TOMS453
Gaussian Quadrature Formulas for Bromwich's Integral

TOMS453 is a FORTRAN90 library which implements ACM TOMS algorithm 453, which computes the abscissas and weights of a Gaussian quadrature formula of given order for Bromwich's integral.

The Bromwich integral is the inverse of the Laplace transform.

While the text of many ACM TOMS algorithms is available online through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html, most of the early algorithms are not available. This is one of them. I typed it in.

Usage:

        call bromin ( n, s, tol, xr, xi, wr, wi, eps, ier )
      

N

the order of the rule;

S

the parameter in the integral;

TOL

an error tolerance;

XR, XI

the real and imaginary parts of the abscissas;

WR, WI

the real and imaginary parts of the weights;

EPS

the relatve accuracy estimate;

IER

the error flag.

Languages:

TOMS453 is available in a FORTRAN90 version.

Reference:

1. Robert Piessens,
   Some Aspects of Gaussian Quadrature Formulas for the Numerical Inversion of the Laplace Transform,
   The Computer Journal,
   November 1971, Volume 14, pages 433-435.
2. Robert Piessens,
   Algorithm 453: Gaussian Quadrature Formulas for Bromwich's Integral,
   Communications of the ACM,
   August 1973, Volume 16, Number 8, pages 486-487.

Source Code:

• toms453.f90, the source code.

Examples and Tests:

• toms453_test.f90, a sample calling program.
• toms453_test.txt, the output from a run of the sample program.

List of Routines:

• BROMIN computes the abscissas and weights for a Gaussian quadrature formula for Bromwich's integral.
• DGAMMA calculates the gamma function for a real argument X.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN90 source codes.