convolution module#

class convolution.convolution#

Bases: object

Computes left or right convolution operator of two functions, with singularity too.

left_convolue_at_x(fq, gq, degree, alpha=0)#

Computes the left convolution at x1, defined as, h(x) := int_{0}^{x1} (x1-t)^alpha f(x1-t) g(t) dt.

Parameters#

x1float: upper terminal of the integral from 0 to x1, x1 in [0,1].
fqarray,float: values of the function f at quadtaure points.
gqarray,float: values of the function g at quadtaure points.
alphafloat: corresponding to singulairty, which also a paramter of Jacobi polynomial, alpha > -1
degreeint: degree of Jacobi polynomial

x1 in [0,1], where x1 is the upper terminal of the integral defined as,

h(x1) := int_{0}^{x1} (x1-t)^alpha f(x1-t) g(t) dt.

right_convolue_at_x(fq, gq, degree, beta=0)#

Computes the right convolution at x1, defined as, h(x) := int_{x1}^{1} (t-x1)^alpha f(t-x1) g(t) dt.

x1float: lower terminal of the integral from x1 to 1, x1 in [0,1].
fqarray,float: values of the function f at quadtaure points.
gqarray,float: values of the function g at quadtaure points.
betafloat: corresponding to singulairty, which also a paramter of Jacobi polynomial, beta > -1
degreeint: degree of Jacobi polynomial

x1 in [0,1], where x1 is the lower terminal of the integral defined as,

h(x1) := int_{x1}^{1} (t-x1)^alpha f(t-x1) g(t) dt.

left_convolue(fq, gq, degree, alpha=0)#

Computes the left convolution at x1, defined as, h(x1) := int_{0}^{x1} (x1-t)^alpha f(x1-t) g(t) dt; for every x1 in x.

xfloat, array: A 1D array of points, as upper terminal of the integral from 0 to x, x in [0,1]. Note, the convolution is compted at supplied x
fqarray,float: values of the function f at quadtaure points in the invertal [0, b], 0 < b <= 1
gqarray,float: values of the function g at quadtaure points in the invertal [0, b], 0 < b <= 1
alphafloat: corresponding to singulairty, which also a paramter of Jacobi polynomial, alpha > -1
degreeint: degree of Jacobi polynomial

V: array, float: Value of left convolution performed from 0 to x1, for every x1 in x.

class convolution.test#

Bases: object