def test_la_roots():
root_func = lambda a: lambda n, mu: orth.la_roots(n, a, mu)
eval_func = lambda a: lambda n, x: orth.eval_genlaguerre(n, a, x)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 5)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 25, atol=1e-13)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 100, atol=1e-12)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 5)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 25, atol=1e-13)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 100, atol=1e-13)
verify_gauss_quad(root_func(1), eval_func(1), 5)
verify_gauss_quad(root_func(1), eval_func(1), 25, atol=1e-13)
verify_gauss_quad(root_func(1), eval_func(1), 100, atol=1e-13)
verify_gauss_quad(root_func(10), eval_func(10), 5)
verify_gauss_quad(root_func(10), eval_func(10), 25, atol=1e-13)
verify_gauss_quad(root_func(10), eval_func(10), 100, atol=1e-12)
verify_gauss_quad(root_func(50), eval_func(50), 5)
verify_gauss_quad(root_func(50), eval_func(50), 25, atol=1e-13)
verify_gauss_quad(root_func(50), eval_func(50), 100, atol=1e-13)
x, w = orth.la_roots(5, 2, False)
y, v, m = orth.la_roots(5, 2, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
assert_raises(ValueError, orth.la_roots, 0, 2)
assert_raises(ValueError, orth.la_roots, 3.3, 2)
assert_raises(ValueError, orth.la_roots, 3, -1.1)
def test_la_roots():
root_func = lambda a: lambda n, mu: orth.la_roots(n, a, mu)
eval_func = lambda a: lambda n, x: orth.eval_genlaguerre(n, a, x)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 5)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 25, atol=1e-13)
verify_gauss_quad(root_func(-0.5), eval_func(-0.5), 100, atol=1e-12)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 5)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 25, atol=1e-13)
verify_gauss_quad(root_func(0.1), eval_func(0.1), 100, atol=1e-13)
verify_gauss_quad(root_func(1), eval_func(1), 5)
verify_gauss_quad(root_func(1), eval_func(1), 25, atol=1e-13)
verify_gauss_quad(root_func(1), eval_func(1), 100, atol=1e-13)
verify_gauss_quad(root_func(10), eval_func(10), 5)
verify_gauss_quad(root_func(10), eval_func(10), 25, atol=1e-13)
verify_gauss_quad(root_func(10), eval_func(10), 100, atol=1e-12)
verify_gauss_quad(root_func(50), eval_func(50), 5)
verify_gauss_quad(root_func(50), eval_func(50), 25, atol=1e-13)
verify_gauss_quad(root_func(50), eval_func(50), 100, atol=1e-13)
x, w = orth.la_roots(5, 2, False)
y, v, m = orth.la_roots(5, 2, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
assert_raises(ValueError, orth.la_roots, 0, 2)
assert_raises(ValueError, orth.la_roots, 3.3, 2)
assert_raises(ValueError, orth.la_roots, 3, -1.1)
def quadrature(self,add=10):
"""base points and weights
number of points = size of basis + add
"""
b,w =so.la_roots(self.n+add,0.)
for i in range(len(b)): w[i]=w[i]*cmath.exp(b[i])
return b/(2.*self.s)+self.o, w
def test_la_roots():
rootf = lambda a: lambda n, mu: orth.la_roots(n, a, mu)
evalf = lambda a: lambda n, x: orth.eval_genlaguerre(n, a, x)
weightf = lambda a: 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=1e-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=1e-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 = orth.la_roots(5, 2, False)
y, v, m = orth.la_roots(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, orth.la_roots, 0, 2)
assert_raises(ValueError, orth.la_roots, 3.3, 2)
assert_raises(ValueError, orth.la_roots, 3, -1.1)
def test_la_roots():
rootf = lambda a: lambda n, mu: orth.la_roots(n, a, mu)
evalf = lambda a: lambda n, x: orth.eval_genlaguerre(n, a, x)
weightf = lambda a: 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=1e-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=1e-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 = orth.la_roots(5, 2, False)
y, v, m = orth.la_roots(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, orth.la_roots, 0, 2)
assert_raises(ValueError, orth.la_roots, 3.3, 2)
assert_raises(ValueError, orth.la_roots, 3, -1.1)
def la_roots(n, alpha=0, method='newton'):
'''
Returns the roots (x) of the nth order generalized (associated) Laguerre
polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature
over [0,inf] with weighting function exp(-x) x**alpha with alpha > -1.
Parameters
----------
n : integer
number of roots
method : 'newton' or 'eigenvalue'
uses Newton Raphson to find zeros of the Laguerre polynomial (Fast)
or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and
weights, respectively.
Returns
-------
x : ndarray
roots
w : ndarray
weights
Example
-------
>>> import numpy as np
>>> [x,w] = h_roots(10)
>>> np.sum(x*w)
1.3352627380516791e-17
See also
--------
qrule, gaussq
References
----------
[1] Golub, G. H. and Welsch, J. H. (1969)
'Calculation of Gaussian Quadrature Rules'
Mathematics of Computation, vol 23,page 221-230,
[2]. Stroud and Secrest (1966), 'gaussian quadrature formulas',
prentice-hall, Englewood cliffs, n.j.
'''
if alpha <= -1:
raise ValueError('alpha must be greater than -1')
if not method.startswith('n'):
return ort.la_roots(n, alpha)
return _la_roots_newton(n, alpha)
def RadialKinetic(N,L,h=1.0):
T = zeros((N,N),'d')
a = 2*L+2
h2 = h*h
#q = Points(N)
q,w = la_roots(N,a)
q = q.real
for i in range(N):
for j in range(N):
if i == j:
T[i,i] = p2jj(N,a,q[i])-L*(L+1)*qm2jj(N,a,q[i])
else:
T[i,j] = pow(-1,i+j)*(p2jk(N,a,q[i],q[j])\
-L*(L+1)*qm2jk(N,a,q[i],q[j]))
return -0.5*T/h2
def RadialKinetic_BH0(N,L):
# Baye + Heenen eq 3.17
a = 2*L+2
x,w = la_roots(N,a)
x = x.real
T = zeros((N,N),'d')
for i in range(N):
xi = x[i]
Sii = Laguerre_S(i,i,x)
T[i,i] = 0.25*pow((a+1.)/xi,2)+Sii
for j in range(i):
xj = x[j]
Sij = Laguerre_S(i,j,x)
T[i,j] = T[j,i] = pow(-1,i-j)*(0.5*(a+1.)*(xi+xj)/pow(xi*xj,1.5)
+Sij)
return x,0.5*T
def __init__(self, order, a=-0.5, options={}):
r"""Initialize a new quadrature rule.
:param order: The order :math:`k` of the Gauss-Laguerre quadrature.
From theory we know that a Gauss quadrature rule
of order :math:`k` is exact for polynomials up to
degree :math:`2 k - 1`.
:param a: The parameter :math:`a > -1` of the generalized Gauss-Laguerre quadrature.
This value defaults to `0` resulting in classical Gauss-Laguerre quadrature.
:raise: :py:class:`ValueError` if order ``order`` is not 1 or above.
.. warning::
This quadrature is made specifically for our needs. Therefore the default
values of :math:`\alpha` is not 0 but set to :math:`-\frac{1}{2}`. There
is hope that this will give less confusion and hidden errors.
"""
# The space dimension of the quadrature rule.
self._dimension = 1
# The order of the Gauss-Laguerre quadrature.
self._order = order
self._a = a
# Quadrature has to have at least a single (node,weight) pair.
if not self._order > 0:
raise ValueError("Quadrature rule has to be of order 1 at least.")
# Set the options
self._options = options
nodes, weights = la_roots(self._order, self._a)
# The number of nodes in this quadrature rule
self._number_nodes = nodes.size
# The quadrature nodes \gamma.
self._nodes = real(nodes).reshape((1,self._number_nodes))
# The quadrature weights \omega.
self._weights = real(weights).reshape((1,self._number_nodes))
def __init__(self, order, a=-0.5, options={}):
r"""Initialize a new quadrature rule.
:param order: The order :math:`k` of the Gauss-Laguerre quadrature.
From theory we know that a Gauss quadrature rule
of order :math:`k` is exact for polynomials up to
degree :math:`2 k - 1`.
:param a: The parameter :math:`a > -1` of the generalized Gauss-Laguerre quadrature.
This value defaults to `0` resulting in classical Gauss-Laguerre quadrature.
:raise: :py:class:`ValueError` if order ``order`` is not 1 or above.
.. warning:: This quadrature is made specifically for our needs. Therefore the default
values of :math:`\alpha` is not 0 but set to :math:`-\frac{1}{2}`. There
is hope that this will give less confusion and hidden errors.
"""
# Quadrature has to have at least a single (node,weight) pair.
if not order > 0:
raise ValueError("Quadrature rule has to be of order 1 at least.")
# The space dimension of the quadrature rule.
self._dimension = 1
# The order of the Gauss-Laguerre quadrature.
self._order = order
self._a = a
# Set the options
self._options = options
nodes, weights = la_roots(self._order, self._a)
# The number of nodes in this quadrature rule
self._number_nodes = nodes.size
# The quadrature nodes \gamma.
self._nodes = real(nodes).reshape((1, self._number_nodes))
# The quadrature weights \omega.
self._weights = real(weights).reshape((1, self._number_nodes))
def la_roots(n, alpha=0, method='newton'):
'''
Returns the roots (x) of the nth order generalized (associated) Laguerre
polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature over
[0,inf] with weighting function exp(-x) x**alpha with alpha > -1.
Parameters
----------
n : integer
number of roots
method : 'newton' or 'eigenvalue'
uses Newton Raphson to find zeros of the Laguerre polynomial (Fast)
or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and
weights, respectively.
Returns
-------
x : ndarray
roots
w : ndarray
weights
Example
-------
>>> import numpy as np
>>> [x,w] = h_roots(10)
>>> np.sum(x*w)
-5.2516042729766621e-19
See also
--------
qrule, gaussq
References
----------
[1] Golub, G. H. and Welsch, J. H. (1969)
'Calculation of Gaussian Quadrature Rules'
Mathematics of Computation, vol 23,page 221-230,
[2]. Stroud and Secrest (1966), 'gaussian quadrature formulas',
prentice-hall, Englewood cliffs, n.j.
'''
if alpha <= -1:
raise ValueError('alpha must be greater than -1')
if not method.startswith('n'):
return ort.la_roots(n, alpha)
else:
max_iter = 10
releps = 3e-14
C = [9.084064e-01, 5.214976e-02, 2.579930e-03, 3.986126e-03]
# Initial approximations to the roots go into z.
anu = 4.0 * n + 2.0 * alpha + 2.0
rhs = np.arange(4 * n - 1, 2, -4) * np.pi / anu
r3 = rhs ** (1. / 3)
r2 = r3 ** 2
theta = r3 * (C[0] + r2 * (C[1] + r2 * (C[2] + r2 * C[3])))
z = anu * np.cos(theta) ** 2
dz = zeros(len(z))
L = zeros((3, len(z)))
Lp = zeros((1, len(z)))
pp = zeros((1, len(z)))
k0 = 0
kp1 = 1
k = slice(len(z))
for _its in xrange(max_iter):
#%Newton's method carried out simultaneously on the roots.
L[k0, k] = 0.
L[kp1, k] = 1.
for jj in xrange(1, n + 1):
# Loop up the recurrence relation to get the Laguerre
# polynomials evaluated at z.
km1 = k0
k0 = kp1
kp1 = np.mod(kp1 + 1, 3)
L[kp1, k] = ((2 * jj - 1 + alpha - z[k]) * L[k0, k] - (jj - 1 + alpha) * L[km1, k]) / jj
#end
#%L now contains the desired Laguerre polynomials.
#%We next compute pp, the derivatives with a standard
#% relation involving the polynomials of one lower order.
Lp[k] = L[k0, k]
pp[k] = (n * L[kp1, k] - (n + alpha) * Lp[k]) / z[k]
dz[k] = L[kp1, k] / pp[k]
z[k] = z[k] - dz[k]# % Newton?s formula.
#%k = find((abs(dz) > releps.*z))
if not np.any(abs(dz) > releps):
break
else:
warnings.warn('too many iterations!')
x = z
w = -np.exp(sp.gammaln(alpha + n) - sp.gammaln(n)) / (pp * n * Lp)
return x, w
def la_roots(n, alpha=0, method='newton'):
'''
Returns the roots (x) of the nth order generalized (associated) Laguerre
polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature
over [0,inf] with weighting function exp(-x) x**alpha with alpha > -1.
Parameters
----------
n : integer
number of roots
method : 'newton' or 'eigenvalue'
uses Newton Raphson to find zeros of the Laguerre polynomial (Fast)
or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and
weights, respectively.
Returns
-------
x : ndarray
roots
w : ndarray
weights
Example
-------
>>> import numpy as np
>>> [x,w] = h_roots(10)
>>> np.sum(x*w)
-5.2516042729766621e-19
See also
--------
qrule, gaussq
References
----------
[1] Golub, G. H. and Welsch, J. H. (1969)
'Calculation of Gaussian Quadrature Rules'
Mathematics of Computation, vol 23,page 221-230,
[2]. Stroud and Secrest (1966), 'gaussian quadrature formulas',
prentice-hall, Englewood cliffs, n.j.
'''
if alpha <= -1:
raise ValueError('alpha must be greater than -1')
if not method.startswith('n'):
return ort.la_roots(n, alpha)
else:
max_iter = 10
releps = 3e-14
C = [9.084064e-01, 5.214976e-02, 2.579930e-03, 3.986126e-03]
# Initial approximations to the roots go into z.
anu = 4.0 * n + 2.0 * alpha + 2.0
rhs = np.arange(4 * n - 1, 2, -4) * np.pi / anu
r3 = rhs ** (1. / 3)
r2 = r3 ** 2
theta = r3 * (C[0] + r2 * (C[1] + r2 * (C[2] + r2 * C[3])))
z = anu * np.cos(theta) ** 2
dz = zeros(len(z))
L = zeros((3, len(z)))
Lp = zeros((1, len(z)))
pp = zeros((1, len(z)))
k0 = 0
kp1 = 1
k = slice(len(z))
for _its in xrange(max_iter):
# Newton's method carried out simultaneously on the roots.
L[k0, k] = 0.
L[kp1, k] = 1.
for jj in xrange(1, n + 1):
# Loop up the recurrence relation to get the Laguerre
# polynomials evaluated at z.
km1 = k0
k0 = kp1
kp1 = np.mod(kp1 + 1, 3)
L[kp1, k] = ((2 * jj - 1 + alpha - z[k]) * L[
k0, k] - (jj - 1 + alpha) * L[km1, k]) / jj
# end
# L now contains the desired Laguerre polynomials.
# We next compute pp, the derivatives with a standard
# relation involving the polynomials of one lower order.
Lp[k] = L[k0, k]
pp[k] = (n * L[kp1, k] - (n + alpha) * Lp[k]) / z[k]
dz[k] = L[kp1, k] / pp[k]
z[k] = z[k] - dz[k] # % Newton?s formula.
# k = find((abs(dz) > releps.*z))
if not np.any(abs(dz) > releps):
break
else:
warnings.warn('too many iterations!')
x = z
w = -np.exp(sp.gammaln(alpha + n) - sp.gammaln(n)) / (pp * n * Lp)
return x, w
def Vmat(N,L):
q = la_roots(N,2*L+2)[0].real
return diag([-1/qi for qi in q])
def quadrature(self,n): return so.la_roots(n,0) class ChebyshevPolynomial(OrthogonalPolynomial):
