gauss_convolution_quad module#
Created on Mon Jul 22 15:26:31 2024
@author: Pavan Pranjivan Mehta
- class gauss_convolution_quad.quad_pts_weights#
Bases:
objectComputes Quadrature Points and weights for convolution operator, with singularity too.
It uses Newton-Cotes formula for compute the weights.
- singular_jacobi_normalised(alpha=0)#
Computes the quadauture weights (and points) for convolution with normalised power-law singularity and an analtyic function,
frac{1}{Gamma (alpha +1)} int_{0}^{1} (1 - x)^{alpha} f(x) dx = sum_{i=0}^{n} w_i f(q_i),
where, |\alpha| < 1, and f is 2n-th degree polynomial.
Parameters#
- degreeint
max degree of polyomial
- alphafloat
|\alpha| < 1. The default is 0.
Returns#
- qfloat, array
quadrutre points
- wfloat, array
weights.
- singular_jacobi(alpha=0)#
Computes the quadauture weights (and points) for convolution with power-law singularity and an analtyic function,
int_{0}^{1} (1 - x)^{alpha} f(x) dx = sum_{i=0}^{n} w_i f(q_i),
where, |\alpha| < 1, and f is 2n-th degree polynomial.
Parameters#
- degreeint
max degree of polyomial
- alphafloat
|\alpha| < 1. The default is 0.
Returns#
- qfloat, array
quadrutre points
- wfloat, array
weights.
- singular_jacobi_arbitary(degree, alpha=0)#
Computes the quadauture weights (and points) for convolution with power-law singularity and an analtyic function, over the arbitray interval, [0, x], x in (0, 1]
int_{0}^{x} (x - s)^{alpha} f(s) dx = sum_{i=0}^{n} w_i f(q_i),
where, |\alpha| < 1, and f is 2n-th degree polynomial.
Parameters#
- xfloat
upper limit of integral over the domian [0, x], x in (0, 1]
- degreeint
max degree of polyomial
- alphafloat
|\alpha| < 1. The default is 0.
Returns#
- qfloat, array
quadrutre points
- wfloat, array
weights.
Reference#
function call to ‘singular_jacobi_normalised’
- singular_jacobi_normalised_arbitary(degree, alpha=0)#
Computes the quadauture weights (and points) for convolution with normailised power-law singularity and an analtyic function, over the arbitray interval, [0, x], x in (0, 1]
frac{1} {Gamma(1 + alpha)} int_{0}^{x} (x - s)^{alpha} f(s) dx = sum_{i=0}^{n} w_i f(q_i),
where, |\alpha| < 1, and f is 2n-th degree polynomial.
Parameters#
- xfloat
upper limit of integral over the domian [0, x], x in (0, 1]
- degreeint
max degree of polyomial
- alphafloat
|\alpha| < 1. The default is 0.
Returns#
- qfloat, array
quadrutre points
- wfloat, array
weights.
Reference#
function call to ‘singular_jacobi’