NAG CL Interface
d01pac (md_simplex)

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.

2 Specification

#include <nag.h>

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.

3 Description

d01pac computes a sequence of approximations

finvls [j - 1]

, for

j = minord + 1, \dots, maxord

, to an integral

\int_{S} f (x_{1}, x_{2}, \dots, x_{n}) d x_{1} d x_{2} \dots d x_{n}

where

S

is an

n

-dimensional simplex defined in terms of its

n + 1

vertices.

finvls [j - 1]

is an approximation which will be exact (except for rounding errors) whenever the integrand is a polynomial of total degree

2 j - 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).

4 References

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

5 Arguments

1: $ndim$ – Integer Input

On entry:

n

, the number of dimensions of the integral.

Constraint:

ndim \geq 2

2: $vert [\dim]$ – double Input/Output

Note: the dimension, dim, of the array vert must be at least

(2 \times (ndim + 1)) \times (ndim + 1)

where

VERT (i, j)

appears in this document, it refers to the array element

vert [(j - 1) \times (ndim + 1) + i - 1]

On entry:

VERT (i, j)

must be set to the

j

th component of the

i

th vertex for the simplex integration region, for

i = 1, 2, \dots, n + 1

and

j = 1, 2, \dots, n

. If

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

minord > 0

. In particular

VERT (n + 1, 2 n + 2)

contains the volume of the simplex.

3: $f$ – 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: $ndim$ – Integer Input

On entry:

n

, the number of dimensions of the integral.

2: $x [ndim]$ – const double Input

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 f.

user – double *
iuser – Integer *
p – Pointer: 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: $minord$ – Integer * Input/Output

On entry: must specify the highest order of the approximations currently available in the array finvls.

$minord = 0$: Indicates an initial call.
$minord > 0$: Indicates that $finvls [0], finvls [1], \dots, finvls [minord - 1]$ have already been computed in a previous call of d01pac.

Constraint:

minord \geq 0

On exit:

minord = maxord

5: $maxord$ – Integer Input

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

Constraint:

maxord > minord

6: $finvls [maxord]$ – double Input/Output

On entry: if

minord > 0

finvls [0], finvls [1], \dots, finvls [minord - 1]

must contain approximations to the integral previously computed by d01pac.

On exit: contains these values unchanged, and the newly computed values

finvls [minord], finvls [minord + 1], \dots, finvls [maxord - 1]

finvls [j - 1]

is an approximation to the integral of polynomial degree

2 j - 1

7: $esterr$ – double * Output

On exit: an absolute error estimate for

finvls [maxord - 1]

8: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

9: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $minord = ⟨ value ⟩$ .
Constraint: $minord \geq 0$ .

On entry, $ndim = ⟨ value ⟩$ .
Constraint: $ndim \geq 2$ .
NE_INT_2: On entry, $maxord = ⟨ value ⟩$ and $minord = ⟨ value ⟩$ .
Constraint: $maxord > minord$ .
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.

7 Accuracy

An absolute error estimate is output through the argument esterr.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

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.

9 Further 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 \times \frac{(maxord + ndim)!}{(maxord - 1)! (ndim + 1)!}

where

T

is the time needed for one call of f.

10 Example

This example demonstrates the use of the function with the integral

\int_{0}^{1} \int_{0}^{1 - x} \int_{0}^{1 - x - y} \exp (x + y + z) \cos (x + y + z) d z d y d x = \frac{1}{4} .

NAG CL Interfaced01pac (md_​simplex)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d01pac (md_simplex)