NAG Library Function Document
nag_quad_md_numth_vec (d01gdc)
1
Purpose
nag_quad_md_numth_vec (d01gdc) calculates an approximation to a definite integral in up to 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 |
(*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
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
and
may be functions of
, for
, with
and
constants. The integral is first of all transformed to an integral over the
-cube
by the change of variables
The method then uses as its basis the number theoretic formula for the
-cube,
:
where
denotes the fractional part of
,
are the so-called optimal coefficients,
is the error, and
is a prime integer. (It is strictly only necessary that
be relatively prime to all
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
which is assumed to have some degree of periodicity. Depending on the choice of
the contributions from certain groups of Fourier coefficients are eliminated from the error,
. Korobov shows that
can be chosen so that the error satisfies
where
and
are real numbers depending on the convergence rate of the Fourier series,
is a constant depending on
, and
is a constant depending on
and
. There are a number of procedures for calculating these optimal coefficients. Korobov imposes the constraint that
and gives a procedure for calculating the argument,
, to satisfy the optimal conditions.
In this function the periodisation is achieved by the simple transformation
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
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),
is replaced by
, for
, where each
, is uniformly distributed over
. Computing the integral for each of a sequence of random vectors
allows a ‘standard error’ to be estimated.
This function provides built-in sets of optimal coefficients, corresponding to six different values of
. 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
is a prime number or
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:
– IntegerInput
-
On entry: , the number of dimensions of the integral.
Constraint:
.
- 2:
– function, 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:
– IntegerInput
-
On entry: , the number of dimensions of the integral.
- 2:
– const doubleInput
-
Note: where appears in this document, it refers to the array element
.
On entry: the coordinates of the points at which the integrand must be evaluated. contains the th coordinate of the th point.
- 3:
– doubleOutput
-
On exit: must contain the value of the integrand of the th point, i.e., , for .
- 4:
– IntegerInput
-
On entry: the number of points at which the integrand is to be evaluated.
- 5:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
vecfun.
- user – double *
- iuser – Integer *
- p – Pointer
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 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: vecfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_quad_md_numth_vec (d01gdc). If your code inadvertently
does return any NaNs or infinities,
nag_quad_md_numth_vec (d01gdc) is likely to produce unexpected results.
- 3:
– function, 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:
– IntegerInput
-
On entry: , the number of dimensions of the integral.
- 2:
– const doubleInput
-
Note: where appears in this document, it refers to the array element
.
On entry: for , , contain the current values of the first coordinates of the th point, which may be used if necessary in calculating the values of and .
- 3:
– IntegerInput
-
On entry: the index for which the limits of the range of integration are required.
- 4:
– doubleOutput
-
On exit: must be set to the lower limit of the range for , for .
- 5:
– doubleOutput
-
On exit: must be set to the upper limit of the range for , for .
- 6:
– IntegerInput
-
On entry: the number of points at which the limits of integration must be specified.
- 7:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
vecreg.
- user – double *
- iuser – Integer *
- p – Pointer
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 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: vecreg should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_quad_md_numth_vec (d01gdc). If your code inadvertently
does return any NaNs or infinities,
nag_quad_md_numth_vec (d01gdc) is likely to produce unexpected results.
- 4:
– IntegerInput
-
On entry: the Korobov rule to be used. There are two alternatives depending on the value of
npts.
(i) |
.
In this case one of six preset rules is chosen using , , , , or points depending on the respective value of npts being , , , , or . |
(ii) |
.
npts is the number of actual points to be used with corresponding optimal coefficients supplied in the array vk. |
Constraint:
.
- 5:
– doubleInput/Output
-
On entry: if
,
vk must contain the
optimal coefficients (which may be calculated using
nag_quad_md_numth_coeff_prime (d01gyc) or
nag_quad_md_numth_coeff_2prime (d01gzc)).
If
,
vk need not be set.
On exit: if
,
vk is unchanged.
If
,
vk contains the
optimal coefficients used by the preset rule.
- 6:
– IntegerInput
-
On entry: the number of random samples to be generated (generally a small value, say
to
, 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
, the total number of integrand evaluations will be
. If
, the number of integrand evaluations will be
, where
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:
.
- 7:
– Nag_BooleanInput
-
On entry: indicates whether the periodising transformation is to be used.
- The transformation is to be used.
- 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:
.
- 8:
– double *Output
-
On exit: the approximation to the integral .
- 9:
– double *Output
-
On exit: the standard error as computed from
nrand sample values. If
,
err contains zero.
- 10:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 11:
– 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_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 had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry,
npts must be at least
:
.
On entry,
nrand must be at least
:
.
- 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
If
, an estimate of the absolute standard error is given by the value, on exit, of
err.
8
Parallelism and Performance
nag_quad_md_numth_vec (d01gdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.
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
, where
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.
10
Example
This example calculates the integral
10.1
Program Text
Program Text (d01gdce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (d01gdce.r)