def test_maijerg_ex_recursive():
"""Test a Meijer-G approximant of a Meijer-G function."""
# e^-x = 1 - x + 1/2 x^2 - 1/6 x^3 + 1/24 x^4 - ...
# First construct Meijer-G from Taylor expansion
tcoeffs = np.array([1, -1, 1. / 2., -1. / 6., 1. / 24.])
borel = borel_trans_coeffs(tcoeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array(
[0.01, 0.1, 0.15, 0.21, 0.23, 0.29, 0.31, 0.32, 0.3, 0.25])
result0 = np.exp(-points)
print result0
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
print "initial Meijer-G ", zs
# Then make a Pade out of the Meijer-G results
gen_pade = GeneralPadeApproximant(points, np.real(zs), 4, 6)
tcoeffs_pade = taylor_coeffs(gen_pade(np.array([0.]), range(6)), 5)
borel1 = borel_trans_coeffs(tcoeffs_pade)
ratios1 = consecutive_ratios_odd(borel)
ps1, qs1 = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps1 = ps1[::-1]
qs1 = qs1[::-1]
qs1 = np.append(qs, np.array(1.))
rootsp1 = find_roots(ps1)
rootsq1 = find_roots(qs1)
xvector1 = np.append(np.array([1]), rootsp1)
yvector1 = rootsq1
pl1 = ps1[0]
ql1 = qs1[0]
#print tcoeffs_pade
zs1 = meijerg_approx_low(xvector1, yvector1, pl1, ql1, points)
print "True result", result0
print "from pade ", gen_pade(points)
print "result 1 ", np.real(zs1)
#y, optz = optimize_pars_meijerg(xvector1, yvector1, pl1, ql1, points, result0)
#print "result value", optz
y = optimize_pars_meijerg(xvector1, yvector1, pl1, ql1, points, result0)
print "result params", y
print "True result", result0
def test_example2():
"""Test Meijer-G approximant for equation 37."""
coeffs = np.array(
[1 / 2., 3 / 4., -21 / 8., 333 / 16., -30885. / 128, 916731 / 256.])
borel = borel_trans_coeffs(coeffs)
print borel
ratios = consecutive_ratios_odd(borel)
print ratios
ps, qs = rational_function_for_ratios(ratios)
print "ps, qs", ps, qs
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
print "xvector", xvector
print "yvector", yvector
pl = ps[0]
ql = qs[0]
points = np.array([1., 2., 50.])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
print zs / 2
def test_maijerg_ex_frompade():
"""Test a Meijer-G approximant of a Meijer-G function."""
# e^x = 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + ...
tcoeffs = np.array([
1, 1, 1. / 2., 1. / 6., 1. / 24. + 1. / 120., 1 / factorial(6.),
1 / factorial(7.)
])
p, q = misc.pade(tcoeffs, 3)
points = np.array([0.8, 0.5, 0.2, -0.5, -0.8, -1.0])
y = np.exp(points)
basic_pade = BasicPadeApproximant(points, y, tcoeffs, 3)
tcoeffs_pade = taylor_coeffs(basic_pade(np.array([0.]), range(4)), 3)
borel = borel_trans_coeffs(tcoeffs_pade)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
#print tcoeffs_pade
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
#print "from pade ", p(points)/q(points)
#print "result 0", y
#print "result 1 ", zs
assert (abs(y - np.real(zs)) < 1e-2).all()
def test_rational_function_for_ratios():
"""Test the rational function fit for the consecutive ratios of Borel coefficients."""
ratios = np.array([-1. / 8., -35. / 96., -11. / 24.])
assert_raises(TypeError, rational_function_for_ratios, [1, 2, 3])
ps, qs = rational_function_for_ratios(ratios)
# Make polynomials to check prediction of ratios
assert np.allclose(ps, [-1. / 8., -113. / 216])
assert np.allclose(qs, [7. / 9])
def test_find_roots():
"""Test of the wrapper function to find the roots of a polynomial."""
ratios = np.array([-1. / 8., -35. / 96., -11. / 24.])
assert_raises(TypeError, rational_function_for_ratios, [1, 2, 3])
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
assert np.allclose(rootsp, [-ps[1] / ps[0]])
assert np.allclose(rootsq, [-1 / qs[0]])
def test_example3():
coeffs = np.array([1, -1, 2.667, -4.667])
borel = borel_trans_coeffs(coeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array([0.6736])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
assert abs(1 - zs) < 1e-3
def test_meijerg():
"""Test Meijer-G approximant, comparing with results from the paper of Mera 2018."""
ratios = np.array([-1. / 8., -35. / 96., -11. / 24.])
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.array([1, rootsp[0]])
yvector = np.array([rootsq[0]])
pl = ps[0]
ql = qs[0]
points = np.array([-1 + 0j, -10 + 0j, -100 + 0j])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
#print zs
assert (zs -
[1.133285 + 0.144952j, 0.744345 + 0.436317j, 0.386356 + 0.321210j]
< 1e-6).all()
def test_meijerg_ex():
"""Test a Meijer-G approximant of a Meijer-G function."""
# e^x = 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + ...
tcoeffs = np.array([1, 1, 1. / 2., 1. / 6., 1. / 24.])
borel = borel_trans_coeffs(tcoeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array([1.2, -1.4])
result0 = np.exp(points)
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
assert (abs(result0 - np.real(zs)) < 1e-7).all()
def test_example5():
coeffs = np.array([1, 6, 210, 13860])
borel = borel_trans_coeffs(coeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array([1, 10, 100])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
result0 = [
0.473794 + 0.368724j, 0.255694 + 0.228610j, 0.144490 + 0.133539j
]
assert (abs(zs - result0) < 1e-6).all()
def test_example4():
coeffs = np.array([1, -0.667, 0.556, -2.056])
borel = borel_trans_coeffs(coeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array([0.5381])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
print "zs", zs
v = 1 / (2. + zs)
result0 = 0.5921
print v, result0
def test_example6():
coeffs = np.array([1, 1 / 2., 9 / 8., 75 / 16.])
borel = borel_trans_coeffs(coeffs)
ratios = consecutive_ratios_odd(borel)
ps, qs = rational_function_for_ratios(ratios)
# Add the constant 1. to the qm coefficients
ps = ps[::-1]
qs = qs[::-1]
qs = np.append(qs, np.array(1.))
rootsp = find_roots(ps)
rootsq = find_roots(qs)
xvector = np.append(np.array([1]), rootsp)
yvector = rootsq
pl = ps[0]
ql = qs[0]
points = np.array([1, 100])
zs = meijerg_approx_low(xvector, yvector, pl, ql, points)
result0 = [
0.990312240887789089 + 0.481308237536857j,
0.13677671640883210679 + 0.23483780883795517888j
]
assert (abs(zs - result0) < 1e-7).all