def test_construct_d_case_5():
"""
This function tests the Case 5 in the step
to calculate coefficients of the d-vector.
Each test case has 3 values -
1. num - Numerator of the rational function a(x).
2. den - Denominator of the rational function a(x).
3. d - The d-vector.
"""
tests = [(Poly(2 * x**3 + x**2 + x - 2, x, extension=True),
Poly(9 * x**3 + 5 * x**2 + 2 * x - 1, x,
extension=True), [[sqrt(2) / 3, -sqrt(2) / 108],
[-sqrt(2) / 3,
sqrt(2) / 108]]),
(Poly(3 * x**5 + x**4 - x**3 + x**2 - 2 * x - 2, x, domain='ZZ'),
Poly(9 * x**5 + 7 * x**4 + 3 * x**3 + 2 * x**2 + 5 * x + 7,
x,
domain='ZZ'), [[sqrt(3) / 3, -2 * sqrt(3) / 27],
[-sqrt(3) / 3, 2 * sqrt(3) / 27]]),
(Poly(x**2 - x + 1, x, domain='ZZ'),
Poly(3 * x**2 + 7 * x + 3, x,
domain='ZZ'), [[sqrt(3) / 3, -5 * sqrt(3) / 9],
[-sqrt(3) / 3, 5 * sqrt(3) / 9]])]
for num, den, d in tests:
# Multiplicity of oo is 0
ser = rational_laurent_series(num, den, x, oo, 0, 1)
assert construct_d_case_5(ser) == d
def test_rational_laurent_series():
"""
This function tests the computation of coefficients
of Laurent series of a rational function.
Each test case has 5 values -
1. num - Numerator of the rational function.
2. den - Denominator of the rational function.
3. x0 - Point about which Laurent series is to
be calculated.
4. mul - Multiplicity of x0 if x0 is a pole of
the rational function (0 otherwise).
5. n - Number of terms upto which the series
is to be calcuated.
"""
tests = [
# Laurent series about simple pole (Multiplicity = 1)
(
Poly(x**2 - 3*x + 9, x, extension=True),
Poly(x**2 - x, x, extension=True),
S(1), 1, 6,
{1: 7, 0: -8, -1: 9, -2: -9, -3: 9, -4: -9}
),
# Laurent series about multiple pole (Multiplicty > 1)
(
Poly(64*x**3 - 1728*x + 1216, x, extension=True),
Poly(64*x**4 - 80*x**3 - 831*x**2 + 1809*x - 972, x, extension=True),
S(9)/8, 2, 3,
{0: S(32177152)/46521675, 2: S(1019)/984, -1: S(11947565056)/28610830125, \
1: S(209149)/75645}
),
(
Poly(1, x, extension=True),
Poly(x**5 + (-4*sqrt(2) - 1)*x**4 + (4*sqrt(2) + 12)*x**3 + (-12 - 8*sqrt(2))*x**2 \
+ (4 + 8*sqrt(2))*x - 4, x, extension=True),
sqrt(2), 4, 6,
{4: 1 + sqrt(2), 3: -3 - 2*sqrt(2), 2: Mul(-1, -3 - 2*sqrt(2), evaluate=False)/(-1 \
+ sqrt(2)), 1: (-3 - 2*sqrt(2))/(-1 + sqrt(2))**2, 0: Mul(-1, -3 - 2*sqrt(2), evaluate=False \
)/(-1 + sqrt(2))**3, -1: (-3 - 2*sqrt(2))/(-1 + sqrt(2))**4}
),
# Laurent series about oo
(
Poly(x**5 - 4*x**3 + 6*x**2 + 10*x - 13, x, extension=True),
Poly(x**2 - 5, x, extension=True),
oo, 3, 6,
{3: 1, 2: 0, 1: 1, 0: 6, -1: 15, -2: 17}
),
# Laurent series at x0 where x0 is not a pole of the function
# Using multiplicity as 0 (as x0 will not be a pole)
(
Poly(3*x**3 + 6*x**2 - 2*x + 5, x, extension=True),
Poly(9*x**4 - x**3 - 3*x**2 + 4*x + 4, x, extension=True),
S(2)/5, 0, 1,
{0: S(3345)/3304, -1: S(399325)/2729104, -2: S(3926413375)/4508479808, \
-3: S(-5000852751875)/1862002160704, -4: S(-6683640101653125)/6152055138966016}
),
(
Poly(-7*x**2 + 2*x - 4, x, extension=True),
Poly(7*x**5 + 9*x**4 + 8*x**3 + 3*x**2 + 6*x + 9, x, extension=True),
oo, 0, 6,
{0: 0, -2: 0, -5: -S(71)/49, -1: 0, -3: -1, -4: S(11)/7}
)]
for num, den, x0, mul, n, ser in tests:
assert ser == rational_laurent_series(num, den, x, x0, mul, n)
def test_construct_d_case_4():
"""
This function tests the Case 4 in the step
to calculate coefficients of the d-vector.
Each test case has 4 values -
1. num - Numerator of the rational function a(x).
2. den - Denominator of the rational function a(x).
3. mul - Multiplicity of oo as a pole.
4. d - The d-vector.
"""
tests = [
# Tests with multiplicity at oo = 2
(
Poly(-x**5 - 2*x**4 + 4*x**3 + 2*x + 5, x, extension=True),
Poly(9*x**3 - 2*x**2 + 10*x - 2, x, extension=True),
2,
[[10*I/27, I/3, -3*I*(S(158)/243 - I/3)/2], \
[-10*I/27, -I/3, 3*I*(S(158)/243 + I/3)/2]]
),
(
Poly(-x**6 + 9*x**5 + 5*x**4 + 6*x**3 + 5*x**2 + 6*x + 7, x, extension=True),
Poly(x**4 + 3*x**3 + 12*x**2 - x + 7, x, extension=True),
2,
[[-6*I, I, -I*(17 - I)/2], [6*I, -I, I*(17 + I)/2]]
),
# Tests with multiplicity at oo = 4
(
Poly(-2*x**6 - x**5 - x**4 - 2*x**3 - x**2 - 3*x - 3, x, extension=True),
Poly(3*x**2 + 10*x + 7, x, extension=True),
4,
[[269*sqrt(6)*I/288, -17*sqrt(6)*I/36, sqrt(6)*I/3, -sqrt(6)*I*(S(16969)/2592 \
- 2*sqrt(6)*I/3)/4], [-269*sqrt(6)*I/288, 17*sqrt(6)*I/36, -sqrt(6)*I/3, \
sqrt(6)*I*(S(16969)/2592 + 2*sqrt(6)*I/3)/4]]
),
(
Poly(-3*x**5 - 3*x**4 - 3*x**3 - x**2 - 1, x, extension=True),
Poly(12*x - 2, x, extension=True),
4,
[[41*I/192, 7*I/24, I/2, -I*(-S(59)/6912 - I)], \
[-41*I/192, -7*I/24, -I/2, I*(-S(59)/6912 + I)]]
),
# Tests with multiplicity at oo = 4
(
Poly(-x**7 - x**5 - x**4 - x**2 - x, x, extension=True),
Poly(x + 2, x, extension=True),
6,
[[-5*I/2, 2*I, -I, I, -I*(-9 - 3*I)/2], [5*I/2, -2*I, I, -I, I*(-9 + 3*I)/2]]
),
(
Poly(-x**7 - x**6 - 2*x**5 - 2*x**4 - x**3 - x**2 + 2*x - 2, x, extension=True),
Poly(2*x - 2, x, extension=True),
6,
[[3*sqrt(2)*I/4, 3*sqrt(2)*I/4, sqrt(2)*I/2, sqrt(2)*I/2, -sqrt(2)*I*(-S(7)/8 - \
3*sqrt(2)*I/2)/2], [-3*sqrt(2)*I/4, -3*sqrt(2)*I/4, -sqrt(2)*I/2, -sqrt(2)*I/2, \
sqrt(2)*I*(-S(7)/8 + 3*sqrt(2)*I/2)/2]]
)]
for num, den, mul, d in tests:
ser = rational_laurent_series(num, den, x, oo, mul, 1)
assert construct_d_case_4(ser, mul // 2) == d