d01fc attempts to evaluate a multidimensional integral (up to

15

dimensions), with constant and finite limits, to a specified relative accuracy, using an adaptive subdivision strategy.

Syntax

C#
public static void d01fc( int ndim, double[] a, double[] b, ref int minpts, int maxpts, D01..::..D01FC_FUNCTN functn, double eps, out double acc, out double finval, out int ifail )

Visual Basic
Public Shared Sub d01fc ( _ ndim As Integer, _ a As Double(), _ b As Double(), _ ByRef minpts As Integer, _ maxpts As Integer, _ functn As D01..::..D01FC_FUNCTN, _ eps As Double, _ <OutAttribute> ByRef acc As Double, _ <OutAttribute> ByRef finval As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub d01fc ( _
	ndim As Integer, _
	a As Double(), _
	b As Double(), _
	ByRef minpts As Integer, _
	maxpts As Integer, _
	functn As D01..::..D01FC_FUNCTN, _
	eps As Double, _
	<OutAttribute> ByRef acc As Double, _
	<OutAttribute> ByRef finval As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void d01fc( int ndim, array<double>^ a, array<double>^ b, int% minpts, int maxpts, D01..::..D01FC_FUNCTN^ functn, double eps, [OutAttribute] double% acc, [OutAttribute] double% finval, [OutAttribute] int% ifail )

Visual C++

public:
static void d01fc(
	int ndim, 
	array<double>^ a, 
	array<double>^ b, 
	int% minpts, 
	int maxpts, 
	D01..::..D01FC_FUNCTN^ functn, 
	double eps, 
	[OutAttribute] double% acc, 
	[OutAttribute] double% finval, 
	[OutAttribute] int% ifail
)

F#
static member d01fc : ndim : int * a : float[] * b : float[] * minpts : int byref * maxpts : int * functn : D01..::..D01FC_FUNCTN * eps : float * acc : float byref * finval : float byref * ifail : int byref -> unit

static member d01fc : 
        ndim : int * 
        a : float[] * 
        b : float[] * 
        minpts : int byref * 
        maxpts : int * 
        functn : D01..::..D01FC_FUNCTN * 
        eps : float * 
        acc : float byref * 
        finval : float byref * 
        ifail : int byref -> unit

Parameters

ndim: Type: System..::..Int32
On entry: $n$ , the number of dimensions of the integral.

Constraint: $2 \leq ndim \leq 15$ .

a: Type: array<System..::..Double>[]()[][]
An array of size [ndim]
On entry: the lower limits of integration, $a_{i}$ , for $i = 1, 2, \dots, n$ .

b: Type: array<System..::..Double>[]()[][]
An array of size [ndim]
On entry: the upper limits of integration, $b_{i}$ , for $i = 1, 2, \dots, n$ .

minpts: Type: System..::..Int32%
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 d01fc.

maxpts

Type: System..::..Int32

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

Constraints:

$maxpts \geq minpts$ ;
$maxpts \geq α$ , where $α = 2^{ndim} + 2 \times {ndim}^{2} + 2 \times ndim + 1$ .

functn: Type: NagLibrary..::..D01..::..D01FC_FUNCTN
functn must return the value of the integrand $f$ at a given point.
A delegate of type D01FC_FUNCTN.

eps: Type: System..::..Double
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: $eps > 0.0$ .

acc: Type: System..::..Double%
On exit: the estimated relative error in finval.

finval: Type: System..::..Double%
On exit: the best estimate obtained for the integral.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

d01fc 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 method argument functn, and also set the minimum and maximum acceptable number of calls to functn (in minpts and maxpts).

The method 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 method 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 [Parameters]. If a smaller value is used, and is exhausted in the course of execution, the method switches to a less efficient mode of operation; only if this mode also breaks down is insufficient storage reported.

d01fc is based on the HALF method developed by van Dooren and de Ridder (1976). It uses a different basic rule, described in Genz and Malik (1980).

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

Error Indicators and Warnings

Note: d01fc may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

$ifail = 1$

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.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

A relative error estimate is output through the parameter acc.

Parallelism and Performance

None.

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.

Example

This example estimates the integral

\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{4 z_{1} z_{3}^{2} \exp (2 z_{1} z_{3})}{{(1 + z_{2} + z_{4})}^{2}} d z_{4} d z_{3} d z_{2} d z_{1} = 0.575364 .

The accuracy requested is one part in

10000

Example program (C#): d01fce.cs

Example program results: d01fce.r