You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
52 lines
1.7 KiB
52 lines
1.7 KiB
5 months ago
|
def finite_diff(expression, variable, increment=1):
|
||
|
|
"""
|
||
|
|
Takes as input a polynomial expression and the variable used to construct
|
||
|
|
it and returns the difference between function's value when the input is
|
||
|
|
incremented to 1 and the original function value. If you want an increment
|
||
|
|
other than one supply it as a third argument.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.abc import x, y, z
|
||
|
|
>>> from sympy.series.kauers import finite_diff
|
||
|
|
>>> finite_diff(x**2, x)
|
||
|
|
2*x + 1
|
||
|
|
>>> finite_diff(y**3 + 2*y**2 + 3*y + 4, y)
|
||
|
|
3*y**2 + 7*y + 6
|
||
|
|
>>> finite_diff(x**2 + 3*x + 8, x, 2)
|
||
|
|
4*x + 10
|
||
|
|
>>> finite_diff(z**3 + 8*z, z, 3)
|
||
|
|
9*z**2 + 27*z + 51
|
||
|
|
"""
|
||
|
|
expression = expression.expand()
|
||
|
|
expression2 = expression.subs(variable, variable + increment)
|
||
|
|
expression2 = expression2.expand()
|
||
|
|
return expression2 - expression
|
||
|
|
|
||
|
|
def finite_diff_kauers(sum):
|
||
|
|
"""
|
||
|
|
Takes as input a Sum instance and returns the difference between the sum
|
||
|
|
with the upper index incremented by 1 and the original sum. For example,
|
||
|
|
if S(n) is a sum, then finite_diff_kauers will return S(n + 1) - S(n).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.series.kauers import finite_diff_kauers
|
||
|
|
>>> from sympy import Sum
|
||
|
|
>>> from sympy.abc import x, y, m, n, k
|
||
|
|
>>> finite_diff_kauers(Sum(k, (k, 1, n)))
|
||
|
|
n + 1
|
||
|
|
>>> finite_diff_kauers(Sum(1/k, (k, 1, n)))
|
||
|
|
1/(n + 1)
|
||
|
|
>>> finite_diff_kauers(Sum((x*y**2), (x, 1, n), (y, 1, m)))
|
||
|
|
(m + 1)**2*(n + 1)
|
||
|
|
>>> finite_diff_kauers(Sum((x*y), (x, 1, m), (y, 1, n)))
|
||
|
|
(m + 1)*(n + 1)
|
||
|
|
"""
|
||
|
|
function = sum.function
|
||
|
|
for l in sum.limits:
|
||
|
|
function = function.subs(l[0], l[- 1] + 1)
|
||
|
|
return function
|