NAG Library Function Document

nag_quad_md_simplex (d01pac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{ndim}$ – IntegerInput

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

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

2:    $\mathbf{vert}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array vert must be at least $\left(2\times \left(\mathbf{ndim}+1\right)\right)\times \left(\mathbf{ndim}+1\right)$.

Where $\mathbf{VERT}\left(i,j\right)$ appears in this document, it refers to the array element $\mathbf{vert}\left[\left(j-1\right)\times \left(\mathbf{ndim}+1\right)+i-1\right]$.

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 nag_quad_md_simplex (d01pac).

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 nag_quad_md_simplex (d01pac) is made with $\mathbf{minord}>0$. In particular $\mathbf{VERT}\left(n+1,2n+2\right)$ contains the volume of the simplex.

3:    $\mathbf{functn}$ – function, supplied by the userExternal Function

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

The specification of functn is:

double  functn (Integer ndim, const double x[], Nag_Comm *comm)

1:    $\mathbf{ndim}$ – IntegerInput

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

2:    $\mathbf{x}\left[\mathbf{ndim}\right]$ – const doubleInput

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

3:    $\mathbf{comm}$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to functn.

user – double *

iuser – Integer *

p – Pointer 

The type Pointer will be void *. Before calling nag_quad_md_simplex (d01pac) you may allocate memory and initialize these pointers with various quantities for use by functn when called from nag_quad_md_simplex (d01pac) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: functn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_quad_md_simplex (d01pac). If your code inadvertently does return any NaNs or infinities, nag_quad_md_simplex (d01pac) is likely to produce unexpected results.

4:    $\mathbf{minord}$ – 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 $\mathbf{finvls}\left[0\right],\mathbf{finvls}\left[1\right],\dots ,\mathbf{finvls}\left[\mathbf{minord}-1\right]$ have already been computed in a previous call of nag_quad_md_simplex (d01pac).

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

On exit: $\mathbf{minord}=\mathbf{maxord}$.

5:    $\mathbf{maxord}$ – IntegerInput

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

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

6:    $\mathbf{finvls}\left[\mathbf{maxord}\right]$ – doubleInput/Output

On entry: if $\mathbf{minord}>0$, $\mathbf{finvls}\left[0\right],\mathbf{finvls}\left[1\right],\dots ,\mathbf{finvls}\left[\mathbf{minord}-1\right]$ must contain approximations to the integral previously computed by nag_quad_md_simplex (d01pac).

On exit: contains these values unchanged, and the newly computed values $\mathbf{finvls}\left[\mathbf{minord}\right],\mathbf{finvls}\left[\mathbf{minord}+1\right],\dots ,\mathbf{finvls}\left[\mathbf{maxord}-1\right]$. $\mathbf{finvls}\left[j-1\right]$ is an approximation to the integral of polynomial degree $2j-1$.

7:    $\mathbf{esterr}$ – double *Output

On exit: an absolute error estimate for $\mathbf{finvls}\left[\mathbf{maxord}-1\right]$.

8:    $\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

9:    $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ACCURACY

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

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

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

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

NE_INT_2

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

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d01pace.c)

10.2

Program Data

10.3

Program Results

Program Results (d01pace.r)