NAG Library Routine Document

D01PAF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NDIM – INTEGERInput

On entry: $n$, the number of dimensions of the integral.

Constraint: $\mathbf{NDIM}\ge 2$.

2:     VERT(LDVERT,SDVERT) – REAL (KIND=nag_wp) arrayInput/Output

On entry: $\mathbf{VERT}\left(\mathit{i},\mathit{j}\right)$ must be set to the $\mathit{j}$th component of the $\mathit{i}$th vertex for the simplex integration region, for $\mathit{i}=1,2,\dots ,n+1$ and $\mathit{j}=1,2,\dots ,n$. If $\mathbf{MINORD}>0$, VERT must be unchanged since the previous call of D01PAF.

On exit: these values are unchanged. The rest of the array VERT is used for workspace and contains information to be used if another call of D01PAF is made with $\mathbf{MINORD}>0$. In particular $\mathbf{VERT}\left(n+1,2n+2\right)$ contains the volume of the simplex.

3:     LDVERT – INTEGERInput

On entry: the first dimension of the array VERT as declared in the (sub)program from which D01PAF is called.

Constraint: $\mathbf{LDVERT}\ge \mathbf{NDIM}+1$.

4:     SDVERT – INTEGERInput

On entry: the second dimension of the array VERT as declared in the (sub)program from which D01PAF is called.

Constraint: $\mathbf{SDVERT}\ge 2\times \left(\mathbf{NDIM}+1\right)$.

5:     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, X) REAL (KIND=nag_wp) FUNCTN INTEGER  NDIM REAL (KIND=nag_wp)  X(NDIM)

1:     NDIM – INTEGERInput

On entry: $n$, the number of dimensions of the integral.

2:     X(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 D01PAF is called. Parameters denoted as Input must not be changed by this procedure.

6:     MINORD – INTEGERInput/Output

On entry: must specify the highest order of the approximations currently available in the array FINVLS. $\mathbf{MINORD}=0$ indicates an initial call; $\mathbf{MINORD}>0$ indicates that $\mathbf{FINVLS}\left(1\right),\mathbf{FINVLS}\left(2\right),\dots ,\mathbf{FINVLS}\left(\mathbf{MINORD}\right)$ have already been computed in a previous call of D01PAF.

Constraint: $\mathbf{MINORD}\ge 0$.

On exit: $\mathbf{MINORD}=\mathbf{MAXORD}$.

7:     MAXORD – INTEGERInput

On entry: the highest order of approximation to the integral to be computed.

Constraint: $\mathbf{MAXORD}>\mathbf{MINORD}$.

8:     FINVLS(MAXORD) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if $\mathbf{MINORD}>0$, $\mathbf{FINVLS}\left(1\right),\mathbf{FINVLS}\left(2\right),\dots ,\mathbf{FINVLS}\left(\mathbf{MINORD}\right)$ must contain approximations to the integral previously computed by D01PAF.

On exit: contains these values unchanged, and the newly computed values $\mathbf{FINVLS}\left(\mathbf{MINORD}+1\right),\mathbf{FINVLS}\left(\mathbf{MINORD}+2\right),\dots ,\mathbf{FINVLS}\left(\mathbf{MAXORD}\right)$. $\mathbf{FINVLS}\left(j\right)$ is an approximation to the integral of polynomial degree $2j-1$.

9:     ESTERR – REAL (KIND=nag_wp)Output

On exit: an absolute error estimate for $\mathbf{FINVLS}\left(\mathbf{MAXORD}\right)$.

10:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{NDIM}<2$,
or	$\mathbf{LDVERT}<\mathbf{NDIM}+1$,
or	$\mathbf{SDVERT}<2\times \left(\mathbf{NDIM}+1\right)$,
or	$\mathbf{MINORD}<0$,
or	$\mathbf{MAXORD}\le \mathbf{MINORD}$.

$\mathbf{IFAIL}=2$

The volume of the simplex integration region (computed as a determinant by a routine from Chapter F03) is too large or too small to be representable in the machine.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d01pafe.f90)

9.2  Program Data

9.3  Program Results

Program Results (d01pafe.r)