CAUCHY_PRINCIPAL_VALUE
Estimate Singular Integrals


CAUCHY_PRINCIPAL_VALUE is a MATLAB 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:

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

Languages:

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:

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

Reference:

  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:

Examples and Tests:

You can go up one level to the MATLAB source codes.


Last modified on 31 March 2015.