import pdb import re import linecache import time from fnmatch import fnmatch from timeit import default_timer as clock import doctest as pdoctest # avoid clashing with our doctest() function from doctest import DocTestFinder, DocTestRunner import random import subprocess import shutil import signal import stat import tempfile import warnings from contextlib import contextmanager from sympy.core.cache import clear_cache from sympy.core.compatibility import (exec_, PY3, unwrap) from sympy.external import import_module IS_WINDOWS = (os.name == 'nt') ON_TRAVIS = os.getenv('TRAVIS_BUILD_NUMBER', None) # emperically generated list of the proportion of time spent running # an even split of tests. This should periodically be regenerated. # A list of [.6, .1, .3] would mean that if the tests are evenly split # into '1/3', '2/3', '3/3', the first split would take 60% of the time, # the second 10% and the third 30%. These lists are normalized to sum # to 1, so [60, 10, 30] has the same behavior as [6, 1, 3] or [.6, .1, .3]. # # This list can be generated with the code: # from time import time # import sympy # import os # os.environ["TRAVIS_BUILD_NUMBER"] = '2' # Mock travis to get more correct densities # delays, num_splits = [], 30 # for i in range(1, num_splits + 1): # tic = time() # sympy.test(split='{}/{}'.format(i, num_splits), time_balance=False) # Add slow=True for slow tests # delays.append(time() - tic) # tot = sum(delays) # print([round(x / tot, 4) for x in delays]) SPLIT_DENSITY = [ 0.0059, 0.0027, 0.0068, 0.0011, 0.0006, 0.0058, 0.0047, 0.0046, 0.004, 0.0257, 0.0017, 0.0026, 0.004, 0.0032, 0.0016, 0.0015, 0.0004, 0.0011, 0.0016, 0.0014, 0.0077, 0.0137, 0.0217, 0.0074, 0.0043, 0.0067, 0.0236, 0.0004, 0.1189, 0.0142, 0.0234, 0.0003, 0.0003, 0.0047, 0.0006, 0.0013, 0.0004, 0.0008, 0.0007, 0.0006, 0.0139, 0.0013, 0.0007, 0.0051, 0.002, 0.0004, 0.0005, 0.0213, 0.0048, 0.0016, 0.0012, 0.0014, 0.0024, 0.0015, 0.0004, 0.0005, 0.0007, 0.011, 0.0062, 0.0015, 0.0021, 0.0049, 0.0006, 0.0006, 0.0011, 0.0006, 0.0019, 0.003, 0.0044, 0.0054, 0.0057, 0.0049, 0.0016, 0.0006, 0.0009, 0.0006, 0.0012, 0.0006, 0.0149, 0.0532, 0.0076, 0.0041, 0.0024, 0.0135, 0.0081, 0.2209, 0.0459, 0.0438, 0.0488, 0.0137, 0.002, 0.0003, 0.0008, 0.0039, 0.0024, 0.0005, 0.0004, 0.003, 0.056, 0.0026] SPLIT_DENSITY_SLOW = [0.0086, 0.0004, 0.0568, 0.0003, 0.0032, 0.0005, 0.0004, 0.0013, 0.0016, 0.0648, 0.0198, 0.1285, 0.098, 0.0005, 0.0064, 0.0003, 0.0004, 0.0026, 0.0007, 0.0051, 0.0089, 0.0024, 0.0033, 0.0057, 0.0005, 0.0003, 0.001, 0.0045, 0.0091, 0.0006, 0.0005, 0.0321, 0.0059, 0.1105, 0.216, 0.1489, 0.0004, 0.0003, 0.0006, 0.0483] class Skipped(Exception): pass class TimeOutError(Exception): pass class DependencyError(Exception): pass # add more flags ?? future_flags = division.compiler_flag def _indent(s, indent=4): """ Add the given number of space characters to the beginning of every non-blank line in ``s``, and return the result. If the string ``s`` is Unicode, it is encoded using the stdout encoding and the ``backslashreplace`` error handler. """ # This regexp matches the start of non-blank lines: return re.sub('(?m)^(?!$)', indent*' ', s) pdoctest._indent = _indent # type: ignore # override reporter to maintain windows and python3 def _report_failure(self, out, test, example, got): """ Report that the given example failed. """ s = self._checker.output_difference(example, got, self.optionflags) s = s.encode('raw_unicode_escape').decode('utf8', 'ignore') out(self._failure_header(test, example) + s) if PY3 and IS_WINDOWS: DocTestRunner.report_failure = _report_failure # type: ignore def convert_to_native_paths(lst): """ Converts a list of '/' separated paths into a list of native (os.sep separated) paths and converts to lowercase if the system is case insensitive. """ newlst = [] for i, rv in enumerate(lst): rv = os.path.join(*rv.split("/")) # on windows the slash after the colon is dropped if sys.platform == "win32": pos = rv.find(':') if pos != -1: if rv[pos + 1] != '\\': rv = rv[:pos + 1] + '\\' + rv[pos + 1:] newlst.append(os.path.normcase(rv)) return newlst def get_sympy_dir(): """ Returns the root sympy directory and set the global value indicating whether the system is case sensitive or not. """ this_file = os.path.abspath(__file__) sympy_dir = os.path.join(os.path.dirname(this_file), "..", "..") sympy_dir = os.path.normpath(sympy_dir) return os.path.normcase(sympy_dir) def setup_pprint(): from sympy import pprint_use_unicode, init_printing import sympy.interactive.printing as interactive_printing # force pprint to be in ascii mode in doctests use_unicode_prev = pprint_use_unicode(False) # hook our nice, hash-stable strprinter init_printing(pretty_print=False) # Prevent init_printing() in doctests from affecting other doctests interactive_printing.NO_GLOBAL = True return use_unicode_prev @contextmanager def raise_on_deprecated(): """Context manager to make DeprecationWarning raise an error This is to catch SymPyDeprecationWarning from library code while running tests and doctests. It is important to use this context manager around each individual test/doctest in case some tests modify the warning filters. """ with warnings.catch_warnings(): warnings.filterwarnings('error', '.*', DeprecationWarning, module='sympy.*') yield def run_in_subprocess_with_hash_randomization( function, function_args=(), function_kwargs=None, command=sys.executable, module='sympy.testing.runtests', force=False): """ Run a function in a Python subprocess with hash randomization enabled. If hash randomization is not supported by the version of Python given, it returns False. Otherwise, it returns the exit value of the command. The function is passed to sys.exit(), so the return value of the function will be the return value. The environment variable PYTHONHASHSEED is used to seed Python's hash randomization. If it is set, this function will return False, because starting a new subprocess is unnecessary in that case. If it is not set, one is set at random, and the tests are run. Note that if this environment variable is set when Python starts, hash randomization is automatically enabled. To force a subprocess to be created even if PYTHONHASHSEED is set, pass ``force=True``. This flag will not force a subprocess in Python versions that do not support hash randomization (see below), because those versions of Python do not support the ``-R`` flag. ``function`` should be a string name of a function that is importable from the module ``module``, like "_test". The default for ``module`` is "sympy.testing.runtests". ``function_args`` and ``function_kwargs`` should be a repr-able tuple and dict, respectively. The default Python command is sys.executable, which is the currently running Python command. This function is necessary because the seed for hash randomization must be set by the environment variable before Python starts. Hence, in order to use a predetermined seed for tests, we must start Python in a separate subprocess. Hash randomization was added in the minor Python versions 2.6.8, 2.7.3, 3.1.5, and 3.2.3, and is enabled by default in all Python versions after and including 3.3.0. Examples ======== >>> from sympy.testing.runtests import ( ... run_in_subprocess_with_hash_randomization) >>> # run the core tests in verbose mode >>> run_in_subprocess_with_hash_randomization("_test", ... function_args=("core",), ... function_kwargs={'verbose': True}) # doctest: +SKIP # Will return 0 if sys.executable supports hash randomization and tests # pass, 1 if they fail, and False if it does not support hash # randomization. """ cwd = get_sympy_dir() # Note, we must return False everywhere, not None, as subprocess.call will # sometimes return None. # First check if the Python version supports hash randomization # If it doesn't have this support, it won't recognize the -R flag p = subprocess.Popen([command, "-RV"], stdout=subprocess.PIPE, stderr=subprocess.STDOUT, cwd=cwd) p.communicate() if p.returncode != 0: return False hash_seed = os.getenv("PYTHONHASHSEED") if not hash_seed: os.environ["PYTHONHASHSEED"] = str(random.randrange(2**32)) else: if not force: return False function_kwargs = function_kwargs or {} # Now run the command commandstring = ("import sys; from %s import %s;sys.exit(%s(*%s, **%s))" % (module, function, function, repr(function_args), repr(function_kwargs))) try: p = subprocess.Popen([command, "-R", "-c", commandstring], cwd=cwd) p.communicate() except KeyboardInterrupt: p.wait() finally: # Put the environment variable back, so that it reads correctly for # the current Python process. if hash_seed is None: del os.environ["PYTHONHASHSEED"] else: os.environ["PYTHONHASHSEED"] = hash_seed return p.returncode def run_all_tests(test_args=(), test_kwargs=None, doctest_args=(), doctest_kwargs=None, examples_args=(), examples_kwargs=None): """ Run all tests. Right now, this runs the regular tests (bin/test), the doctests (bin/doctest), the examples (examples/all.py), and the sage tests (see sympy/external/tests/test_sage.py). This is what ``setup.py test`` uses. You can pass arguments and keyword arguments to the test functions that support them (for now, test, doctest, and the examples). See the docstrings of those functions for a description of the available options. For example, to run the solvers tests with colors turned off: >>> from sympy.testing.runtests import run_all_tests >>> run_all_tests(test_args=("solvers",), ... test_kwargs={"colors:False"}) # doctest: +SKIP """ cwd = get_sympy_dir() tests_successful = True test_kwargs = test_kwargs or {} doctest_kwargs = doctest_kwargs or {} examples_kwargs = examples_kwargs or {'quiet': True} try: # Regular tests if not test(*test_args, **test_kwargs): # some regular test fails, so set the tests_successful # flag to false and continue running the doctests tests_successful = False # Doctests print() if not doctest(*doctest_args, **doctest_kwargs): tests_successful = False # Examples print() sys.path.append("examples") # examples/all.py from all import run_examples # type: ignore if not run_examples(*examples_args, **examples_kwargs): tests_successful = False # Sage tests if sys.platform != "win32" and not PY3 and os.path.exists("bin/test"): # run Sage tests; Sage currently doesn't support Windows or Python 3 # Only run Sage tests if 'bin/test' is present (it is missing from # our release because everything in the 'bin' directory gets # installed). dev_null = open(os.devnull, 'w') if subprocess.call("sage -v", shell=True, stdout=dev_null, stderr=dev_null) == 0: if subprocess.call("sage -python bin/test " "sympy/external/tests/test_sage.py", shell=True, cwd=cwd) != 0: tests_successful = False if tests_successful: return else: # Return nonzero exit code sys.exit(1) except KeyboardInterrupt: print() print("DO *NOT* COMMIT!") sys.exit(1) def test(*paths, subprocess=True, rerun=0, **kwargs): """ Run tests in the specified test_*.py files. Tests in a particular test_*.py file are run if any of the given strings in ``paths`` matches a part of the test file's path. If ``paths=[]``, tests in all test_*.py files are run. Notes: - If sort=False, tests are run in random order (not default). - Paths can be entered in native system format or in unix, forward-slash format. - Files that are on the blacklist can be tested by providing their path; they are only excluded if no paths are given. **Explanation of test results** ====== =============================================================== Output Meaning ====== =============================================================== . passed F failed X XPassed (expected to fail but passed) f XFAILed (expected to fail and indeed failed) s skipped w slow T timeout (e.g., when ``--timeout`` is used) K KeyboardInterrupt (when running the slow tests with ``--slow``, you can interrupt one of them without killing the test runner) ====== =============================================================== Colors have no additional meaning and are used just to facilitate interpreting the output. Examples ======== >>> import sympy Run all tests: >>> sympy.test() # doctest: +SKIP Run one file: >>> sympy.test("sympy/core/tests/test_basic.py") # doctest: +SKIP >>> sympy.test("_basic") # doctest: +SKIP Run all tests in sympy/functions/ and some particular file: >>> sympy.test("sympy/core/tests/test_basic.py", ... "sympy/functions") # doctest: +SKIP Run all tests in sympy/core and sympy/utilities: >>> sympy.test("/core", "/util") # doctest: +SKIP Run specific test from a file: >>> sympy.test("sympy/core/tests/test_basic.py", ... kw="test_equality") # doctest: +SKIP Run specific test from any file: >>> sympy.test(kw="subs") # doctest: +SKIP Run the tests with verbose mode on: >>> sympy.test(verbose=True) # doctest: +SKIP Don't sort the test output: >>> sympy.test(sort=False) # doctest: +SKIP Turn on post-mortem pdb: >>> sympy.test(pdb=True) # doctest: +SKIP Turn off colors: >>> sympy.test(colors=False) # doctest: +SKIP Force colors, even when the output is not to a terminal (this is useful, e.g., if you are piping to ``less -r`` and you still want colors) >>> sympy.test(force_colors=False) # doctest: +SKIP The traceback verboseness can be set to "short" or "no" (default is "short") >>> sympy.test(tb='no') # doctest: +SKIP The ``split`` option can be passed to split the test run into parts. The split currently only splits the test files, though this may change in the future. ``split`` should be a string of the form 'a/b', which will run part ``a`` of ``b``. For instance, to run the first half of the test suite: >>> sympy.test(split='1/2') # doctest: +SKIP The ``time_balance`` option can be passed in conjunction with ``split``. If ``time_balance=True`` (the default for ``sympy.test``), sympy will attempt to split the tests such that each split takes equal time. This heuristic for balancing is based on pre-recorded test data. >>> sympy.test(split='1/2', time_balance=True) # doctest: +SKIP You can disable running the tests in a separate subprocess using ``subprocess=False``. This is done to support seeding hash randomization, which is enabled by default in the Python versions where it is supported. If subprocess=False, hash randomization is enabled/disabled according to whether it has been enabled or not in the calling Python process. However, even if it is enabled, the seed cannot be printed unless it is called from a new Python process. Hash randomization was added in the minor Python versions 2.6.8, 2.7.3, 3.1.5, and 3.2.3, and is enabled by default in all Python versions after and including 3.3.0. If hash randomization is not supported ``subprocess=False`` is used automatically. >>> sympy.test(subprocess=False) # doctest: +SKIP To set the hash randomization seed, set the environment variable ``PYTHONHASHSEED`` before running the tests. This can be done from within Python using >>> import os >>> os.environ['PYTHONHASHSEED'] = '42' # doctest: +SKIP Or from the command line using $ PYTHONHASHSEED=42 ./bin/test If the seed is not set, a random seed will be chosen. Note that to reproduce the same hash values, you must use both the same seed as well as the same architecture (32-bit vs. 64-bit). """ # count up from 0, do not print 0 print_counter = lambda i : (print("rerun %d" % (rerun-i)) if rerun-i else None) if subprocess: # loop backwards so last i is 0 for i in range(rerun, -1, -1): print_counter(i) ret = run_in_subprocess_with_hash_randomization("_test", function_args=paths, function_kwargs=kwargs) if ret is False: break val = not bool(ret) # exit on the first failure or if done if not val or i == 0: return val # rerun even if hash randomization is not supported for i in range(rerun, -1, -1): print_counter(i) val = not bool(_test(*paths, **kwargs)) if not val or i == 0: return val def _test(*paths, verbose=False, tb="short", kw=None, pdb=False, colors=True, force_colors=False, sort=True, seed=None, timeout=False, fail_on_timeout=False, slow=False, enhance_asserts=False, split=None, time_balance=True, blacklist=('sympy/integrals/rubi/rubi_tests/tests',), fast_threshold=None, slow_threshold=None): """ Internal function that actually runs the tests. All keyword arguments from ``test()`` are passed to this function except for ``subprocess``. Returns 0 if tests passed and 1 if they failed. See the docstring of ``test()`` for more information. """ kw = kw or () # ensure that kw is a tuple if isinstance(kw, str): kw = (kw,) post_mortem = pdb if seed is None: seed = random.randrange(100000000) if ON_TRAVIS and timeout is False: # Travis times out if no activity is seen for 10 minutes. timeout = 595 fail_on_timeout = True if ON_TRAVIS: # pyglet does not work on Travis blacklist = list(blacklist) + ['sympy/plotting/pygletplot/tests'] blacklist = convert_to_native_paths(blacklist) r = PyTestReporter(verbose=verbose, tb=tb, colors=colors, force_colors=force_colors, split=split) t = SymPyTests(r, kw, post_mortem, seed, fast_threshold=fast_threshold, slow_threshold=slow_threshold) test_files = t.get_test_files('sympy') not_blacklisted = [f for f in test_files if not any(b in f for b in blacklist)] if len(paths) == 0: matched = not_blacklisted else: paths = convert_to_native_paths(paths) matched = [] for f in not_blacklisted: basename = os.path.basename(f) for p in paths: if p in f or fnmatch(basename, p): matched.append(f) break density = None if time_balance: if slow: density = SPLIT_DENSITY_SLOW else: density = SPLIT_DENSITY if split: matched = split_list(matched, split, density=density) t._testfiles.extend(matched) return int(not t.test(sort=sort, timeout=timeout, slow=slow, enhance_asserts=enhance_asserts, fail_on_timeout=fail_on_timeout)) def doctest(*paths, subprocess=True, rerun=0, **kwargs): r""" Runs doctests in all \*.py files in the sympy directory which match any of the given strings in ``paths`` or all tests if paths=[]. Notes: - Paths can be entered in native system format or in unix, forward-slash format. - Files that are on the blacklist can be tested by providing their path; they are only excluded if no paths are given. Examples ======== >>> import sympy Run all tests: >>> sympy.doctest() # doctest: +SKIP Run one file: >>> sympy.doctest("sympy/core/basic.py") # doctest: +SKIP >>> sympy.doctest("polynomial.rst") # doctest: +SKIP Run all tests in sympy/functions/ and some particular file: >>> sympy.doctest("/functions", "basic.py") # doctest: +SKIP Run any file having polynomial in its name, doc/src/modules/polynomial.rst, sympy/functions/special/polynomials.py, and sympy/polys/polynomial.py: >>> sympy.doctest("polynomial") # doctest: +SKIP The ``split`` option can be passed to split the test run into parts. The split currently only splits the test files, though this may change in the future. ``split`` should be a string of the form 'a/b', which will run part ``a`` of ``b``. Note that the regular doctests and the Sphinx doctests are split independently. For instance, to run the first half of the test suite: >>> sympy.doctest(split='1/2') # doctest: +SKIP The ``subprocess`` and ``verbose`` options are the same as with the function ``test()``. See the docstring of that function for more information. """ # count up from 0, do not print 0 print_counter = lambda i : (print("rerun %d" % (rerun-i)) if rerun-i else None) if subprocess: # loop backwards so last i is 0 for i in range(rerun, -1, -1): print_counter(i) ret = run_in_subprocess_with_hash_randomization("_doctest", function_args=paths, function_kwargs=kwargs) if ret is False: break val = not bool(ret) # exit on the first failure or if done if not val or i == 0: return val # rerun even if hash randomization is not supported for i in range(rerun, -1, -1): print_counter(i) val = not bool(_doctest(*paths, **kwargs)) if not val or i == 0: return val def _get_doctest_blacklist(): '''Get the default blacklist for the doctests''' blacklist = [] blacklist.extend([ "doc/src/modules/plotting.rst", # generates live plots "doc/src/modules/physics/mechanics/autolev_parser.rst", "sympy/galgebra.py", # no longer part of SymPy "sympy/this.py", # prints text "sympy/physics/gaussopt.py", # raises deprecation warning "sympy/matrices/densearith.py", # raises deprecation warning "sympy/matrices/densesolve.py", # raises deprecation warning "sympy/matrices/densetools.py", # raises deprecation warning "sympy/printing/ccode.py", # backwards compatibility shim, importing it breaks the codegen doctests "sympy/printing/fcode.py", # backwards compatibility shim, importing it breaks the codegen doctests "sympy/printing/cxxcode.py", # backwards compatibility shim, importing it breaks the codegen doctests "sympy/parsing/autolev/_antlr/autolevlexer.py", # generated code "sympy/parsing/autolev/_antlr/autolevparser.py", # generated code "sympy/parsing/autolev/_antlr/autolevlistener.py", # generated code "sympy/parsing/latex/_antlr/latexlexer.py", # generated code "sympy/parsing/latex/_antlr/latexparser.py", # generated code "sympy/integrals/rubi/rubi.py", "sympy/plotting/pygletplot/__init__.py", # crashes on some systems "sympy/plotting/pygletplot/plot.py", # crashes on some systems ]) # autolev parser tests num = 12 for i in range (1, num+1): blacklist.append("sympy/parsing/autolev/test-examples/ruletest" + str(i) + ".py") blacklist.extend(["sympy/parsing/autolev/test-examples/pydy-example-repo/mass_spring_damper.py", "sympy/parsing/autolev/test-examples/pydy-example-repo/chaos_pendulum.py", "sympy/parsing/autolev/test-examples/pydy-example-repo/double_pendulum.py", "sympy/parsing/autolev/test-examples/pydy-example-repo/non_min_pendulum.py"]) if import_module('numpy') is None: blacklist.extend([ "sympy/plotting/experimental_lambdify.py", "sympy/plotting/plot_implicit.py", "examples/advanced/autowrap_integrators.py", "examples/advanced/autowrap_ufuncify.py", "examples/intermediate/sample.py", "examples/intermediate/mplot2d.py", "examples/intermediate/mplot3d.py", "doc/src/modules/numeric-computation.rst" ]) else: if import_module('matplotlib') is None: blacklist.extend([ "examples/intermediate/mplot2d.py", "examples/intermediate/mplot3d.py" ]) else: # Use a non-windowed backend, so that the tests work on Travis import matplotlib matplotlib.use('Agg') if ON_TRAVIS or import_module('pyglet') is None: blacklist.extend(["sympy/plotting/pygletplot"]) if import_module('theano') is None: blacklist.extend([ "sympy/printing/theanocode.py", "doc/src/modules/numeric-computation.rst", ]) if import_module('antlr4') is None: blacklist.extend([ "sympy/parsing/autolev/__init__.py", "sympy/parsing/latex/_parse_latex_antlr.py", ]) if import_module('lfortran') is None: #throws ImportError when lfortran not installed blacklist.extend([ "sympy/parsing/sym_expr.py", ]) # disabled because of doctest failures in asmeurer's bot blacklist.extend([ "sympy/utilities/autowrap.py", "examples/advanced/autowrap_integrators.py", "examples/advanced/autowrap_ufuncify.py" ]) # blacklist these modules until issue 4840 is resolved blacklist.extend([ "sympy/conftest.py", # Python 2.7 issues "sympy/testing/benchmarking.py", ]) # These are deprecated stubs to be removed: blacklist.extend([ "sympy/utilities/benchmarking.py", "sympy/utilities/tmpfiles.py", "sympy/utilities/pytest.py", "sympy/utilities/runtests.py", "sympy/utilities/quality_unicode.py", "sympy/utilities/randtest.py", ]) blacklist = convert_to_native_paths(blacklist) return blacklist def _doctest(*paths, **kwargs): """ Internal function that actually runs the doctests. All keyword arguments from ``doctest()`` are passed to this function except for ``subprocess``. Returns 0 if tests passed and 1 if they failed. See the docstrings of ``doctest()`` and ``test()`` for more information. """ from sympy import pprint_use_unicode normal = kwargs.get("normal", False) verbose = kwargs.get("verbose", False) colors = kwargs.get("colors", True) force_colors = kwargs.get("force_colors", False) blacklist = kwargs.get("blacklist", []) split = kwargs.get('split', None) blacklist.extend(_get_doctest_blacklist()) # Use a non-windowed backend, so that the tests work on Travis if import_module('matplotlib') is not None: import matplotlib matplotlib.use('Agg') # Disable warnings for external modules import sympy.external sympy.external.importtools.WARN_OLD_VERSION = False sympy.external.importtools.WARN_NOT_INSTALLED = False # Disable showing up of plots from sympy.plotting.plot import unset_show unset_show() r = PyTestReporter(verbose, split=split, colors=colors,\ force_colors=force_colors) t = SymPyDocTests(r, normal) test_files = t.get_test_files('sympy') test_files.extend(t.get_test_files('examples', init_only=False)) not_blacklisted = [f for f in test_files if not any(b in f for b in blacklist)] if len(paths) == 0: matched = not_blacklisted else: # take only what was requested...but not blacklisted items # and allow for partial match anywhere or fnmatch of name paths = convert_to_native_paths(paths) matched = [] for f in not_blacklisted: basename = os.path.basename(f) for p in paths: if p in f or fnmatch(basename, p): matched.append(f) break if split: matched = split_list(matched, split) t._testfiles.extend(matched) # run the tests and record the result for this *py portion of the tests if t._testfiles: failed = not t.test() else: failed = False # N.B. # -------------------------------------------------------------------- # Here we test *.rst files at or below doc/src. Code from these must # be self supporting in terms of imports since there is no importing # of necessary modules by doctest.testfile. If you try to pass *.py # files through this they might fail because they will lack the needed # imports and smarter parsing that can be done with source code. # test_files = t.get_test_files('doc/src', '*.rst', init_only=False) test_files.sort() not_blacklisted = [f for f in test_files if not any(b in f for b in blacklist)] if len(paths) == 0: matched = not_blacklisted else: # Take only what was requested as long as it's not on the blacklist. # Paths were already made native in *py tests so don't repeat here. # There's no chance of having a *py file slip through since we # only have *rst files in test_files. matched = [] for f in not_blacklisted: basename = os.path.basename(f) for p in paths: if p in f or fnmatch(basename, p): matched.append(f) break if split: matched = split_list(matched, split) first_report = True for rst_file in matched: if not os.path.isfile(rst_file): continue old_displayhook = sys.displayhook try: use_unicode_prev = setup_pprint() out = sympytestfile( rst_file, module_relative=False, encoding='utf-8', optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE | pdoctest.IGNORE_EXCEPTION_DETAIL) finally: # make sure we return to the original displayhook in case some # doctest has changed that sys.displayhook = old_displayhook # The NO_GLOBAL flag overrides the no_global flag to init_printing # if True import sympy.interactive.printing as interactive_printing interactive_printing.NO_GLOBAL = False pprint_use_unicode(use_unicode_prev) rstfailed, tested = out if tested: failed = rstfailed or failed if first_report: first_report = False msg = 'rst doctests start' if not t._testfiles: r.start(msg=msg) else: r.write_center(msg) print() # use as the id, everything past the first 'sympy' file_id = rst_file[rst_file.find('sympy') + len('sympy') + 1:] print(file_id, end=" ") # get at least the name out so it is know who is being tested wid = r.terminal_width - len(file_id) - 1 # update width test_file = '[%s]' % (tested) report = '[%s]' % (rstfailed or 'OK') print(''.join( [test_file, ' '*(wid - len(test_file) - len(report)), report]) ) # the doctests for *py will have printed this message already if there was # a failure, so now only print it if there was intervening reporting by # testing the *rst as evidenced by first_report no longer being True. if not first_report and failed: print() print("DO *NOT* COMMIT!") return int(failed) sp = re.compile(r'([0-9]+)/([1-9][0-9]*)') def split_list(l, split, density=None): """ Splits a list into part a of b split should be a string of the form 'a/b'. For instance, '1/3' would give the split one of three. If the length of the list is not divisible by the number of splits, the last split will have more items. `density` may be specified as a list. If specified, tests will be balanced so that each split has as equal-as-possible amount of mass according to `density`. >>> from sympy.testing.runtests import split_list >>> a = list(range(10)) >>> split_list(a, '1/3') [0, 1, 2] >>> split_list(a, '2/3') [3, 4, 5] >>> split_list(a, '3/3') [6, 7, 8, 9] """ m = sp.match(split) if not m: raise ValueError("split must be a string of the form a/b where a and b are ints") i, t = map(int, m.groups()) if not density: return l[(i - 1)*len(l)//t : i*len(l)//t] # normalize density tot = sum(density) density = [x / tot for x in density] def density_inv(x): """Interpolate the inverse to the cumulative distribution function given by density""" if x <= 0: return 0 if x >= sum(density): return 1 # find the first time the cumulative sum surpasses x # and linearly interpolate cumm = 0 for i, d in enumerate(density): cumm += d if cumm >= x: break frac = (d - (cumm - x)) / d return (i + frac) / len(density) lower_frac = density_inv((i - 1) / t) higher_frac = density_inv(i / t) return l[int(lower_frac*len(l)) : int(higher_frac*len(l))] from collections import namedtuple SymPyTestResults = namedtuple('SymPyTestResults', 'failed attempted') def sympytestfile(filename, module_relative=True, name=None, package=None, globs=None, verbose=None, report=True, optionflags=0, extraglobs=None, raise_on_error=False, parser=pdoctest.DocTestParser(), encoding=None): """ Test examples in the given file. Return (#failures, #tests). Optional keyword arg ``module_relative`` specifies how filenames should be interpreted: - If ``module_relative`` is True (the default), then ``filename`` specifies a module-relative path. By default, this path is relative to the calling module's directory; but if the ``package`` argument is specified, then it is relative to that package. To ensure os-independence, ``filename`` should use "/" characters to separate path segments, and should not be an absolute path (i.e., it may not begin with "/"). - If ``module_relative`` is False, then ``filename`` specifies an os-specific path. The path may be absolute or relative (to the current working directory). Optional keyword arg ``name`` gives the name of the test; by default use the file's basename. Optional keyword argument ``package`` is a Python package or the name of a Python package whose directory should be used as the base directory for a module relative filename. If no package is specified, then the calling module's directory is used as the base directory for module relative filenames. It is an error to specify ``package`` if ``module_relative`` is False. Optional keyword arg ``globs`` gives a dict to be used as the globals when executing examples; by default, use {}. A copy of this dict is actually used for each docstring, so that each docstring's examples start with a clean slate. Optional keyword arg ``extraglobs`` gives a dictionary that should be merged into the globals that are used to execute examples. By default, no extra globals are used. Optional keyword arg ``verbose`` prints lots of stuff if true, prints only failures if false; by default, it's true iff "-v" is in sys.argv. Optional keyword arg ``report`` prints a summary at the end when true, else prints nothing at the end. In verbose mode, the summary is detailed, else very brief (in fact, empty if all tests passed). Optional keyword arg ``optionflags`` or's together module constants, and defaults to 0. Possible values (see the docs for details): - DONT_ACCEPT_TRUE_FOR_1 - DONT_ACCEPT_BLANKLINE - NORMALIZE_WHITESPACE - ELLIPSIS - SKIP - IGNORE_EXCEPTION_DETAIL - REPORT_UDIFF - REPORT_CDIFF - REPORT_NDIFF - REPORT_ONLY_FIRST_FAILURE Optional keyword arg ``raise_on_error`` raises an exception on the first unexpected exception or failure. This allows failures to be post-mortem debugged. Optional keyword arg ``parser`` specifies a DocTestParser (or subclass) that should be used to extract tests from the files. Optional keyword arg ``encoding`` specifies an encoding that should be used to convert the file to unicode. Advanced tomfoolery: testmod runs methods of a local instance of class doctest.Tester, then merges the results into (or creates) global Tester instance doctest.master. Methods of doctest.master can be called directly too, if you want to do something unusual. Passing report=0 to testmod is especially useful then, to delay displaying a summary. Invoke doctest.master.summarize(verbose) when you're done fiddling. """ if package and not module_relative: raise ValueError("Package may only be specified for module-" "relative paths.") # Relativize the path if not PY3: text, filename = pdoctest._load_testfile( filename, package, module_relative) if encoding is not None: text = text.decode(encoding) else: text, filename = pdoctest._load_testfile( filename, package, module_relative, encoding) # If no name was given, then use the file's name. if name is None: name = os.path.basename(filename) # Assemble the globals. if globs is None: globs = {} else: globs = globs.copy() if extraglobs is not None: globs.update(extraglobs) if '__name__' not in globs: globs['__name__'] = '__main__' if raise_on_error: runner = pdoctest.DebugRunner(verbose=verbose, optionflags=optionflags) else: runner = SymPyDocTestRunner(verbose=verbose, optionflags=optionflags) runner._checker = SymPyOutputChecker() # Read the file, convert it to a test, and run it. test = parser.get_doctest(text, globs, name, filename, 0) runner.run(test, compileflags=future_flags) if report: runner.summarize() if pdoctest.master is None: pdoctest.master = runner else: pdoctest.master.merge(runner) return SymPyTestResults(runner.failures, runner.tries) class SymPyTests: def __init__(self, reporter, kw="", post_mortem=False, seed=None, fast_threshold=None, slow_threshold=None): self._post_mortem = post_mortem self._kw = kw self._count = 0 self._root_dir = get_sympy_dir() self._reporter = reporter self._reporter.root_dir(self._root_dir) self._testfiles = [] self._seed = seed if seed is not None else random.random() # Defaults in seconds, from human / UX design limits # http://www.nngroup.com/articles/response-times-3-important-limits/ # # These defaults are *NOT* set in stone as we are measuring different # things, so others feel free to come up with a better yardstick :) if fast_threshold: self._fast_threshold = float(fast_threshold) else: self._fast_threshold = 8 if slow_threshold: self._slow_threshold = float(slow_threshold) else: self._slow_threshold = 10 def test(self, sort=False, timeout=False, slow=False, enhance_asserts=False, fail_on_timeout=False): """ Runs the tests returning True if all tests pass, otherwise False. If sort=False run tests in random order. """ if sort: self._testfiles.sort() elif slow: pass else: random.seed(self._seed) random.shuffle(self._testfiles) self._reporter.start(self._seed) for f in self._testfiles: try: self.test_file(f, sort, timeout, slow, enhance_asserts, fail_on_timeout) except KeyboardInterrupt: print(" interrupted by user") self._reporter.finish() raise return self._reporter.finish() def _enhance_asserts(self, source): from ast import (NodeTransformer, Compare, Name, Store, Load, Tuple, Assign, BinOp, Str, Mod, Assert, parse, fix_missing_locations) ops = {"Eq": '==', "NotEq": '!=', "Lt": '<', "LtE": '<=', "Gt": '>', "GtE": '>=', "Is": 'is', "IsNot": 'is not', "In": 'in', "NotIn": 'not in'} class Transform(NodeTransformer): def visit_Assert(self, stmt): if isinstance(stmt.test, Compare): compare = stmt.test values = [compare.left] + compare.comparators names = [ "_%s" % i for i, _ in enumerate(values) ] names_store = [ Name(n, Store()) for n in names ] names_load = [ Name(n, Load()) for n in names ] target = Tuple(names_store, Store()) value = Tuple(values, Load()) assign = Assign([target], value) new_compare = Compare(names_load[0], compare.ops, names_load[1:]) msg_format = "\n%s " + "\n%s ".join([ ops[op.__class__.__name__] for op in compare.ops ]) + "\n%s" msg = BinOp(Str(msg_format), Mod(), Tuple(names_load, Load())) test = Assert(new_compare, msg, lineno=stmt.lineno, col_offset=stmt.col_offset) return [assign, test] else: return stmt tree = parse(source) new_tree = Transform().visit(tree) return fix_missing_locations(new_tree) def test_file(self, filename, sort=True, timeout=False, slow=False, enhance_asserts=False, fail_on_timeout=False): reporter = self._reporter funcs = [] try: gl = {'__file__': filename} try: if PY3: open_file = lambda: open(filename, encoding="utf8") else: open_file = lambda: open(filename) with open_file() as f: source = f.read() if self._kw: for l in source.splitlines(): if l.lstrip().startswith('def '): if any(l.find(k) != -1 for k in self._kw): break else: return if enhance_asserts: try: source = self._enhance_asserts(source) except ImportError: pass code = compile(source, filename, "exec", flags=0, dont_inherit=True) exec_(code, gl) except (SystemExit, KeyboardInterrupt): raise except ImportError: reporter.import_error(filename, sys.exc_info()) return except Exception: reporter.test_exception(sys.exc_info()) clear_cache() self._count += 1 random.seed(self._seed) disabled = gl.get("disabled", False) if not disabled: # we need to filter only those functions that begin with 'test_' # We have to be careful about decorated functions. As long as # the decorator uses functools.wraps, we can detect it. funcs = [] for f in gl: if (f.startswith("test_") and (inspect.isfunction(gl[f]) or inspect.ismethod(gl[f]))): func = gl[f] # Handle multiple decorators while hasattr(func, '__wrapped__'): func = func.__wrapped__ if inspect.getsourcefile(func) == filename: funcs.append(gl[f]) if slow: funcs = [f for f in funcs if getattr(f, '_slow', False)] # Sorting of XFAILed functions isn't fixed yet :-( funcs.sort(key=lambda x: inspect.getsourcelines(x)[1]) i = 0 while i < len(funcs): if inspect.isgeneratorfunction(funcs[i]): # some tests can be generators, that return the actual # test functions. We unpack it below: f = funcs.pop(i) for fg in f(): func = fg[0] args = fg[1:] fgw = lambda: func(*args) funcs.insert(i, fgw) i += 1 else: i += 1 # drop functions that are not selected with the keyword expression: funcs = [x for x in funcs if self.matches(x)] if not funcs: return except Exception: reporter.entering_filename(filename, len(funcs)) raise reporter.entering_filename(filename, len(funcs)) if not sort: random.shuffle(funcs) for f in funcs: start = time.time() reporter.entering_test(f) try: if getattr(f, '_slow', False) and not slow: raise Skipped("Slow") with raise_on_deprecated(): if timeout: self._timeout(f, timeout, fail_on_timeout) else: random.seed(self._seed) f() except KeyboardInterrupt: if getattr(f, '_slow', False): reporter.test_skip("KeyboardInterrupt") else: raise except Exception: if timeout: signal.alarm(0) # Disable the alarm. It could not be handled before. t, v, tr = sys.exc_info() if t is AssertionError: reporter.test_fail((t, v, tr)) if self._post_mortem: pdb.post_mortem(tr) elif t.__name__ == "Skipped": reporter.test_skip(v) elif t.__name__ == "XFail": reporter.test_xfail() elif t.__name__ == "XPass": reporter.test_xpass(v) else: reporter.test_exception((t, v, tr)) if self._post_mortem: pdb.post_mortem(tr) else: reporter.test_pass() taken = time.time() - start if taken > self._slow_threshold: filename = os.path.relpath(filename, reporter._root_dir) reporter.slow_test_functions.append( (filename + "::" + f.__name__, taken)) if getattr(f, '_slow', False) and slow: if taken < self._fast_threshold: filename = os.path.relpath(filename, reporter._root_dir) reporter.fast_test_functions.append( (filename + "::" + f.__name__, taken)) reporter.leaving_filename() def _timeout(self, function, timeout, fail_on_timeout): def callback(x, y): signal.alarm(0) if fail_on_timeout: raise TimeOutError("Timed out after %d seconds" % timeout) else: raise Skipped("Timeout") signal.signal(signal.SIGALRM, callback) signal.alarm(timeout) # Set an alarm with a given timeout function() signal.alarm(0) # Disable the alarm def matches(self, x): """ Does the keyword expression self._kw match "x"? Returns True/False. Always returns True if self._kw is "". """ if not self._kw: return True for kw in self._kw: if x.__name__.find(kw) != -1: return True return False def get_test_files(self, dir, pat='test_*.py'): """ Returns the list of test_*.py (default) files at or below directory ``dir`` relative to the sympy home directory. """ dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0]) g = [] for path, folders, files in os.walk(dir): g.extend([os.path.join(path, f) for f in files if fnmatch(f, pat)]) return sorted([os.path.normcase(gi) for gi in g]) class SymPyDocTests: def __init__(self, reporter, normal): self._count = 0 self._root_dir = get_sympy_dir() self._reporter = reporter self._reporter.root_dir(self._root_dir) self._normal = normal self._testfiles = [] def test(self): """ Runs the tests and returns True if all tests pass, otherwise False. """ self._reporter.start() for f in self._testfiles: try: self.test_file(f) except KeyboardInterrupt: print(" interrupted by user") self._reporter.finish() raise return self._reporter.finish() def test_file(self, filename): clear_cache() from sympy.core.compatibility import StringIO import sympy.interactive.printing as interactive_printing from sympy import pprint_use_unicode rel_name = filename[len(self._root_dir) + 1:] dirname, file = os.path.split(filename) module = rel_name.replace(os.sep, '.')[:-3] if rel_name.startswith("examples"): # Examples files do not have __init__.py files, # So we have to temporarily extend sys.path to import them sys.path.insert(0, dirname) module = file[:-3] # remove ".py" try: module = pdoctest._normalize_module(module) tests = SymPyDocTestFinder().find(module) except (SystemExit, KeyboardInterrupt): raise except ImportError: self._reporter.import_error(filename, sys.exc_info()) return finally: if rel_name.startswith("examples"): del sys.path[0] tests = [test for test in tests if len(test.examples) > 0] # By default tests are sorted by alphabetical order by function name. # We sort by line number so one can edit the file sequentially from # bottom to top. However, if there are decorated functions, their line # numbers will be too large and for now one must just search for these # by text and function name. tests.sort(key=lambda x: -x.lineno) if not tests: return self._reporter.entering_filename(filename, len(tests)) for test in tests: assert len(test.examples) != 0 if self._reporter._verbose: self._reporter.write("\n{} ".format(test.name)) # check if there are external dependencies which need to be met if '_doctest_depends_on' in test.globs: try: self._check_dependencies(**test.globs['_doctest_depends_on']) except DependencyError as e: self._reporter.test_skip(v=str(e)) continue runner = SymPyDocTestRunner(optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE | pdoctest.IGNORE_EXCEPTION_DETAIL) runner._checker = SymPyOutputChecker() old = sys.stdout new = StringIO() sys.stdout = new # If the testing is normal, the doctests get importing magic to # provide the global namespace. If not normal (the default) then # then must run on their own; all imports must be explicit within # a function's docstring. Once imported that import will be # available to the rest of the tests in a given function's # docstring (unless clear_globs=True below). if not self._normal: test.globs = {} # if this is uncommented then all the test would get is what # comes by default with a "from sympy import *" #exec('from sympy import *') in test.globs test.globs['print_function'] = print_function old_displayhook = sys.displayhook use_unicode_prev = setup_pprint() try: f, t = runner.run(test, compileflags=future_flags, out=new.write, clear_globs=False) except KeyboardInterrupt: raise finally: sys.stdout = old if f > 0: self._reporter.doctest_fail(test.name, new.getvalue()) else: self._reporter.test_pass() sys.displayhook = old_displayhook interactive_printing.NO_GLOBAL = False pprint_use_unicode(use_unicode_prev) self._reporter.leaving_filename() def get_test_files(self, dir, pat='*.py', init_only=True): r""" Returns the list of \*.py files (default) from which docstrings will be tested which are at or below directory ``dir``. By default, only those that have an __init__.py in their parent directory and do not start with ``test_`` will be included. """ def importable(x): """ Checks if given pathname x is an importable module by checking for __init__.py file. Returns True/False. Currently we only test if the __init__.py file exists in the directory with the file "x" (in theory we should also test all the parent dirs). """ init_py = os.path.join(os.path.dirname(x), "__init__.py") return os.path.exists(init_py) dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0]) g = [] for path, folders, files in os.walk(dir): g.extend([os.path.join(path, f) for f in files if not f.startswith('test_') and fnmatch(f, pat)]) if init_only: # skip files that are not importable (i.e. missing __init__.py) g = [x for x in g if importable(x)] return [os.path.normcase(gi) for gi in g] def _check_dependencies(self, executables=(), modules=(), disable_viewers=(), python_version=(3, 5)): """ Checks if the dependencies for the test are installed. Raises ``DependencyError`` it at least one dependency is not installed. """ for executable in executables: if not shutil.which(executable): raise DependencyError("Could not find %s" % executable) for module in modules: if module == 'matplotlib': matplotlib = import_module( 'matplotlib', import_kwargs={'fromlist': ['pyplot', 'cm', 'collections']}, min_module_version='1.0.0', catch=(RuntimeError,)) if matplotlib is None: raise DependencyError("Could not import matplotlib") else: if not import_module(module): raise DependencyError("Could not import %s" % module) if disable_viewers: tempdir = tempfile.mkdtemp() os.environ['PATH'] = '%s:%s' % (tempdir, os.environ['PATH']) vw = ('#!/usr/bin/env {}\n' 'import sys\n' 'if len(sys.argv) <= 1:\n' ' exit("wrong number of args")\n').format( 'python3' if PY3 else 'python') for viewer in disable_viewers: with open(os.path.join(tempdir, viewer), 'w') as fh: fh.write(vw) # make the file executable os.chmod(os.path.join(tempdir, viewer), stat.S_IREAD | stat.S_IWRITE | stat.S_IXUSR) if python_version: if sys.version_info < python_version: raise DependencyError("Requires Python >= " + '.'.join(map(str, python_version))) if 'pyglet' in modules: # monkey-patch pyglet s.t. it does not open a window during # doctesting import pyglet class DummyWindow: def __init__(self, *args, **kwargs): self.has_exit = True self.width = 600 self.height = 400 def set_vsync(self, x): pass def switch_to(self): pass def push_handlers(self, x): pass def close(self): pass pyglet.window.Window = DummyWindow class SymPyDocTestFinder(DocTestFinder): """ A class used to extract the DocTests that are relevant to a given object, from its docstring and the docstrings of its contained objects. Doctests can currently be extracted from the following object types: modules, functions, classes, methods, staticmethods, classmethods, and properties. Modified from doctest's version to look harder for code that appears comes from a different module. For example, the @vectorize decorator makes it look like functions come from multidimensional.py even though their code exists elsewhere. """ def _find(self, tests, obj, name, module, source_lines, globs, seen): """ Find tests for the given object and any contained objects, and add them to ``tests``. """ if self._verbose: print('Finding tests in %s' % name) # If we've already processed this object, then ignore it. if id(obj) in seen: return seen[id(obj)] = 1 # Make sure we don't run doctests for classes outside of sympy, such # as in numpy or scipy. if inspect.isclass(obj): if obj.__module__.split('.')[0] != 'sympy': return # Find a test for this object, and add it to the list of tests. test = self._get_test(obj, name, module, globs, source_lines) if test is not None: tests.append(test) if not self._recurse: return # Look for tests in a module's contained objects. if inspect.ismodule(obj): for rawname, val in obj.__dict__.items(): # Recurse to functions & classes. if inspect.isfunction(val) or inspect.isclass(val): # Make sure we don't run doctests functions or classes # from different modules if val.__module__ != module.__name__: continue assert self._from_module(module, val), \ "%s is not in module %s (rawname %s)" % (val, module, rawname) try: valname = '%s.%s' % (name, rawname) self._find(tests, val, valname, module, source_lines, globs, seen) except KeyboardInterrupt: raise # Look for tests in a module's __test__ dictionary. for valname, val in getattr(obj, '__test__', {}).items(): if not isinstance(valname, str): raise ValueError("SymPyDocTestFinder.find: __test__ keys " "must be strings: %r" % (type(valname),)) if not (inspect.isfunction(val) or inspect.isclass(val) or inspect.ismethod(val) or inspect.ismodule(val) or isinstance(val, str)): raise ValueError("SymPyDocTestFinder.find: __test__ values " "must be strings, functions, methods, " "classes, or modules: %r" % (type(val),)) valname = '%s.__test__.%s' % (name, valname) self._find(tests, val, valname, module, source_lines, globs, seen) # Look for tests in a class's contained objects. if inspect.isclass(obj): for valname, val in obj.__dict__.items(): # Special handling for staticmethod/classmethod. if isinstance(val, staticmethod): val = getattr(obj, valname) if isinstance(val, classmethod): val = getattr(obj, valname).__func__ # Recurse to methods, properties, and nested classes. if ((inspect.isfunction(unwrap(val)) or inspect.isclass(val) or isinstance(val, property)) and self._from_module(module, val)): # Make sure we don't run doctests functions or classes # from different modules if isinstance(val, property): if hasattr(val.fget, '__module__'): if val.fget.__module__ != module.__name__: continue else: if val.__module__ != module.__name__: continue assert self._from_module(module, val), \ "%s is not in module %s (valname %s)" % ( val, module, valname) valname = '%s.%s' % (name, valname) self._find(tests, val, valname, module, source_lines, globs, seen) def _get_test(self, obj, name, module, globs, source_lines): """ Return a DocTest for the given object, if it defines a docstring; otherwise, return None. """ lineno = None # Extract the object's docstring. If it doesn't have one, # then return None (no test for this object). if isinstance(obj, str): # obj is a string in the case for objects in the polys package. # Note that source_lines is a binary string (compiled polys # modules), which can't be handled by _find_lineno so determine # the line number here. docstring = obj matches = re.findall(r"line \d+", name) assert len(matches) == 1, \ "string '%s' does not contain lineno " % name # NOTE: this is not the exact linenumber but its better than no # lineno ;) lineno = int(matches[0][5:]) else: try: if obj.__doc__ is None: docstring = '' else: docstring = obj.__doc__ if not isinstance(docstring, str): docstring = str(docstring) except (TypeError, AttributeError): docstring = '' # Don't bother if the docstring is empty. if self._exclude_empty and not docstring: return None # check that properties have a docstring because _find_lineno # assumes it if isinstance(obj, property): if obj.fget.__doc__ is None: return None # Find the docstring's location in the file. if lineno is None: obj = unwrap(obj) # handling of properties is not implemented in _find_lineno so do # it here if hasattr(obj, 'func_closure') and obj.func_closure is not None: tobj = obj.func_closure[0].cell_contents elif isinstance(obj, property): tobj = obj.fget else: tobj = obj lineno = self._find_lineno(tobj, source_lines) if lineno is None: return None # Return a DocTest for this object. if module is None: filename = None else: filename = getattr(module, '__file__', module.__name__) if filename[-4:] in (".pyc", ".pyo"): filename = filename[:-1] globs['_doctest_depends_on'] = getattr(obj, '_doctest_depends_on', {}) return self._parser.get_doctest(docstring, globs, name, filename, lineno) class SymPyDocTestRunner(DocTestRunner): """ A class used to run DocTest test cases, and accumulate statistics. The ``run`` method is used to process a single DocTest case. It returns a tuple ``(f, t)``, where ``t`` is the number of test cases tried, and ``f`` is the number of test cases that failed. Modified from the doctest version to not reset the sys.displayhook (see issue 5140). See the docstring of the original DocTestRunner for more information. """ def run(self, test, compileflags=None, out=None, clear_globs=True): """ Run the examples in ``test``, and display the results using the writer function ``out``. The examples are run in the namespace ``test.globs``. If ``clear_globs`` is true (the default), then this namespace will be cleared after the test runs, to help with garbage collection. If you would like to examine the namespace after the test completes, then use ``clear_globs=False``. ``compileflags`` gives the set of flags that should be used by the Python compiler when running the examples. If not specified, then it will default to the set of future-import flags that apply to ``globs``. The output of each example is checked using ``SymPyDocTestRunner.check_output``, and the results are formatted by the ``SymPyDocTestRunner.report_*`` methods. """ self.test = test if compileflags is None: compileflags = pdoctest._extract_future_flags(test.globs) save_stdout = sys.stdout if out is None: out = save_stdout.write sys.stdout = self._fakeout # Patch pdb.set_trace to restore sys.stdout during interactive # debugging (so it's not still redirected to self._fakeout). # Note that the interactive output will go to *our* # save_stdout, even if that's not the real sys.stdout; this # allows us to write test cases for the set_trace behavior. save_set_trace = pdb.set_trace self.debugger = pdoctest._OutputRedirectingPdb(save_stdout) self.debugger.reset() pdb.set_trace = self.debugger.set_trace # Patch linecache.getlines, so we can see the example's source # when we're inside the debugger. self.save_linecache_getlines = pdoctest.linecache.getlines linecache.getlines = self.__patched_linecache_getlines # Fail for deprecation warnings with raise_on_deprecated(): try: test.globs['print_function'] = print_function return self.__run(test, compileflags, out) finally: sys.stdout = save_stdout pdb.set_trace = save_set_trace linecache.getlines = self.save_linecache_getlines if clear_globs: test.globs.clear() # We have to override the name mangled methods. monkeypatched_methods = [ 'patched_linecache_getlines', 'run', 'record_outcome' ] for method in monkeypatched_methods: oldname = '_DocTestRunner__' + method newname = '_SymPyDocTestRunner__' + method setattr(SymPyDocTestRunner, newname, getattr(DocTestRunner, oldname)) class SymPyOutputChecker(pdoctest.OutputChecker): """ Compared to the OutputChecker from the stdlib our OutputChecker class supports numerical comparison of floats occurring in the output of the doctest examples """ def __init__(self): # NOTE OutputChecker is an old-style class with no __init__ method, # so we can't call the base class version of __init__ here got_floats = r'(\d+\.\d*|\.\d+)' # floats in the 'want' string may contain ellipses want_floats = got_floats + r'(\.{3})?' front_sep = r'\s|\+|\-|\*|,' back_sep = front_sep + r'|j|e' fbeg = r'^%s(?=%s|$)' % (got_floats, back_sep) fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, got_floats, back_sep) self.num_got_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend)) fbeg = r'^%s(?=%s|$)' % (want_floats, back_sep) fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, want_floats, back_sep) self.num_want_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend)) def check_output(self, want, got, optionflags): """ Return True iff the actual output from an example (`got`) matches the expected output (`want`). These strings are always considered to match if they are identical; but depending on what option flags the test runner is using, several non-exact match types are also possible. See the documentation for `TestRunner` for more information about option flags. """ # Handle the common case first, for efficiency: # if they're string-identical, always return true. if got == want: return True # TODO parse integers as well ? # Parse floats and compare them. If some of the parsed floats contain # ellipses, skip the comparison. matches = self.num_got_rgx.finditer(got) numbers_got = [match.group(1) for match in matches] # list of strs matches = self.num_want_rgx.finditer(want) numbers_want = [match.group(1) for match in matches] # list of strs if len(numbers_got) != len(numbers_want): return False if len(numbers_got) > 0: nw_ = [] for ng, nw in zip(numbers_got, numbers_want): if '...' in nw: nw_.append(ng) continue else: nw_.append(nw) if abs(float(ng)-float(nw)) > 1e-5: return False got = self.num_got_rgx.sub(r'%s', got) got = got % tuple(nw_) # <BLANKLINE> can be used as a special sequence to signify a # blank line, unless the DONT_ACCEPT_BLANKLINE flag is used. if not (optionflags & pdoctest.DONT_ACCEPT_BLANKLINE): # Replace <BLANKLINE> in want with a blank line. want = re.sub(r'(?m)^%s\s*?$' % re.escape(pdoctest.BLANKLINE_MARKER), '', want) # If a line in got contains only spaces, then remove the # spaces. got = re.sub(r'(?m)^\s*?$', '', got) if got == want: return True # This flag causes doctest to ignore any differences in the # contents of whitespace strings. Note that this can be used # in conjunction with the ELLIPSIS flag. if optionflags & pdoctest.NORMALIZE_WHITESPACE: got = ' '.join(got.split()) want = ' '.join(want.split()) if got == want: return True # The ELLIPSIS flag says to let the sequence "..." in `want` # match any substring in `got`. if optionflags & pdoctest.ELLIPSIS: if pdoctest._ellipsis_match(want, got): return True # We didn't find any match; return false. return False class Reporter: """ Parent class for all reporters. """ pass class PyTestReporter(Reporter): """ Py.test like reporter. Should produce output identical to py.test. """ def __init__(self, verbose=False, tb="short", colors=True, force_colors=False, split=None): self._verbose = verbose self._tb_style = tb self._colors = colors self._force_colors = force_colors self._xfailed = 0 self._xpassed = [] self._failed = [] self._failed_doctest = [] self._passed = 0 self._skipped = 0 self._exceptions = [] self._terminal_width = None self._default_width = 80 self._split = split self._active_file = '' self._active_f = None # TODO: Should these be protected? self.slow_test_functions = [] self.fast_test_functions = [] # this tracks the x-position of the cursor (useful for positioning # things on the screen), without the need for any readline library: self._write_pos = 0 self._line_wrap = False def root_dir(self, dir): self._root_dir = dir @property def terminal_width(self): if self._terminal_width is not None: return self._terminal_width def findout_terminal_width(): if sys.platform == "win32": # Windows support is based on: # # http://code.activestate.com/recipes/ # 440694-determine-size-of-console-window-on-windows/ from ctypes import windll, create_string_buffer h = windll.kernel32.GetStdHandle(-12) csbi = create_string_buffer(22) res = windll.kernel32.GetConsoleScreenBufferInfo(h, csbi) if res: import struct (_, _, _, _, _, left, _, right, _, _, _) = \ struct.unpack("hhhhHhhhhhh", csbi.raw) return right - left else: return self._default_width if hasattr(sys.stdout, 'isatty') and not sys.stdout.isatty(): return self._default_width # leave PIPEs alone try: process = subprocess.Popen(['stty', '-a'], stdout=subprocess.PIPE, stderr=subprocess.PIPE) stdout = process.stdout.read() if PY3: stdout = stdout.decode("utf-8") except OSError: pass else: # We support the following output formats from stty: # # 1) Linux -> columns 80 # 2) OS X -> 80 columns # 3) Solaris -> columns = 80 re_linux = r"columns\s+(?P<columns>\d+);" re_osx = r"(?P<columns>\d+)\s*columns;" re_solaris = r"columns\s+=\s+(?P<columns>\d+);" for regex in (re_linux, re_osx, re_solaris): match = re.search(regex, stdout) if match is not None: columns = match.group('columns') try: width = int(columns) except ValueError: pass if width != 0: return width return self._default_width width = findout_terminal_width() self._terminal_width = width return width def write(self, text, color="", align="left", width=None, force_colors=False): """ Prints a text on the screen. It uses sys.stdout.write(), so no readline library is necessary. Parameters ========== color : choose from the colors below, "" means default color align : "left"/"right", "left" is a normal print, "right" is aligned on the right-hand side of the screen, filled with spaces if necessary width : the screen width """ color_templates = ( ("Black", "0;30"), ("Red", "0;31"), ("Green", "0;32"), ("Brown", "0;33"), ("Blue", "0;34"), ("Purple", "0;35"), ("Cyan", "0;36"), ("LightGray", "0;37"), ("DarkGray", "1;30"), ("LightRed", "1;31"), ("LightGreen", "1;32"), ("Yellow", "1;33"), ("LightBlue", "1;34"), ("LightPurple", "1;35"), ("LightCyan", "1;36"), ("White", "1;37"), ) colors = {} for name, value in color_templates: colors[name] = value c_normal = '\033[0m' c_color = '\033[%sm' if width is None: width = self.terminal_width if align == "right": if self._write_pos + len(text) > width: # we don't fit on the current line, create a new line self.write("\n") self.write(" "*(width - self._write_pos - len(text))) if not self._force_colors and hasattr(sys.stdout, 'isatty') and not \ sys.stdout.isatty(): # the stdout is not a terminal, this for example happens if the # output is piped to less, e.g. "bin/test | less". In this case, # the terminal control sequences would be printed verbatim, so # don't use any colors. color = "" elif sys.platform == "win32": # Windows consoles don't support ANSI escape sequences color = "" elif not self._colors: color = "" if self._line_wrap: if text[0] != "\n": sys.stdout.write("\n") # Avoid UnicodeEncodeError when printing out test failures if PY3 and IS_WINDOWS: text = text.encode('raw_unicode_escape').decode('utf8', 'ignore') elif PY3 and not sys.stdout.encoding.lower().startswith('utf'): text = text.encode(sys.stdout.encoding, 'backslashreplace' ).decode(sys.stdout.encoding) if color == "": sys.stdout.write(text) else: sys.stdout.write("%s%s%s" % (c_color % colors[color], text, c_normal)) sys.stdout.flush() l = text.rfind("\n") if l == -1: self._write_pos += len(text) else: self._write_pos = len(text) - l - 1 self._line_wrap = self._write_pos >= width self._write_pos %= width def write_center(self, text, delim="="): width = self.terminal_width if text != "": text = " %s " % text idx = (width - len(text)) // 2 t = delim*idx + text + delim*(width - idx - len(text)) self.write(t + "\n") def write_exception(self, e, val, tb): # remove the first item, as that is always runtests.py tb = tb.tb_next t = traceback.format_exception(e, val, tb) self.write("".join(t)) def start(self, seed=None, msg="test process starts"): self.write_center(msg) executable = sys.executable v = tuple(sys.version_info) python_version = "%s.%s.%s-%s-%s" % v implementation = platform.python_implementation() if implementation == 'PyPy': implementation += " %s.%s.%s-%s-%s" % sys.pypy_version_info self.write("executable: %s (%s) [%s]\n" % (executable, python_version, implementation)) from sympy.utilities.misc import ARCH self.write("architecture: %s\n" % ARCH) from sympy.core.cache import USE_CACHE self.write("cache: %s\n" % USE_CACHE) from sympy.core.compatibility import GROUND_TYPES, HAS_GMPY version = '' if GROUND_TYPES =='gmpy': if HAS_GMPY == 1: import gmpy elif HAS_GMPY == 2: import gmpy2 as gmpy version = gmpy.version() self.write("ground types: %s %s\n" % (GROUND_TYPES, version)) numpy = import_module('numpy') self.write("numpy: %s\n" % (None if not numpy else numpy.__version__)) if seed is not None: self.write("random seed: %d\n" % seed) from sympy.utilities.misc import HASH_RANDOMIZATION self.write("hash randomization: ") hash_seed = os.getenv("PYTHONHASHSEED") or '0' if HASH_RANDOMIZATION and (hash_seed == "random" or int(hash_seed)): self.write("on (PYTHONHASHSEED=%s)\n" % hash_seed) else: self.write("off\n") if self._split: self.write("split: %s\n" % self._split) self.write('\n') self._t_start = clock() def finish(self): self._t_end = clock() self.write("\n") global text, linelen text = "tests finished: %d passed, " % self._passed linelen = len(text) def add_text(mytext): global text, linelen """Break new text if too long.""" if linelen + len(mytext) > self.terminal_width: text += '\n' linelen = 0 text += mytext linelen += len(mytext) if len(self._failed) > 0: add_text("%d failed, " % len(self._failed)) if len(self._failed_doctest) > 0: add_text("%d failed, " % len(self._failed_doctest)) if self._skipped > 0: add_text("%d skipped, " % self._skipped) if self._xfailed > 0: add_text("%d expected to fail, " % self._xfailed) if len(self._xpassed) > 0: add_text("%d expected to fail but passed, " % len(self._xpassed)) if len(self._exceptions) > 0: add_text("%d exceptions, " % len(self._exceptions)) add_text("in %.2f seconds" % (self._t_end - self._t_start)) if self.slow_test_functions: self.write_center('slowest tests', '_') sorted_slow = sorted(self.slow_test_functions, key=lambda r: r[1]) for slow_func_name, taken in sorted_slow: print('%s - Took %.3f seconds' % (slow_func_name, taken)) if self.fast_test_functions: self.write_center('unexpectedly fast tests', '_') sorted_fast = sorted(self.fast_test_functions, key=lambda r: r[1]) for fast_func_name, taken in sorted_fast: print('%s - Took %.3f seconds' % (fast_func_name, taken)) if len(self._xpassed) > 0: self.write_center("xpassed tests", "_") for e in self._xpassed: self.write("%s: %s\n" % (e[0], e[1])) self.write("\n") if self._tb_style != "no" and len(self._exceptions) > 0: for e in self._exceptions: filename, f, (t, val, tb) = e self.write_center("", "_") if f is None: s = "%s" % filename else: s = "%s:%s" % (filename, f.__name__) self.write_center(s, "_") self.write_exception(t, val, tb) self.write("\n") if self._tb_style != "no" and len(self._failed) > 0: for e in self._failed: filename, f, (t, val, tb) = e self.write_center("", "_") self.write_center("%s:%s" % (filename, f.__name__), "_") self.write_exception(t, val, tb) self.write("\n") if self._tb_style != "no" and len(self._failed_doctest) > 0: for e in self._failed_doctest: filename, msg = e self.write_center("", "_") self.write_center("%s" % filename, "_") self.write(msg) self.write("\n") self.write_center(text) ok = len(self._failed) == 0 and len(self._exceptions) == 0 and \ len(self._failed_doctest) == 0 if not ok: self.write("DO *NOT* COMMIT!\n") return ok def entering_filename(self, filename, n): rel_name = filename[len(self._root_dir) + 1:] self._active_file = rel_name self._active_file_error = False self.write(rel_name) self.write("[%d] " % n) def leaving_filename(self): self.write(" ") if self._active_file_error: self.write("[FAIL]", "Red", align="right") else: self.write("[OK]", "Green", align="right") self.write("\n") if self._verbose: self.write("\n") def entering_test(self, f): self._active_f = f if self._verbose: self.write("\n" + f.__name__ + " ") def test_xfail(self): self._xfailed += 1 self.write("f", "Green") def test_xpass(self, v): message = str(v) self._xpassed.append((self._active_file, message)) self.write("X", "Green") def test_fail(self, exc_info): self._failed.append((self._active_file, self._active_f, exc_info)) self.write("F", "Red") self._active_file_error = True def doctest_fail(self, name, error_msg): # the first line contains "******", remove it: error_msg = "\n".join(error_msg.split("\n")[1:]) self._failed_doctest.append((name, error_msg)) self.write("F", "Red") self._active_file_error = True def test_pass(self, char="."): self._passed += 1 if self._verbose: self.write("ok", "Green") else: self.write(char, "Green") def test_skip(self, v=None): char = "s" self._skipped += 1 if v is not None: message = str(v) if message == "KeyboardInterrupt": char = "K" elif message == "Timeout": char = "T" elif message == "Slow": char = "w" if self._verbose: if v is not None: self.write(message + ' ', "Blue") else: self.write(" - ", "Blue") self.write(char, "Blue") def test_exception(self, exc_info): self._exceptions.append((self._active_file, self._active_f, exc_info)) if exc_info[0] is TimeOutError: self.write("T", "Red") else: self.write("E", "Red") self._active_file_error = True def import_error(self, filename, exc_info): self._exceptions.append((filename, None, exc_info)) rel_name = filename[len(self._root_dir) + 1:] self.write(rel_name) self.write("[?] Failed to import", "Red") self.write(" ") self.write("[FAIL]", "Red", align="right") self.write("\n")
bdc1324386af359dbbe4ff6b4e623f74eb6c028f8309dd3d0cc5dc9533c93e35
""" Helpers for randomized testing """ from random import uniform, Random, randrange, randint from sympy.core.compatibility import is_sequence, as_int from sympy.core.containers import Tuple from sympy.core.numbers import comp, I from sympy.core.symbol import Symbol from sympy.simplify.simplify import nsimplify def random_complex_number(a=2, b=-1, c=3, d=1, rational=False, tolerance=None): """ Return a random complex number. To reduce chance of hitting branch cuts or anything, we guarantee b <= Im z <= d, a <= Re z <= c When rational is True, a rational approximation to a random number is obtained within specified tolerance, if any. """ A, B = uniform(a, c), uniform(b, d) if not rational: return A + I*B return (nsimplify(A, rational=True, tolerance=tolerance) + I*nsimplify(B, rational=True, tolerance=tolerance)) def verify_numerically(f, g, z=None, tol=1.0e-6, a=2, b=-1, c=3, d=1): """ Test numerically that f and g agree when evaluated in the argument z. If z is None, all symbols will be tested. This routine does not test whether there are Floats present with precision higher than 15 digits so if there are, your results may not be what you expect due to round- off errors. Examples ======== >>> from sympy import sin, cos >>> from sympy.abc import x >>> from sympy.testing.randtest import verify_numerically as tn >>> tn(sin(x)**2 + cos(x)**2, 1, x) True """ f, g, z = Tuple(f, g, z) z = [z] if isinstance(z, Symbol) else (f.free_symbols | g.free_symbols) reps = list(zip(z, [random_complex_number(a, b, c, d) for _ in z])) z1 = f.subs(reps).n() z2 = g.subs(reps).n() return comp(z1, z2, tol) def test_derivative_numerically(f, z, tol=1.0e-6, a=2, b=-1, c=3, d=1): """ Test numerically that the symbolically computed derivative of f with respect to z is correct. This routine does not test whether there are Floats present with precision higher than 15 digits so if there are, your results may not be what you expect due to round-off errors. Examples ======== >>> from sympy import sin >>> from sympy.abc import x >>> from sympy.testing.randtest import test_derivative_numerically as td >>> td(sin(x), x) True """ from sympy.core.function import Derivative z0 = random_complex_number(a, b, c, d) f1 = f.diff(z).subs(z, z0) f2 = Derivative(f, z).doit_numerically(z0) return comp(f1.n(), f2.n(), tol) def _randrange(seed=None): """Return a randrange generator. ``seed`` can be o None - return randomly seeded generator o int - return a generator seeded with the int o list - the values to be returned will be taken from the list in the order given; the provided list is not modified. Examples ======== >>> from sympy.testing.randtest import _randrange >>> rr = _randrange() >>> rr(1000) # doctest: +SKIP 999 >>> rr = _randrange(3) >>> rr(1000) # doctest: +SKIP 238 >>> rr = _randrange([0, 5, 1, 3, 4]) >>> rr(3), rr(3) (0, 1) """ if seed is None: return randrange elif isinstance(seed, int): return Random(seed).randrange elif is_sequence(seed): seed = list(seed) # make a copy seed.reverse() def give(a, b=None, seq=seed): if b is None: a, b = 0, a a, b = as_int(a), as_int(b) w = b - a if w < 1: raise ValueError('_randrange got empty range') try: x = seq.pop() except IndexError: raise ValueError('_randrange sequence was too short') if a <= x < b: return x else: return give(a, b, seq) return give else: raise ValueError('_randrange got an unexpected seed') def _randint(seed=None): """Return a randint generator. ``seed`` can be o None - return randomly seeded generator o int - return a generator seeded with the int o list - the values to be returned will be taken from the list in the order given; the provided list is not modified. Examples ======== >>> from sympy.testing.randtest import _randint >>> ri = _randint() >>> ri(1, 1000) # doctest: +SKIP 999 >>> ri = _randint(3) >>> ri(1, 1000) # doctest: +SKIP 238 >>> ri = _randint([0, 5, 1, 2, 4]) >>> ri(1, 3), ri(1, 3) (1, 2) """ if seed is None: return randint elif isinstance(seed, int): return Random(seed).randint elif is_sequence(seed): seed = list(seed) # make a copy seed.reverse() def give(a, b, seq=seed): a, b = as_int(a), as_int(b) w = b - a if w < 0: raise ValueError('_randint got empty range') try: x = seq.pop() except IndexError: raise ValueError('_randint sequence was too short') if a <= x <= b: return x else: return give(a, b, seq) return give else: raise ValueError('_randint got an unexpected seed')
6c5f24b4a0bd943852eb981a841f6139ec2c677efc52484b82add01e2a966175
""" This module adds context manager for temporary files generated by the tests. """ import shutil import os class TmpFileManager: """ A class to track record of every temporary files created by the tests. """ tmp_files = set('') tmp_folders = set('') @classmethod def tmp_file(cls, name=''): cls.tmp_files.add(name) return name @classmethod def tmp_folder(cls, name=''): cls.tmp_folders.add(name) return name @classmethod def cleanup(cls): while cls.tmp_files: file = cls.tmp_files.pop() if os.path.isfile(file): os.remove(file) while cls.tmp_folders: folder = cls.tmp_folders.pop() shutil.rmtree(folder) def cleanup_tmp_files(test_func): """ A decorator to help test codes remove temporary files after the tests. """ def wrapper_function(): try: test_func() finally: TmpFileManager.cleanup() return wrapper_function
45ead2e335568d421758fcad26831db19331a301a09e758ad90a82f807210a29
from sympy import S, simplify from sympy.core import Basic, diff from sympy.matrices import Matrix from sympy.vector import (CoordSys3D, Vector, ParametricRegion, parametric_region_list, ImplicitRegion) from sympy.vector.operators import _get_coord_sys_from_expr from sympy.integrals import Integral, integrate from sympy.utilities.iterables import topological_sort, default_sort_key from sympy.geometry.entity import GeometryEntity class ParametricIntegral(Basic): """ Represents integral of a scalar or vector field over a Parametric Region Examples ======== >>> from sympy import cos, sin, pi >>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral >>> from sympy.abc import r, t, theta, phi >>> C = CoordSys3D('C') >>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2)) >>> ParametricIntegral(C.x, curve) 5*sqrt(10)/2 >>> length = ParametricIntegral(1, curve) >>> length sqrt(10) >>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\ (theta, 0, 2*pi), (phi, 0, pi/2)) >>> ParametricIntegral(C.z, semisphere) 8*pi >>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta)) 0 """ def __new__(cls, field, parametricregion): coord_set = _get_coord_sys_from_expr(field) if len(coord_set) == 0: coord_sys = CoordSys3D('C') elif len(coord_set) > 1: raise ValueError else: coord_sys = next(iter(coord_set)) if parametricregion.dimensions == 0: return S.Zero base_vectors = coord_sys.base_vectors() base_scalars = coord_sys.base_scalars() parametricfield = field r = Vector.zero for i in range(len(parametricregion.definition)): r += base_vectors[i]*parametricregion.definition[i] if len(coord_set) != 0: for i in range(len(parametricregion.definition)): parametricfield = parametricfield.subs(base_scalars[i], parametricregion.definition[i]) if parametricregion.dimensions == 1: parameter = parametricregion.parameters[0] r_diff = diff(r, parameter) lower, upper = parametricregion.limits[parameter][0], parametricregion.limits[parameter][1] if isinstance(parametricfield, Vector): integrand = simplify(r_diff.dot(parametricfield)) else: integrand = simplify(r_diff.magnitude()*parametricfield) result = integrate(integrand, (parameter, lower, upper)) elif parametricregion.dimensions == 2: u, v = cls._bounds_case(parametricregion.parameters, parametricregion.limits) r_u = diff(r, u) r_v = diff(r, v) normal_vector = simplify(r_u.cross(r_v)) if isinstance(parametricfield, Vector): integrand = parametricfield.dot(normal_vector) else: integrand = parametricfield*normal_vector.magnitude() integrand = simplify(integrand) lower_u, upper_u = parametricregion.limits[u][0], parametricregion.limits[u][1] lower_v, upper_v = parametricregion.limits[v][0], parametricregion.limits[v][1] result = integrate(integrand, (u, lower_u, upper_u), (v, lower_v, upper_v)) else: variables = cls._bounds_case(parametricregion.parameters, parametricregion.limits) coeff = Matrix(parametricregion.definition).jacobian(variables).det() integrand = simplify(parametricfield*coeff) l = [(var, parametricregion.limits[var][0], parametricregion.limits[var][1]) for var in variables] result = integrate(integrand, *l) if not isinstance(result, Integral): return result else: return super().__new__(cls, field, parametricregion) @classmethod def _bounds_case(cls, parameters, limits): V = list(limits.keys()) E = list() for p in V: lower_p = limits[p][0] upper_p = limits[p][1] lower_p = lower_p.atoms() upper_p = upper_p.atoms() for q in V: if p == q: continue if lower_p.issuperset({q}) or upper_p.issuperset({q}): E.append((p, q)) if not E: return parameters else: return topological_sort((V, E), key=default_sort_key) @property def field(self): return self.args[0] @property def parametricregion(self): return self.args[1] def vector_integrate(field, *region): """ Compute the integral of a vector/scalar field over a a region or a set of parameters. Examples ======== >>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate >>> from sympy.abc import x, y, t >>> C = CoordSys3D('C') >>> region = ParametricRegion((t, t**2), (t, 1, 5)) >>> vector_integrate(C.x*C.i, region) 12 Integrals over some objects of geometry module can also be calculated. >>> from sympy.geometry import Point, Circle, Triangle >>> c = Circle(Point(0, 2), 5) >>> vector_integrate(C.x**2 + C.y**2, c) 290*pi >>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5)) >>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle) -8 Integrals over some simple implicit regions can be computed. But in most cases, it takes too long to compute over them. This is due to the expressions of parametric representation becoming large. >>> from sympy.vector import ImplicitRegion >>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9) >>> vector_integrate(1, c2) 12*pi Integral of fields with respect to base scalars: >>> vector_integrate(12*C.y**3, (C.y, 1, 3)) 240 >>> vector_integrate(C.x**2*C.z, C.x) C.x**3*C.z/3 >>> vector_integrate(C.x*C.i - C.y*C.k, C.x) (Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k >>> _.doit() C.x**2/2*C.i + (-C.x*C.y)*C.k """ if len(region) == 1: if isinstance(region[0], ParametricRegion): return ParametricIntegral(field, region[0]) if isinstance(region[0], ImplicitRegion): region = parametric_region_list(region[0])[0] return vector_integrate(field, region) if isinstance(region[0], GeometryEntity): regions_list = parametric_region_list(region[0]) result = 0 for reg in regions_list: result += vector_integrate(field, reg) return result return integrate(field, *region)
a0f901321d56a81dd8326bd3574f263041ee8bd28c5380bdf3248bd30094e5fe
from functools import singledispatch from sympy import pi, tan from sympy.simplify import trigsimp from sympy.core import Basic, Tuple from sympy.core.symbol import _symbol from sympy.solvers import solve from sympy.geometry import Point, Segment, Curve, Ellipse, Polygon from sympy.vector import ImplicitRegion class ParametricRegion(Basic): """ Represents a parametric region in space. Examples ======== >>> from sympy import cos, sin, pi >>> from sympy.abc import r, theta, t, a, b, x, y >>> from sympy.vector import ParametricRegion >>> ParametricRegion((t, t**2), (t, -1, 2)) ParametricRegion((t, t**2), (t, -1, 2)) >>> ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) >>> ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) >>> ParametricRegion((a*cos(t), b*sin(t)), t) ParametricRegion((a*cos(t), b*sin(t)), t) >>> circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, pi)) >>> circle.parameters (r, theta) >>> circle.definition (r*cos(theta), r*sin(theta)) >>> circle.limits {theta: (0, pi)} Dimension of a parametric region determines whether a region is a curve, surface or volume region. It does not represent its dimensions in space. >>> circle.dimensions 1 Parameters ========== definition : tuple to define base scalars in terms of parameters. bounds : Parameter or a tuple of length 3 to define parameter and corresponding lower and upper bound. """ def __new__(cls, definition, *bounds): parameters = () limits = {} if not isinstance(bounds, Tuple): bounds = Tuple(*bounds) for bound in bounds: if isinstance(bound, tuple) or isinstance(bound, Tuple): if len(bound) != 3: raise ValueError("Tuple should be in the form (parameter, lowerbound, upperbound)") parameters += (bound[0],) limits[bound[0]] = (bound[1], bound[2]) else: parameters += (bound,) if not (isinstance(definition, tuple) or isinstance(definition, Tuple)): definition = (definition,) obj = super().__new__(cls, Tuple(*definition), *bounds) obj._parameters = parameters obj._limits = limits return obj @property def definition(self): return self.args[0] @property def limits(self): return self._limits @property def parameters(self): return self._parameters @property def dimensions(self): return len(self.limits) @singledispatch def parametric_region_list(reg): """ Returns a list of ParametricRegion objects representing the geometric region. Examples ======== >>> from sympy.abc import t >>> from sympy.vector import parametric_region_list >>> from sympy.geometry import Point, Curve, Ellipse, Segment, Polygon >>> p = Point(2, 5) >>> parametric_region_list(p) [ParametricRegion((2, 5))] >>> c = Curve((t**3, 4*t), (t, -3, 4)) >>> parametric_region_list(c) [ParametricRegion((t**3, 4*t), (t, -3, 4))] >>> e = Ellipse(Point(1, 3), 2, 3) >>> parametric_region_list(e) [ParametricRegion((2*cos(t) + 1, 3*sin(t) + 3), (t, 0, 2*pi))] >>> s = Segment(Point(1, 3), Point(2, 6)) >>> parametric_region_list(s) [ParametricRegion((t + 1, 3*t + 3), (t, 0, 1))] >>> p1, p2, p3, p4 = [(0, 1), (2, -3), (5, 3), (-2, 3)] >>> poly = Polygon(p1, p2, p3, p4) >>> parametric_region_list(poly) [ParametricRegion((2*t, 1 - 4*t), (t, 0, 1)), ParametricRegion((3*t + 2, 6*t - 3), (t, 0, 1)),\ ParametricRegion((5 - 7*t, 3), (t, 0, 1)), ParametricRegion((2*t - 2, 3 - 2*t), (t, 0, 1))] """ raise ValueError("SymPy cannot determine parametric representation of the region.") @parametric_region_list.register(Point) def _(obj): return [ParametricRegion(obj.args)] @parametric_region_list.register(Curve) # type: ignore def _(obj): definition = obj.arbitrary_point(obj.parameter).args bounds = obj.limits return [ParametricRegion(definition, bounds)] @parametric_region_list.register(Ellipse) # type: ignore def _(obj, parameter='t'): definition = obj.arbitrary_point(parameter).args t = _symbol(parameter, real=True) bounds = (t, 0, 2*pi) return [ParametricRegion(definition, bounds)] @parametric_region_list.register(Segment) # type: ignore def _(obj, parameter='t'): t = _symbol(parameter, real=True) definition = obj.arbitrary_point(t).args for i in range(0, 3): lower_bound = solve(definition[i] - obj.points[0].args[i], t) upper_bound = solve(definition[i] - obj.points[1].args[i], t) if len(lower_bound) == 1 and len(upper_bound) == 1: bounds = t, lower_bound[0], upper_bound[0] break definition_tuple = obj.arbitrary_point(parameter).args return [ParametricRegion(definition_tuple, bounds)] @parametric_region_list.register(Polygon) # type: ignore def _(obj, parameter='t'): l = [parametric_region_list(side, parameter)[0] for side in obj.sides] return l @parametric_region_list.register(ImplicitRegion) # type: ignore def _(obj, parameters=('t', 's')): definition = obj.rational_parametrization(parameters) bounds = [] for i in range(len(obj.variables) - 1): # Each parameter is replaced by its tangent to simplify intergation parameter = _symbol(parameters[i], real=True) definition = [trigsimp(elem.subs(parameter, tan(parameter))) for elem in definition] bounds.append((parameter, 0, 2*pi),) definition = Tuple(*definition) return [ParametricRegion(definition, *bounds)]
a0e1ea6649a5d05352810ad3addb46b941f79f9fec28af814c6e7ad6419b1611
"""Curves in 2-dimensional Euclidean space. Contains ======== Curve """ from sympy import sqrt from sympy.core import sympify, diff from sympy.core.compatibility import is_sequence from sympy.core.containers import Tuple from sympy.core.symbol import _symbol from sympy.geometry.entity import GeometryEntity, GeometrySet from sympy.geometry.point import Point from sympy.integrals import integrate class Curve(GeometrySet): """A curve in space. A curve is defined by parametric functions for the coordinates, a parameter and the lower and upper bounds for the parameter value. Parameters ========== function : list of functions limits : 3-tuple Function parameter and lower and upper bounds. Attributes ========== functions parameter limits Raises ====== ValueError When `functions` are specified incorrectly. When `limits` are specified incorrectly. Examples ======== >>> from sympy import sin, cos, interpolate >>> from sympy.abc import t, a >>> from sympy.geometry import Curve >>> C = Curve((sin(t), cos(t)), (t, 0, 2)) >>> C.functions (sin(t), cos(t)) >>> C.limits (t, 0, 2) >>> C.parameter t >>> C = Curve((t, interpolate([1, 4, 9, 16], t)), (t, 0, 1)); C Curve((t, t**2), (t, 0, 1)) >>> C.subs(t, 4) Point2D(4, 16) >>> C.arbitrary_point(a) Point2D(a, a**2) See Also ======== sympy.core.function.Function sympy.polys.polyfuncs.interpolate """ def __new__(cls, function, limits): fun = sympify(function) if not is_sequence(fun) or len(fun) != 2: raise ValueError("Function argument should be (x(t), y(t)) " "but got %s" % str(function)) if not is_sequence(limits) or len(limits) != 3: raise ValueError("Limit argument should be (t, tmin, tmax) " "but got %s" % str(limits)) return GeometryEntity.__new__(cls, Tuple(*fun), Tuple(*limits)) def __call__(self, f): return self.subs(self.parameter, f) def _eval_subs(self, old, new): if old == self.parameter: return Point(*[f.subs(old, new) for f in self.functions]) def arbitrary_point(self, parameter='t'): """A parameterized point on the curve. Parameters ========== parameter : str or Symbol, optional Default value is 't'. The Curve's parameter is selected with None or self.parameter otherwise the provided symbol is used. Returns ======= Point : Returns a point in parametric form. Raises ====== ValueError When `parameter` already appears in the functions. Examples ======== >>> from sympy import Symbol >>> from sympy.abc import s >>> from sympy.geometry import Curve >>> C = Curve([2*s, s**2], (s, 0, 2)) >>> C.arbitrary_point() Point2D(2*t, t**2) >>> C.arbitrary_point(C.parameter) Point2D(2*s, s**2) >>> C.arbitrary_point(None) Point2D(2*s, s**2) >>> C.arbitrary_point(Symbol('a')) Point2D(2*a, a**2) See Also ======== sympy.geometry.point.Point """ if parameter is None: return Point(*self.functions) tnew = _symbol(parameter, self.parameter, real=True) t = self.parameter if (tnew.name != t.name and tnew.name in (f.name for f in self.free_symbols)): raise ValueError('Symbol %s already appears in object ' 'and cannot be used as a parameter.' % tnew.name) return Point(*[w.subs(t, tnew) for w in self.functions]) @property def free_symbols(self): """Return a set of symbols other than the bound symbols used to parametrically define the Curve. Returns ======= set : Set of all non-parameterized symbols. Examples ======== >>> from sympy.abc import t, a >>> from sympy.geometry import Curve >>> Curve((t, t**2), (t, 0, 2)).free_symbols set() >>> Curve((t, t**2), (t, a, 2)).free_symbols {a} """ free = set() for a in self.functions + self.limits[1:]: free |= a.free_symbols free = free.difference({self.parameter}) return free @property def ambient_dimension(self): """The dimension of the curve. Returns ======= int : the dimension of curve. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.ambient_dimension 2 """ return len(self.args[0]) @property def functions(self): """The functions specifying the curve. Returns ======= functions : list of parameterized coordinate functions. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve((t, t**2), (t, 0, 2)) >>> C.functions (t, t**2) See Also ======== parameter """ return self.args[0] @property def limits(self): """The limits for the curve. Returns ======= limits : tuple Contains parameter and lower and upper limits. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve([t, t**3], (t, -2, 2)) >>> C.limits (t, -2, 2) See Also ======== plot_interval """ return self.args[1] @property def parameter(self): """The curve function variable. Returns ======= Symbol : returns a bound symbol. Examples ======== >>> from sympy.abc import t >>> from sympy.geometry import Curve >>> C = Curve([t, t**2], (t, 0, 2)) >>> C.parameter t See Also ======== functions """ return self.args[1][0] @property def length(self): """The curve length. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import t >>> Curve((t, t), (t, 0, 1)).length sqrt(2) """ integrand = sqrt(sum(diff(func, self.limits[0])**2 for func in self.functions)) return integrate(integrand, self.limits) def plot_interval(self, parameter='t'): """The plot interval for the default geometric plot of the curve. Parameters ========== parameter : str or Symbol, optional Default value is 't'; otherwise the provided symbol is used. Returns ======= List : the plot interval as below: [parameter, lower_bound, upper_bound] Examples ======== >>> from sympy import Curve, sin >>> from sympy.abc import x, s >>> Curve((x, sin(x)), (x, 1, 2)).plot_interval() [t, 1, 2] >>> Curve((x, sin(x)), (x, 1, 2)).plot_interval(s) [s, 1, 2] See Also ======== limits : Returns limits of the parameter interval """ t = _symbol(parameter, self.parameter, real=True) return [t] + list(self.limits[1:]) def rotate(self, angle=0, pt=None): """This function is used to rotate a curve along given point ``pt`` at given angle(in radian). Parameters ========== angle : the angle at which the curve will be rotated(in radian) in counterclockwise direction. default value of angle is 0. pt : Point the point along which the curve will be rotated. If no point given, the curve will be rotated around origin. Returns ======= Curve : returns a curve rotated at given angle along given point. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> from sympy import pi >>> Curve((x, x), (x, 0, 1)).rotate(pi/2) Curve((-x, x), (x, 0, 1)) """ from sympy.matrices import Matrix, rot_axis3 if pt: pt = -Point(pt, dim=2) else: pt = Point(0,0) rv = self.translate(*pt.args) f = list(rv.functions) f.append(0) f = Matrix(1, 3, f) f *= rot_axis3(angle) rv = self.func(f[0, :2].tolist()[0], self.limits) pt = -pt return rv.translate(*pt.args) def scale(self, x=1, y=1, pt=None): """Override GeometryEntity.scale since Curve is not made up of Points. Returns ======= Curve : returns scaled curve. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).scale(2) Curve((2*x, x), (x, 0, 1)) """ if pt: pt = Point(pt, dim=2) return self.translate(*(-pt).args).scale(x, y).translate(*pt.args) fx, fy = self.functions return self.func((fx*x, fy*y), self.limits) def translate(self, x=0, y=0): """Translate the Curve by (x, y). Returns ======= Curve : returns a translated curve. Examples ======== >>> from sympy.geometry.curve import Curve >>> from sympy.abc import x >>> Curve((x, x), (x, 0, 1)).translate(1, 2) Curve((x + 1, x + 2), (x, 0, 1)) """ fx, fy = self.functions return self.func((fx + x, fy + y), self.limits)
1d66c71f1b9864623b8ecd5e6570eb45fe060b0603fdbb4387ac776b736ee39d
"""Recurrence Operators""" from sympy import symbols, Symbol, S from sympy.printing import sstr from sympy.core.sympify import sympify def RecurrenceOperators(base, generator): """ Returns an Algebra of Recurrence Operators and the operator for shifting i.e. the `Sn` operator. The first argument needs to be the base polynomial ring for the algebra and the second argument must be a generator which can be either a noncommutative Symbol or a string. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') """ ring = RecurrenceOperatorAlgebra(base, generator) return (ring, ring.shift_operator) class RecurrenceOperatorAlgebra: """ A Recurrence Operator Algebra is a set of noncommutative polynomials in intermediate `Sn` and coefficients in a base ring A. It follows the commutation rule: Sn * a(n) = a(n + 1) * Sn This class represents a Recurrence Operator Algebra and serves as the parent ring for Recurrence Operators. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') >>> R Univariate Recurrence Operator Algebra in intermediate Sn over the base ring ZZ[n] See Also ======== RecurrenceOperator """ def __init__(self, base, generator): # the base ring for the algebra self.base = base # the operator representing shift i.e. `Sn` self.shift_operator = RecurrenceOperator( [base.zero, base.one], self) if generator is None: self.gen_symbol = symbols('Sn', commutative=False) else: if isinstance(generator, str): self.gen_symbol = symbols(generator, commutative=False) elif isinstance(generator, Symbol): self.gen_symbol = generator def __str__(self): string = 'Univariate Recurrence Operator Algebra in intermediate '\ + sstr(self.gen_symbol) + ' over the base ring ' + \ (self.base).__str__() return string __repr__ = __str__ def __eq__(self, other): if self.base == other.base and self.gen_symbol == other.gen_symbol: return True else: return False def _add_lists(list1, list2): if len(list1) <= len(list2): sol = [a + b for a, b in zip(list1, list2)] + list2[len(list1):] else: sol = [a + b for a, b in zip(list1, list2)] + list1[len(list2):] return sol class RecurrenceOperator: """ The Recurrence Operators are defined by a list of polynomials in the base ring and the parent ring of the Operator. Explanation =========== Takes a list of polynomials for each power of Sn and the parent ring which must be an instance of RecurrenceOperatorAlgebra. A Recurrence Operator can be created easily using the operator `Sn`. See examples below. Examples ======== >>> from sympy.holonomic.recurrence import RecurrenceOperator, RecurrenceOperators >>> from sympy.polys.domains import ZZ >>> from sympy import symbols >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n),'Sn') >>> RecurrenceOperator([0, 1, n**2], R) (1)Sn + (n**2)Sn**2 >>> Sn*n (n + 1)Sn >>> n*Sn*n + 1 - Sn**2*n (1) + (n**2 + n)Sn + (-n - 2)Sn**2 See Also ======== DifferentialOperatorAlgebra """ _op_priority = 20 def __init__(self, list_of_poly, parent): # the parent ring for this operator # must be an RecurrenceOperatorAlgebra object self.parent = parent # sequence of polynomials in n for each power of Sn # represents the operator # convert the expressions into ring elements using from_sympy if isinstance(list_of_poly, list): for i, j in enumerate(list_of_poly): if isinstance(j, int): list_of_poly[i] = self.parent.base.from_sympy(S(j)) elif not isinstance(j, self.parent.base.dtype): list_of_poly[i] = self.parent.base.from_sympy(j) self.listofpoly = list_of_poly self.order = len(self.listofpoly) - 1 def __mul__(self, other): """ Multiplies two Operators and returns another RecurrenceOperator instance using the commutation rule Sn * a(n) = a(n + 1) * Sn """ listofself = self.listofpoly base = self.parent.base if not isinstance(other, RecurrenceOperator): if not isinstance(other, self.parent.base.dtype): listofother = [self.parent.base.from_sympy(sympify(other))] else: listofother = [other] else: listofother = other.listofpoly # multiply a polynomial `b` with a list of polynomials def _mul_dmp_diffop(b, listofother): if isinstance(listofother, list): sol = [] for i in listofother: sol.append(i * b) return sol else: return [b * listofother] sol = _mul_dmp_diffop(listofself[0], listofother) # compute Sn^i * b def _mul_Sni_b(b): sol = [base.zero] if isinstance(b, list): for i in b: j = base.to_sympy(i).subs(base.gens[0], base.gens[0] + S.One) sol.append(base.from_sympy(j)) else: j = b.subs(base.gens[0], base.gens[0] + S.One) sol.append(base.from_sympy(j)) return sol for i in range(1, len(listofself)): # find Sn^i * b in ith iteration listofother = _mul_Sni_b(listofother) # solution = solution + listofself[i] * (Sn^i * b) sol = _add_lists(sol, _mul_dmp_diffop(listofself[i], listofother)) return RecurrenceOperator(sol, self.parent) def __rmul__(self, other): if not isinstance(other, RecurrenceOperator): if isinstance(other, int): other = S(other) if not isinstance(other, self.parent.base.dtype): other = (self.parent.base).from_sympy(other) sol = [] for j in self.listofpoly: sol.append(other * j) return RecurrenceOperator(sol, self.parent) def __add__(self, other): if isinstance(other, RecurrenceOperator): sol = _add_lists(self.listofpoly, other.listofpoly) return RecurrenceOperator(sol, self.parent) else: if isinstance(other, int): other = S(other) list_self = self.listofpoly if not isinstance(other, self.parent.base.dtype): list_other = [((self.parent).base).from_sympy(other)] else: list_other = [other] sol = [] sol.append(list_self[0] + list_other[0]) sol += list_self[1:] return RecurrenceOperator(sol, self.parent) __radd__ = __add__ def __sub__(self, other): return self + (-1) * other def __rsub__(self, other): return (-1) * self + other def __pow__(self, n): if n == 1: return self if n == 0: return RecurrenceOperator([self.parent.base.one], self.parent) # if self is `Sn` if self.listofpoly == self.parent.shift_operator.listofpoly: sol = [] for i in range(0, n): sol.append(self.parent.base.zero) sol.append(self.parent.base.one) return RecurrenceOperator(sol, self.parent) else: if n % 2 == 1: powreduce = self**(n - 1) return powreduce * self elif n % 2 == 0: powreduce = self**(n / 2) return powreduce * powreduce def __str__(self): listofpoly = self.listofpoly print_str = '' for i, j in enumerate(listofpoly): if j == self.parent.base.zero: continue if i == 0: print_str += '(' + sstr(j) + ')' continue if print_str: print_str += ' + ' if i == 1: print_str += '(' + sstr(j) + ')Sn' continue print_str += '(' + sstr(j) + ')' + 'Sn**' + sstr(i) return print_str __repr__ = __str__ def __eq__(self, other): if isinstance(other, RecurrenceOperator): if self.listofpoly == other.listofpoly and self.parent == other.parent: return True else: return False else: if self.listofpoly[0] == other: for i in self.listofpoly[1:]: if i is not self.parent.base.zero: return False return True else: return False class HolonomicSequence: """ A Holonomic Sequence is a type of sequence satisfying a linear homogeneous recurrence relation with Polynomial coefficients. Alternatively, A sequence is Holonomic if and only if its generating function is a Holonomic Function. """ def __init__(self, recurrence, u0=[]): self.recurrence = recurrence if not isinstance(u0, list): self.u0 = [u0] else: self.u0 = u0 if len(self.u0) == 0: self._have_init_cond = False else: self._have_init_cond = True self.n = recurrence.parent.base.gens[0] def __repr__(self): str_sol = 'HolonomicSequence(%s, %s)' % ((self.recurrence).__repr__(), sstr(self.n)) if not self._have_init_cond: return str_sol else: cond_str = '' seq_str = 0 for i in self.u0: cond_str += ', u(%s) = %s' % (sstr(seq_str), sstr(i)) seq_str += 1 sol = str_sol + cond_str return sol __str__ = __repr__ def __eq__(self, other): if self.recurrence == other.recurrence: if self.n == other.n: if self._have_init_cond and other._have_init_cond: if self.u0 == other.u0: return True else: return False else: return True else: return False else: return False