D01GDF 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
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
The method then uses as its basis the number theoretic formula for the
n-cube,
0,1
n
:
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
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
and gives a procedure for calculating the parameter,
a, to satisfy the optimal conditions.
In this routine 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
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 routine provides built-in sets of optimal coefficients, corresponding to six different values of
p. Alternatively, the optimal coefficients may be supplied by you. Routines
D01GYF and
D01GZF compute the optimal coefficients for the cases where
p is a prime number or
p is a product of two primes, respectively.
This routine is designed to be particularly efficient on vector processors, although it is very important that you also code
VECFUN and
VECREG efficiently.
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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
If
NRAND>1, an estimate of the absolute standard error is given by the value, on exit, of
ERR.
D01GDF performs the same computation as
D01GCF. However, the interface has been modified so that it can perform more efficiently on machines with vector processing capabilities. In particular,
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 subroutines, which must be supplied by you, may account for a significant part of the total computation time. For this reason it is vital that you consider the possibilities for vectorization in the code supplied for these two subroutines.
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 D01GDF by calls to
G05SAF. Separate runs will produce identical answers.
This example calculates the integral
None.