# 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.

### 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.

Last revised on 11 July 2008.