def test_roots_gegenbauer(): rootf = lambda a: lambda n, mu: sc.roots_gegenbauer(n, a, mu) evalf = lambda a: lambda n, x: sc.eval_gegenbauer(n, a, x) weightf = lambda a: 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=1e-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_gegenbauer(): rootf = lambda a: lambda n, mu: sc.roots_gegenbauer(n, a, mu) evalf = lambda a: lambda n, x: orth.eval_gegenbauer(n, a, x) weightf = lambda a: 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) # 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), orth.eval_chebyt, weightf(0), -1., 1., 5) vgq(rootf(0), orth.eval_chebyt, weightf(0), -1., 1., 25) vgq(rootf(0), orth.eval_chebyt, weightf(0), -1., 1., 100) 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 gauss_gegenbauer(n, alpha, lower=-1, upper=1): ''' Gauss-Gegenbauer quadrature: A rule of order 2*n-1 on the interval [-1, 1] with respect to the weight function w(x) = (1-x**2)**(alpha-1/2). ''' nodes, weights = special.roots_gegenbauer(n, alpha) if lower != -1 or upper != 1: nodes = (upper+lower)/2 + (upper-lower)/2*nodes weights = (upper-lower)/2*weights return nodes, weights
def test_roots_jacobi(): rf = lambda a, b: lambda n, mu: sc.roots_jacobi(n, a, b, mu) ef = lambda a, b: lambda n, x: orth.eval_jacobi(n, a, b, x) wf = lambda a, b: 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=1e-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=2e-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) 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(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_jacobi(): rf = lambda a, b: lambda n, mu: sc.roots_jacobi(n, a, b, mu) ef = lambda a, b: lambda n, x: sc.eval_jacobi(n, a, b, x) wf = lambda a, b: 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(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)