jburkardt

CAUCHY_PRINCIPAL_VALUE\ Estimate Singular Integrals {#cauchy_principal_value-estimate-singular-integrals align=”center”} ===========================

CAUCHY_PRINCIPAL_VALUE is a C++ library which uses Gauss-Legendre quadrature to estimate the Cauchy Principal Value of certain singular integrals.

The singular integrals to be considered will have the form:

        Integral ( a <= t <= b ) f(t) / ( t - x ) dt

The Cauchy Principal Value is defined as

        CPV = limit ( s --> x ) Integral ( a <= t <= s ) f(t) / ( t - x ) dt
            + limit ( x <-- s ) Integral ( s <= t <= b ) f(t) / ( t - x ) dt

We suppose that our singular integral is posed on an interval that is symmetric with respect to the location of the singularity:

        Integral ( x-d <= t <= x+d ) f(t) / ( t - x ) dt

and we propose to estimate the integral using a Gauss-Legendre rule of even order N:

        CPV approx sum ( 1 <= i <= N ) w(i) * f(xi(i)*d+x) / xi(i)

where xi(i) and w(i) are the points and weights, respectively, of the Gauss-Legendre rule.

Licensing: {#licensing align=”center”}

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

CAUCHY_PRINCIPAL_VALUE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs: {#related-data-and-programs align=”center”}

QUADRULE, a C++ library which defines quadrature rules for approximating an integral over a 1D domain.

Reference: {#reference align=”center”}

1. Julian Noble,\ Gauss-Legendre Principal Value Integration,\ Computing in Science and Engineering,\ Volume 2, Number 1, January-February 2000, pages 92-95.

Source Code: {#source-code align=”center”}

• cpv.cpp, the source code.
• cpv.hpp, the include file.

Examples and Tests: {#examples-and-tests align=”center”}

• cpv_prb.cpp, a sample calling program.
• cpv_prb_output.txt, the output file.

List of Routines: {#list-of-routines align=”center”}

• CPV estimates the Cauchy Principal Value of an integral.
• LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last modified on 01 April 2015.