NAG Library Function Document

nag_quad_md_simplex (d01pac)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     ndim – IntegerInput

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

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

2:     vert[$\mathit{dim}$] – 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:     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:     ndim – IntegerInput

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

2:     x[ndim] – const doubleInput

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

3:     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 3.2.1.1 in the Essential Introduction).

4:     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:     maxord – IntegerInput

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

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

6:     finvls[maxord] – 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:     esterr – double *Output

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

8:     comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

9:     fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

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_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.

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)