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535 lines
17 KiB
Python
535 lines
17 KiB
Python
"""
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Created on 20. aug. 2015
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@author: pab
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"""
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from __future__ import division
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from collections import namedtuple
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import warnings
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import numdifftools as nd
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import numdifftools.nd_algopy as nda
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from numdifftools.extrapolation import dea3
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from numdifftools.limits import Limit
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import numpy as np
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from numpy import linalg
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from numpy.polynomial.chebyshev import chebval, Chebyshev
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from numpy.polynomial import polynomial
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from wafo.misc import piecewise, findcross, ecross
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_FINFO = np.finfo(float)
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EPS = _FINFO(float).eps
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_EPS = EPS
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_TINY = _FINFO.tiny
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_HUGE = _FINFO.max
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def _assert(cond, msg):
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if not cond:
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raise ValueError(msg)
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def _assert_warn(cond, msg):
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if not cond:
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warnings.warn(msg)
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class PolyBasis(object):
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@staticmethod
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def _derivative(c, m):
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return polynomial.polyder(c, m)
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@staticmethod
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def eval(t, c):
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return polynomial.polyval(t, c)
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@staticmethod
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def _coefficients(k):
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c = np.zeros(k + 1)
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c[k] = 1
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return c
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def derivative(self, t, k, n=1):
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c = self._coefficients(k)
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d_c = self._derivative(c, m=n)
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return self.eval(t, d_c)
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def __call__(self, t, k):
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return t**k
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poly_basis = PolyBasis()
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class ChebyshevBasis(PolyBasis):
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@staticmethod
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def _derivative(c, m):
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cheb = Chebyshev(c)
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dcheb = cheb.deriv(m=m)
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return dcheb.coef
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@staticmethod
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def eval(t, c):
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return chebval(t, c)
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def __call__(self, t, k):
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c = self._coefficients(k)
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return self.eval(t, c)
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chebyshev_basis = ChebyshevBasis()
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def richardson(q_val, k):
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# license BSD
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# Richardson extrapolation with parameter estimation
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c = np.real((q_val[k - 1] - q_val[k - 2]) / (q_val[k] - q_val[k - 1])) - 1.
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# The lower bound 0.07 admits the singularity x.^-0.9
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c = max(c, 0.07)
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return q_val[k] + (q_val[k] - q_val[k - 1]) / c
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def evans_webster_weights(omega, g, d_g, x, basis, *args, **kwds):
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def psi(t, k):
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return d_g(t, *args, **kwds) * basis(t, k)
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j_w = 1j * omega
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n = len(x)
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a_matrix = np.zeros((n, n), dtype=complex)
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rhs = np.zeros((n,), dtype=complex)
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dbasis = basis.derivative
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lim_g = Limit(g)
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b_1 = np.exp(j_w*lim_g(1, *args, **kwds))
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if np.isnan(b_1):
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b_1 = 0.0
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a_1 = np.exp(j_w*lim_g(-1, *args, **kwds))
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if np.isnan(a_1):
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a_1 = 0.0
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lim_psi = Limit(psi)
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for k in range(n):
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rhs[k] = basis(1, k) * b_1 - basis(-1, k) * a_1
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a_matrix[k] = (dbasis(x, k, n=1) + j_w * lim_psi(x, k))
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solution = linalg.lstsq(a_matrix, rhs)
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return solution[0]
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def osc_weights(omega, g, d_g, x, basis, a_b, *args, **kwds):
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def _g(t):
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return g(scale * t + offset, *args, **kwds)
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def _d_g(t):
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return scale * d_g(scale * t + offset, *args, **kwds)
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w = []
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for a, b in zip(a_b[::2], a_b[1::2]):
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scale = (b - a) / 2
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offset = (a + b) / 2
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w.append(evans_webster_weights(omega, _g, _d_g, x, basis))
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return np.asarray(w).ravel()
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class _Integrator(object):
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info = namedtuple('info', ['error_estimate', 'n'])
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def __init__(self, f, g, dg=None, a=-1, b=1, basis=chebyshev_basis, s=1,
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precision=10, endpoints=True, full_output=False):
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self.f = f
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self.g = g
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self.dg = nd.Derivative(g) if dg is None else dg
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self.basis = basis
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self.a = a
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self.b = b
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self.s = s
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self.endpoints = endpoints
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self.precision = precision
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self.full_output = full_output
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class QuadOsc(_Integrator):
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def __init__(self, f, g, dg=None, a=-1, b=1, basis=chebyshev_basis, s=15,
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precision=10, endpoints=False, full_output=False, maxiter=17):
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self.maxiter = maxiter
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super(QuadOsc, self).__init__(f, g, dg=dg, a=a, b=b, basis=basis, s=s,
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precision=precision, endpoints=endpoints,
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full_output=full_output)
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@staticmethod
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def _change_interval_to_0_1(f, g, d_g, a, _b):
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def f_01(t, *args, **kwds):
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den = 1-t
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return f(a + t / den, *args, **kwds) / den ** 2
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def g_01(t, *args, **kwds):
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return g(a + t / (1 - t), *args, **kwds)
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def d_g_01(t, *args, **kwds):
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den = 1-t
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return d_g(a + t / den, *args, **kwds) / den ** 2
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return f_01, g_01, d_g_01, 0., 1.
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@staticmethod
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def _change_interval_to_m1_0(f, g, d_g, _a, b):
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def f_m10(t, *args, **kwds):
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den = 1 + t
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return f(b + t / den, *args, **kwds) / den ** 2
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def g_m10(t, *args, **kwds):
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return g(b + t / (1 + t), *args, **kwds)
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def d_g_m10(t, *args, **kwds):
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den = 1 + t
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return d_g(b + t / den, *args, **kwds) / den ** 2
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return f_m10, g_m10, d_g_m10, -1.0, 0.0
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@staticmethod
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def _change_interval_to_m1_1(f, g, d_g, _a, _b):
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def f_m11(t, *args, **kwds):
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den = (1 - t**2)
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return f(t / den, *args, **kwds) * (1+t**2) / den ** 2
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def g_m11(t, *args, **kwds):
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den = (1 - t**2)
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return g(t / den, *args, **kwds)
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def d_g_m11(t, *args, **kwds):
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den = (1 - t**2)
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return d_g(t / den, *args, **kwds) * (1+t**2) / den ** 2
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return f_m11, g_m11, d_g_m11, -1., 1.
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def _get_functions(self):
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a, b = self.a, self.b
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reverse = np.real(a) > np.real(b)
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if reverse:
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a, b = b, a
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f, g, dg = self.f, self.g, self.dg
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if a == b:
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pass
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elif np.isinf(a) | np.isinf(b):
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# Check real limits
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if ~np.isreal(a) | ~np.isreal(b) | np.isnan(a) | np.isnan(b):
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raise ValueError('Infinite intervals must be real.')
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# Change of variable
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if np.isfinite(a) & np.isinf(b):
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f, g, dg, a, b = self._change_interval_to_0_1(f, g, dg, a, b)
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elif np.isinf(a) & np.isfinite(b):
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f, g, dg, a, b = self._change_interval_to_m1_0(f, g, dg, a, b)
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else: # -inf to inf
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f, g, dg, a, b = self._change_interval_to_m1_1(f, g, dg, a, b)
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return f, g, dg, a, b, reverse
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def __call__(self, omega, *args, **kwds):
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f, g, dg, a, b, reverse = self._get_functions()
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val, err = self._quad_osc(f, g, dg, a, b, omega, *args, **kwds)
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if reverse:
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val = -val
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if self.full_output:
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return val, err
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return val
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@staticmethod
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def _get_best_estimate(k, q_0, q_1, q_2):
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if k >= 5:
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q_v = np.hstack((q_0[k], q_1[k], q_2[k]))
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q_w = np.hstack((q_0[k - 1], q_1[k - 1], q_2[k - 1]))
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elif k >= 3:
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q_v = np.hstack((q_0[k], q_1[k]))
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q_w = np.hstack((q_0[k - 1], q_1[k - 1]))
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else:
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q_v = np.atleast_1d(q_0[k])
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q_w = q_0[k - 1]
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errors = np.atleast_1d(abs(q_v - q_w))
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j = np.nanargmin(errors)
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return q_v[j], errors[j]
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def _extrapolate(self, k, q_0, q_1, q_2):
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if k >= 4:
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q_1[k] = dea3(q_0[k - 2], q_0[k - 1], q_0[k])[0]
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q_2[k] = dea3(q_1[k - 2], q_1[k - 1], q_1[k])[0]
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elif k >= 2:
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q_1[k] = dea3(q_0[k - 2], q_0[k - 1], q_0[k])[0]
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# # Richardson extrapolation
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# if k >= 4:
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# q_1[k] = richardson(q_0, k)
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# q_2[k] = richardson(q_1, k)
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# elif k >= 2:
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# q_1[k] = richardson(q_0, k)
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q, err = self._get_best_estimate(k, q_0, q_1, q_2)
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return q, err
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def _quad_osc(self, f, g, dg, a, b, omega, *args, **kwds):
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if a == b:
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q_val = b - a
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err = np.abs(b - a)
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return q_val, err
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abseps = 10**-self.precision
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max_iter = self.maxiter
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basis = self.basis
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if self.endpoints:
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x_n = chebyshev_extrema(self.s)
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else:
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x_n = chebyshev_roots(self.s)
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# x_n = tanh_sinh_open_nodes(self.s)
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# One interval
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hh = (b - a) / 2
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x = (a + b) / 2 + hh * x_n # Nodes
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dtype = complex
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val0 = np.zeros((max_iter, 1), dtype=dtype) # Quadrature
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val1 = np.zeros((max_iter, 1), dtype=dtype) # First extrapolation
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val2 = np.zeros((max_iter, 1), dtype=dtype) # Second extrapolation
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lim_f = Limit(f)
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a_b = np.hstack([a, b])
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wq = osc_weights(omega, g, dg, x_n, basis, a_b, *args, **kwds)
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val0[0] = hh * np.sum(wq * lim_f(x, *args, **kwds))
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# Successive bisection of intervals
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nq = len(x_n)
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n = nq
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for k in range(1, max_iter):
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n += nq * 2**k
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hh = hh / 2
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x = np.hstack([x + a, x + b]) / 2
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a_b = np.hstack([a_b + a, a_b + b]) / 2
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wq = osc_weights(omega, g, dg, x_n, basis, a_b, *args, **kwds)
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val0[k] = hh * np.sum(wq * lim_f(x, *args, **kwds))
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q_val, err = self._extrapolate(k, val0, val1, val2)
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converged = (err <= abseps) | ~np.isfinite(q_val)
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if converged:
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break
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_assert_warn(converged, 'Max number of iterations reached '
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'without convergence.')
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_assert_warn(np.isfinite(q_val),
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'Integral approximation is Infinite or NaN.')
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# The error estimate should not be zero
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err += 2 * np.finfo(q_val).eps
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return q_val, self.info(err, n)
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def adaptive_levin_points(m, delta):
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m_1 = m - 1
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prm = 0.5
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while prm * m_1 / delta >= 1:
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delta = 2 * delta
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k = np.arange(m)
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x = piecewise([k < prm * m_1, k == np.ceil(prm * m_1)],
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[-1 + k / delta, 0 * k, 1 - (m_1 - k) / delta])
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return x
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def open_levin_points(m, delta):
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return adaptive_levin_points(m+2, delta)[1:-1]
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def chebyshev_extrema(m, delta=None):
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k = np.arange(m)
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x = np.cos(k * np.pi / (m-1))
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return x
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def tanh_sinh_nodes(m, delta=None, tol=_EPS):
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tmax = np.arcsinh(np.arctanh(1-_EPS)*2/np.pi)
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# tmax = 3.18
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m_1 = int(np.floor(-np.log2(tmax/max(m-1, 1)))) - 1
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h = 2.0**-m_1
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t = np.arange((m+1)//2+1)*h
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x = np.tanh(np.pi/2*np.sinh(t))
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k = np.flatnonzero(np.abs(x - 1) <= 10*tol)
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y = x[:k[0]+1] if len(k) else x
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return np.hstack((-y[:0:-1], y))
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def tanh_sinh_open_nodes(m, delta=None, tol=_EPS):
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return tanh_sinh_nodes(m+1, delta, tol)[1:-1]
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def chebyshev_roots(m, delta=None):
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k = np.arange(1, 2*m, 2) * 0.5
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x = np.cos(k * np.pi / m)
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return x
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class AdaptiveLevin(_Integrator):
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"""Return integral for the Levin-type and adaptive Levin-type methods"""
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@staticmethod
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def _a_levin(omega, f, g, d_g, x, s, basis, *args, **kwds):
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def psi(t, k):
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return d_g(t, *args, **kwds) * basis(t, k)
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j_w = 1j * omega
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nu = np.ones((len(x),), dtype=int)
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nu[0] = nu[-1] = s
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S = np.cumsum(np.hstack((nu, 0)))
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S[-1] = 0
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n = int(S[-2])
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a_matrix = np.zeros((n, n), dtype=complex)
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rhs = np.zeros((n,))
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dff = Limit(nda.Derivative(f))
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d_psi = Limit(nda.Derivative(psi))
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dbasis = basis.derivative
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for r, t in enumerate(x):
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for j in range(S[r - 1], S[r]):
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order = ((j - S[r - 1]) % nu[r]) # derivative order
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dff.fun.n = order
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rhs[j] = dff(t, *args, **kwds)
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d_psi.fun.n = order
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for k in range(n):
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a_matrix[j, k] = (dbasis(t, k, n=order+1) +
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j_w * d_psi(t, k))
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k1 = np.flatnonzero(1-np.isfinite(rhs))
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if k1.size > 0: # Remove singularities
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warnings.warn('Singularities detected! ')
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a_matrix[k1] = 0
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rhs[k1] = 0
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solution = linalg.lstsq(a_matrix, rhs)
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v = basis.eval([-1, 1], solution[0])
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lim_g = Limit(g)
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g_b = np.exp(j_w * lim_g(1, *args, **kwds))
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if np.isnan(g_b):
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g_b = 0
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g_a = np.exp(j_w * lim_g(-1, *args, **kwds))
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if np.isnan(g_a):
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g_a = 0
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return v[1] * g_b - v[0] * g_a
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def _get_integration_limits(self, omega, args, kwds):
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a, b = self.a, self.b
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M = 30
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ab = [a]
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scale = (b - a) / 2
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n = 30
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x = np.linspace(a, b, n + 1)
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dg_x = np.asarray([scale * omega * self.dg(xi, *args, **kwds)
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for xi in x])
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i10 = findcross(dg_x, M)
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i1 = findcross(dg_x, 1)
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i0 = findcross(dg_x, 0)
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im1 = findcross(dg_x, -1)
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im10 = findcross(dg_x, -M)
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x10 = ecross(x, dg_x, i10, M) if len(i10) else ()
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x1 = ecross(x, dg_x, i1, 1) if len(i1) else ()
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x0 = ecross(x, dg_x, i0, 0) if len(i0) else ()
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xm1 = ecross(x, dg_x, im1, -1) if len(im1) else ()
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xm10 = ecross(x, dg_x, im10, -M) if len(im10) else ()
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for i in np.unique(np.hstack((x10, x1, x0, xm1, xm10))):
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if x[0] < i < x[n]:
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ab.append(i)
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ab.append(b)
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return ab
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def __call__(self, omega, *args, **kwds):
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ab = self._get_integration_limits(omega, args, kwds)
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s = self.s
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val = 0
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n = 0
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err = 0
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for ai, bi in zip(ab[:-1], ab[1:]):
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vali, infoi = self._QaL(s, ai, bi, omega, *args, **kwds)
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val += vali
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err += infoi.error_estimate
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n += infoi.n
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if self.full_output:
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info = self.info(err, n)
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|
return val, info
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|
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_val = 1e+300
|
|
num_function_evaluations = 0
|
|
n = 0
|
|
for num_collocation_points in num_collocation_point_list:
|
|
n_old = n
|
|
q_old = q_val
|
|
x = points(num_collocation_points, betam)
|
|
n = len(x)
|
|
if n > n_old:
|
|
q_val = self._a_levin(omega, ff, gg, dgg, x, s, basis, *args,
|
|
**kwds)
|
|
num_function_evaluations += n
|
|
err = np.abs(q_val-q_old)
|
|
if err <= abseps:
|
|
break
|
|
info = self.info(err, num_function_evaluations)
|
|
return q_val, 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):
|
|
super(EvansWebster,
|
|
self).__init__(f, g, dg=dg, a=a, b=b, basis=basis, s=s,
|
|
precision=precision, endpoints=endpoints,
|
|
full_output=full_output)
|
|
|
|
def _a_levin(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)
|
|
return np.sum(f*w)
|
|
|
|
def _get_num_points(self, s, prec, betam):
|
|
return 8 if s > 1 else int(prec / max(np.log10(betam + 1), 1) + 1)
|
|
|
|
|
|
if __name__ == '__main__':
|
|
tanh_sinh_nodes(16)
|