You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
586 lines
18 KiB
Python
586 lines
18 KiB
Python
'''
|
|
Created on 20. aug. 2015
|
|
|
|
@author: pab
|
|
'''
|
|
from __future__ import division
|
|
import numpy as np
|
|
import warnings
|
|
import numdifftools as nd # @UnresolvedImport
|
|
import numdifftools.nd_algopy as nda # @UnresolvedImport
|
|
from numdifftools.limits import Limit # @UnresolvedImport
|
|
from numpy import linalg
|
|
from numpy.polynomial.chebyshev import chebval, Chebyshev
|
|
from numpy.polynomial import polynomial
|
|
from wafo.misc import piecewise, findcross, ecross
|
|
from collections import namedtuple
|
|
|
|
EPS = np.finfo(float).eps
|
|
_EPS = EPS
|
|
finfo = np.finfo(float)
|
|
_TINY = finfo.tiny
|
|
_HUGE = finfo.max
|
|
dea3 = nd.dea3
|
|
|
|
|
|
class PolyBasis(object):
|
|
@staticmethod
|
|
def derivative(t, k, n=1):
|
|
c = np.zeros(k + 1)
|
|
c[k] = 1
|
|
dc = polynomial.polyder(c, m=n)
|
|
return polynomial.polyval(t, dc)
|
|
|
|
@staticmethod
|
|
def eval(t, c):
|
|
return polynomial.polyval(t, c)
|
|
|
|
def __call__(self, t, k):
|
|
return t**k
|
|
poly_basis = PolyBasis()
|
|
|
|
|
|
class ChebyshevBasis(object):
|
|
@staticmethod
|
|
def derivative(t, k, n=1):
|
|
c = np.zeros(k + 1)
|
|
c[k] = 1
|
|
cheb = Chebyshev(c)
|
|
dcheb = cheb.deriv(m=n)
|
|
return chebval(t, dcheb.coef)
|
|
|
|
@staticmethod
|
|
def eval(t, c):
|
|
return chebval(t, c)
|
|
|
|
def __call__(self, t, k):
|
|
c = np.zeros(k + 1)
|
|
c[k] = 1
|
|
return chebval(t, c)
|
|
chebyshev_basis = ChebyshevBasis()
|
|
|
|
|
|
def richardson(Q, k):
|
|
# license BSD
|
|
# Richardson extrapolation with parameter estimation
|
|
c = np.real((Q[k - 1] - Q[k - 2]) / (Q[k] - Q[k - 1])) - 1.
|
|
# The lower bound 0.07 admits the singularity x.^-0.9
|
|
c = max(c, 0.07)
|
|
R = Q[k] + (Q[k] - Q[k - 1]) / c
|
|
return R
|
|
|
|
|
|
def evans_webster_weights(omega, gg, dgg, x, basis, *args, **kwds):
|
|
|
|
def Psi(t, k):
|
|
return dgg(t, *args, **kwds) * basis(t, k)
|
|
|
|
j_w = 1j * omega
|
|
nn = len(x)
|
|
A = np.zeros((nn, nn), dtype=complex)
|
|
F = np.zeros((nn,), dtype=complex)
|
|
|
|
dbasis = basis.derivative
|
|
lim_gg = Limit(gg)
|
|
b1 = np.exp(j_w*lim_gg(1, *args, **kwds))
|
|
if np.isnan(b1):
|
|
b1 = 0.0
|
|
a1 = np.exp(j_w*lim_gg(-1, *args, **kwds))
|
|
if np.isnan(a1):
|
|
a1 = 0.0
|
|
|
|
lim_Psi = Limit(Psi)
|
|
for k in range(nn):
|
|
F[k] = basis(1, k)*b1 - basis(-1, k)*a1
|
|
A[k] = (dbasis(x, k, n=1) + j_w * lim_Psi(x, k))
|
|
|
|
LS = linalg.lstsq(A, F)
|
|
return LS[0]
|
|
|
|
|
|
def osc_weights(omega, g, dg, x, basis, ab, *args, **kwds):
|
|
def gg(t):
|
|
return g(scale * t + offset, *args, **kwds)
|
|
|
|
def dgg(t):
|
|
return scale * dg(scale * t + offset, *args, **kwds)
|
|
|
|
w = []
|
|
|
|
for a, b in zip(ab[::2], ab[1::2]):
|
|
scale = (b - a) / 2
|
|
offset = (a + b) / 2
|
|
|
|
w.append(evans_webster_weights(omega, gg, dgg, x, basis))
|
|
|
|
return np.asarray(w).ravel()
|
|
|
|
|
|
class QuadOsc(object):
|
|
info = namedtuple('info', ['error_estimate', 'n'])
|
|
|
|
def __init__(self, f, g, dg=None, a=-1, b=1, basis=chebyshev_basis, s=15,
|
|
precision=10, endpoints=False, maxiter=17, full_output=False):
|
|
self.f = f
|
|
self.g = g
|
|
self.dg = nd.Derivative(g) if dg is None else dg
|
|
self.basis = basis
|
|
self.a = a
|
|
self.b = b
|
|
self.s = s
|
|
self.endpoints = endpoints
|
|
self.maxiter = maxiter
|
|
self.precision = precision
|
|
self.full_output = full_output
|
|
|
|
@staticmethod
|
|
def _change_interval_to_0_1(f, g, dg, a, b):
|
|
def f1(t, *args, **kwds):
|
|
den = 1-t
|
|
return f(a + t / den, *args, **kwds) / den ** 2
|
|
|
|
def g1(t, *args, **kwds):
|
|
return g(a + t / (1 - t), *args, **kwds)
|
|
|
|
def dg1(t, *args, **kwds):
|
|
den = 1-t
|
|
return dg(a + t / den, *args, **kwds) / den ** 2
|
|
return f1, g1, dg1, 0., 1.
|
|
|
|
@staticmethod
|
|
def _change_interval_to_m1_0(f, g, dg, a, b):
|
|
def f2(t, *args, **kwds):
|
|
den = 1 + t
|
|
return f(b + t / den, *args, **kwds) / den ** 2
|
|
|
|
def g2(t, *args, **kwds):
|
|
return g(b + t / (1 + t), *args, **kwds)
|
|
|
|
def dg2(t, *args, **kwds):
|
|
den = 1 + t
|
|
return dg(b + t / den, *args, **kwds) / den ** 2
|
|
return f2, g2, dg2, -1.0, 0.0
|
|
|
|
@staticmethod
|
|
def _change_interval_to_m1_1(f, g, dg, a, b):
|
|
def f2(t, *args, **kwds):
|
|
den = (1 - t**2)
|
|
return f(t / den, *args, **kwds) * (1+t**2) / den ** 2
|
|
|
|
def g2(t, *args, **kwds):
|
|
den = (1 - t**2)
|
|
return g(t / den, *args, **kwds)
|
|
|
|
def dg2(t, *args, **kwds):
|
|
den = (1 - t**2)
|
|
return dg(t / den, *args, **kwds) * (1+t**2) / den ** 2
|
|
return f2, g2, dg2, -1., 1.
|
|
|
|
def _get_functions(self):
|
|
a, b = self.a, self.b
|
|
reverse = np.real(a) > np.real(b)
|
|
if reverse:
|
|
a, b = b, a
|
|
f, g, dg = self.f, self.g, self.dg
|
|
|
|
if a == b:
|
|
pass
|
|
elif np.isinf(a) | np.isinf(b):
|
|
# Check real limits
|
|
if ~np.isreal(a) | ~np.isreal(b) | np.isnan(a) | np.isnan(b):
|
|
raise ValueError('Infinite intervals must be real.')
|
|
# Change of variable
|
|
if np.isfinite(a) & np.isinf(b):
|
|
f, g, dg, a, b = self._change_interval_to_0_1(f, g, dg, a, b)
|
|
elif np.isinf(a) & np.isfinite(b):
|
|
f, g, dg, a, b = self._change_interval_to_m1_0(f, g, dg, a, b)
|
|
else: # -inf to inf
|
|
f, g, dg, a, b = self._change_interval_to_m1_1(f, g, dg, a, b)
|
|
|
|
return f, g, dg, a, b, reverse
|
|
|
|
def __call__(self, omega, *args, **kwds):
|
|
f, g, dg, a, b, reverse = self._get_functions()
|
|
|
|
val, err = self._quad_osc(f, g, dg, a, b, omega, *args, **kwds)
|
|
|
|
if reverse:
|
|
val = -val
|
|
if self.full_output:
|
|
return val, err
|
|
return val
|
|
|
|
@staticmethod
|
|
def _get_best_estimate(k, q0, q1, q2):
|
|
if k >= 5:
|
|
qv = np.hstack((q0[k], q1[k], q2[k]))
|
|
qw = np.hstack((q0[k - 1], q1[k - 1], q2[k - 1]))
|
|
elif k >= 3:
|
|
qv = np.hstack((q0[k], q1[k]))
|
|
qw = np.hstack((q0[k - 1], q1[k - 1]))
|
|
else:
|
|
qv = np.atleast_1d(q0[k])
|
|
qw = q0[k - 1]
|
|
errors = np.atleast_1d(abs(qv - qw))
|
|
j = np.nanargmin(errors)
|
|
return qv[j], errors[j]
|
|
|
|
def _extrapolate(self, k, q0, q1, q2):
|
|
if k >= 4:
|
|
q1[k], _err1 = dea3(q0[k - 2], q0[k - 1], q0[k])
|
|
q2[k], _err2 = dea3(q1[k - 2], q1[k - 1], q1[k])
|
|
elif k >= 2:
|
|
q1[k], _err1 = dea3(q0[k - 2], q0[k - 1], q0[k])
|
|
# # Richardson extrapolation
|
|
# if k >= 4:
|
|
# q1[k] = richardson(q0, k)
|
|
# q2[k] = richardson(q1, k)
|
|
# elif k >= 2:
|
|
# q1[k] = richardson(q0, k)
|
|
q, err = self._get_best_estimate(k, q0, q1, q2)
|
|
return q, err
|
|
|
|
def _quad_osc(self, f, g, dg, a, b, omega, *args, **kwds):
|
|
if a == b:
|
|
Q = b - a
|
|
err = b - a
|
|
return Q, err
|
|
|
|
abseps = 10**-self.precision
|
|
max_iter = self.maxiter
|
|
basis = self.basis
|
|
if self.endpoints:
|
|
xq = chebyshev_extrema(self.s)
|
|
else:
|
|
xq = chebyshev_roots(self.s)
|
|
# xq = tanh_sinh_open_nodes(self.s)
|
|
|
|
# One interval
|
|
hh = (b - a) / 2
|
|
x = (a + b) / 2 + hh * xq # Nodes
|
|
|
|
dtype = complex
|
|
Q0 = np.zeros((max_iter, 1), dtype=dtype) # Quadrature
|
|
Q1 = np.zeros((max_iter, 1), dtype=dtype) # First extrapolation
|
|
Q2 = np.zeros((max_iter, 1), dtype=dtype) # Second extrapolation
|
|
|
|
lim_f = Limit(f)
|
|
ab = np.hstack([a, b])
|
|
wq = osc_weights(omega, g, dg, xq, basis, ab, *args, **kwds)
|
|
Q0[0] = hh * np.sum(wq * lim_f(x, *args, **kwds))
|
|
|
|
# Successive bisection of intervals
|
|
nq = len(xq)
|
|
n = nq
|
|
for k in range(1, max_iter):
|
|
n += nq*2**k
|
|
|
|
hh = hh / 2
|
|
x = np.hstack([x + a, x + b]) / 2
|
|
ab = np.hstack([ab + a, ab + b]) / 2
|
|
wq = osc_weights(omega, g, dg, xq, basis, ab, *args, **kwds)
|
|
|
|
Q0[k] = hh * np.sum(wq * lim_f(x, *args, **kwds))
|
|
|
|
Q, err = self._extrapolate(k, Q0, Q1, Q2)
|
|
|
|
convergence = (err <= abseps) | ~np.isfinite(Q)
|
|
if convergence:
|
|
break
|
|
else:
|
|
warnings.warn('Max number of iterations reached '
|
|
'without convergence.')
|
|
|
|
if ~np.isfinite(Q):
|
|
warnings.warn('Integral approximation is Infinite or NaN.')
|
|
|
|
# The error estimate should not be zero
|
|
err += 2 * np.finfo(Q).eps
|
|
return Q, self.info(err, n)
|
|
|
|
|
|
def adaptive_levin_points(M, delta):
|
|
m = M - 1
|
|
prm = 0.5
|
|
while prm * m / delta >= 1:
|
|
delta = 2 * delta
|
|
k = np.arange(M)
|
|
x = piecewise([k < prm * m, k == np.ceil(prm * m)],
|
|
[-1 + k / delta, 0 * k, 1 - (m - k) / delta])
|
|
return x
|
|
|
|
|
|
def open_levin_points(M, delta):
|
|
return adaptive_levin_points(M+2, delta)[1:-1]
|
|
|
|
|
|
def chebyshev_extrema(M, delta=None):
|
|
k = np.arange(M)
|
|
x = np.cos(k * np.pi / (M-1))
|
|
return x
|
|
|
|
_EPS = np.finfo(float).eps
|
|
|
|
|
|
def tanh_sinh_nodes(M, delta=None, tol=_EPS):
|
|
tmax = np.arcsinh(np.arctanh(1-_EPS)*2/np.pi)
|
|
# tmax = 3.18
|
|
m = int(np.floor(-np.log2(tmax/max(M-1, 1)))) - 1
|
|
h = 2.0**-m
|
|
t = np.arange((M+1)//2+1)*h
|
|
x = np.tanh(np.pi/2*np.sinh(t))
|
|
k = np.flatnonzero(np.abs(x - 1) <= 10*tol)
|
|
y = x[:k[0]+1] if len(k) else x
|
|
return np.hstack((-y[:0:-1], y))
|
|
|
|
|
|
def tanh_sinh_open_nodes(M, delta=None, tol=_EPS):
|
|
return tanh_sinh_nodes(M+1, delta, tol)[1:-1]
|
|
|
|
|
|
def chebyshev_roots(M, delta=None):
|
|
k = np.arange(1, 2*M, 2) * 0.5
|
|
x = np.cos(k * np.pi / M)
|
|
return x
|
|
|
|
|
|
class AdaptiveLevin(object):
|
|
'''Return integral for the Levin-type and adaptive Levin-type methods'''
|
|
info = namedtuple('info', ['error_estimate', 'n'])
|
|
|
|
def __init__(self, f, g, dg=None, a=-1, b=1, basis=chebyshev_basis, s=1,
|
|
precision=10, endpoints=True, full_output=False):
|
|
self.f = f
|
|
self.g = g
|
|
self.dg = nd.Derivative(g) if dg is None else dg
|
|
self.basis = basis
|
|
self.a = a
|
|
self.b = b
|
|
self.s = s
|
|
self.precision = precision
|
|
self.endpoints = endpoints
|
|
self.full_output = full_output
|
|
|
|
@staticmethod
|
|
def aLevinTQ(omega, ff, gg, dgg, x, s, basis, *args, **kwds):
|
|
|
|
def Psi(t, k):
|
|
return dgg(t, *args, **kwds) * basis(t, k)
|
|
|
|
j_w = 1j * omega
|
|
nu = np.ones((len(x),), dtype=int)
|
|
nu[0] = nu[-1] = s
|
|
S = np.cumsum(np.hstack((nu, 0)))
|
|
S[-1] = 0
|
|
nn = int(S[-2])
|
|
A = np.zeros((nn, nn), dtype=complex)
|
|
F = np.zeros((nn,))
|
|
dff = Limit(nda.Derivative(ff))
|
|
dPsi = Limit(nda.Derivative(Psi))
|
|
dbasis = basis.derivative
|
|
for r, t in enumerate(x):
|
|
for j in range(S[r - 1], S[r]):
|
|
order = ((j - S[r - 1]) % nu[r]) # derivative order
|
|
dff.f.n = order
|
|
F[j] = dff(t, *args, **kwds)
|
|
dPsi.f.n = order
|
|
for k in range(nn):
|
|
A[j, k] = (dbasis(t, k, n=order+1) + j_w * dPsi(t, k))
|
|
k1 = np.flatnonzero(1-np.isfinite(F))
|
|
if k1.size > 0: # Remove singularities
|
|
warnings.warn('Singularities detected! ')
|
|
A[k1] = 0
|
|
F[k1] = 0
|
|
LS = linalg.lstsq(A, F)
|
|
v = basis.eval([-1, 1], LS[0])
|
|
|
|
lim_gg = Limit(gg)
|
|
gb = np.exp(j_w * lim_gg(1, *args, **kwds))
|
|
if np.isnan(gb):
|
|
gb = 0
|
|
ga = np.exp(j_w * lim_gg(-1, *args, **kwds))
|
|
if np.isnan(ga):
|
|
ga = 0
|
|
NR = (v[1] * gb - v[0] * ga)
|
|
return NR
|
|
|
|
def _get_integration_limits(self, omega, args, kwds):
|
|
a, b = self.a, self.b
|
|
M = 30
|
|
ab = [a]
|
|
scale = (b - a) / 2
|
|
n = 30
|
|
x = np.linspace(a, b, n + 1)
|
|
dg_x = np.asarray([scale * omega * self.dg(xi, *args, **kwds)
|
|
for xi in x])
|
|
i10 = findcross(dg_x, M)
|
|
i1 = findcross(dg_x, 1)
|
|
i0 = findcross(dg_x, 0)
|
|
im1 = findcross(dg_x, -1)
|
|
im10 = findcross(dg_x, -M)
|
|
x10 = ecross(x, dg_x, i10, M) if len(i10) else ()
|
|
x1 = ecross(x, dg_x, i1, 1) if len(i1) else ()
|
|
x0 = ecross(x, dg_x, i0, 0) if len(i0) else ()
|
|
xm1 = ecross(x, dg_x, im1, -1) if len(im1) else ()
|
|
xm10 = ecross(x, dg_x, im10, -M) if len(im10) else ()
|
|
|
|
for i in np.unique(np.hstack((x10, x1, x0, xm1, xm10))):
|
|
if x[0] < i < x[n]:
|
|
ab.append(i)
|
|
ab.append(b)
|
|
return ab
|
|
|
|
def __call__(self, omega, *args, **kwds):
|
|
ab = self._get_integration_limits(omega, args, kwds)
|
|
s = self.s
|
|
val = 0
|
|
n = 0
|
|
err = 0
|
|
for ai, bi in zip(ab[:-1], ab[1:]):
|
|
vali, infoi = self._QaL(s, ai, bi, omega, *args, **kwds)
|
|
val += vali
|
|
err += infoi.error_estimate
|
|
n += infoi.n
|
|
if self.full_output:
|
|
info = self.info(err, n)
|
|
return val, info
|
|
return val
|
|
|
|
@staticmethod
|
|
def _get_num_points(s, prec, betam):
|
|
return 1 if s > 1 else int(prec / max(np.log10(betam + 1), 1) + 1)
|
|
|
|
def _QaL(self, s, a, b, omega, *args, **kwds):
|
|
'''if s>1,the integral is computed by Q_s^L'''
|
|
scale = (b - a) / 2
|
|
offset = (a + b) / 2
|
|
prec = self.precision # desired precision
|
|
|
|
def ff(t, *args, **kwds):
|
|
return scale * self.f(scale * t + offset, *args, **kwds)
|
|
|
|
def gg(t, *args, **kwds):
|
|
return self.g(scale * t + offset, *args, **kwds)
|
|
|
|
def dgg(t, *args, **kwds):
|
|
return scale * self.dg(scale * t + offset, *args, **kwds)
|
|
dg_a = abs(omega * dgg(-1, *args, **kwds))
|
|
dg_b = abs(omega * dgg(1, *args, **kwds))
|
|
g_a = abs(omega * gg(-1, *args, **kwds))
|
|
g_b = abs(omega * gg(1, *args, **kwds))
|
|
delta, alpha = min(dg_a, dg_b), min(g_a, g_b)
|
|
|
|
betam = delta # * scale
|
|
if self.endpoints:
|
|
if (delta < 10 or alpha <= 10 or s > 1):
|
|
points = chebyshev_extrema
|
|
else:
|
|
points = adaptive_levin_points
|
|
elif (delta < 10 or alpha <= 10 or s > 1):
|
|
points = chebyshev_roots
|
|
else:
|
|
points = open_levin_points # tanh_sinh_open_nodes
|
|
|
|
m = self._get_num_points(s, prec, betam)
|
|
abseps = 10*10.0**-prec
|
|
num_collocation_point_list = m*2**np.arange(1, 5) + 1
|
|
basis = self.basis
|
|
|
|
Q = 1e+300
|
|
n = 0
|
|
ni = 0
|
|
for num_collocation_points in num_collocation_point_list:
|
|
ni_old = ni
|
|
Q_old = Q
|
|
x = points(num_collocation_points, betam)
|
|
ni = len(x)
|
|
if ni > ni_old:
|
|
Q = self.aLevinTQ(omega, ff, gg, dgg, x, s, basis, *args,
|
|
**kwds)
|
|
n += ni
|
|
err = np.abs(Q-Q_old)
|
|
if err <= abseps:
|
|
break
|
|
info = self.info(err, n)
|
|
return Q, info
|
|
|
|
|
|
class EvansWebster(AdaptiveLevin):
|
|
'''Return integral for the Evans Webster method'''
|
|
|
|
def __init__(self, f, g, dg=None, a=-1, b=1, basis=chebyshev_basis, s=8,
|
|
precision=10, endpoints=False, full_output=False):
|
|
self.f = f
|
|
self.g = g
|
|
self.dg = nd.Derivative(g) if dg is None else dg
|
|
self.basis = basis
|
|
self.a = a
|
|
self.b = b
|
|
self.s = s
|
|
self.endpoints = endpoints
|
|
self.precision = precision
|
|
self.full_output = full_output
|
|
|
|
def aLevinTQ(self, omega, ff, gg, dgg, x, s, basis, *args, **kwds):
|
|
w = evans_webster_weights(omega, gg, dgg, x, basis, *args, **kwds)
|
|
|
|
f = Limit(ff)(x, *args, **kwds)
|
|
NR = np.sum(f*w)
|
|
return NR
|
|
|
|
def _get_num_points(self, s, prec, betam):
|
|
return 8 if s > 1 else int(prec / max(np.log10(betam + 1), 1) + 1)
|
|
|
|
# def _QaL(self, s, a, b, omega, *args, **kwds):
|
|
# '''if s>1,the integral is computed by Q_s^L'''
|
|
# scale = (b - a) / 2
|
|
# offset = (a + b) / 2
|
|
# prec = self.precision # desired precision
|
|
#
|
|
# def ff(t, *args, **kwds):
|
|
# return scale * self.f(scale * t + offset, *args, **kwds)
|
|
#
|
|
# def gg(t, *args, **kwds):
|
|
# return self.g(scale * t + offset, *args, **kwds)
|
|
#
|
|
# def dgg(t, *args, **kwds):
|
|
# return scale * self.dg(scale * t + offset, *args, **kwds)
|
|
# dg_a = abs(omega * dgg(-1, *args, **kwds))
|
|
# dg_b = abs(omega * dgg(1, *args, **kwds))
|
|
# g_a = abs(omega * gg(-1, *args, **kwds))
|
|
# g_b = abs(omega * gg(1, *args, **kwds))
|
|
# delta, alpha = min(dg_a, dg_b), min(g_a, g_b)
|
|
#
|
|
# betam = delta # * scale
|
|
# if self.endpoints:
|
|
# if (delta < 10 or alpha <= 10 or s > 1):
|
|
# points = chebyshev_extrema
|
|
# else:
|
|
# points = adaptive_levin_points
|
|
# elif (delta < 10 or alpha <= 10 or s > 1):
|
|
# points = chebyshev_roots
|
|
# else:
|
|
# points = tanh_sinh_open_nodes
|
|
#
|
|
# m = 8 if s > 1 else int(prec / max(np.log10(betam + 1), 1) + 1)
|
|
# abseps = 10*10.0**-prec
|
|
# num_collocation_point_list = 2*m*np.arange(1, 6, 2)
|
|
# # range(num_points, num_points+3, 2)
|
|
# basis = self.basis
|
|
# Q = 1e+300
|
|
# n = 0
|
|
# for num_collocation_points in num_collocation_point_list:
|
|
# Q_old = Q
|
|
# x = points(num_collocation_points-1, betam)
|
|
# Q = self.aLevinTQ(omega, ff, gg, dgg, x, s, basis, *args, **kwds)
|
|
# n += num_collocation_points
|
|
# err = np.abs(Q-Q_old)
|
|
# if err <= abseps:
|
|
# break
|
|
# info = self.info(err, n)
|
|
# return Q, info
|
|
|
|
|
|
if __name__ == '__main__':
|
|
tanh_sinh_nodes(16)
|