TOMS468
Algorithm for Automatic Numerical Integration Over a Finite Interval

TOMS468 is a FORTRAN77 program, using single precision arithmetic, which implements ACM TOMS algorithm 468, for estimating the integral of a function over a finite interval.

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.

Related Data and Programs:

INTLIB is a FORTRAN90 library which includes many routines for estimating the integral of a function.

QUADPACK is a FORTRAN90 library which includes many routines for estimating the integral of a function, including weight functions, singularities, and infinite intervals.

QUADRULE is a FORTRAN90 library which defines many simple quadrature schemes.

STROUD is a FORTRAN90 library which defines quadrature schemes for a number of geometrically interesting regions.

TEST_INT is a FORTRAN90 library which contains routines which define integrands suitable for testing integration software.

TOMS351 is a FORTRAN77 library which carries out Romberg integration of a function.

TOMS379 is a FORTRAN77 library which carries out adaptive integration of a function.

Usage:

call quad ( a, b, result, k, epsil, npts, icheck, f )

where A and B are the limits of the interval, RESULT is an array of length 8, which on output will contain the integral estimate in entry RESULT(K), K is set on output to indicate the appropriate entry of RESULT, EPSIL is a requested relative accuracy, NPTS is returned with the number of function evaluations, ICHECK is an error flag which is 0 for normal return, and F is the name of a user-supplied function which evaluates the integrand at an arbitrary point X.

Reference:

1. TNL Patterson,
   The Optimal Addition of Points to Quadrature Formulae,
   Mathematics of Computation,
   Volume 22, 1968, pages 847-856.
2. TNL Patterson,
   Algorithm 468: Algorithm for Automatic Numerical Integration Over a Finite Interval,
   Communications of the ACM,
   November 1973, Volume 16, Number 11, pages 694-699.

Source Code:

• toms468.f, the source code.
• toms468.csh, commands to compile the source code.

Examples and Tests:

• toms468_prb.f, a sample calling program.
• toms468_prb.csh, commands to compile and run the sample program.
• toms468_prb.out, the output from a run of the sample program.

List of Routines:

• QUAD is the basis quadrature routine.
• QSUB uses QUAD in a nonadaptive subdivision scheme.
• QSUBA uses QUAD in an adaptive subdivision scheme.

You can go up one level to the FORTRAN77 source codes.