D01FBF (PDF version)
D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

D01FBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

D01FBF computes an estimate of a multidimensional integral (from 1 to 20 dimensions), given the analytic form of the integrand and suitable Gaussian weights and abscissae.

2  Specification

FUNCTION D01FBF ( NDIM, NPTVEC, LWA, WEIGHT, ABSCIS, FUN, IFAIL)
REAL (KIND=nag_wp) D01FBF
INTEGER  NDIM, NPTVEC(NDIM), LWA, IFAIL
REAL (KIND=nag_wp)  WEIGHT(LWA), ABSCIS(LWA), FUN
EXTERNAL  FUN

3  Description

D01FBF approximates a multidimensional integral by evaluating the summation
i1=1l1 w 1,i1 i2=1l2 w2,i2 in=1ln wn,in f x 1 , i1 , x 2 , i2 ,, x n , in  
given the weights wj,ij and abscissae xj,ij for a multidimensional product integration rule (see Davis and Rabinowitz (1975)). The number of dimensions may be anything from 1 to 20.
The weights and abscissae for each dimension must have been placed in successive segments of the arrays WEIGHT and ABSCIS; for example, by calling D01BCF or D01TBF once for each dimension using a quadrature formula and number of abscissae appropriate to the range of each xj and to the functional dependence of f on xj.
If normal weights are used, the summation will approximate the integral
w1x1w2x2wnxnf x1,x2,,xn dxndx2dx1  
where wjx is the weight function associated with the quadrature formula chosen for the jth dimension; while if adjusted weights are used, the summation will approximate the integral
fx1,x2,,xndxndx2dx1.  
You must supply a subroutine to evaluate
fx1,x2,,xn  
at any values of x1,x2,,xn within the range of integration.

4  References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press

5  Arguments

1:     NDIM – INTEGERInput
On entry: n, the number of dimensions of the integral.
Constraint: 1NDIM20.
2:     NPTVECNDIM – INTEGER arrayInput
On entry: NPTVECj must specify the number of points in the jth dimension of the summation, for j=1,2,,n.
3:     LWA – INTEGERInput
On entry: the dimension of the arrays WEIGHT and ABSCIS as declared in the (sub)program from which D01FBF is called.
Constraint: LWANPTVEC1+NPTVEC2++NPTVECNDIM.
4:     WEIGHTLWA – REAL (KIND=nag_wp) arrayInput
On entry: must contain in succession the weights for the various dimensions, i.e., WEIGHTk contains the ith weight for the jth dimension, with
k=NPTVEC1+NPTVEC2++NPTVECj-1+i.  
5:     ABSCISLWA – REAL (KIND=nag_wp) arrayInput
On entry: must contain in succession the abscissae for the various dimensions, i.e., ABSCISk contains the ith abscissa for the jth dimension, with
k=NPTVEC1+NPTVEC2++NPTVECj-1+i.  
6:     FUN – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
FUN must return the value of the integrand f at a specified point.
The specification of FUN is:
FUNCTION FUN ( NDIM, X)
REAL (KIND=nag_wp) FUN
INTEGER  NDIM
REAL (KIND=nag_wp)  X(NDIM)
1:     NDIM – INTEGERInput
On entry: n, the number of dimensions of the integral.
2:     XNDIM – REAL (KIND=nag_wp) arrayInput
On entry: the coordinates of the point at which the integrand f must be evaluated.
FUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01FBF is called. Arguments denoted as Input must not be changed by this procedure.
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=1
On entry,NDIM<1,
orNDIM>20,
orLWA<NPTVEC1+NPTVEC2++NPTVECNDIM.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

The accuracy of the computed multidimensional sum depends on the weights and the integrand values at the abscissae. If these numbers vary significantly in size and sign then considerable accuracy could be lost. If these numbers are all positive, then little accuracy will be lost in computing the sum.

8  Parallelism and Performance

D01FBF is not threaded in any implementation.

9  Further Comments

The total time taken by D01FBF will be proportional to
T×NPTVEC1×NPTVEC2××NPTVECNDIM,  
where T is the time taken for one evaluation of FUN.

10  Example

This example evaluates the integral
120-1 x1x2x3 6 x4+2 8e-2x2e-0.5x32dx4dx3dx2dx1  
using adjusted weights. The quadrature formulae chosen are:
Four points are sufficient in each dimension, as this integral is in fact a product of four one-dimensional integrals, for each of which the chosen four-point formula is exact.

10.1  Program Text

Program Text (d01fbfe.f90)

10.2  Program Data

None.

10.3  Program Results

Program Results (d01fbfe.r)


D01FBF (PDF version)
D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016