NAG Library Routine Document

D01FCF

D01FCF returns an estimate of a multidimensional integral over a hyper-rectangle (i.e., with constant limits), and also an estimate of the relative error. You set the relative accuracy required, return values for the integrand via a routine argument FUNCTN, and also set the minimum and maximum acceptable number of calls to FUNCTN (in MINPTS and MAXPTS).

The routine operates by repeated subdivision of the hyper-rectangular region into smaller hyper-rectangles. In each subregion, the integral is estimated using a seventh-degree rule, and an error estimate is obtained by comparison with a fifth-degree rule which uses a subset of the same points. The fourth differences of the integrand along each coordinate axis are evaluated, and the subregion is marked for possible future subdivision in half along that coordinate axis which has the largest absolute fourth difference.

If the estimated errors, totalled over the subregions, exceed the requested relative error (or if fewer than MINPTS calls to FUNCTN have been made), further subdivision is necessary, and is performed on the subregion with the largest estimated error, that subregion being halved along the appropriate coordinate axis.

The routine will fail if the requested relative error level has not been attained by the time MAXPTS calls to FUNCTN have been made; or, if the amount LENWRK of working storage is insufficient. A formula for the recommended value of LENWRK is given in Section 5. If a smaller value is used, and is exhausted in the course of execution, the routine switches to a less efficient mode of operation; only if this mode also breaks down is insufficient storage reported.

D01FCF is based on the HALF subroutine developed by van Dooren and de Ridder (1976). It uses a different basic rule, described in Genz and Malik (1980).

4 References

Genz A C and Malik A A (1980) An adaptive algorithm for numerical integration over an N-dimensional rectangular region J. Comput. Appl. Math. 6 295–302

van Dooren P and de Ridder L (1976) An adaptive algorithm for numerical integration over an N-dimensional cube J. Comput. Appl. Math. 2 207–217

5 Parameters

1: $NDIM$ – INTEGERInput

On entry:

n

, the number of dimensions of the integral.

Constraint:

2 \leq NDIM \leq 15

2: $A (NDIM)$ – REAL (KIND=nag_wp) arrayInput

On entry: the lower limits of integration,

a_{i}

, for

i = 1, 2, \dots, n

3: $B (NDIM)$ – REAL (KIND=nag_wp) arrayInput

On entry: the upper limits of integration,

b_{i}

, for

i = 1, 2, \dots, n

4: $MINPTS$ – INTEGERInput/Output

On entry: must be set to the minimum number of integrand evaluations to be allowed.

On exit: contains the actual number of integrand evaluations used by D01FCF.

5: $MAXPTS$ – INTEGERInput

On entry: the maximum number of integrand evaluations to be allowed.

Constraints:

$MAXPTS \geq MINPTS$ ;
$MAXPTS \geq α$ , where $α = 2^{NDIM} + 2 \times {NDIM}^{2} + 2 \times NDIM + 1$ .

6: $FUNCTN$ – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

FUNCTN must return the value of the integrand

f

at a given point.

The specification of FUNCTN is:

FUNCTION FUNCTN (

NDIM, Z)

REAL (KIND=nag_wp) FUNCTN

INTEGER	NDIM
REAL (KIND=nag_wp)	Z(NDIM)

1: $NDIM$ – INTEGERInput: On entry: $n$ , the number of dimensions of the integral.
2: $Z (NDIM)$ – REAL (KIND=nag_wp) arrayInput: On entry: the coordinates of the point at which the integrand $f$ must be evaluated.

FUNCTN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01FCF is called. Parameters denoted as Input must not be changed by this procedure.

7: $EPS$ – REAL (KIND=nag_wp)Input

On entry: the relative error acceptable to you. When the solution is zero or very small relative accuracy may not be achievable but you may still set EPS to a reasonable value and check for the error exit

IFAIL = 2

Constraint:

EPS > 0.0

8: $ACC$ – REAL (KIND=nag_wp)Output

On exit: the estimated relative error in FINVAL.

9: $LENWRK$ – INTEGERInput

On entry: the dimension of the array WRKSTR as declared in the (sub)program from which D01FCF is called.

Suggested value: for maximum efficiency,

LENWRK \geq (NDIM + 2) \times (1 + MAXPTS / α)

(see parameter MAXPTS for

α

If LENWRK is less than this, D01FCF will usually run less efficiently and may fail.

Constraint:

LENWRK \geq 2 \times NDIM + 4

10: $WRKSTR (LENWRK)$ – REAL (KIND=nag_wp) arrayWorkspace

11: $FINVAL$ – REAL (KIND=nag_wp)Output

On exit: the best estimate obtained for the integral.

12: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: D01FCF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NDIM < 2$ ,
or	$NDIM > 15$ ,
or	MAXPTS is too small,
or	$LENWRK < 2 \times NDIM + 4$ ,
or	$EPS \leq 0.0$ .

$IFAIL = 2$: MAXPTS was too small to obtain the required relative accuracy EPS. On soft failure, FINVAL and ACC contain estimates of the integral and the relative error, but ACC will be greater than EPS.

$IFAIL = 3$: LENWRK was too small. On soft failure, FINVAL and ACC contain estimates of the integral and the relative error, but ACC will be greater than EPS.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

A relative error estimate is output through the parameter ACC.

8 Parallelism and Performance

D01FCF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

D01FCF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Execution time will usually be dominated by the time taken to evaluate FUNCTN, and hence the maximum time that could be taken will be proportional to MAXPTS.