D120: Adaptive Quadrature for Multiple Integrals over N-Dimensional Rectangular Regions

Author(s): A.C. Genz, A.A. Malik	Library: MATHLIB
Submitter: K.S. Kölbig	Submitted: 15.11.1995
Language: Fortran	Revised:

Subroutine subprograms RADMUL and DADMUL compute, to an attempted specified accuracy, the value of the integral

over an n-dimensional rectangular region, where , ( ) are constants.

On computers other than CDC and Cray, only the double-precision version DADMUL is available. On CDC and Cray computers, only the single-precision version RADMUL is available.

Structure:

SUBROUTINE subprograms
User Entry Names : RADMUL, DADMUL
External References: User-supplied SUBROUTINE subprogram

Usage:

For (type REAL), (type DOUBLE PRECISION),

    CALL tADMUL(F,N,A,B,MINPTS,MAXPTS,EPS,WK,IWK,RESULT,RELERR,NFNEVL,IFAIL)

F

(type according to t) Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program.

N

(INTEGER) Number n of dimensions ( ).

A,B

(type according to t) One-dimensional arrays of length . On entry, and ,
( ), contain the lower and upper limits of integration, respectively. Note that correspond to .

MINPTS

(INTEGER) Minimum number of function evaluations requested. Must not exceed MAXPTS.

MAXPTS

(INTEGER) Maximum number ( 2**N + 2N(N + 1) + 1) of function evaluations to be allowed.

EPS

(type according to t) Specified relative accuracy.

WK

(type according to t) One-dimensional array of length IWK, used as working space.

IWK

(INTEGER) Length ( (2N + 3) * (1 + MAXPTS/(2**N + 2N(N + 1) + 1))/2) of WK.

RESULT

(type according to t) Contains, on exit, an approximate value of the integral .

RELERR

(type according to t) Contains, on exit, an estimation of the relative accuray of RESULT.

NFNEVL

(INTEGER) Contains, on exit, the number of function evaluations performed.

IFAIL

(INTEGER) On exit:

0

Normal exit. . At least MINPTS and at most MAXPTS calls to the function F were performed.

1

MAXPTS is too small for the specified accuracy EPS. RESULT and RELERR contain the values obtainable for the specified value of MAXPTS.

2

IWK is too small for the specified number MAXPTS of function evaluations.
RESULT and RELERR contain the values obtainable for the specified value of IRK.

3

, or , or , or MAXPTS < 2**N + 2N(N + 1) + 1.
RESULT and RELERR are set equal to zero.

The user-supplied FUNCTION subprogram F should be of the form

FUNCTION F(N,X)

DIMENSION X(*)

...

.

RETURN

END

where X and F are of type t.

Method:

An integration rule of degree seven is used together with a certain strategy of subdivision. For a more detailed description of the method see References.

Error handling:

See description of argument IFAIL.

Notes:

1. Multi-dimensional integration is time-consuming. For each rectangular subregion, the routine requires function evaluations. Careful programming of the integrand might result in substantial saving of time.
2. Numerical integration usually works best for smooth functions. Some analysis or suitable transformations of the integral prior to numerical work may contribute to numerical efficiency.

This subroutine is an adapted version of Fortran program ADAPT published in Ref. 1.

References:

1. A.C. Genz and A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980) 295-302.
2. A. van Doren and L. de Ridder, An adaptive algorithm for numerical integration over an n-dimensional cube, J. Comput. Appl. Math. 2 (1976) 207-217.

A copy of the text part of the References is available.