# NAG CL Interfaced01pac (md_​simplex)

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## 1Purpose

d01pac returns a sequence of approximations to the integral of a function over a multidimensional simplex, together with an error estimate for the last approximation.

## 2Specification

 #include
void  d01pac (Integer ndim, double vert[],
 double (*f)(Integer ndim, const double x[], Nag_Comm *comm),
Integer *minord, Integer maxord, double finvls[], double *esterr, Nag_Comm *comm, NagError *fail)
The function may be called by the names: d01pac or nag_quad_md_simplex.

## 3Description

d01pac computes a sequence of approximations ${\mathbf{finvls}}\left[\mathit{j}-1\right]$, for $\mathit{j}={\mathbf{minord}}+1,\dots ,{\mathbf{maxord}}$, to an integral
 $∫Sf(x1,x2,…,xn)dx1dx2⋯dxn$
where $S$ is an $n$-dimensional simplex defined in terms of its $n+1$ vertices. ${\mathbf{finvls}}\left[j-1\right]$ is an approximation which will be exact (except for rounding errors) whenever the integrand is a polynomial of total degree $2j-1$ or less.
The type of method used has been described in Grundmann and Moller (1978), and is implemented in an extrapolated form using the theory from de Doncker (1979).

## 4References

de Doncker E (1979) New Euler–Maclaurin Expansions and their application to quadrature over the $s$-dimensional simplex Math. Comput. 33 1003–1018
Grundmann A and Moller H M (1978) Invariant integration formulas for the $n$-simplex by combinatorial methods SIAM J. Numer. Anal. 15 282–290

## 5Arguments

1: $\mathbf{ndim}$Integer Input
On entry: $n$, the number of dimensions of the integral.
Constraint: ${\mathbf{ndim}}\ge 2$.
2: $\mathbf{vert}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array vert must be at least $\left(2×\left({\mathbf{ndim}}+1\right)\right)×\left({\mathbf{ndim}}+1\right)$.
where ${\mathbf{VERT}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{vert}}\left[\left(j-1\right)×\left({\mathbf{ndim}}+1\right)+i-1\right]$.
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 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 d01pac is made with ${\mathbf{minord}}>0$. In particular ${\mathbf{VERT}}\left(n+1,2n+2\right)$ contains the volume of the simplex.
3: $\mathbf{f}$function, supplied by the user External Function
f must return the value of the integrand at a given point.
The specification of f is:
 double f (Integer ndim, const double x[], Nag_Comm *comm)
1: $\mathbf{ndim}$Integer Input
On entry: $n$, the number of dimensions of the integral.
2: $\mathbf{x}\left[{\mathbf{ndim}}\right]$const double Input
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 f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d01pac you may allocate memory and initialize these pointers with various quantities for use by f when called from d01pac (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01pac. If your code inadvertently does return any NaNs or infinities, 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 d01pac.
Constraint: ${\mathbf{minord}}\ge 0$.
On exit: ${\mathbf{minord}}={\mathbf{maxord}}$.
5: $\mathbf{maxord}$Integer Input
On entry: the highest order of approximation to the integral to be computed.
Constraint: ${\mathbf{maxord}}>{\mathbf{minord}}$.
6: $\mathbf{finvls}\left[{\mathbf{maxord}}\right]$double Input/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 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 3.1.1 in the Introduction to the NAG Library CL Interface).
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

An absolute error estimate is output through the argument esterr.

## 8Parallelism and Performance

d01pac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d01pac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The running time for d01pac will usually be dominated by the time used to evaluate the integrand f. The maximum time that could be used by d01pac will be approximately given by
 $T×(maxord+ndim)! (maxord-1)!(ndim+1)!$
where $T$ is the time needed for one call of f.

## 10Example

This example demonstrates the use of the function with the integral
 $∫01 ∫01-x ∫01-x-y exp(x+y+z) cos(x+y+z) dz dy dx = 14 .$

### 10.1Program Text

Program Text (d01pace.c)

### 10.2Program Data

Program Data (d01pace.d)

### 10.3Program Results

Program Results (d01pace.r)