TOMS453
Gaussian Quadrature Formulas for Bromwich's Integral

TOMS453 is a FORTRAN77 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 sometimes called the Fourier-Mellin integral or the Mellin integral. It is the inverse of the Laplace transform. Thus, the quadrature rule, applied to a complex function G(z), which is the Laplace transform of the real function f(t), can be used to approximate the value of f(t) at a point.

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.

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 FORTRAN77 version and 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.f, the source code.
• toms453.sh, commands to compile the source code.

Examples and Tests:

• toms453_prb.f, a sample calling program.
• toms453_prb.sh, commands to compile and run the sample program.
• toms453_prb_output.txt, the output file.

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 FORTRAN77 source codes.