CAUCHY_PRINCIPAL_VALUE
Estimate Singular Integrals

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
      

        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
      

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

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 FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

QUADRULE, a C++ 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:

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

Examples and Tests:

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

List of Routines:

• 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.