MLPY/Lib/site-packages/sympy/solvers/ode/nonhomogeneous.py

r"""
This File contains helper functions for nth_linear_constant_coeff_undetermined_coefficients,
nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients,
nth_linear_constant_coeff_variation_of_parameters,
and nth_linear_euler_eq_nonhomogeneous_variation_of_parameters.

All the functions in this file are used by more than one solvers so, instead of creating
instances in other classes for using them it is better to keep it here as separate helpers.

"""
from collections import defaultdict
from sympy.core import Add, S
from sympy.core.function import diff, expand, _mexpand, expand_mul
from sympy.core.relational import Eq
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Dummy, Wild
from sympy.functions import exp, cos, cosh, im, log, re, sin, sinh, \
    atan2, conjugate
from sympy.integrals import Integral
from sympy.polys import (Poly, RootOf, rootof, roots)
from sympy.simplify import collect, simplify, separatevars, powsimp, trigsimp # type: ignore
from sympy.utilities import numbered_symbols
from sympy.solvers.solvers import solve
from sympy.matrices import wronskian
from .subscheck import sub_func_doit
from sympy.solvers.ode.ode import get_numbered_constants


def _test_term(coeff, func, order):
    r"""
    Linear Euler ODEs have the form  K*x**order*diff(y(x), x, order) = F(x),
    where K is independent of x and y(x), order>= 0.
    So we need to check that for each term, coeff == K*x**order from
    some K.  We have a few cases, since coeff may have several
    different types.
    """
    x = func.args[0]
    f = func.func
    if order < 0:
        raise ValueError("order should be greater than 0")
    if coeff == 0:
        return True
    if order == 0:
        if x in coeff.free_symbols:
            return False
        return True
    if coeff.is_Mul:
        if coeff.has(f(x)):
            return False
        return x**order in coeff.args
    elif coeff.is_Pow:
        return coeff.as_base_exp() == (x, order)
    elif order == 1:
        return x == coeff
    return False


def _get_euler_characteristic_eq_sols(eq, func, match_obj):
    r"""
    Returns the solution of homogeneous part of the linear euler ODE and
    the list of roots of characteristic equation.

    The parameter ``match_obj`` is a dict of order:coeff terms, where order is the order
    of the derivative on each term, and coeff is the coefficient of that derivative.

    """
    x = func.args[0]
    f = func.func

    # First, set up characteristic equation.
    chareq, symbol = S.Zero, Dummy('x')

    for i in match_obj:
        if i >= 0:
            chareq += (match_obj[i]*diff(x**symbol, x, i)*x**-symbol).expand()

    chareq = Poly(chareq, symbol)
    chareqroots = [rootof(chareq, k) for k in range(chareq.degree())]
    collectterms = []

    # A generator of constants
    constants = list(get_numbered_constants(eq, num=chareq.degree()*2))
    constants.reverse()

    # Create a dict root: multiplicity or charroots
    charroots = defaultdict(int)
    for root in chareqroots:
        charroots[root] += 1
    gsol = S.Zero
    ln = log
    for root, multiplicity in charroots.items():
        for i in range(multiplicity):
            if isinstance(root, RootOf):
                gsol += (x**root) * constants.pop()
                if multiplicity != 1:
                    raise ValueError("Value should be 1")
                collectterms = [(0, root, 0)] + collectterms
            elif root.is_real:
                gsol += ln(x)**i*(x**root) * constants.pop()
                collectterms = [(i, root, 0)] + collectterms
            else:
                reroot = re(root)
                imroot = im(root)
                gsol += ln(x)**i * (x**reroot) * (
                    constants.pop() * sin(abs(imroot)*ln(x))
                    + constants.pop() * cos(imroot*ln(x)))
                collectterms = [(i, reroot, imroot)] + collectterms

    gsol = Eq(f(x), gsol)

    gensols = []
    # Keep track of when to use sin or cos for nonzero imroot
    for i, reroot, imroot in collectterms:
        if imroot == 0:
            gensols.append(ln(x)**i*x**reroot)
        else:
            sin_form = ln(x)**i*x**reroot*sin(abs(imroot)*ln(x))
            if sin_form in gensols:
                cos_form = ln(x)**i*x**reroot*cos(imroot*ln(x))
                gensols.append(cos_form)
            else:
                gensols.append(sin_form)
    return gsol, gensols


def _solve_variation_of_parameters(eq, func, roots, homogen_sol, order, match_obj, simplify_flag=True):
    r"""
    Helper function for the method of variation of parameters and nonhomogeneous euler eq.

    See the
    :py:meth:`~sympy.solvers.ode.single.NthLinearConstantCoeffVariationOfParameters`
    docstring for more information on this method.

    The parameter are ``match_obj`` should be a dictionary that has the following
    keys:

    ``list``
    A list of solutions to the homogeneous equation.

    ``sol``
    The general solution.

    """
    f = func.func
    x = func.args[0]
    r = match_obj
    psol = 0
    wr = wronskian(roots, x)

    if simplify_flag:
        wr = simplify(wr)  # We need much better simplification for
                        # some ODEs. See issue 4662, for example.
        # To reduce commonly occurring sin(x)**2 + cos(x)**2 to 1
        wr = trigsimp(wr, deep=True, recursive=True)
    if not wr:
        # The wronskian will be 0 iff the solutions are not linearly
        # independent.
        raise NotImplementedError("Cannot find " + str(order) +
        " solutions to the homogeneous equation necessary to apply " +
        "variation of parameters to " + str(eq) + " (Wronskian == 0)")
    if len(roots) != order:
        raise NotImplementedError("Cannot find " + str(order) +
        " solutions to the homogeneous equation necessary to apply " +
        "variation of parameters to " +
        str(eq) + " (number of terms != order)")
    negoneterm = S.NegativeOne**(order)
    for i in roots:
        psol += negoneterm*Integral(wronskian([sol for sol in roots if sol != i], x)*r[-1]/wr, x)*i/r[order]
        negoneterm *= -1

    if simplify_flag:
        psol = simplify(psol)
        psol = trigsimp(psol, deep=True)
    return Eq(f(x), homogen_sol.rhs + psol)


def _get_const_characteristic_eq_sols(r, func, order):
    r"""
    Returns the roots of characteristic equation of constant coefficient
    linear ODE and list of collectterms which is later on used by simplification
    to use collect on solution.

    The parameter `r` is a dict of order:coeff terms, where order is the order of the
    derivative on each term, and coeff is the coefficient of that derivative.

    """
    x = func.args[0]
    # First, set up characteristic equation.
    chareq, symbol = S.Zero, Dummy('x')

    for i in r.keys():
        if isinstance(i, str) or i < 0:
            pass
        else:
            chareq += r[i]*symbol**i

    chareq = Poly(chareq, symbol)
    # Can't just call roots because it doesn't return rootof for unsolveable
    # polynomials.
    chareqroots = roots(chareq, multiple=True)
    if len(chareqroots) != order:
        chareqroots = [rootof(chareq, k) for k in range(chareq.degree())]

    chareq_is_complex = not all(i.is_real for i in chareq.all_coeffs())

    # Create a dict root: multiplicity or charroots
    charroots = defaultdict(int)
    for root in chareqroots:
        charroots[root] += 1
    # We need to keep track of terms so we can run collect() at the end.
    # This is necessary for constantsimp to work properly.
    collectterms = []
    gensols = []
    conjugate_roots = [] # used to prevent double-use of conjugate roots
    # Loop over roots in theorder provided by roots/rootof...
    for root in chareqroots:
        # but don't repoeat multiple roots.
        if root not in charroots:
            continue
        multiplicity = charroots.pop(root)
        for i in range(multiplicity):
            if chareq_is_complex:
                gensols.append(x**i*exp(root*x))
                collectterms = [(i, root, 0)] + collectterms
                continue
            reroot = re(root)
            imroot = im(root)
            if imroot.has(atan2) and reroot.has(atan2):
                # Remove this condition when re and im stop returning
                # circular atan2 usages.
                gensols.append(x**i*exp(root*x))
                collectterms = [(i, root, 0)] + collectterms
            else:
                if root in conjugate_roots:
                    collectterms = [(i, reroot, imroot)] + collectterms
                    continue
                if imroot == 0:
                    gensols.append(x**i*exp(reroot*x))
                    collectterms = [(i, reroot, 0)] + collectterms
                    continue
                conjugate_roots.append(conjugate(root))
                gensols.append(x**i*exp(reroot*x) * sin(abs(imroot) * x))
                gensols.append(x**i*exp(reroot*x) * cos(    imroot  * x))

                # This ordering is important
                collectterms = [(i, reroot, imroot)] + collectterms
    return gensols, collectterms


# Ideally these kind of simplification functions shouldn't be part of solvers.
# odesimp should be improved to handle these kind of specific simplifications.
def _get_simplified_sol(sol, func, collectterms):
    r"""
    Helper function which collects the solution on
    collectterms. Ideally this should be handled by odesimp.It is used
    only when the simplify is set to True in dsolve.

    The parameter ``collectterms`` is a list of tuple (i, reroot, imroot) where `i` is
    the multiplicity of the root, reroot is real part and imroot being the imaginary part.

    """
    f = func.func
    x = func.args[0]
    collectterms.sort(key=default_sort_key)
    collectterms.reverse()
    assert len(sol) == 1 and sol[0].lhs == f(x)
    sol = sol[0].rhs
    sol = expand_mul(sol)
    for i, reroot, imroot in collectterms:
        sol = collect(sol, x**i*exp(reroot*x)*sin(abs(imroot)*x))
        sol = collect(sol, x**i*exp(reroot*x)*cos(imroot*x))
    for i, reroot, imroot in collectterms:
        sol = collect(sol, x**i*exp(reroot*x))
    sol = powsimp(sol)
    return Eq(f(x), sol)


def _undetermined_coefficients_match(expr, x, func=None, eq_homogeneous=S.Zero):
    r"""
    Returns a trial function match if undetermined coefficients can be applied
    to ``expr``, and ``None`` otherwise.

    A trial expression can be found for an expression for use with the method
    of undetermined coefficients if the expression is an
    additive/multiplicative combination of constants, polynomials in `x` (the
    independent variable of expr), `\sin(a x + b)`, `\cos(a x + b)`, and
    `e^{a x}` terms (in other words, it has a finite number of linearly
    independent derivatives).

    Note that you may still need to multiply each term returned here by
    sufficient `x` to make it linearly independent with the solutions to the
    homogeneous equation.

    This is intended for internal use by ``undetermined_coefficients`` hints.

    SymPy currently has no way to convert `\sin^n(x) \cos^m(y)` into a sum of
    only `\sin(a x)` and `\cos(b x)` terms, so these are not implemented.  So,
    for example, you will need to manually convert `\sin^2(x)` into `[1 +
    \cos(2 x)]/2` to properly apply the method of undetermined coefficients on
    it.

    Examples
    ========

    >>> from sympy import log, exp
    >>> from sympy.solvers.ode.nonhomogeneous import _undetermined_coefficients_match
    >>> from sympy.abc import x
    >>> _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x)
    {'test': True, 'trialset': {x*exp(x), exp(-x), exp(x)}}
    >>> _undetermined_coefficients_match(log(x), x)
    {'test': False}

    """
    a = Wild('a', exclude=[x])
    b = Wild('b', exclude=[x])
    expr = powsimp(expr, combine='exp')  # exp(x)*exp(2*x + 1) => exp(3*x + 1)
    retdict = {}

    def _test_term(expr, x):
        r"""
        Test if ``expr`` fits the proper form for undetermined coefficients.
        """
        if not expr.has(x):
            return True
        elif expr.is_Add:
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Mul:
            if expr.has(sin, cos):
                foundtrig = False
                # Make sure that there is only one trig function in the args.
                # See the docstring.
                for i in expr.args:
                    if i.has(sin, cos):
                        if foundtrig:
                            return False
                        else:
                            foundtrig = True
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Function:
            if expr.func in (sin, cos, exp, sinh, cosh):
                if expr.args[0].match(a*x + b):
                    return True
                else:
                    return False
            else:
                return False
        elif expr.is_Pow and expr.base.is_Symbol and expr.exp.is_Integer and \
                expr.exp >= 0:
            return True
        elif expr.is_Pow and expr.base.is_number:
            if expr.exp.match(a*x + b):
                return True
            else:
                return False
        elif expr.is_Symbol or expr.is_number:
            return True
        else:
            return False

    def _get_trial_set(expr, x, exprs=set()):
        r"""
        Returns a set of trial terms for undetermined coefficients.

        The idea behind undetermined coefficients is that the terms expression
        repeat themselves after a finite number of derivatives, except for the
        coefficients (they are linearly dependent).  So if we collect these,
        we should have the terms of our trial function.
        """
        def _remove_coefficient(expr, x):
            r"""
            Returns the expression without a coefficient.

            Similar to expr.as_independent(x)[1], except it only works
            multiplicatively.
            """
            term = S.One
            if expr.is_Mul:
                for i in expr.args:
                    if i.has(x):
                        term *= i
            elif expr.has(x):
                term = expr
            return term

        expr = expand_mul(expr)
        if expr.is_Add:
            for term in expr.args:
                if _remove_coefficient(term, x) in exprs:
                    pass
                else:
                    exprs.add(_remove_coefficient(term, x))
                    exprs = exprs.union(_get_trial_set(term, x, exprs))
        else:
            term = _remove_coefficient(expr, x)
            tmpset = exprs.union({term})
            oldset = set()
            while tmpset != oldset:
                # If you get stuck in this loop, then _test_term is probably
                # broken
                oldset = tmpset.copy()
                expr = expr.diff(x)
                term = _remove_coefficient(expr, x)
                if term.is_Add:
                    tmpset = tmpset.union(_get_trial_set(term, x, tmpset))
                else:
                    tmpset.add(term)
            exprs = tmpset
        return exprs

    def is_homogeneous_solution(term):
        r""" This function checks whether the given trialset contains any root
            of homogeneous equation"""
        return expand(sub_func_doit(eq_homogeneous, func, term)).is_zero

    retdict['test'] = _test_term(expr, x)
    if retdict['test']:
        # Try to generate a list of trial solutions that will have the
        # undetermined coefficients. Note that if any of these are not linearly
        # independent with any of the solutions to the homogeneous equation,
        # then they will need to be multiplied by sufficient x to make them so.
        # This function DOES NOT do that (it doesn't even look at the
        # homogeneous equation).
        temp_set = set()
        for i in Add.make_args(expr):
            act = _get_trial_set(i, x)
            if eq_homogeneous is not S.Zero:
                while any(is_homogeneous_solution(ts) for ts in act):
                    act = {x*ts for ts in act}
            temp_set = temp_set.union(act)

        retdict['trialset'] = temp_set
    return retdict


def _solve_undetermined_coefficients(eq, func, order, match, trialset):
    r"""
    Helper function for the method of undetermined coefficients.

    See the
    :py:meth:`~sympy.solvers.ode.single.NthLinearConstantCoeffUndeterminedCoefficients`
    docstring for more information on this method.

    The parameter ``trialset`` is the set of trial functions as returned by
    ``_undetermined_coefficients_match()['trialset']``.

    The parameter ``match`` should be a dictionary that has the following
    keys:

    ``list``
    A list of solutions to the homogeneous equation.

    ``sol``
    The general solution.

    """
    r = match
    coeffs = numbered_symbols('a', cls=Dummy)
    coefflist = []
    gensols = r['list']
    gsol = r['sol']
    f = func.func
    x = func.args[0]

    if len(gensols) != order:
        raise NotImplementedError("Cannot find " + str(order) +
        " solutions to the homogeneous equation necessary to apply" +
        " undetermined coefficients to " + str(eq) +
        " (number of terms != order)")

    trialfunc = 0
    for i in trialset:
        c = next(coeffs)
        coefflist.append(c)
        trialfunc += c*i

    eqs = sub_func_doit(eq, f(x), trialfunc)

    coeffsdict = dict(list(zip(trialset, [0]*(len(trialset) + 1))))

    eqs = _mexpand(eqs)

    for i in Add.make_args(eqs):
        s = separatevars(i, dict=True, symbols=[x])
        if coeffsdict.get(s[x]):
            coeffsdict[s[x]] += s['coeff']
        else:
            coeffsdict[s[x]] = s['coeff']

    coeffvals = solve(list(coeffsdict.values()), coefflist)

    if not coeffvals:
        raise NotImplementedError(
            "Could not solve `%s` using the "
            "method of undetermined coefficients "
            "(unable to solve for coefficients)." % eq)

    psol = trialfunc.subs(coeffvals)

    return Eq(f(x), gsol.rhs + psol)