def __init__(self, polygon, center=0, n_node=8, method='krylov'):
"""
polygon = target polygon of SC transformation
center = point in polygon where disk center is mapped to
n_node = number of nodes for gaussian quadrature
method = used for root finding
"""
p = polygon.copy()
p.roll()
w = p.vertex
beta = p.angle
n = len(w)
k_rat = p.FiniteEdge([0, n - 1], n - 3)
k_fix = p.InfVertex()
if len(k_fix) == 0: k_fix = [1]
node = np.empty([n + 1, n_node])
weight = np.empty_like(node)
for k in range(n):
if np.isinf(w[k]): continue
(node[k], weight[k]) = roots_jacobi(n_node, 0, -beta[k])
(node[n], weight[n]) = roots_legendre(n_node)
self.prevertex = np.empty(n, dtype=np.complex)
self.vertex = w
self.angle = beta
self.node = node
self.weight = weight
self.A = center
self.map = np.vectorize(self.map)
self.invmap = np.vectorize(self.invmap)
y = np.zeros(n - 1)
f = np.empty_like(y)
def scfun(y):
z = self.yztran(y)
C = (center - w[-1]) / self.zquad(1, 0, n - 1)
i = 0
for k in k_fix:
q = w[k - 1] - center + C * self.zquad(z[k - 1], 0, k - 1)
f[i] = np.real(q)
f[i + 1] = np.imag(q)
i += 2
for k in k_rat:
q = self.zquad(z[k], z[k + 1], k, k + 1)
f[i] = np.abs(w[k + 1] - w[k]) - np.abs(C * q)
i += 1
self.C = C
return f
sol = root(scfun, y, method=method, options={'disp': True})
self.yztran(sol.x)
def zernike_quad(nmax, mmax):
'''
Construct a list of quadrature points and associated weights. These must be adapted to compute
scalar products involved in the analysis (Zernike transform), hence must have >= 2mmax+1 radial branches,
and >= nmax+1 radial points (which allows integration of polynomials up to degree 2nmax+1). Note that
the radial measure is proportional to radius for Zernike polynomials orthogonality, hence the shifted Jacobi polynomial
P_{nmax+1}^{0,1}(2r-1) is chosen for the radial quadrature.
'''
# First define radial points and weights
from scipy.special import roots_jacobi
xx, w8 = roots_jacobi(nmax + 1, 0, 1) # xx in [-1,1]
r = (xx + 1.) / 2.
theta = np.arange(2 * mmax + 1) / (2. * mmax + 1.) * 2. * np.pi
ctheta = np.cos(theta)
stheta = np.sin(theta)
X = np.outer(r, ctheta).flatten()
Y = np.outer(r, stheta).flatten()
# Normalize weights: factor of 4 for shifted polynomial variable change, factor of (2*mmax+1)/2. for angular integration
w8 /= 2. * (2 * mmax + 1.)
W = np.outer(w8, np.ones_like(ctheta)).flatten()
return X, Y, W
def jacobi_quad(nmax):
# Here 2 = d-1 = 3-1 in 3D
xx, ww = ss.roots_jacobi(nmax + 1, 0, 2)
t = (xx + 1.) / 2.
return (t, ww)
def get_GLquadrature(k, a=0., b=1., quad='Gauss-Legendre'):
"""Get the weights and points of the k-th order Gauss-Legendre quadrature.
Return the points shifted in (a, b)."""
bmao2 = .5 * (b - a)
if quad == 'Gauss-Legendre':
# Gauss-Legendre (default interval is [-1, 1])
x, w = np.polynomial.legendre.leggauss(k)
elif quad == 'Gauss-Jacobi':
# Gauss-Jacobi, overcoming the singularity of Ki3 at 0 (tangent)
# warning: weights and points differ from Hebert's book!
# the first parameters used by Hebert is unknown
x, w = roots_jacobi(k, 1, 0)
elif quad == 'Rectangular':
x, w = np.linspace(-1., 1., k + 1), np.full(k, 2. / k)
x = (x[1:] + x[:-1]) / 2.
else:
raise ValueError("Unknown type of numerical quadrature.")
# Translate x values from the interval [-1, 1] to [a, b]
## return (x + 1) * bmao2 + a, w * bmao2
c = 1 - x
if quad == 'Gauss-Jacobi':
c *= .5 * (1 - x)
return b - c * bmao2, w * bmao2
def __init__(self, polygon, n_node=8, method='krylov'):
"""
polygon = target polygon of SC transformation
n_node = number of nodes for gaussian quadrature
method = used for root finding
"""
if np.any(np.isinf(polygon.vertex)):
print('infinite vertex is not allowed'); exit()
p = polygon.copy()
p.flip()
p.roll(1)
w = p.vertex
b = p.angle
b[b==-1] = 1
n = len(w)
node = np.empty([n+1, n_node])
weight = np.empty_like(node)
for k in range(n):
if np.isfinite(w[k]):
(node[k], weight[k]) = roots_jacobi(n_node, 0, b[k])
(node[n], weight[n]) = roots_legendre(n_node)
self.prevertex = np.empty(n, dtype=np.complex)
self.vertex = w
self.angle = b
self.node = node
self.weight = weight
self.map = np.vectorize(self.map)
y = np.zeros(n-1)
f = np.empty_like(y)
def scfun(y):
z = self.yztran(y)
C = (w[-1] - w[0])/self.zquad(z[0], z[-1], 0, n-1)
for k in range(n-3):
q = self.zquad(z[k], z[k+1], k, k+1)
f[k] = np.abs(w[k+1] - w[k]) - np.abs(C*q)
r = np.sum(b/z)
f[n-3] = np.real(r)
f[n-2] = np.imag(r)
self.C = C
return f
sol = root(scfun, y, method=method, options={'disp': True})
self.yztran(sol.x)
def __init__(self, z, n_node=8, method='krylov'):
"""
z = vertices as complex numbers on boundary
n_node = number of nodes for gaussian quadrature
method = used for root finding
"""
if len(z)<2:
print('there must be 2 or more vertices'); exit()
if len(z)==2:
z.insert(1, (z[0]+z[1])/2)
n = len(z) - 1
beta = np.empty(n+1)
node = np.empty([n+1, n_node])
weight = np.empty_like(node)
(node[n], weight[n]) = roots_legendre(n_node)
(node[0], weight[0]) = (node[n], weight[n])
gamma = np.angle(z[1] - z[0])/np.pi
for k in range(1,n):
beta[k] = np.angle(z[k+1] - z[k])/np.pi - gamma
beta[k] = (beta[k] + 1)%2 - 1
if beta[k]-1 > 1.e-8: beta[k] = -1
gamma += beta[k]
(node[k], weight[k]) = roots_jacobi(n_node, 0, -beta[k])
self.prevertex = np.empty(n+1, dtype=np.complex)
self.prevertex[0] = -1
self.vertex = np.array(z, dtype=np.complex)
self.angle = beta
self.node = node
self.weight = weight
self.reflect = False
self.map = np.vectorize(self.map)
y = np.zeros(n-1)
f = np.empty_like(y)
def kfun(y):
x = self.yxtran(y)
C = (z[1] - z[0])/self.xquad(x[0], x[1], 0, 1)
for k in range(1,n):
q = self.xquad(x[k], x[k+1], k, k+1)
f[k-1] = np.abs(z[k+1] - z[k]) - np.abs(C*q)
self.C = C
return f
sol = root(kfun, y, method=method, options={'disp': True})
self.yxtran(sol.x)
def points_and_weights(self,
N=None,
map_true_domain=False,
weighted=True,
**kw):
if N is None:
N = self.N
assert self.quad == "JG"
points, weights = roots_jacobi(N, 0, 0)
if map_true_domain is True:
points = self.map_true_domain(points)
return points, weights
def jacobz(n, alpha, beta):
"""
compute zeros jacobi polynomial P_n^{alpha,beta}
:param n: order of jacobi polynomial, integer
:param alpha: parameter, alpha > -1
:param beta: parameter, beta > -1
:return: zeros of jacobi polynomials
"""
if n <= 0:
return None
z, _ = roots_jacobi(n, alpha, beta)
return z
def points_and_weights(self,
N=None,
map_true_domain=False,
weighted=True,
**kw):
if N is None:
N = self.shape(False)
assert self.quad == "JG"
points, weights = roots_jacobi(N, self.alpha + 1, self.beta + 1)
if map_true_domain is True:
points = self.map_true_domain(points)
return points, weights
def zwgj(n, alpha, beta):
"""
compute Gauss-Jacobi quadrature points and weights associated with
polynomial P_n^{alpha,beta}
:param n: order of Jacobi polynomial
:param alpha: parameter, alpha > -1
:param beta: parameter, beta > -1
:return: tuple (z,w), where z contains quadrature points, w contains weights
"""
if n <= 0:
return None, None
else:
return roots_jacobi(n, alpha, beta)
def gauss_jacobi(n, alpha, beta, lower=-1, upper=1):
'''
Gauss-Jacobi quadrature:
A rule of order 2*n-1 on the interval [-1, 1] with respect to
the weight function w(x) = (1-x)**alpha*(1+x)**beta.
'''
nodes, weights = special.roots_jacobi(n, alpha, beta)
if lower != -1 or upper != 1:
nodes = (upper+lower)/2 + (upper-lower)/2*nodes
weights = (upper-lower)/2*weights
return nodes, weights
def cumulative_sfs(k1: int,
k2: int,
n: int,
c: float,
t: float,
theta: float,
m_max: int,
order: int = None) -> float:
"""Compute the CDF of the expected joint site frequency spectrum.
Parameters
----------
k1 : int
the allele count in population 1
k2 : int
the allele count in population 2
n : int
the sample size in each population
c : float
the admixture proportion in the present
t : float
the population-scaled split time
theta : float
the population-scaled mutation rate
m_max : int
the largest basis function to compute
order : int, optional
the quadrature order (default=2n)
Returns
-------
float
"""
m = np.arange(m_max + 1)
lambdas = jacobi_eigenval(m, theta)
norms = jacobi_norm(m, theta)
# The jacobi_norm(0, theta) scales the density to integrate to 1
weights = np.exp(-2 * lambdas * t) / norms / jacobi_norm(0, theta)
if order is None:
order = 2 * n
x, w = roots_jacobi(order, theta - 1, theta - 1)
x = (x + 1) / 2
w /= 2**(2 * theta - 1)
W = w * w[:, np.newaxis]
evals = __integrand(x, x[:, np.newaxis], k1, k2, n, c, t, theta, m,
weights)
return np.sum(W * evals)
def gll(n):
"""Compute the points and weights of the Gouss-Lenendre-Lobbato quadrature
Parameters
----------
n, int
The number of points.
"""
# Special case
if n == 2:
return np.array([-1, 1]), np.array([1, 1])
x, w = roots_jacobi(n - 2, 1, 1)
for i in range(x.size):
w[i] /= 1 - x[i]**2
x = np.append(-1, np.append(x, 1))
w = np.append(2 / (n * (n - 1)), np.append(w, 2 / (n * (n - 1))))
return x, w
def gauss_lobatto_jacobi_weights(n, a, b):
X = roots_jacobi(n - 2, a + 1, b + 1)[0]
Wl = (b + 1) * 2**(a + b + 1) * gamma(a + n) * gamma(b + n) / (
(n - 1) * gamma(n) * gamma(a + b + n + 1) *
(jacobi_polynomial(n - 1, a, b, -1)**2))
W = 2**(a + b + 1) * gamma(a + n) * gamma(b + n) / (
(n - 1) * gamma(n) * gamma(a + b + n + 1) *
(jacobi_polynomial(n - 1, a, b, X)**2))
Wr = (a + 1) * 2**(a + b + 1) * gamma(a + n) * gamma(b + n) / (
(n - 1) * gamma(n) * gamma(a + b + n + 1) *
(jacobi_polynomial(n - 1, a, b, 1)**2))
W = np.append(W, Wr)
W = np.append(Wl, W)
X = np.append(-1, X)
X = np.append(X, 1)
return [X, W]
def GaussLobattoJacobiWeights(Q: int, a, b):
W = []
X = roots_jacobi(Q - 2, a + 1, b + 1)[0]
if a == 0 and b == 0:
W = 2 / ((Q - 1) * (Q) * (Jacobi(Q - 1, 0, 0, X)**2))
Wl = 2 / ((Q - 1) * (Q) * (Jacobi(Q - 1, 0, 0, -1)**2))
Wr = 2 / ((Q - 1) * (Q) * (Jacobi(Q - 1, 0, 0, 1)**2))
else:
W = 2**(a + b + 1) * gamma(a + Q) * gamma(b + Q) / (
(Q - 1) * gamma(Q) * gamma(a + b + Q + 1) *
(Jacobi(Q - 1, a, b, X)**2))
Wl = (b + 1) * 2**(a + b + 1) * gamma(a + Q) * gamma(b + Q) / (
(Q - 1) * gamma(Q) * gamma(a + b + Q + 1) *
(Jacobi(Q - 1, a, b, -1)**2))
Wr = (a + 1) * 2**(a + b + 1) * gamma(a + Q) * gamma(b + Q) / (
(Q - 1) * gamma(Q) * gamma(a + b + Q + 1) *
(Jacobi(Q - 1, a, b, 1)**2))
W = np.append(W, Wr)
W = np.append(Wl, W)
X = np.append(X, 1)
X = np.append(-1, X)
return [X, W]
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)
def JacobiGQ(alpha, beta, N):
x, w = sci.roots_jacobi(N, alpha, beta)
return x, w
import numpy as np from numpy import sqrt from numpy.testing import assert_allclose import scipy.special as sc import scipy.special.orthogonal as orth 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) rtol = 1e-15 atol = 1e-14 N = 25 root_func = rf(18.24, 27.3) eval_func = ef(18.24, 27.3) x, w, mu = root_func(N, True) # test orthogonality. Note that the results need to be normalized, # otherwise the huge values that can arise from fast growing # functions like Laguerre can be very confusing. n = np.arange(N) 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)
def __init__(self, z, alpha, n_node=8, method='krylov'):
"""
z = vertices as complex numbers on boundary
alpha = interior angles between edges
n_node = number of nodes for gaussian quadrature
method = used for root finding
"""
if len(z) < 3:
print('there must be 3 or more vertices')
exit()
if z.count(np.inf) != 1:
print('there must be one Infinity')
exit()
if np.isinf(z[0]) or np.isinf(z[-1]):
print('z[0], and z[-1] must be finite')
exit()
if np.isinf(z[1]):
z.insert(1, z[0] + np.exp(-np.pi * alpha[0] * 1j))
alpha[0] = 1
n = len(z) - 1
L = z.index(np.inf)
beta = np.empty(n + 1)
gamma = np.angle(z[1] - z[0]) / np.pi
for k in range(1, n):
if k == L or k + 1 == L:
beta[k] = 1 - alpha[k + 1 - L]
else:
beta[k] = np.angle(z[k + 1] - z[k]) / np.pi - gamma
beta[k] = (beta[k] + 1) % 2 - 1
if beta[k] - 1 > 1.e-8: beta[k] = -1
gamma += beta[k]
beta[L] -= 1
node = np.empty([n + 1, n_node])
weight = np.empty_like(node)
for k in range(1, n):
if k == L: continue
(node[k], weight[k]) = roots_jacobi(n_node, 0, -beta[k])
(node[0], weight[0]) = roots_jacobi(n_node, 0, 1)
(node[n], weight[n]) = roots_legendre(n_node)
self.prevertex = np.empty(n + 1, dtype=np.complex)
self.prevertex[0] = -1
self.vertex = np.array(z, dtype=np.complex)
self.angle = beta
self.node = node
self.weight = weight
self.L = L
self.map = np.vectorize(self.map)
y = np.zeros(n - 1)
f = np.empty_like(y)
def jfun(y):
s = self.ystran(y)
C = (z[1] - z[0]) / self.squad(s[0], s[1], 0, 1)
q = self.squad(s[0], 0.5j, 0) - self.squad(s[n], 0.5j, n)
q = z[n] - z[0] - C * q
f[L - 2] = np.real(q)
f[L - 1] = np.imag(q)
for k in range(1, n):
if k == L or k + 1 == L: continue
q = self.squad(s[k], s[k + 1], k, k + 1)
f[k - 1] = np.abs(z[k + 1] - z[k]) - np.abs(C * q)
self.C = C
return f
sol = root(jfun, y, method=method, options={'disp': True})
self.ystran(sol.x)
s = np.real(self.prevertex[1:-1])
t = 1 / (1 + s[L - 1]**2)
self.q = np.abs(self.C) * np.pi * t
self.theta = np.angle(self.C)\
- np.dot(beta[1:-1], np.pi/2 + 2 * np.arctan(s))
t *= 2 * s[L - 1]
self.w1 = self.q * (np.log((1 + t) / (1 - t)) / np.pi + 1j)
self.sigma_L = t
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 GaussJacobiWeights(Q: int, a, b):
[X, W] = roots_jacobi(Q, a, b)
return [X, W]
def __init__(self, polygon, n_node=8, method='krylov'):
"""
polygon = target polygon of SC transformation
n_node = number of nodes for gaussian quadrature
method = used for root finding
"""
p = polygon.copy()
p.roll()
w = p.vertex
beta = p.angle
n = len(w)
node = np.empty([n + 1, n_node])
weight = np.empty_like(node)
for k in range(n):
if np.isinf(w[k]): continue
(node[k], weight[k]) = roots_jacobi(n_node, 0, -beta[k])
(node[n], weight[n]) = roots_legendre(n_node)
self.prevertex = np.empty(n, dtype=np.complex)
self.prevertex[[0, -1, -2]] = [1, 0, np.inf]
self.vertex = w
self.angle = beta
self.node = node
self.weight = weight
self.map = np.vectorize(self.map)
self.invmap = np.vectorize(self.invmap)
if n == 3:
self.C = (w[0] - w[-1]) / self.zquad(0, 1, n - 1, 0)
return
k_rat = p.FiniteEdge([0, n - 3])
k_fix = p.InfVertex([1, n - 3])
y = np.zeros(n - 3)
f = np.empty_like(y)
def scfun(y):
z = self.yztran(y)
C = (w[0] - w[-1]) / self.zquad(0, 1, n - 1, 0)
i = 0
for k in k_rat:
q = self.zquad(z[k], z[k + 1], k, k + 1)
f[i] = np.abs(w[k + 1] - w[k]) - np.abs(C * q)
i += 1
for k in k_fix:
zm = (z[k - 1] + z[k + 1] + (z[k + 1] - z[k - 1]) * 1j) / 2
q = self.zquad(z[k - 1], zm, k - 1) - self.zquad(
z[k + 1], zm, k + 1)
q = w[k + 1] - w[k - 1] - C * q
f[i] = np.real(q)
f[i + 1] = np.imag(q)
i += 2
self.C = C
return f
sol = root(scfun, y, method=method, options={'disp': True})
self.yztran(sol.x)
#1.double the legendre
n = 5
position_x, weight_x = np.polynomial.legendre.leggauss(n)
position_y = position_x
weight_y = weight_x
sum = 0
for i in range(len(position_x)):
for j in range(len(position_y)):
sum += weight_x[i] * weight_y[j] * f(position_x[i], position_y[j])
print(sum)
#2.half jacobi- half legendre - such as i get it
position_y, weight_y = sp.roots_jacobi(n, 0, 0)
sum = 0
for i in range(len(position_x)):
for j in range(len(position_y)):
sum += weight_x[i] * weight_y[j] * f(position_x[i], position_y[j]) * position_x[i]
print(sum)
#5.
import sobol_seq
def f(x, y):
return math.exp(-(x * x + y * y))
answer = 0.25 * math.pi * sp.erf(1) ** 2
n = 2000
arr = sobol_seq.i4_sobol_generate(2, n)
def JacobiGQ(alpha, beta, N):
"""Compute N'th order Gauss quadrature points and weights"""
x, w = sci.roots_jacobi(N, alpha, beta)
return x, w