import numpy as np from numpy import array, sqrt from numpy.testing import (assert_array_almost_equal, assert_equal, assert_almost_equal, assert_allclose) from pytest import raises as assert_raises from scipy import integrate import scipy.special as sc from scipy.special import gamma import scipy.special._orthogonal as orth class TestCheby: def test_chebyc(self): C0 = orth.chebyc(0) C1 = orth.chebyc(1) with np.errstate(all='ignore'): C2 = orth.chebyc(2) C3 = orth.chebyc(3) C4 = orth.chebyc(4) C5 = orth.chebyc(5) assert_array_almost_equal(C0.c,[2],13) assert_array_almost_equal(C1.c,[1,0],13) assert_array_almost_equal(C2.c,[1,0,-2],13) assert_array_almost_equal(C3.c,[1,0,-3,0],13) assert_array_almost_equal(C4.c,[1,0,-4,0,2],13) assert_array_almost_equal(C5.c,[1,0,-5,0,5,0],13) def test_chebys(self): S0 = orth.chebys(0) S1 = orth.chebys(1) S2 = orth.chebys(2) S3 = orth.chebys(3) S4 = orth.chebys(4) S5 = orth.chebys(5) assert_array_almost_equal(S0.c,[1],13) assert_array_almost_equal(S1.c,[1,0],13) assert_array_almost_equal(S2.c,[1,0,-1],13) assert_array_almost_equal(S3.c,[1,0,-2,0],13) assert_array_almost_equal(S4.c,[1,0,-3,0,1],13) assert_array_almost_equal(S5.c,[1,0,-4,0,3,0],13) def test_chebyt(self): T0 = orth.chebyt(0) T1 = orth.chebyt(1) T2 = orth.chebyt(2) T3 = orth.chebyt(3) T4 = orth.chebyt(4) T5 = orth.chebyt(5) assert_array_almost_equal(T0.c,[1],13) assert_array_almost_equal(T1.c,[1,0],13) assert_array_almost_equal(T2.c,[2,0,-1],13) assert_array_almost_equal(T3.c,[4,0,-3,0],13) assert_array_almost_equal(T4.c,[8,0,-8,0,1],13) assert_array_almost_equal(T5.c,[16,0,-20,0,5,0],13) def test_chebyu(self): U0 = orth.chebyu(0) U1 = orth.chebyu(1) U2 = orth.chebyu(2) U3 = orth.chebyu(3) U4 = orth.chebyu(4) U5 = orth.chebyu(5) assert_array_almost_equal(U0.c,[1],13) assert_array_almost_equal(U1.c,[2,0],13) assert_array_almost_equal(U2.c,[4,0,-1],13) assert_array_almost_equal(U3.c,[8,0,-4,0],13) assert_array_almost_equal(U4.c,[16,0,-12,0,1],13) assert_array_almost_equal(U5.c,[32,0,-32,0,6,0],13) class TestGegenbauer: def test_gegenbauer(self): a = 5*np.random.random() - 0.5 if np.any(a == 0): a = -0.2 Ca0 = orth.gegenbauer(0,a) Ca1 = orth.gegenbauer(1,a) Ca2 = orth.gegenbauer(2,a) Ca3 = orth.gegenbauer(3,a) Ca4 = orth.gegenbauer(4,a) Ca5 = orth.gegenbauer(5,a) assert_array_almost_equal(Ca0.c,array([1]),13) assert_array_almost_equal(Ca1.c,array([2*a,0]),13) assert_array_almost_equal(Ca2.c,array([2*a*(a+1),0,-a]),13) assert_array_almost_equal(Ca3.c,array([4*sc.poch(a,3),0,-6*a*(a+1), 0])/3.0,11) assert_array_almost_equal(Ca4.c,array([4*sc.poch(a,4),0,-12*sc.poch(a,3), 0,3*a*(a+1)])/6.0,11) assert_array_almost_equal(Ca5.c,array([4*sc.poch(a,5),0,-20*sc.poch(a,4), 0,15*sc.poch(a,3),0])/15.0,11) class TestHermite: def test_hermite(self): H0 = orth.hermite(0) H1 = orth.hermite(1) H2 = orth.hermite(2) H3 = orth.hermite(3) H4 = orth.hermite(4) H5 = orth.hermite(5) assert_array_almost_equal(H0.c,[1],13) assert_array_almost_equal(H1.c,[2,0],13) assert_array_almost_equal(H2.c,[4,0,-2],13) assert_array_almost_equal(H3.c,[8,0,-12,0],13) assert_array_almost_equal(H4.c,[16,0,-48,0,12],12) assert_array_almost_equal(H5.c,[32,0,-160,0,120,0],12) def test_hermitenorm(self): # He_n(x) = 2**(-n/2) H_n(x/sqrt(2)) psub = np.poly1d([1.0/sqrt(2),0]) H0 = orth.hermitenorm(0) H1 = orth.hermitenorm(1) H2 = orth.hermitenorm(2) H3 = orth.hermitenorm(3) H4 = orth.hermitenorm(4) H5 = orth.hermitenorm(5) he0 = orth.hermite(0)(psub) he1 = orth.hermite(1)(psub) / sqrt(2) he2 = orth.hermite(2)(psub) / 2.0 he3 = orth.hermite(3)(psub) / (2*sqrt(2)) he4 = orth.hermite(4)(psub) / 4.0 he5 = orth.hermite(5)(psub) / (4.0*sqrt(2)) assert_array_almost_equal(H0.c,he0.c,13) assert_array_almost_equal(H1.c,he1.c,13) assert_array_almost_equal(H2.c,he2.c,13) assert_array_almost_equal(H3.c,he3.c,13) assert_array_almost_equal(H4.c,he4.c,13) assert_array_almost_equal(H5.c,he5.c,13) class TestShLegendre: def test_sh_legendre(self): # P*_n(x) = P_n(2x-1) psub = np.poly1d([2,-1]) Ps0 = orth.sh_legendre(0) Ps1 = orth.sh_legendre(1) Ps2 = orth.sh_legendre(2) Ps3 = orth.sh_legendre(3) Ps4 = orth.sh_legendre(4) Ps5 = orth.sh_legendre(5) pse0 = orth.legendre(0)(psub) pse1 = orth.legendre(1)(psub) pse2 = orth.legendre(2)(psub) pse3 = orth.legendre(3)(psub) pse4 = orth.legendre(4)(psub) pse5 = orth.legendre(5)(psub) assert_array_almost_equal(Ps0.c,pse0.c,13) assert_array_almost_equal(Ps1.c,pse1.c,13) assert_array_almost_equal(Ps2.c,pse2.c,13) assert_array_almost_equal(Ps3.c,pse3.c,13) assert_array_almost_equal(Ps4.c,pse4.c,12) assert_array_almost_equal(Ps5.c,pse5.c,12) class TestShChebyt: def test_sh_chebyt(self): # T*_n(x) = T_n(2x-1) psub = np.poly1d([2,-1]) Ts0 = orth.sh_chebyt(0) Ts1 = orth.sh_chebyt(1) Ts2 = orth.sh_chebyt(2) Ts3 = orth.sh_chebyt(3) Ts4 = orth.sh_chebyt(4) Ts5 = orth.sh_chebyt(5) tse0 = orth.chebyt(0)(psub) tse1 = orth.chebyt(1)(psub) tse2 = orth.chebyt(2)(psub) tse3 = orth.chebyt(3)(psub) tse4 = orth.chebyt(4)(psub) tse5 = orth.chebyt(5)(psub) assert_array_almost_equal(Ts0.c,tse0.c,13) assert_array_almost_equal(Ts1.c,tse1.c,13) assert_array_almost_equal(Ts2.c,tse2.c,13) assert_array_almost_equal(Ts3.c,tse3.c,13) assert_array_almost_equal(Ts4.c,tse4.c,12) assert_array_almost_equal(Ts5.c,tse5.c,12) class TestShChebyu: def test_sh_chebyu(self): # U*_n(x) = U_n(2x-1) psub = np.poly1d([2,-1]) Us0 = orth.sh_chebyu(0) Us1 = orth.sh_chebyu(1) Us2 = orth.sh_chebyu(2) Us3 = orth.sh_chebyu(3) Us4 = orth.sh_chebyu(4) Us5 = orth.sh_chebyu(5) use0 = orth.chebyu(0)(psub) use1 = orth.chebyu(1)(psub) use2 = orth.chebyu(2)(psub) use3 = orth.chebyu(3)(psub) use4 = orth.chebyu(4)(psub) use5 = orth.chebyu(5)(psub) assert_array_almost_equal(Us0.c,use0.c,13) assert_array_almost_equal(Us1.c,use1.c,13) assert_array_almost_equal(Us2.c,use2.c,13) assert_array_almost_equal(Us3.c,use3.c,13) assert_array_almost_equal(Us4.c,use4.c,12) assert_array_almost_equal(Us5.c,use5.c,11) class TestShJacobi: def test_sh_jacobi(self): # G^(p,q)_n(x) = n! gamma(n+p)/gamma(2*n+p) * P^(p-q,q-1)_n(2*x-1) def conv(n, p): return gamma(n + 1) * gamma(n + p) / gamma(2 * n + p) psub = np.poly1d([2,-1]) q = 4 * np.random.random() p = q-1 + 2*np.random.random() # print("shifted jacobi p,q = ", p, q) G0 = orth.sh_jacobi(0,p,q) G1 = orth.sh_jacobi(1,p,q) G2 = orth.sh_jacobi(2,p,q) G3 = orth.sh_jacobi(3,p,q) G4 = orth.sh_jacobi(4,p,q) G5 = orth.sh_jacobi(5,p,q) ge0 = orth.jacobi(0,p-q,q-1)(psub) * conv(0,p) ge1 = orth.jacobi(1,p-q,q-1)(psub) * conv(1,p) ge2 = orth.jacobi(2,p-q,q-1)(psub) * conv(2,p) ge3 = orth.jacobi(3,p-q,q-1)(psub) * conv(3,p) ge4 = orth.jacobi(4,p-q,q-1)(psub) * conv(4,p) ge5 = orth.jacobi(5,p-q,q-1)(psub) * conv(5,p) assert_array_almost_equal(G0.c,ge0.c,13) assert_array_almost_equal(G1.c,ge1.c,13) assert_array_almost_equal(G2.c,ge2.c,13) assert_array_almost_equal(G3.c,ge3.c,13) assert_array_almost_equal(G4.c,ge4.c,13) assert_array_almost_equal(G5.c,ge5.c,13) class TestCall: def test_call(self): poly = [] for n in range(5): poly.extend([x.strip() for x in (""" orth.jacobi(%(n)d,0.3,0.9) orth.sh_jacobi(%(n)d,0.3,0.9) orth.genlaguerre(%(n)d,0.3) orth.laguerre(%(n)d) orth.hermite(%(n)d) orth.hermitenorm(%(n)d) orth.gegenbauer(%(n)d,0.3) orth.chebyt(%(n)d) orth.chebyu(%(n)d) orth.chebyc(%(n)d) orth.chebys(%(n)d) orth.sh_chebyt(%(n)d) orth.sh_chebyu(%(n)d) orth.legendre(%(n)d) orth.sh_legendre(%(n)d) """ % dict(n=n)).split() ]) with np.errstate(all='ignore'): for pstr in poly: p = eval(pstr) assert_almost_equal(p(0.315), np.poly1d(p.coef)(0.315), err_msg=pstr) class TestGenlaguerre: def test_regression(self): assert_equal(orth.genlaguerre(1, 1, monic=False)(0), 2.) assert_equal(orth.genlaguerre(1, 1, monic=True)(0), -2.) assert_equal(orth.genlaguerre(1, 1, monic=False), np.poly1d([-1, 2])) assert_equal(orth.genlaguerre(1, 1, monic=True), np.poly1d([1, -2])) def verify_gauss_quad(root_func, eval_func, weight_func, a, b, N, rtol=1e-15, atol=5e-14): # this test is copied from numpy's TestGauss in test_hermite.py x, w, mu = root_func(N, True) n = np.arange(N, dtype=np.dtype("long")) v = eval_func(n[:,np.newaxis], x) vv = np.dot(v*w, v.T) vd = 1 / np.sqrt(vv.diagonal()) vv = vd[:, np.newaxis] * vv * vd assert_allclose(vv, np.eye(N), rtol, atol) # check that the integral of 1 is correct assert_allclose(w.sum(), mu, rtol, atol) # compare the results of integrating a function with quad. def f(x): return x ** 3 - 3 * x ** 2 + x - 2 resI = integrate.quad(lambda x: f(x)*weight_func(x), a, b) resG = np.vdot(f(x), w) rtol = 1e-6 if 1e-6 < resI[1] else resI[1] * 10 assert_allclose(resI[0], resG, rtol=rtol) def test_roots_jacobi(): def rf(a, b): return lambda n, mu: sc.roots_jacobi(n, a, b, mu) def ef(a, b): return lambda n, x: sc.eval_jacobi(n, a, b, x) def wf(a, b): return lambda x: (1 - x) ** a * (1 + x) ** b vgq = verify_gauss_quad vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., 5) vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., 25, atol=1e-12) vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., 100, atol=1e-11) vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 5) vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 25, atol=1.5e-13) vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 100, atol=2e-12) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 5, atol=2e-13) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 25, atol=2e-13) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 100, atol=1e-12) vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 5) vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 25, atol=1e-13) vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 100, atol=3e-13) vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., 5) vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., 25, atol=1.1e-14) vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., 100, atol=1e-13) vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., 5, atol=1e-13) vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., 25, atol=2e-13) vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., 100, atol=1e-11) vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 5, atol=2e-13) vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 25, atol=1e-12) vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 100, atol=1e-11) vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 250, atol=1e-11) vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 5, atol=1e-12) vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 25, atol=1e-11) vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 100, atol=1e-10) vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 5, atol=1e-12) vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 25, atol=1e-11) vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 100, atol=1e-10) vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 5, atol=1e-12) vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 25, atol=1e-11) vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 100, atol=1e-10) vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 5) vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 25, atol=1e-13) vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 100, atol=1e-13) # when alpha == beta == 0, P_n^{a,b}(x) == P_n(x) xj, wj = sc.roots_jacobi(6, 0.0, 0.0) xl, wl = sc.roots_legendre(6) assert_allclose(xj, xl, 1e-14, 1e-14) assert_allclose(wj, wl, 1e-14, 1e-14) # when alpha == beta != 0, P_n^{a,b}(x) == C_n^{alpha+0.5}(x) xj, wj = sc.roots_jacobi(6, 4.0, 4.0) xc, wc = sc.roots_gegenbauer(6, 4.5) assert_allclose(xj, xc, 1e-14, 1e-14) assert_allclose(wj, wc, 1e-14, 1e-14) x, w = sc.roots_jacobi(5, 2, 3, False) y, v, m = sc.roots_jacobi(5, 2, 3, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(wf(2,3), -1, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_jacobi, 0, 1, 1) assert_raises(ValueError, sc.roots_jacobi, 3.3, 1, 1) assert_raises(ValueError, sc.roots_jacobi, 3, -2, 1) assert_raises(ValueError, sc.roots_jacobi, 3, 1, -2) assert_raises(ValueError, sc.roots_jacobi, 3, -2, -2) def test_roots_sh_jacobi(): def rf(a, b): return lambda n, mu: sc.roots_sh_jacobi(n, a, b, mu) def ef(a, b): return lambda n, x: sc.eval_sh_jacobi(n, a, b, x) def wf(a, b): return lambda x: (1.0 - x) ** (a - b) * x ** (b - 1.0) vgq = verify_gauss_quad vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., 5) vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., 25, atol=1e-12) vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., 100, atol=1e-11) vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 5) vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 25, atol=1e-13) vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 100, atol=1e-12) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 5) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 25, atol=1.5e-13) vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 100, atol=2e-12) vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 5) vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 25, atol=1e-13) vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 100, atol=1e-12) vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., 5) vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., 25) vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., 100, atol=1e-13) vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 5, atol=1e-12) vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 25, atol=1e-11) vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 100, atol=1e-10) vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., 5, atol=3.5e-14) vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., 25, atol=2e-13) vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., 100, atol=1e-12) x, w = sc.roots_sh_jacobi(5, 3, 2, False) y, v, m = sc.roots_sh_jacobi(5, 3, 2, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(wf(3,2), 0, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_sh_jacobi, 0, 1, 1) assert_raises(ValueError, sc.roots_sh_jacobi, 3.3, 1, 1) assert_raises(ValueError, sc.roots_sh_jacobi, 3, 1, 2) # p - q <= -1 assert_raises(ValueError, sc.roots_sh_jacobi, 3, 2, -1) # q <= 0 assert_raises(ValueError, sc.roots_sh_jacobi, 3, -2, -1) # both def test_roots_hermite(): rootf = sc.roots_hermite evalf = sc.eval_hermite weightf = orth.hermite(5).weight_func verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5) verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13) verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12) # Golub-Welsch branch x, w = sc.roots_hermite(5, False) y, v, m = sc.roots_hermite(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -np.inf, np.inf) assert_allclose(m, muI, rtol=muI_err) # Asymptotic branch (switch over at n >= 150) x, w = sc.roots_hermite(200, False) y, v, m = sc.roots_hermite(200, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) assert_allclose(sum(v), m, 1e-14, 1e-14) assert_raises(ValueError, sc.roots_hermite, 0) assert_raises(ValueError, sc.roots_hermite, 3.3) def test_roots_hermite_asy(): # Recursion for Hermite functions def hermite_recursion(n, nodes): H = np.zeros((n, nodes.size)) H[0,:] = np.pi**(-0.25) * np.exp(-0.5*nodes**2) if n > 1: H[1,:] = sqrt(2.0) * nodes * H[0,:] for k in range(2, n): H[k,:] = sqrt(2.0/k) * nodes * H[k-1,:] - sqrt((k-1.0)/k) * H[k-2,:] return H # This tests only the nodes def test(N, rtol=1e-15, atol=1e-14): x, w = orth._roots_hermite_asy(N) H = hermite_recursion(N+1, x) assert_allclose(H[-1,:], np.zeros(N), rtol, atol) assert_allclose(sum(w), sqrt(np.pi), rtol, atol) test(150, atol=1e-12) test(151, atol=1e-12) test(300, atol=1e-12) test(301, atol=1e-12) test(500, atol=1e-12) test(501, atol=1e-12) test(999, atol=1e-12) test(1000, atol=1e-12) test(2000, atol=1e-12) test(5000, atol=1e-12) def test_roots_hermitenorm(): rootf = sc.roots_hermitenorm evalf = sc.eval_hermitenorm weightf = orth.hermitenorm(5).weight_func verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5) verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13) verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12) x, w = sc.roots_hermitenorm(5, False) y, v, m = sc.roots_hermitenorm(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -np.inf, np.inf) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_hermitenorm, 0) assert_raises(ValueError, sc.roots_hermitenorm, 3.3) def test_roots_gegenbauer(): def rootf(a): return lambda n, mu: sc.roots_gegenbauer(n, a, mu) def evalf(a): return lambda n, x: sc.eval_gegenbauer(n, a, x) def weightf(a): return lambda x: (1 - x ** 2) ** (a - 0.5) vgq = verify_gauss_quad vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 5) vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 25, atol=1e-12) vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 100, atol=1e-11) vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 5) vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 25, atol=1e-13) vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 100, atol=1e-12) vgq(rootf(1), evalf(1), weightf(1), -1., 1., 5) vgq(rootf(1), evalf(1), weightf(1), -1., 1., 25, atol=1e-13) vgq(rootf(1), evalf(1), weightf(1), -1., 1., 100, atol=1e-12) vgq(rootf(10), evalf(10), weightf(10), -1., 1., 5) vgq(rootf(10), evalf(10), weightf(10), -1., 1., 25, atol=1e-13) vgq(rootf(10), evalf(10), weightf(10), -1., 1., 100, atol=1e-12) vgq(rootf(50), evalf(50), weightf(50), -1., 1., 5, atol=1e-13) vgq(rootf(50), evalf(50), weightf(50), -1., 1., 25, atol=1e-12) vgq(rootf(50), evalf(50), weightf(50), -1., 1., 100, atol=1e-11) # Alpha=170 is where the approximation used in roots_gegenbauer changes vgq(rootf(170), evalf(170), weightf(170), -1., 1., 5, atol=1e-13) vgq(rootf(170), evalf(170), weightf(170), -1., 1., 25, atol=1e-12) vgq(rootf(170), evalf(170), weightf(170), -1., 1., 100, atol=1e-11) vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 5, atol=1.25e-13) vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 25, atol=1e-12) vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 100, atol=1e-11) # Test for failures, e.g. overflows, resulting from large alphas vgq(rootf(238), evalf(238), weightf(238), -1., 1., 5, atol=1e-13) vgq(rootf(238), evalf(238), weightf(238), -1., 1., 25, atol=1e-12) vgq(rootf(238), evalf(238), weightf(238), -1., 1., 100, atol=1e-11) vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 5, atol=1e-12) vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 25, atol=1e-11) vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 100, atol=1e-10) # this is a special case that the old code supported. # when alpha = 0, the gegenbauer polynomial is uniformly 0. but it goes # to a scaled down copy of T_n(x) there. vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 5) vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 25) vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 100, atol=1e-12) x, w = sc.roots_gegenbauer(5, 2, False) y, v, m = sc.roots_gegenbauer(5, 2, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf(2), -1, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_gegenbauer, 0, 2) assert_raises(ValueError, sc.roots_gegenbauer, 3.3, 2) assert_raises(ValueError, sc.roots_gegenbauer, 3, -.75) def test_roots_chebyt(): weightf = orth.chebyt(5).weight_func verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 5) verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 25) verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 100, atol=1e-12) x, w = sc.roots_chebyt(5, False) y, v, m = sc.roots_chebyt(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -1, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_chebyt, 0) assert_raises(ValueError, sc.roots_chebyt, 3.3) def test_chebyt_symmetry(): x, w = sc.roots_chebyt(21) pos, neg = x[:10], x[11:] assert_equal(neg, -pos[::-1]) assert_equal(x[10], 0) def test_roots_chebyu(): weightf = orth.chebyu(5).weight_func verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 5) verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 25) verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 100) x, w = sc.roots_chebyu(5, False) y, v, m = sc.roots_chebyu(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -1, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_chebyu, 0) assert_raises(ValueError, sc.roots_chebyu, 3.3) def test_roots_chebyc(): weightf = orth.chebyc(5).weight_func verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 5) verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 25) verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 100, atol=1e-12) x, w = sc.roots_chebyc(5, False) y, v, m = sc.roots_chebyc(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -2, 2) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_chebyc, 0) assert_raises(ValueError, sc.roots_chebyc, 3.3) def test_roots_chebys(): weightf = orth.chebys(5).weight_func verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 5) verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 25) verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 100) x, w = sc.roots_chebys(5, False) y, v, m = sc.roots_chebys(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -2, 2) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_chebys, 0) assert_raises(ValueError, sc.roots_chebys, 3.3) def test_roots_sh_chebyt(): weightf = orth.sh_chebyt(5).weight_func verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., 5) verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., 25) verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., 100, atol=1e-13) x, w = sc.roots_sh_chebyt(5, False) y, v, m = sc.roots_sh_chebyt(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, 0, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_sh_chebyt, 0) assert_raises(ValueError, sc.roots_sh_chebyt, 3.3) def test_roots_sh_chebyu(): weightf = orth.sh_chebyu(5).weight_func verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., 5) verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., 25) verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., 100, atol=1e-13) x, w = sc.roots_sh_chebyu(5, False) y, v, m = sc.roots_sh_chebyu(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, 0, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_sh_chebyu, 0) assert_raises(ValueError, sc.roots_sh_chebyu, 3.3) def test_roots_legendre(): weightf = orth.legendre(5).weight_func verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., 5) verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., 25, atol=1e-13) verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., 100, atol=1e-12) x, w = sc.roots_legendre(5, False) y, v, m = sc.roots_legendre(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, -1, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_legendre, 0) assert_raises(ValueError, sc.roots_legendre, 3.3) def test_roots_sh_legendre(): weightf = orth.sh_legendre(5).weight_func verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., 5) verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., 25, atol=1e-13) verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., 100, atol=1e-12) x, w = sc.roots_sh_legendre(5, False) y, v, m = sc.roots_sh_legendre(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, 0, 1) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_sh_legendre, 0) assert_raises(ValueError, sc.roots_sh_legendre, 3.3) def test_roots_laguerre(): weightf = orth.laguerre(5).weight_func verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, 5) verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, 25, atol=1e-13) verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, 100, atol=1e-12) x, w = sc.roots_laguerre(5, False) y, v, m = sc.roots_laguerre(5, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf, 0, np.inf) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_laguerre, 0) assert_raises(ValueError, sc.roots_laguerre, 3.3) def test_roots_genlaguerre(): def rootf(a): return lambda n, mu: sc.roots_genlaguerre(n, a, mu) def evalf(a): return lambda n, x: sc.eval_genlaguerre(n, a, x) def weightf(a): return lambda x: x ** a * np.exp(-x) vgq = verify_gauss_quad vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 5) vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 25, atol=1e-13) vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 100, atol=1e-12) vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 5) vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 25, atol=1e-13) vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 100, atol=1.6e-13) vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 5) vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 25, atol=1e-13) vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 100, atol=1.03e-13) vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 5) vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 25, atol=1e-13) vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 100, atol=1e-12) vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 5) vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 25, atol=1e-13) vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 100, rtol=1e-14, atol=2e-13) x, w = sc.roots_genlaguerre(5, 2, False) y, v, m = sc.roots_genlaguerre(5, 2, True) assert_allclose(x, y, 1e-14, 1e-14) assert_allclose(w, v, 1e-14, 1e-14) muI, muI_err = integrate.quad(weightf(2.), 0., np.inf) assert_allclose(m, muI, rtol=muI_err) assert_raises(ValueError, sc.roots_genlaguerre, 0, 2) assert_raises(ValueError, sc.roots_genlaguerre, 3.3, 2) assert_raises(ValueError, sc.roots_genlaguerre, 3, -1.1) def test_gh_6721(): # Regression test for gh_6721. This should not raise. sc.chebyt(65)(0.2)