def test_roots_sh_jacobi():
rf = lambda a, b: lambda n, mu: sc.roots_sh_jacobi(n, a, b, mu)
ef = lambda a, b: lambda n, x: sc.eval_sh_jacobi(n, a, b, x)
wf = lambda a, b: lambda x: (1. - x)**(a - b) * (x)**(b - 1.)
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_sh_jacobi():
rf = lambda a, b: lambda n, mu: sc.roots_sh_jacobi(n, a, b, mu)
ef = lambda a, b: lambda n, x: orth.eval_sh_jacobi(n, a, b, x)
wf = lambda a, b: lambda x: (1. - x)**(a - b) * (x)**(b - 1.)
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=1e-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 recur_colloc_symm(n_pts, a):
"""
For symmetric collocation problems, produce:
rootsx = polynomial root locations
Ax = 1st derivative operator
Lx = 1-D Laplacian
W = quadrature weights
Reference: Michelsen and Villadsen 1971
sheet: a=1
cylinder: a=2
sphere: a=3
"""
n_pts_m = n_pts - 1 # number of interior collocation points
rootsu = np.zeros([n_pts])
rootsu[-1] = 1 # right bounary
# Define optimal Jacobi polynomial for interior points
p = 1.5 + (a - 1) / 2
q = p - 1
rootsu[:-1] = roots_sh_jacobi(n_pts_m, p, q)[0] # roots in u domain
rootsx = np.sqrt(rootsu) # roots in x domain
# define 3 by n_pts matrix of polynomial derivatives at collocation points
# row index (k) is the k + 1 derivative
px_mat = np.zeros((3, n_pts))
for ii in range(n_pts):
px_mat[:, ii] = calc_px_column(rootsu[ii], rootsu)
# Calculate derivative operators (A and B) in u domain
Amat = np.zeros((n_pts, n_pts))
Bmat = np.zeros((n_pts, n_pts))
for ii in range(n_pts):
for jj in range(n_pts):
if ii == jj: # Eq. 12
Amat[ii, jj] = (1 / 2) * px_mat[1, ii] / px_mat[0, ii]
Bmat[ii, jj] = (1 / 3) * px_mat[2, ii] / px_mat[0, ii]
else: # Eqs. 13, 14
G = rootsu[ii] - rootsu[jj]
Amat[ii, jj] = (1 / G) * px_mat[0, ii] / px_mat[0, jj]
Bmat[ii, jj] = (1 / G) * (px_mat[1, ii] / px_mat[0, jj] -
2 * Amat[ii, jj])
Ax = np.zeros((n_pts, n_pts)) # First derivative operator in x domain
Bx = np.zeros((n_pts, n_pts)) # 1-D Laplacian operator in x domain
for ii in range(n_pts):
for jj in range(n_pts):
# Following Eq. 33
Ax[ii, jj] = 2 * rootsx[ii] * Amat[ii, jj]
Bx[ii, jj] = 2 * a * Amat[ii, jj] + rootsu[ii] * 4 * Bmat[ii, jj]
W = np.zeros(n_pts)
for ii in range(n_pts):
# Eq. 23 - 25 but omitting root at zero for nodal polynomial.
wp_sum = (1 / (rootsu[:] * px_mat[0, :]**2)).sum()
W[ii] = (1 / (rootsu[ii] * px_mat[0, ii]**2)) / wp_sum / (a)
return (rootsx, Ax, Bx, W)
