Spaces:
Sleeping
Sleeping
File size: 5,557 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 |
"""Gosper's algorithm for hypergeometric summation. """
from sympy.core import S, Dummy, symbols
from sympy.polys import Poly, parallel_poly_from_expr, factor
from sympy.utilities.iterables import is_sequence
def gosper_normal(f, g, n, polys=True):
r"""
Compute the Gosper's normal form of ``f`` and ``g``.
Explanation
===========
Given relatively prime univariate polynomials ``f`` and ``g``,
rewrite their quotient to a normal form defined as follows:
.. math::
\frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}
where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are
monic polynomials in ``n`` with the following properties:
1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}`
2. `\gcd(B(n), C(n+1)) = 1`
3. `\gcd(A(n), C(n)) = 1`
This normal form, or rational factorization in other words, is a
crucial step in Gosper's algorithm and in solving of difference
equations. It can be also used to decide if two hypergeometric
terms are similar or not.
This procedure will return a tuple containing elements of this
factorization in the form ``(Z*A, B, C)``.
Examples
========
>>> from sympy.concrete.gosper import gosper_normal
>>> from sympy.abc import n
>>> gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False)
(1/4, n + 3/2, n + 1/4)
"""
(p, q), opt = parallel_poly_from_expr(
(f, g), n, field=True, extension=True)
a, A = p.LC(), p.monic()
b, B = q.LC(), q.monic()
C, Z = A.one, a/b
h = Dummy('h')
D = Poly(n + h, n, h, domain=opt.domain)
R = A.resultant(B.compose(D))
roots = set(R.ground_roots().keys())
for r in set(roots):
if not r.is_Integer or r < 0:
roots.remove(r)
for i in sorted(roots):
d = A.gcd(B.shift(+i))
A = A.quo(d)
B = B.quo(d.shift(-i))
for j in range(1, i + 1):
C *= d.shift(-j)
A = A.mul_ground(Z)
if not polys:
A = A.as_expr()
B = B.as_expr()
C = C.as_expr()
return A, B, C
def gosper_term(f, n):
r"""
Compute Gosper's hypergeometric term for ``f``.
Explanation
===========
Suppose ``f`` is a hypergeometric term such that:
.. math::
s_n = \sum_{k=0}^{n-1} f_k
and `f_k` does not depend on `n`. Returns a hypergeometric
term `g_n` such that `g_{n+1} - g_n = f_n`.
Examples
========
>>> from sympy.concrete.gosper import gosper_term
>>> from sympy import factorial
>>> from sympy.abc import n
>>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
(-n - 1/2)/(n + 1/4)
"""
from sympy.simplify import hypersimp
r = hypersimp(f, n)
if r is None:
return None # 'f' is *not* a hypergeometric term
p, q = r.as_numer_denom()
A, B, C = gosper_normal(p, q, n)
B = B.shift(-1)
N = S(A.degree())
M = S(B.degree())
K = S(C.degree())
if (N != M) or (A.LC() != B.LC()):
D = {K - max(N, M)}
elif not N:
D = {K - N + 1, S.Zero}
else:
D = {K - N + 1, (B.nth(N - 1) - A.nth(N - 1))/A.LC()}
for d in set(D):
if not d.is_Integer or d < 0:
D.remove(d)
if not D:
return None # 'f(n)' is *not* Gosper-summable
d = max(D)
coeffs = symbols('c:%s' % (d + 1), cls=Dummy)
domain = A.get_domain().inject(*coeffs)
x = Poly(coeffs, n, domain=domain)
H = A*x.shift(1) - B*x - C
from sympy.solvers.solvers import solve
solution = solve(H.coeffs(), coeffs)
if solution is None:
return None # 'f(n)' is *not* Gosper-summable
x = x.as_expr().subs(solution)
for coeff in coeffs:
if coeff not in solution:
x = x.subs(coeff, 0)
if x.is_zero:
return None # 'f(n)' is *not* Gosper-summable
else:
return B.as_expr()*x/C.as_expr()
def gosper_sum(f, k):
r"""
Gosper's hypergeometric summation algorithm.
Explanation
===========
Given a hypergeometric term ``f`` such that:
.. math ::
s_n = \sum_{k=0}^{n-1} f_k
and `f(n)` does not depend on `n`, returns `g_{n} - g(0)` where
`g_{n+1} - g_n = f_n`, or ``None`` if `s_n` cannot be expressed
in closed form as a sum of hypergeometric terms.
Examples
========
>>> from sympy.concrete.gosper import gosper_sum
>>> from sympy import factorial
>>> from sympy.abc import n, k
>>> f = (4*k + 1)*factorial(k)/factorial(2*k + 1)
>>> gosper_sum(f, (k, 0, n))
(-factorial(n) + 2*factorial(2*n + 1))/factorial(2*n + 1)
>>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2])
True
>>> gosper_sum(f, (k, 3, n))
(-60*factorial(n) + factorial(2*n + 1))/(60*factorial(2*n + 1))
>>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5])
True
References
==========
.. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B,
AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100
"""
indefinite = False
if is_sequence(k):
k, a, b = k
else:
indefinite = True
g = gosper_term(f, k)
if g is None:
return None
if indefinite:
result = f*g
else:
result = (f*(g + 1)).subs(k, b) - (f*g).subs(k, a)
if result is S.NaN:
try:
result = (f*(g + 1)).limit(k, b) - (f*g).limit(k, a)
except NotImplementedError:
result = None
return factor(result)
|