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 argument,

a

, to satisfy the optimal conditions.

In this function 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 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: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

Constraint:

1 \leq ndim \leq 20

2: $vecfun$ – 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: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

2: $x [m \times ndim]$ – const doubleInput

Note: where

X (i, j)

appears in this document, it refers to the array element

x [(j - 1) \times m + i - 1]

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: $fv [m]$ – doubleOutput

On exit:

fv [i - 1]

must contain the value of the integrand of the

i

th point, i.e.,

fv [i - 1] = 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.

5: $comm$ – 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 2.3.1.1 in How to Use the NAG Library and its Documentation).

3: $vecreg$ – 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: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

2: $x [m \times ndim]$ – const doubleInput

Note: where

X (i, j)

appears in this document, it refers to the array element

x [(j - 1) \times m + i - 1]

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: $c [m]$ – doubleOutput

On exit:

c [i - 1]

must be set to the lower limit of the range for

X (i, j)

, for

i = 1, 2, \dots, m

5: $d [m]$ – doubleOutput

On exit:

d [i - 1]

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.

7: $comm$ – 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 2.3.1.1 in How to Use the NAG Library and its Documentation).

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: $vk [ndim]$ – 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)).

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 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 \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: $transform$ – Nag_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: $res$ – double *Output

On exit: the approximation to the integral

I

9: $err$ – double *Output

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

nrand = 1

, then err contains zero.

10: $comm$ – Nag_Comm *

The NAG communication argument (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 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 $〈value〉$ had an illegal value.
NE_INT: On entry, $ndim = 〈value〉$ .
Constraint: $1 \leq ndim \leq 20$ .

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.

An unexpected error has been triggered by this function. Please contact NAG.
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

nrand > 1

, 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.

9 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 \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 nag_quad_md_numth_vec (d01gdc) by calls to nag_rand_basic (g05sac). Separate runs will produce identical answers.