def test_pade_complex():
# Test sequence with known solutions - see page 6 of 10.1109/PESGM.2012.6344759.
# Variable x is parameter - these tests will work with any complex number.
x = 0.2 + 0.6j
an = [
1.0, x, -x * x.conjugate(),
x.conjugate() * (x**2) + x * (x.conjugate()**2),
-(x**3) * x.conjugate() - 3 * (x * x.conjugate())**2 - x *
(x.conjugate()**3)
]
nump, denomp = pade(an, 1, 1)
assert_array_almost_equal(nump.c, [x + x.conjugate(), 1.0])
assert_array_almost_equal(denomp.c, [x.conjugate(), 1.0])
nump, denomp = pade(an, 1, 2)
assert_array_almost_equal(nump.c, [x**2, 2 * x + x.conjugate(), 1.0])
assert_array_almost_equal(denomp.c, [x + x.conjugate(), 1.0])
nump, denomp = pade(an, 2, 2)
assert_array_almost_equal(nump.c, [
x**2 + x * x.conjugate() + x.conjugate()**2, 2 *
(x + x.conjugate()), 1.0
])
assert_array_almost_equal(denomp.c,
[x.conjugate()**2, x + 2 * x.conjugate(), 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])
nump, denomp = pade([1.0], 0, 0)
assert_array_equal(nump.c, [1.0])
assert_array_equal(denomp.c, [1.0])
def test_pade_ints():
# Simple test sequences (one of ints, one of floats).
an_int = [1, 2, 3, 4]
an_flt = [1.0, 2.0, 3.0, 4.0]
# Make sure integer arrays give the same result as float arrays with same values.
for i in range(0, len(an_int)):
for j in range(0, len(an_int) - i):
# Create float and int pade approximation for given order.
nump_int, denomp_int = pade(an_int, i, j)
nump_flt, denomp_flt = pade(an_flt, i, j)
# Check that they are the same.
assert_array_equal(nump_int.c, nump_flt.c)
assert_array_equal(denomp_int.c, denomp_flt.c)
def se_pade(m, n, s, ell):
order = m + n
#power
p = np.arange(order + 1)
#intermediate variable
c = np.power((-np.square(ell) / 2), p) / factorial(p)
#simo's implementation of pade seems different to the scipy one
#it seems that simo take account negativity of c in his pade_approx.m
#I dont know which is one correct
pade_res = pade(c, n, m)
a = pade_res[0].coefficients
b = pade_res[1].coefficients
A = np.zeros(2 * a.shape[0] - 1)
B = np.zeros(2 * b.shape[0] - 1)
A[::2] = a
B[::2] = b
A = A * np.square(s) * np.sqrt(2 * np.pi) * ell
return np.poly1d(A), np.poly1d(B)
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, dtype=np.float64)
taylor = np.append([1.0], (-c)**i / factorial(i))
num, den = pade(taylor, q)
return LinearSystem((num, den), analog=True)
def fit(zs,us,m=10,n=10):
"""Fit to a Pade approximant"""
us = 1./us
e_exp = np.polyfit(zs,us,(m+n)*2)
p, q = pade(np.flip(e_exp), m,n)
# e_exp = np.flip(e_exp)
print(e_exp)
e_poly = np.poly1d(e_exp)
def f(z): return 1/(p(z)/q(z))
return f
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 pade_vector_rational_function(taylor_polynomial, pade_power, **kwargs):
taylor_polynomial = VectorPolynomial(
taylor_polynomial) # shape = (N, taylor_order)
pp = list()
qq = list()
for ii in range(taylor_polynomial.N):
pii, qii = pade(taylor_polynomial.coeffs[ii, :], pade_power, **kwargs)
pp.append(pii.coeffs[::-1])
qq.append(qii.coeffs[::-1])
numerator = VectorPolynomial(np.array(pp))
denominator = VectorPolynomial(np.array(qq))
return numerator / denominator
def exp_coeff(Y, L, nu, n_stop):
"""
Computes the coefficient for the exponential approximation of the convolution
coefficients.
"""
print "*** Starting Pade approximation"
# defining the formal serie
# with the real part of the coefficients
f = [[Y[i,n].real for n in range(nu, n_stop)] for i in range(8)]
# matrix to store q_{i,l} and b_{i,l}
q = np.zeros((8, L), dtype=complex)
b = np.zeros((8, L), dtype=complex)
for i in range(8):
P, Q = pade(f[i], L)
assert(P.o == L-1 and Q.o == L)
root = Q.roots
for l in range(L):
q[i,l] = root[l]
# assert(abs(q[i,l]) > 1)
b[i,l] = -(P(q[i,l])/Q.deriv()(q[i,l]))*(q[i,l]**(nu-1))
print " * max(b) = {}, max(q) = {}".format(np.amax(abs(b)), np.amax(abs(q)))
print "*** Pade approximation --> done\n"
print "*** Checking Pade approximation"
Y_appr = np.zeros((8, n_stop), dtype=complex)
for i in range(8):
for n in range(nu):
Y_appr[i,n] = Y[i,n]
for n in range(nu, n_stop):
Y_appr[i,n] = np.sum([b[i,l]*(q[i,l]**(-n)) for l in range(L)])
# testing the approximation
test = True
for i in range(8):
error = np.linalg.norm(abs(Y[i,:] - Y_appr[i,:]))
print " * i = {} --> |Y-Y_appr| = {}".format(i, error)
if error > 10**(-3):
test = False
assert(test == True)
print "*** Pade approximation --> checked\n"
return q, b
def pade_propagator_coefs_m(*, pade_order, diff2, k0, dx, spe=False, alpha=0):
if spe:
def sqrt_1plus(x):
return 1 + x / 2
elif alpha == 0:
def sqrt_1plus(x):
return np.sqrt(1 + x)
else:
raise Exception('alpha not supported')
def propagator_func(s):
return np.exp(1j * k0 * dx * (sqrt_1plus(diff2(s)) - 1))
taylor_coefs = approximate_taylor_polynomial(
propagator_func, 0, pade_order[0] + pade_order[1] + 5, 0.01)
p, q = pade(taylor_coefs, pade_order[0], pade_order[1])
pade_coefs = list(
zip_longest([-1 / complex(v) for v in np.roots(p)],
[-1 / complex(v) for v in np.roots(q)],
fillvalue=0.0j))
return pade_coefs
def se_pade(m, n, s, ell):
order = m + n
#power
p = np.arange(order, -1, -1)
#intermediate variable
c = np.power((np.square(ell) / 2), p) / factorial(p)
pade_res = pade(c, m, n)
b = pade_res[0].coefficients
a = pade_res[1].coefficients
A = np.zeros(2 * a.shape[0] - 1)
B = np.zeros(2 * b.shape[0] - 1)
A[::2] = a
B[::2] = b
B = B * np.square(s) * np.sqrt(np.pi) * ell
return A, B
y = asymptote_nan(f(x), x, x0list=[0])
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(1, 1, 1)
#ax.plot(x,y)
ax.set_ylim(-4, 4)
ax.grid('on')
# ax.plot(x,tay(x),markersize=8)
# ax.plot(x,np.sin(x),markersize=8)
x0 = 1.0 #1.0
T10poly = scipy.interpolate.approximate_taylor_polynomial(f,
x=x0,
degree=10,
scale=0.05)
P5poly, Q5poly = pade(T10poly.coeffs[::-1], 5)
T10 = lambda x: T10poly(x - x0)
R5_5 = lambda x: P5poly(x - x0) / Q5poly(x - x0)
print(T10poly)
print("P5 poly {}".format(P5poly))
print(Q5poly)
#fig = plt.figure(figsize=(8,6))
#ax = fig.add_subplot(1,1,1)
ax.plot(x, y, label='f(x)')
ax.plot(x0, f(x0), '.k', markersize=8)
yR5 = R5_5(x)
# use real roots of Q5(x-x0) for finding asymptotes
yR5 = asymptote_nan(
yR5, x, [r + x0 for r in np.roots(Q5poly) if np.abs(np.imag(r)) < 1e-8])
ax.plot(x, T10(x), label='T10(x)')
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])
# Testing inclusion of optional parameter
nump, denomp = pade(an, 0, 3)
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, 2)
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, 1)
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, 0)
assert_array_almost_equal(nump.c, [1.0])
assert_array_almost_equal(denomp.c, [-1.0 / 6, 0.5, -1.0, 1.0])
# Testing reducing array.
nump, denomp = pade(an, 0, 2)
assert_array_almost_equal(nump.c, [0.5, 1.0, 1.0])
assert_array_almost_equal(denomp.c, [1.0])
nump, denomp = pade(an, 1, 1)
assert_array_almost_equal(nump.c, [1.0 / 2, 1.0])
assert_array_almost_equal(denomp.c, [-1.0 / 2, 1.0])
nump, denomp = pade(an, 2, 0)
assert_array_almost_equal(nump.c, [1.0])
assert_array_almost_equal(denomp.c, [1.0 / 2, -1.0, 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])