nag_quad_md_numth_vec (d01gdc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_quad_md_numth_vec (d01gdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_quad_md_numth_vec (d01gdc) calculates an approximation to a definite integral in up to 20 dimensions, using the Korobov–Conroy number theoretic method.

2  Specification

#include <nag.h>
#include <nagd01.h>
void  nag_quad_md_numth_vec (Integer ndim,
void (*vecfun)(Integer ndim, const double x[], double fv[], Integer m, Nag_Comm *comm),
void (*vecreg)(Integer ndim, const double x[], Integer j, double c[], double d[], Integer m, Nag_Comm *comm),
Integer npts, double vk[], Integer nrand, Nag_Boolean transform, double *res, double *err, Nag_Comm *comm, NagError *fail)

3  Description

nag_quad_md_numth_vec (d01gdc) calculates an approximation to the integral
I= c1 d1 cn dn f x1,,xn dxn dx1 (1)
using the Korobov–Conroy number theoretic method (see Korobov (1957), Korobov (1963) and Conroy (1967)). The region of integration defined in (1) is such that generally ci  and di  may be functions of x1 , x2 ,, xi-1 , for i= 2 , 3 ,, n , with c1  and d1  constants. The integral is first of all transformed to an integral over the n-cube 0,1 n  by the change of variables
xi = ci + di - ci yi ,   i= 1 , 2 ,, n .
The method then uses as its basis the number theoretic formula for the n-cube, 0,1 n :
01 01 g x1,,xn dxn dx1 = 1p k=1p g k a1p ,, k anp - E (2)
where x  denotes the fractional part of x, a1 ,, an  are the so-called optimal coefficients, E is the error, and p is a prime integer. (It is strictly only necessary that p be relatively prime to all a1 ,, an  and is in fact chosen to be even for some cases in Conroy (1967).) The method makes use of properties of the Fourier expansion of g x1,,xn  which is assumed to have some degree of periodicity. Depending on the choice of a1 ,, an  the contributions from certain groups of Fourier coefficients are eliminated from the error, E. Korobov shows that a1 ,, an  can be chosen so that the error satisfies
ECK p-α ln αβ p (3)
where α and C are real numbers depending on the convergence rate of the Fourier series, β is a constant depending on n, and K is a constant depending on α and n. There are a number of procedures for calculating these optimal coefficients. Korobov imposes the constraint that
a1 = 1   and   ai = ai-1 mod p (4)
and gives a procedure for calculating the argument, a, to satisfy the optimal conditions.
In this function the periodisation is achieved by the simple transformation
xi = yi2 3-2yi ,   i= 1 , 2 ,, n .
More sophisticated periodisation procedures are available but in practice the degree of periodisation does not appear to be a critical requirement of the method.
An easily calculable error estimate is not available apart from repetition with an increasing sequence of values of p which can yield erratic results. The difficulties have been studied by Cranley and Patterson (1976) who have proposed a Monte–Carlo error estimate arising from converting (2) into a stochastic integration rule by the inclusion of a random origin shift which leaves the form of the error (3) unchanged; i.e., in the formula (2), k ai p  is replaced by αi+k ai p , for i= 1 , 2 ,, n , where each αi , is uniformly distributed over 0,1 . Computing the integral for each of a sequence of random vectors α allows a ‘standard error’ to be estimated.
This function provides built-in sets of optimal coefficients, corresponding to six different values of p. Alternatively, the optimal coefficients may be supplied by you. Functions nag_quad_md_numth_coeff_prime (d01gyc) and nag_quad_md_numth_coeff_2prime (d01gzc) compute the optimal coefficients for the cases where p is a prime number or p is a product of two primes, respectively.

4  References

Conroy H (1967) Molecular Shroedinger equation VIII. A new method for evaluting multi-dimensional integrals J. Chem. Phys. 47 5307–5318
Cranley R and Patterson T N L (1976) Randomisation of number theoretic methods for mulitple integration SIAM J. Numer. Anal. 13 904–914
Korobov N M (1957) The approximate calculation of multiple integrals using number theoretic methods Dokl. Acad. Nauk SSSR 115 1062–1065
Korobov N M (1963) Number Theoretic Methods in Approximate Analysis Fizmatgiz, Moscow

5  Arguments

1:     ndimIntegerInput
On entry: n, the number of dimensions of the integral.
Constraint: 1ndim20.
2:     vecfunfunction, supplied by the userExternal Function
vecfun must evaluate the integrand at a specified set of points.
The specification of vecfun is:
void  vecfun (Integer ndim, const double x[], double fv[], Integer m, Nag_Comm *comm)
1:     ndimIntegerInput
On entry: n, the number of dimensions of the integral.
2:     x[m×ndim]const doubleInput
Note: where Xi,j appears in this document, it refers to the array element x[j-1×m+i-1].
On entry: the coordinates of the m points at which the integrand must be evaluated. Xi,j contains the jth coordinate of the ith point.
3:     fv[m]doubleOutput
On exit: fv[i-1] must contain the value of the integrand of the ith point, i.e., fv[i-1]=f Xi,1,Xi,2,,Xi,ndim, for i=1,2,,m.
4:     mIntegerInput
On entry: the number of points m at which the integrand is to be evaluated.
5:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to vecfun.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_quad_md_numth_vec (d01gdc) you may allocate memory and initialize these pointers with various quantities for use by vecfun when called from nag_quad_md_numth_vec (d01gdc) (see Section 3.2.1 in the Essential Introduction).
3:     vecregfunction, supplied by the userExternal Function
vecreg must evaluate the limits of integration in any dimension for a set of points.
The specification of vecreg is:
void  vecreg (Integer ndim, const double x[], Integer j, double c[], double d[], Integer m, Nag_Comm *comm)
1:     ndimIntegerInput
On entry: n, the number of dimensions of the integral.
2:     x[m×ndim]const doubleInput
Note: where Xi,j appears in this document, it refers to the array element x[j-1×m+i-1].
On entry: for i=1,2,,m, Xi,1, Xi,2,,Xi,j-1 contain the current values of the first j-1 coordinates of the ith point, which may be used if necessary in calculating the m values of cj and dj.
3:     jIntegerInput
On entry: the index j for which the limits of the range of integration are required.
4:     c[m]doubleOutput
On exit: c[i-1] must be set to the lower limit of the range for Xi,j, for i=1,2,,m.
5:     d[m]doubleOutput
On exit: d[i-1] must be set to the upper limit of the range for Xi,j, for i=1,2,,m.
6:     mIntegerInput
On entry: the number of points m at which the limits of integration must be specified.
7:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to vecreg.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_quad_md_numth_vec (d01gdc) you may allocate memory and initialize these pointers with various quantities for use by vecreg when called from nag_quad_md_numth_vec (d01gdc) (see Section 3.2.1 in the Essential Introduction).
4:     nptsIntegerInput
On entry: the Korobov rule to be used. There are two alternatives depending on the value of npts.
(i) 1npts6.
In this case one of six preset rules is chosen using 2129, 5003, 10007, 20011, 40009 or 80021 points depending on the respective value of npts being 1, 2, 3, 4, 5 or 6.
(ii) npts>6.
npts is the number of actual points to be used with corresponding optimal coefficients supplied in the array vk.
Constraint: npts1.
5:     vk[ndim]doubleInput/Output
On entry: if npts>6, vk must contain the n optimal coefficients (which may be calculated using nag_quad_md_numth_coeff_prime (d01gyc) or nag_quad_md_numth_coeff_2prime (d01gzc)).
If npts6, vk need not be set.
On exit: if npts>6, vk is unchanged.
If npts6, vk contains the n optimal coefficients used by the preset rule.
6:     nrandIntegerInput
On entry: the number of random samples to be generated (generally a small value, say 3 to 5, is sufficient). The estimate, res, of the value of the integral returned by the function is then the average of nrand calculations with different random origin shifts. If npts>6, the total number of integrand evaluations will be nrand×npts. If 1npts6, then the number of integrand evaluations will be nrand×p, where p is the number of points corresponding to the six preset rules. For reasons of efficiency, these values are calculated a number at a time in vecfun.
Constraint: nrand1.
7:     transformNag_BooleanInput
On entry: indicates whether the periodising transformation is to be used.
transform=Nag_TRUE
The transformation is to be used.
transform=Nag_FALSE
The transformation is to be suppressed (to cover cases where the integrand may already be periodic or where you want to specify a particular transformation in the definition of vecfun).
Suggested value: transform=Nag_TRUE.
8:     resdouble *Output
On exit: the approximation to the integral I.
9:     errdouble *Output
On exit: the standard error as computed from nrand sample values. If nrand=1, then err contains zero.
10:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, ndim=value.
Constraint: 1ndim20.
On entry, npts must be at least 1: npts=value.
On entry, nrand must be at least 1: nrand=value.
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

If nrand>1, an estimate of the absolute standard error is given by the value, on exit, of err.

8  Further Comments

vecfun and vecreg must calculate the integrand and limits of integration at a set of points. For some problems the amount of time spent in these two functions, which must be supplied by you, may account for a significant part of the total computation time.
The time taken will be approximately proportional to nrand×p, where p is the number of points used, but may depend significantly on the efficiency of the code provided by you in vecfun and vecreg.
The exact values of res and err on return will depend (within statistical limits) on the sequence of random numbers generated within nag_quad_md_numth_vec (d01gdc) by calls to nag_rand_basic (g05sac). Separate runs will produce identical answers.

9  Example

This example calculates the integral
01 01 01 01 cos 0.5+2 x1 + x2 + x3 + x4 - 4 dx1 dx2 dx3 dx4 .

9.1  Program Text

Program Text (d01gdce.c)

9.2  Program Data

Program Data (d01gdce.d)

9.3  Program Results

Program Results (d01gdce.r)


nag_quad_md_numth_vec (d01gdc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012