from __future__ import print_function, division
from sympy.core import S, Dummy, pi
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.trigonometric import sin, cos
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.gamma_functions import gamma
from sympy.polys.orthopolys import (legendre_poly, laguerre_poly,
hermite_poly, jacobi_poly)
from sympy.polys.rootoftools import RootOf
from sympy.core.compatibility import range
[docs]def gauss_legendre(n, n_digits):
r"""
Computes the Gauss-Legendre quadrature [1]_ points and weights.
The Gauss-Legendre quadrature approximates the integral:
.. math::
\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `P_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{2}{\left(1-x_i^2\right) \left(P'_n(x_i)\right)^2}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy.integrals.quadrature import gauss_legendre
>>> x, w = gauss_legendre(3, 5)
>>> x
[-0.7746, 0, 0.7746]
>>> w
[0.55556, 0.88889, 0.55556]
>>> x, w = gauss_legendre(4, 5)
>>> x
[-0.86114, -0.33998, 0.33998, 0.86114]
>>> w
[0.34785, 0.65215, 0.65215, 0.34785]
See Also
========
gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule/legendre_rule.html
"""
x = Dummy("x")
p = legendre_poly(n, x, polys=True)
pd = p.diff(x)
xi = []
w = []
for r in p.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((2/((1-r**2) * pd.subs(x, r)**2)).n(n_digits))
return xi, w
[docs]def gauss_laguerre(n, n_digits):
r"""
Computes the Gauss-Laguerre quadrature [1]_ points and weights.
The Gauss-Laguerre quadrature approximates the integral:
.. math::
\int_0^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `L_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy.integrals.quadrature import gauss_laguerre
>>> x, w = gauss_laguerre(3, 5)
>>> x
[0.41577, 2.2943, 6.2899]
>>> w
[0.71109, 0.27852, 0.010389]
>>> x, w = gauss_laguerre(6, 5)
>>> x
[0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983]
>>> w
[0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7]
See Also
========
gauss_legendre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/laguerre_rule/laguerre_rule.html
"""
x = Dummy("x")
p = laguerre_poly(n, x, polys=True)
p1 = laguerre_poly(n+1, x, polys=True)
xi = []
w = []
for r in p.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((r/((n+1)**2 * p1.subs(x, r)**2)).n(n_digits))
return xi, w
[docs]def gauss_hermite(n, n_digits):
r"""
Computes the Gauss-Hermite quadrature [1]_ points and weights.
The Gauss-Hermite quadrature approximates the integral:
.. math::
\int_{-\infty}^{\infty} e^{-x^2} f(x)\,dx \approx
\sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `H_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 \left(H_{n-1}(x_i)\right)^2}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy.integrals.quadrature import gauss_hermite
>>> x, w = gauss_hermite(3, 5)
>>> x
[-1.2247, 0, 1.2247]
>>> w
[0.29541, 1.1816, 0.29541]
>>> x, w = gauss_hermite(6, 5)
>>> x
[-2.3506, -1.3358, -0.43608, 0.43608, 1.3358, 2.3506]
>>> w
[0.00453, 0.15707, 0.72463, 0.72463, 0.15707, 0.00453]
See Also
========
gauss_legendre, gauss_laguerre, gauss_gen_laguerre, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Gauss-Hermite_Quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/hermite_rule/hermite_rule.html
.. [3] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html
"""
x = Dummy("x")
p = hermite_poly(n, x, polys=True)
p1 = hermite_poly(n-1, x, polys=True)
xi = []
w = []
for r in p.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append(((2**(n-1) * factorial(n) * sqrt(pi)) /
(n**2 * p1.subs(x, r)**2)).n(n_digits))
return xi, w
[docs]def gauss_gen_laguerre(n, alpha, n_digits):
r"""
Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights.
The generalized Gauss-Laguerre quadrature approximates the integral:
.. math::
\int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx
\sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of
`L^{\alpha}_n` and the weights `w_i` are given by:
.. math::
w_i = \frac{\Gamma(\alpha+n)}
{n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}
Parameters
==========
n : the order of quadrature
alpha : the exponent of the singularity, `\alpha > -1`
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_gen_laguerre
>>> x, w = gauss_gen_laguerre(3, -S.Half, 5)
>>> x
[0.19016, 1.7845, 5.5253]
>>> w
[1.4493, 0.31413, 0.00906]
>>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5)
>>> x
[0.97851, 2.9904, 6.3193, 11.712]
>>> w
[0.53087, 0.67721, 0.11895, 0.0023152]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html
"""
x = Dummy("x")
p = laguerre_poly(n, x, alpha=alpha, polys=True)
p1 = laguerre_poly(n-1, x, alpha=alpha, polys=True)
p2 = laguerre_poly(n-1, x, alpha=alpha+1, polys=True)
xi = []
w = []
for r in p.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((gamma(alpha+n) /
(n*gamma(n)*p1.subs(x, r)*p2.subs(x, r))).n(n_digits))
return xi, w
[docs]def gauss_chebyshev_t(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the first kind.
The Gauss-Chebyshev quadrature of the first kind approximates the integral:
.. math::
\int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx
\sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `T_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_t
>>> x, w = gauss_chebyshev_t(3, 5)
>>> x
[0.86602, 0, -0.86602]
>>> w
[1.0472, 1.0472, 1.0472]
>>> x, w = gauss_chebyshev_t(6, 5)
>>> x
[0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
>>> w
[0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html
"""
xi = []
w = []
for i in range(1, n+1):
xi.append((cos((2*i-S.One)/(2*n)*S.Pi)).n(n_digits))
w.append((S.Pi/n).n(n_digits))
return xi, w
[docs]def gauss_chebyshev_u(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the second kind.
The Gauss-Chebyshev quadrature of the second kind approximates the
integral:
.. math::
\int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `U_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_u
>>> x, w = gauss_chebyshev_u(3, 5)
>>> x
[0.70711, 0, -0.70711]
>>> w
[0.3927, 0.7854, 0.3927]
>>> x, w = gauss_chebyshev_u(6, 5)
>>> x
[0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
>>> w
[0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_jacobi, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html
"""
xi = []
w = []
for i in range(1, n+1):
xi.append((cos(i/(n+S.One)*S.Pi)).n(n_digits))
w.append((S.Pi/(n+S.One)*sin(i*S.Pi/(n+S.One))**2).n(n_digits))
return xi, w
[docs]def gauss_jacobi(n, alpha, beta, n_digits):
r"""
Computes the Gauss-Jacobi quadrature [1]_ points and weights.
The Gauss-Jacobi quadrature of the first kind approximates the integral:
.. math::
\int_{-1}^1 (1-x)^\alpha (1+x)^\beta f(x)\,dx \approx
\sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of
`P^{(\alpha,\beta)}_n` and the weights `w_i` are given by:
.. math::
w_i = -\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1}
\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}
{\Gamma(n+\alpha+\beta+1)(n+1)!}
\frac{2^{\alpha+\beta}}{P'_n(x_i)
P^{(\alpha,\beta)}_{n+1}(x_i)}
Parameters
==========
n : the order of quadrature
alpha : the first parameter of the Jacobi Polynomial, `\alpha > -1`
beta : the second parameter of the Jacobi Polynomial, `\beta > -1`
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_jacobi
>>> x, w = gauss_jacobi(3, S.Half, -S.Half, 5)
>>> x
[-0.90097, -0.22252, 0.62349]
>>> w
[1.7063, 1.0973, 0.33795]
>>> x, w = gauss_jacobi(6, 1, 1, 5)
>>> x
[-0.87174, -0.5917, -0.2093, 0.2093, 0.5917, 0.87174]
>>> w
[0.050584, 0.22169, 0.39439, 0.39439, 0.22169, 0.050584]
See Also
========
gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_chebyshev_u, gauss_lobatto
References
==========
.. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/jacobi_rule/jacobi_rule.html
.. [3] http://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html
"""
x = Dummy("x")
p = jacobi_poly(n, alpha, beta, x, polys=True)
pd = p.diff(x)
pn = jacobi_poly(n+1, alpha, beta, x, polys=True)
xi = []
w = []
for r in p.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((
- (2*n+alpha+beta+2) / (n+alpha+beta+S.One) *
(gamma(n+alpha+1)*gamma(n+beta+1)) /
(gamma(n+alpha+beta+S.One)*gamma(n+2)) *
2**(alpha+beta) / (pd.subs(x, r) * pn.subs(x, r))).n(n_digits))
return xi, w
[docs]def gauss_lobatto(n, n_digits):
r"""
Computes the Gauss-Lobatto quadrature [1]_ points and weights.
The Gauss-Lobatto quadrature approximates the integral:
.. math::
\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `P'_(n-1)`
and the weights `w_i` are given by:
.. math::
&w_i = \frac{2}{n(n-1) \left[P_{n-1}(x_i)\right]^2},\quad x\neq\pm 1\\
&w_i = \frac{2}{n(n-1)},\quad x=\pm 1
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from sympy.integrals.quadrature import gauss_lobatto
>>> x, w = gauss_lobatto(3, 5)
>>> x
[-1, 0, 1]
>>> w
[0.33333, 1.3333, 0.33333]
>>> x, w = gauss_lobatto(4, 5)
>>> x
[-1, -0.44721, 0.44721, 1]
>>> w
[0.16667, 0.83333, 0.83333, 0.16667]
See Also
========
gauss_legendre,gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi
References
==========
.. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules
.. [2] http://people.math.sfu.ca/~cbm/aands/page_888.htm
"""
x = Dummy("x")
p = legendre_poly(n-1, x, polys=True)
pd = p.diff(x)
xi = []
w = []
for r in pd.real_roots():
if isinstance(r, RootOf):
r = r.eval_rational(S(1)/10**(n_digits+2))
xi.append(r.n(n_digits))
w.append((2/(n*(n-1) * p.subs(x, r)**2)).n(n_digits))
xi.insert(0, -1)
xi.append(1)
w.insert(0, (S(2)/(n*(n-1))).n(n_digits))
w.append((S(2)/(n*(n-1))).n(n_digits))
return xi, w
