NAG Library Routine Document

D01GDF

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

c_{i}

and

d_{i}

may be functions of

x_{1}, x_{2}, \dots, x_{i - 1}

, for

i = 2, 3, \dots, n

, with

c_{1}

and

d_{1}

constants. The integral is first of all transformed to an integral over the

n

-cube

{[0, 1]}^{n}

by the change of variables

x_{i} = c_{i} + (d_{i} - c_{i}) y_{i}, i = 1, 2, \dots, n .

The method then uses as its basis the number theoretic formula for the

n

-cube,

{[0, 1]}^{n}

\int_{0}^{1} \dots \int_{0}^{1} g (x_{1}, \dots, x_{n}) d x_{n} \dots d x_{1} = \frac{1}{p} \sum_{k = 1}^{p} g (\{k \frac{a_{1}}{p}\}, \dots, \{k \frac{a_{n}}{p}\}) - E

(2)

where

\{x\}

denotes the fractional part of

x

a_{1}, \dots, a_{n}

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

a_{1}, \dots, a_{n}

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 (x_{1}, \dots, x_{n})

which is assumed to have some degree of periodicity. Depending on the choice of

a_{1}, \dots, a_{n}

the contributions from certain groups of Fourier coefficients are eliminated from the error,

E

. Korobov shows that

a_{1}, \dots, a_{n}

can be chosen so that the error satisfies

E \leq C K p^{- α} \ln^{α β} 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

a_{1} = 1 and a_{i} = a^{i - 1} (\mod p)

(4)

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

x_{i} = y_{i}^{2} (3 - 2 y_{i}), i = 1, 2, \dots, 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 \frac{a_{i}}{p}\}

is replaced by

\{α_{i} + k \frac{a_{i}}{p}\}

, for

i = 1, 2, \dots, n

, where each

α_{i}

, 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.

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 Parameters

1: NDIM – INTEGERInput

On entry:

n

, the number of dimensions of the integral.

Constraint:

1 \leq NDIM \leq 20

2: VECFUN – SUBROUTINE, supplied by the user.External Procedure

VECFUN must evaluate the integrand at a specified set of points.

The specification of VECFUN is:

SUBROUTINE VECFUN (

NDIM, X, FV, M)

INTEGER	NDIM, M
REAL (KIND=nag_wp)	X(M,NDIM), FV(M)

1: NDIM – INTEGERInput: On entry: $n$ , the number of dimensions of the integral.
2: X(M,NDIM) – REAL (KIND=nag_wp) arrayInput: On entry: the coordinates of the $m$ points at which the integrand must be evaluated. $X (i, j)$ contains the $j$ th coordinate of the $i$ th point.
3: FV(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $FV (i)$ must contain the value of the integrand of the $i$ th point, i.e., $FV (i) = f (X (i, 1), X (i, 2), \dots, X (i, NDIM))$ , for $i = 1, 2, \dots, M$ .
4: M – INTEGERInput: On entry: the number of points $m$ at which the integrand is to be evaluated.

VECFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01GDF is called. Parameters denoted as Input must not be changed by this procedure.

3: VECREG – SUBROUTINE, supplied by the user.External Procedure

VECREG must evaluate the limits of integration in any dimension for a set of points.

The specification of VECREG is:

SUBROUTINE VECREG (

NDIM, X, J, C, D, M)

INTEGER	NDIM, J, M
REAL (KIND=nag_wp)	X(M,NDIM), C(M), D(M)

1: NDIM – INTEGERInput: On entry: $n$ , the number of dimensions of the integral.
2: X(M,NDIM) – REAL (KIND=nag_wp) arrayInput: On entry: for $i = 1, 2, \dots, m$ , $X (i, 1)$ , $X (i, 2), \dots, X (i, j - 1)$ contain the current values of the first $(j - 1)$ coordinates of the $i$ th point, which may be used if necessary in calculating the $m$ values of $c_{j}$ and $d_{j}$ .
3: J – INTEGERInput: On entry: the index $j$ for which the limits of the range of integration are required.
4: C(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $C (i)$ must be set to the lower limit of the range for $X (i, j)$ , for $i = 1, 2, \dots, m$ .
5: D(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $D (i)$ must be set to the upper limit of the range for $X (i, j)$ , for $i = 1, 2, \dots, m$ .
6: M – INTEGERInput: On entry: the number of points $m$ at which the limits of integration must be specified.

VECREG must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01GDF is called. Parameters denoted as Input must not be changed by this procedure.

4: NPTS – INTEGERInput

On entry: the Korobov rule to be used. There are two alternatives depending on the value of NPTS.

(i)	$1 \leq NPTS \leq 6$ . 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:

NPTS \geq 1

5: VK(NDIM) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

NPTS > 6

, VK must contain the

n

optimal coefficients (which may be calculated using D01GYF or D01GZF).

NPTS \leq 6

, VK need not be set.

On exit: if

NPTS > 6

, VK is unchanged.

NPTS \leq 6

, VK contains the

n

optimal coefficients used by the preset rule.

6: NRAND – INTEGERInput

On entry: the number of random samples to be generated (generally a small value, say

3

5

, is sufficient). The estimate, RES, of the value of the integral returned by the routine is then the average of NRAND calculations with different random origin shifts. If

NPTS > 6

, the total number of integrand evaluations will be

NRAND \times NPTS

. If

1 \leq NPTS \leq 6

, then the number of integrand evaluations will be

NRAND \times 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:

NRAND \geq 1

7: ITRANS – INTEGERInput

On entry: indicates whether the periodising transformation is to be used.

$ITRANS = 0$: The transformation is to be used.
$ITRANS \neq 0$: 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:

ITRANS = 0

8: RES – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral

I

9: ERR – REAL (KIND=nag_wp)Output

On exit: the standard error as computed from NRAND sample values. If

NRAND = 1

, then ERR contains zero.

10: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NDIM < 1$ ,
or	$NDIM > 20$ .

$IFAIL = 2$

On entry,

NPTS < 1

$IFAIL = 3$

On entry,

NRAND < 1

7 Accuracy

NRAND > 1

, an estimate of the absolute standard error is given by the value, on exit, of ERR.

8 Further Comments

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 \times 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.