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'''
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Created on 31. aug. 2015
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@author: pab
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'''
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from __future__ import division
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import numpy as np
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import mpmath as mp
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import unittest
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from wafo.integrate_oscillating import (adaptive_levin_points,
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chebyshev_extrema,
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chebyshev_roots, tanh_sinh_nodes,
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tanh_sinh_open_nodes,
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AdaptiveLevin, poly_basis,
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chebyshev_basis,
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EvansWebster, QuadOsc)
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# import numdifftools as nd
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from numpy.testing import assert_allclose
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from scipy.special import gamma, digamma
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_EPS = np.finfo(float).eps
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class TestBasis(unittest.TestCase):
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def test_poly(self):
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t = 1
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vals = [poly_basis.derivative(t, k, n=1) for k in range(3)]
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assert_allclose(vals, range(3))
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vals = [poly_basis.derivative(0, k, n=1) for k in range(3)]
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assert_allclose(vals, [0, 1, 0])
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vals = [poly_basis.derivative(0, k, n=2) for k in range(3)]
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assert_allclose(vals, [0, 0, 2])
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def test_chebyshev(self):
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t = 1
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vals = [chebyshev_basis.derivative(t, k, n=1) for k in range(3)]
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assert_allclose(vals, np.arange(3)**2)
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vals = [chebyshev_basis.derivative(0, k, n=1) for k in range(3)]
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assert_allclose(vals, [0, 1, 0])
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vals = [chebyshev_basis.derivative(0, k, n=2) for k in range(3)]
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assert_allclose(vals, [0, 0, 4])
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class TestLevinPoints(unittest.TestCase):
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def test_adaptive(self):
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M = 11
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delta = 100
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x = adaptive_levin_points(M, delta)
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true_x = [-1., -0.99, -0.98, -0.97, -0.96, 0.,
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0.96, 0.97, 0.98, 0.99, 1.]
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assert_allclose(x, true_x)
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def test_chebyshev_extrema(self):
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M = 11
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delta = 100
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x = chebyshev_extrema(M, delta)
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true_x = [1.000000e+00, 9.510565e-01, 8.090170e-01, 5.877853e-01,
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3.090170e-01, 6.123234e-17, -3.090170e-01, -5.877853e-01,
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-8.090170e-01, -9.510565e-01, -1.000000e+00]
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assert_allclose(x, true_x)
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def test_chebyshev_roots(self):
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M = 11
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delta = 100
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x = chebyshev_roots(M, delta)
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true_x = [9.89821442e-01, 9.09631995e-01, 7.55749574e-01,
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5.40640817e-01, 2.81732557e-01, 2.83276945e-16,
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-2.81732557e-01, -5.40640817e-01, -7.55749574e-01,
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-9.09631995e-01, -9.89821442e-01]
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assert_allclose(x, true_x)
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def test_tanh_sinh_nodes(self):
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for n in 2**np.arange(1, 5) + 1:
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x = tanh_sinh_nodes(n)
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# self.assertEqual(n, len(x))
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def test_tanh_sinh_open_nodes(self):
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for n in 2**np.arange(1, 5) + 1:
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x = tanh_sinh_open_nodes(n)
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# self.assertEqual(n, len(x))
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class LevinQuadrature(unittest.TestCase):
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def test_exp_4t_exp_jw_gamma_t_exp_4t(self):
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def f(t):
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return np.exp(4 * t) # amplitude function
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def g(t):
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return t + np.exp(4 * t) * gamma(t) # phase function
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def dg(t):
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return 1 + (4 + digamma(t)) * np.exp(4 * t) * gamma(t)
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a = 1
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b = 2
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omega = 100
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def ftot(t):
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exp4t = mp.exp(4*t)
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return exp4t * mp.exp(1j * omega * (t+exp4t*mp.gamma(t)))
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_true_val, _err = mp.quadts(ftot, [a, (a+b)/2, b], error=True)
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true_val = 0.00435354129735323908804 + 0.00202865398517716214366j
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# quad = AdaptiveLevin(f, g, dg, a=a, b=b, s=1, full_output=True)
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for quadfun in [EvansWebster, QuadOsc, AdaptiveLevin]:
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quad = quadfun(f, g, dg, a=a, b=b, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-11)
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# assert_allclose(info.n, 9)
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def test_exp_jw_t(self):
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def g(t):
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return t
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def dg(t):
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return np.ones(np.shape(t))
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def true_F(t):
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return np.exp(1j*omega*g(t))/(1j*omega)
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val, _err = mp.quadts(g, [0, 1], error=True)
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a = 1
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b = 2
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omega = 1
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true_val = true_F(b)-true_F(a)
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for quadfun in [QuadOsc, AdaptiveLevin, EvansWebster]:
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quad = quadfun(dg, g, dg, a, b, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-12)
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# assert_allclose(info.n, 21)
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def test_I1_1_p_ln_x_exp_jw_xlnx(self):
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def g(t):
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return t*np.log(t)
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def dg(t):
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return 1 + np.log(t)
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def true_F(t):
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return np.exp(1j*(omega*g(t)))/(1j*omega)
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a = 100
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b = 200
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omega = 1
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true_val = true_F(b)-true_F(a)
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for quadfun in [AdaptiveLevin, QuadOsc, EvansWebster]:
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quad = quadfun(dg, g, dg, a, b, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-10)
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# assert_allclose(info.n, 11)
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def test_I4_ln_x_exp_jw_30x(self):
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n = 7
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def g(t):
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return t**n
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def dg(t):
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return n*t**(n-1)
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def f(t):
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return dg(t)*np.log(g(t))
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a = 0
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b = (2 * np.pi)**(1./n)
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omega = 30
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def ftot(t):
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return n*t**(n-1)*mp.log(t**n) * mp.exp(1j * omega * t**n)
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_true_val, _err = mp.quadts(ftot, [a, b], error=True, maxdegree=8)
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# true_val = (-0.052183048684992 - 0.193877275099872j)
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true_val = (-0.0521830486849921 - 0.193877275099871j)
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for quadfun in [QuadOsc, EvansWebster, AdaptiveLevin]:
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quad = quadfun(f, g, dg, a, b, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-5)
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def test_I5_coscost_sint_exp_jw_sint(self):
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a = 0
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b = np.pi/2
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omega = 100
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def f(t):
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return np.cos(np.cos(t))*np.sin(t)
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def g(t):
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return np.sin(t)
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def dg(t):
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return np.cos(t)
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def ftot(t):
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return mp.cos(mp.cos(t)) * mp.sin(t) * mp.exp(1j * omega *
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mp.sin(t))
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_true_val, _err = mp.quadts(ftot, [a, 0.5, 1, b], maxdegree=9,
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error=True)
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true_val = 0.0325497765499959-0.121009052128827j
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for quadfun in [QuadOsc, EvansWebster, AdaptiveLevin]:
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quad = quadfun(f, g, dg, a, b, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-9)
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def test_I6_exp_jw_td_1_m_t(self):
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a = 0
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b = 1
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omega = 1
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def f(t):
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return np.ones(np.shape(t))
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def g(t):
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return t/(1-t)
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def dg(t):
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return 1./(1-t)**2
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def ftot(t):
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return mp.exp(1j * omega * t/(1-t))
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true_val = (0.3785503757641866423607342717846606761068353230802945830 +
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0.3433779615564270328325330038583124340012440194999075192j)
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for quadfun in [QuadOsc, EvansWebster, AdaptiveLevin]:
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quad = quadfun(f, g, dg, a, b, endpoints=False, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-10)
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def test_I8_cos_47pix2d4_exp_jw_x(self):
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def f(t):
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return np.cos(47*np.pi/4*t**2)
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def g(t):
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return t
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def dg(t):
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return 1
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a = -1
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b = 1
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omega = 451*np.pi/4
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true_val = 2.3328690362927e-3
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s = 15
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for quadfun in [QuadOsc, EvansWebster]: # , AdaptiveLevin]:
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quad = quadfun(f, g, dg, a, b, s=s, endpoints=False,
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full_output=True)
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val, _info = quad(omega)
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assert_allclose(val.real, true_val)
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s = 1 if s <= 2 else s // 2
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# self.assert_(info.error_estimate < 1e-10)
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# assert_allclose(info.n, 11)
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def test_I9_exp_tant_sec2t_exp_jw_tant(self):
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a = 0
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b = np.pi/2
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omega = 100
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def f(t):
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return np.exp(-np.tan(t))/np.cos(t)**2
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def g(t):
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return np.tan(t)
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def dg(t):
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return 1./np.cos(t)**2
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true_val = (0.0000999900009999000099990000999900009999000099990000999 +
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0.009999000099990000999900009999000099990000999900009999j)
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for quadfun in [QuadOsc, EvansWebster, AdaptiveLevin]:
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quad = quadfun(f, g, dg, a, b, endpoints=False, full_output=True)
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val, info = quad(omega)
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assert_allclose(val, true_val)
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self.assert_(info.error_estimate < 1e-8)
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def test_exp_zdcos2t_dcos2t_exp_jw_cos_t_b_dcos2t(self):
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x1 = -20
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y1 = 20
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z1 = 20
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beta = np.abs(np.arctan(y1/x1))
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R = np.sqrt(x1**2+y1**2)
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def f(t, beta, z1):
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cos2t = np.cos(t)**2
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return np.where(cos2t == 0, 0, np.exp(-z1/cos2t)/cos2t)
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def g(t, beta, z1):
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return np.cos(t-beta)/np.cos(t)**2
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def dg(t, beta, z1=0):
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cos3t = np.cos(t)**3
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return 0.5*(3*np.sin(beta)-np.sin(beta-2*t))/cos3t
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def append_dg_zero(zeros, g1, beta):
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signs = [1, ] if np.abs(g1) <= _EPS else [-1, 1]
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for sgn1 in signs:
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tn = np.arccos(sgn1 * g1)
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if -np.pi / 2 <= tn <= np.pi / 2:
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for sgn2 in [-1, 1]:
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t = sgn2 * tn
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if np.abs(dg(t, beta)) < 10*_EPS:
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zeros.append(t)
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return zeros
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def zeros_dg(beta):
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k0 = (9*np.cos(2*beta)-7)
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if k0 < 0: # No stationary points
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return ()
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k1 = 3*np.cos(2*beta)-5
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g0 = np.sqrt(2)*np.sqrt(np.cos(beta)**2*k0)
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zeros = []
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if g0+k1 < _EPS:
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g1 = 1./2*np.sqrt(-g0-k1)
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zeros = append_dg_zero(zeros, g1, beta)
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if _EPS < g0-k1:
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g2 = 1./2*np.sqrt(g0-k1)
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zeros = append_dg_zero(zeros, g2, beta)
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if np.abs(g0+k1) <= _EPS or np.abs(g0-k1) <= _EPS:
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zeros = append_dg_zero(zeros, 0, beta)
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return tuple(zeros)
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|
a = -np.pi/2
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|
b = np.pi/2
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|
|
omega = R
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|
def ftot(t):
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|
cos2t = mp.cos(t)**2
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|
return (mp.exp(-z1/cos2t) / cos2t *
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|
|
mp.exp(1j * omega * mp.cos(t-beta)/cos2t))
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|
|
zdg = zeros_dg(beta)
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|
|
ab = (a, ) + zdg + (b, )
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|
|
true_val, _err = mp.quadts(ftot, ab, maxdegree=9, error=True)
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|
|
if False:
|
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|
|
|
import matplotlib.pyplot as plt
|
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|
|
|
t = np.linspace(a, b, 5*513)
|
|
|
|
|
plt.subplot(2, 1, 1)
|
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|
|
|
f2 = f(t, beta, z1)*np.exp(1j*R*g(t, beta, z1))
|
|
|
|
|
|
|
|
|
|
true_val2 = np.trapz(f2, t)
|
|
|
|
|
plt.plot(t, f2.real, t, f2.imag, 'r')
|
|
|
|
|
plt.title('integral=%g+1j%g, '
|
|
|
|
|
'(%g+1j%g)' % (true_val2.real, true_val2.imag,
|
|
|
|
|
true_val.real, true_val.imag))
|
|
|
|
|
plt.subplot(2, 1, 2)
|
|
|
|
|
plt.plot(t, dg(t, beta, z1), 'r')
|
|
|
|
|
plt.plot(t, g(t, beta, z1))
|
|
|
|
|
plt.hlines(0, a, b)
|
|
|
|
|
plt.axis([a, b, -5, 5])
|
|
|
|
|
plt.title('beta=%g' % beta)
|
|
|
|
|
print(np.trapz(f2, t))
|
|
|
|
|
plt.show('hold')
|
|
|
|
|
# true_val = 0.00253186684281+0.004314054498j
|
|
|
|
|
# s = 15
|
|
|
|
|
for quadfun in [QuadOsc, EvansWebster, AdaptiveLevin]:
|
|
|
|
|
# EvansWebster]: # , AdaptiveLevin, ]:
|
|
|
|
|
quad = quadfun(f, g, dg, a, b, precision=10, endpoints=False,
|
|
|
|
|
full_output=True)
|
|
|
|
|
val, _info = quad(omega, beta, z1) # @UnusedVariable
|
|
|
|
|
# assert_allclose(val, complex(true_val), rtol=1e-3)
|
|
|
|
|
# s = 1 if s<=1 else s//2
|
|
|
|
|
pass
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
if __name__ == "__main__":
|
|
|
|
|
# import sys;sys.argv = ['', 'Test.testName']
|
|
|
|
|
unittest.main()
|
