NAG FL Interface
d01paf (md_​simplex)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{ndim}$ – Integer Input

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

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

2: $\mathbf{vert}\left(\mathbf{ldvert},\mathbf{sdvert}\right)$ – Real (Kind=nag_wp) array Input/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: $\mathbf{ldvert}$ – Integer Input

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: $\mathbf{sdvert}$ – Integer Input

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: $\mathbf{f}$ – real (Kind=nag_wp) Function, supplied by the user. External Procedure

f must return the value of the integrand at a given point.

The specification of f is:

Fortran Interface

Function f ( ndim, x) Real (Kind=nag_wp) :: f Integer, Intent (In) :: ndim Real (Kind=nag_wp), Intent (In) :: x(ndim)

C Header Interface

double  f_ (const Integer *ndim, const double x[])

C++ Header Interface

#include <nag.h> extern "C" { double  f_ (const Integer &ndim, const double x[]) }

1: $\mathbf{ndim}$ – Integer Input

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

2: $\mathbf{x}\left(\mathbf{ndim}\right)$ – Real (Kind=nag_wp) array Input

On entry: the coordinates of the point at which the integrand $f$ must be evaluated.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01paf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01paf. If your code inadvertently does return any NaNs or infinities, d01paf is likely to produce unexpected results.

6: $\mathbf{minord}$ – Integer Input/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: $\mathbf{maxord}$ – Integer Input

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

Constraint: $\mathbf{maxord}>\mathbf{minord}$.

8: $\mathbf{finvls}\left(\mathbf{maxord}\right)$ – Real (Kind=nag_wp) array Input/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: $\mathbf{esterr}$ – Real (Kind=nag_wp) Output

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

10: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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 argument, 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, $\mathbf{ldvert}=〈\mathit{\text{value}}〉$ and $\mathbf{ndim}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldvert}\ge \mathbf{ndim}+1$.

On entry, $\mathbf{maxord}=〈\mathit{\text{value}}〉$ and $\mathbf{minord}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxord}>\mathbf{minord}$.

On entry, $\mathbf{minord}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{minord}\ge 0$.

On entry, $\mathbf{ndim}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ndim}\ge 2$.

On entry, $\mathbf{sdvert}=〈\mathit{\text{value}}〉$ and $\mathbf{ndim}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{sdvert}\ge 2\times \left(\mathbf{ndim}+1\right)$.

$\mathbf{ifail}=2$

The volume of the simplex integration region is too large or too small to be represented on the machine.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results