Spaces:
Sleeping
Sleeping
File size: 7,760 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 |
from itertools import product
from sympy.core.function import (Function, diff)
from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import exp
from sympy.calculus.finite_diff import (
apply_finite_diff, differentiate_finite, finite_diff_weights,
_as_finite_diff
)
from sympy.testing.pytest import raises, warns_deprecated_sympy
def test_apply_finite_diff():
x, h = symbols('x h')
f = Function('f')
assert (apply_finite_diff(1, [x-h, x+h], [f(x-h), f(x+h)], x) -
(f(x+h)-f(x-h))/(2*h)).simplify() == 0
assert (apply_finite_diff(1, [5, 6, 7], [f(5), f(6), f(7)], 5) -
(Rational(-3, 2)*f(5) + 2*f(6) - S.Half*f(7))).simplify() == 0
raises(ValueError, lambda: apply_finite_diff(1, [x, h], [f(x)]))
def test_finite_diff_weights():
d = finite_diff_weights(1, [5, 6, 7], 5)
assert d[1][2] == [Rational(-3, 2), 2, Rational(-1, 2)]
# Table 1, p. 702 in doi:10.1090/S0025-5718-1988-0935077-0
# --------------------------------------------------------
xl = [0, 1, -1, 2, -2, 3, -3, 4, -4]
# d holds all coefficients
d = finite_diff_weights(4, xl, S.Zero)
# Zeroeth derivative
for i in range(5):
assert d[0][i] == [S.One] + [S.Zero]*8
# First derivative
assert d[1][0] == [S.Zero]*9
assert d[1][2] == [S.Zero, S.Half, Rational(-1, 2)] + [S.Zero]*6
assert d[1][4] == [S.Zero, Rational(2, 3), Rational(-2, 3), Rational(-1, 12), Rational(1, 12)] + [S.Zero]*4
assert d[1][6] == [S.Zero, Rational(3, 4), Rational(-3, 4), Rational(-3, 20), Rational(3, 20),
Rational(1, 60), Rational(-1, 60)] + [S.Zero]*2
assert d[1][8] == [S.Zero, Rational(4, 5), Rational(-4, 5), Rational(-1, 5), Rational(1, 5),
Rational(4, 105), Rational(-4, 105), Rational(-1, 280), Rational(1, 280)]
# Second derivative
for i in range(2):
assert d[2][i] == [S.Zero]*9
assert d[2][2] == [-S(2), S.One, S.One] + [S.Zero]*6
assert d[2][4] == [Rational(-5, 2), Rational(4, 3), Rational(4, 3), Rational(-1, 12), Rational(-1, 12)] + [S.Zero]*4
assert d[2][6] == [Rational(-49, 18), Rational(3, 2), Rational(3, 2), Rational(-3, 20), Rational(-3, 20),
Rational(1, 90), Rational(1, 90)] + [S.Zero]*2
assert d[2][8] == [Rational(-205, 72), Rational(8, 5), Rational(8, 5), Rational(-1, 5), Rational(-1, 5),
Rational(8, 315), Rational(8, 315), Rational(-1, 560), Rational(-1, 560)]
# Third derivative
for i in range(3):
assert d[3][i] == [S.Zero]*9
assert d[3][4] == [S.Zero, -S.One, S.One, S.Half, Rational(-1, 2)] + [S.Zero]*4
assert d[3][6] == [S.Zero, Rational(-13, 8), Rational(13, 8), S.One, -S.One,
Rational(-1, 8), Rational(1, 8)] + [S.Zero]*2
assert d[3][8] == [S.Zero, Rational(-61, 30), Rational(61, 30), Rational(169, 120), Rational(-169, 120),
Rational(-3, 10), Rational(3, 10), Rational(7, 240), Rational(-7, 240)]
# Fourth derivative
for i in range(4):
assert d[4][i] == [S.Zero]*9
assert d[4][4] == [S(6), -S(4), -S(4), S.One, S.One] + [S.Zero]*4
assert d[4][6] == [Rational(28, 3), Rational(-13, 2), Rational(-13, 2), S(2), S(2),
Rational(-1, 6), Rational(-1, 6)] + [S.Zero]*2
assert d[4][8] == [Rational(91, 8), Rational(-122, 15), Rational(-122, 15), Rational(169, 60), Rational(169, 60),
Rational(-2, 5), Rational(-2, 5), Rational(7, 240), Rational(7, 240)]
# Table 2, p. 703 in doi:10.1090/S0025-5718-1988-0935077-0
# --------------------------------------------------------
xl = [[j/S(2) for j in list(range(-i*2+1, 0, 2))+list(range(1, i*2+1, 2))]
for i in range(1, 5)]
# d holds all coefficients
d = [finite_diff_weights({0: 1, 1: 2, 2: 4, 3: 4}[i], xl[i], 0) for
i in range(4)]
# Zeroth derivative
assert d[0][0][1] == [S.Half, S.Half]
assert d[1][0][3] == [Rational(-1, 16), Rational(9, 16), Rational(9, 16), Rational(-1, 16)]
assert d[2][0][5] == [Rational(3, 256), Rational(-25, 256), Rational(75, 128), Rational(75, 128),
Rational(-25, 256), Rational(3, 256)]
assert d[3][0][7] == [Rational(-5, 2048), Rational(49, 2048), Rational(-245, 2048), Rational(1225, 2048),
Rational(1225, 2048), Rational(-245, 2048), Rational(49, 2048), Rational(-5, 2048)]
# First derivative
assert d[0][1][1] == [-S.One, S.One]
assert d[1][1][3] == [Rational(1, 24), Rational(-9, 8), Rational(9, 8), Rational(-1, 24)]
assert d[2][1][5] == [Rational(-3, 640), Rational(25, 384), Rational(-75, 64),
Rational(75, 64), Rational(-25, 384), Rational(3, 640)]
assert d[3][1][7] == [Rational(5, 7168), Rational(-49, 5120),
Rational(245, 3072), Rational(-1225, 1024),
Rational(1225, 1024), Rational(-245, 3072),
Rational(49, 5120), Rational(-5, 7168)]
# Reasonably the rest of the table is also correct... (testing of that
# deemed excessive at the moment)
raises(ValueError, lambda: finite_diff_weights(-1, [1, 2]))
raises(ValueError, lambda: finite_diff_weights(1.2, [1, 2]))
x = symbols('x')
raises(ValueError, lambda: finite_diff_weights(x, [1, 2]))
def test_as_finite_diff():
x = symbols('x')
f = Function('f')
dx = Function('dx')
_as_finite_diff(f(x).diff(x), [x-2, x-1, x, x+1, x+2])
# Use of undefined functions in ``points``
df_true = -f(x+dx(x)/2-dx(x+dx(x)/2)/2) / dx(x+dx(x)/2) \
+ f(x+dx(x)/2+dx(x+dx(x)/2)/2) / dx(x+dx(x)/2)
df_test = diff(f(x), x).as_finite_difference(points=dx(x), x0=x+dx(x)/2)
assert (df_test - df_true).simplify() == 0
def test_differentiate_finite():
x, y, h = symbols('x y h')
f = Function('f')
with warns_deprecated_sympy():
res0 = differentiate_finite(f(x, y) + exp(42), x, y, evaluate=True)
xm, xp, ym, yp = [v + sign*S.Half for v, sign in product([x, y], [-1, 1])]
ref0 = f(xm, ym) + f(xp, yp) - f(xm, yp) - f(xp, ym)
assert (res0 - ref0).simplify() == 0
g = Function('g')
with warns_deprecated_sympy():
res1 = differentiate_finite(f(x)*g(x) + 42, x, evaluate=True)
ref1 = (-f(x - S.Half) + f(x + S.Half))*g(x) + \
(-g(x - S.Half) + g(x + S.Half))*f(x)
assert (res1 - ref1).simplify() == 0
res2 = differentiate_finite(f(x) + x**3 + 42, x, points=[x-1, x+1])
ref2 = (f(x + 1) + (x + 1)**3 - f(x - 1) - (x - 1)**3)/2
assert (res2 - ref2).simplify() == 0
raises(TypeError, lambda: differentiate_finite(f(x)*g(x), x,
pints=[x-1, x+1]))
res3 = differentiate_finite(f(x)*g(x).diff(x), x)
ref3 = (-g(x) + g(x + 1))*f(x + S.Half) - (g(x) - g(x - 1))*f(x - S.Half)
assert res3 == ref3
res4 = differentiate_finite(f(x)*g(x).diff(x).diff(x), x)
ref4 = -((g(x - Rational(3, 2)) - 2*g(x - S.Half) + g(x + S.Half))*f(x - S.Half)) \
+ (g(x - S.Half) - 2*g(x + S.Half) + g(x + Rational(3, 2)))*f(x + S.Half)
assert res4 == ref4
res5_expr = f(x).diff(x)*g(x).diff(x)
res5 = differentiate_finite(res5_expr, points=[x-h, x, x+h])
ref5 = (-2*f(x)/h + f(-h + x)/(2*h) + 3*f(h + x)/(2*h))*(-2*g(x)/h + g(-h + x)/(2*h) \
+ 3*g(h + x)/(2*h))/(2*h) - (2*f(x)/h - 3*f(-h + x)/(2*h) - \
f(h + x)/(2*h))*(2*g(x)/h - 3*g(-h + x)/(2*h) - g(h + x)/(2*h))/(2*h)
assert res5 == ref5
res6 = res5.limit(h, 0).doit()
ref6 = diff(res5_expr, x)
assert res6 == ref6
|