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 _pade_delay(p, q, c):
"""Numerically evaluated state-space using Pade approximants.
This may have numerical issues for large values of p or q.
"""
i = np.arange(1, p+q+1)
taylor = np.append([1.0], (-c)**i / factorial(i))
num, den = pade(taylor, q)
return LinearFilter(num, den)
def test_pade_4term_exp():
# First four Taylor coefficients of exp(x).
# Unlike poly1d, the first array element is the zero-order term.
an = [1.0, 1.0, 0.5, 1.0 / 6]
nump, denomp = pade(an, 0)
assert_array_almost_equal(nump.c, [1.0 / 6, 0.5, 1.0, 1.0])
assert_array_almost_equal(denomp.c, [1.0])
nump, denomp = pade(an, 1)
assert_array_almost_equal(nump.c, [1.0 / 6, 2.0 / 3, 1.0])
assert_array_almost_equal(denomp.c, [-1.0 / 3, 1.0])
nump, denomp = pade(an, 2)
assert_array_almost_equal(nump.c, [1.0 / 3, 1.0])
assert_array_almost_equal(denomp.c, [1.0 / 6, -2.0 / 3, 1.0])
nump, denomp = pade(an, 3)
assert_array_almost_equal(nump.c, [1.0])
assert_array_almost_equal(denomp.c, [-1.0 / 6, 0.5, -1.0, 1.0])
def test_pade_4term_exp():
# First four Taylor coefficients of exp(x).
# Unlike poly1d, the first array element is the zero-order term.
an = [1.0, 1.0, 0.5, 1.0/6]
nump, denomp = pade(an, 0)
assert_array_almost_equal(nump.c, [1.0/6, 0.5, 1.0, 1.0])
assert_array_almost_equal(denomp.c, [1.0])
nump, denomp = pade(an, 1)
assert_array_almost_equal(nump.c, [1.0/6, 2.0/3, 1.0])
assert_array_almost_equal(denomp.c, [-1.0/3, 1.0])
nump, denomp = pade(an, 2)
assert_array_almost_equal(nump.c, [1.0/3, 1.0])
assert_array_almost_equal(denomp.c, [1.0/6, -2.0/3, 1.0])
nump, denomp = pade(an, 3)
assert_array_almost_equal(nump.c, [1.0])
assert_array_almost_equal(denomp.c, [-1.0/6, 0.5, -1.0, 1.0])
def lambert_delay(delay, sub_delay, tau, p, q):
"""Returns F = p/q s.t. F((tau*s+1)/e^(-sb)) = e^(-sa)."""
a, b = delay, sub_delay
r = a / b
c = np.exp(a / tau)
d = (b / tau) * np.exp(b / tau)
i = np.arange(1, p + q + 1)
taylor = np.append([1./r], (i+r)**(i-1) / factorial(i))
tf = pade(taylor, q)
nds = np.poly1d([-d, 0]) # -ds
return LinearSystem((c*r*tf[0](nds), tf[1](nds)), analog=True)
def series_evaluate(series, deta, order=1, series_type="taylor"):
""" Return evaluated polynomial using the data frame """
assert order <= series_to_num_order(series)
coeffs = series["b2"].values[:(order + 1)]
if series_type == "pade" or series_type == "q" or series_type == "p":
p_pade, q_pade = misc.pade(coeffs, 1)
if series_type == "pade":
return p_pade(deta) / q_pade(deta)
if series_type == "q":
return q_pade(deta)
return p_pade(deta)
poly = np.poly1d(coeffs[::-1])
return poly(deta)
def test_pade_trivial():
nump, denomp = pade([1.0], 0)
assert_array_equal(nump.c, [1.0])
assert_array_equal(denomp.c, [1.0])
def test_pade_trivial():
nump, denomp = pade([1.0], 0)
assert_array_equal(nump.c, [1.0])
assert_array_equal(denomp.c, [1.0])
def test_pade():
# make sure scipy.misc.pade exists
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, "`pade` is deprecated")
pade([1, 2], 1)
def test_pade():
# make sure scipy.misc.pade exists
with warnings.catch_warnings():
warnings.simplefilter('ignore', DeprecationWarning)
pade([1, 2], 1)
def test_pade():
# make sure scipy.misc.pade exists
with warnings.catch_warnings():
warnings.simplefilter('ignore', DeprecationWarning)
pade([1, 2], 1)
