NAG Library Routine Document

G05YMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05YMF generates a uniformly distributed low-discrepancy sequence as proposed by Sobol, Faure or Niederreiter. It must be preceded by a call to one of the initialization routines G05YLF or G05YNF.

2 Specification

SUBROUTINE G05YMF (

N, RCORD, QUAS, LDQUAS, IREF, IFAIL)

INTEGER	N, RCORD, LDQUAS, IREF( $liref$ ), IFAIL
REAL (KIND=nag_wp)	QUAS(LDQUAS, $tdquas$ )

3 Description

Low discrepancy (quasi-random) sequences are used in numerical integration, simulation and optimization. Like pseudorandom numbers they are uniformly distributed but they are not statistically independent, rather they are designed to give more even distribution in multidimensional space (uniformity). Therefore they are often more efficient than pseudorandom numbers in multidimensional Monte–Carlo methods.

G05YMF generates a set of points

x^{1}, x^{2}, \dots, x^{N}

with high uniformity in the

S

-dimensional unit cube

I^{S} = {[0, 1]}^{S}

Let

G

be a subset of

I^{S}

and define the counting function

S_{N} (G)

as the number of points

x^{i} \in G

. For each

x = (x_{1}, x_{2}, \dots, x_{S}) \in I^{S}

, let

G_{x}

be the rectangular

S

-dimensional region

G_{x} = [0, x_{1}) \times [0, x_{2}) \times \dots \times [0, x_{S})

with volume

x_{1}, x_{2}, \dots, x_{S}

. Then one measure of the uniformity of the points

x^{1}, x^{2}, \dots, x^{N}

is the discrepancy:

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) = \sup_{x \in I^{S}} |S_{N} (G_{x}) - N x_{1}, x_{2}, \dots, x_{S}| .

which has the form

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) \leq C_{S} {(\log N)}^{S} + O ({(\log N)}^{S - 1}) for all N \geq 2 .

The principal aim in the construction of low-discrepancy sequences is to find sequences of points in

I^{S}

with a bound of this form where the constant

C_{S}

is as small as possible.

The type of low-discrepancy sequence generated by G05YMF depends on the initialization routine called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization routine G05YNF was used then the sequence will be scrambled (see Section 3 in G05YNF for details).

4 References

Bratley P and Fox B L (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software 14(1) 88–100

Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376

5 Parameters

Note: the following variables are used in the parameter descriptions:

$idim = IDIM$ , the number of dimensions required, see G05YLF or G05YNF
$liref = LIREF$ , the length of IREF as supplied to the initialization routine G05YLF or G05YNF
$tdquas = N$ if $RCORD = 1$ ; otherwise $tdquas = idim$

1: N – INTEGERInput

On entry: the number of quasi-random numbers required.

Constraint:

N \geq 0

and

N + previous number of generated values \leq 2^{31} - 1

2: RCORD – INTEGERInput

On entry: the order in which the generated values are returned.

Constraint:

RCORD = 1

2

3: QUAS(LDQUAS, $tdquas$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the N quasi-random numbers of dimension idim.

RCORD = 1

QUAS (i, j)

holds the

j

th value for the

i

th dimension.

RCORD = 2

QUAS (i, j)

holds the

i

th value for the

j

th dimension.

4: LDQUAS – INTEGERInput

On entry: the first dimension of the array QUAS as declared in the (sub)program from which G05YMF is called.

Constraints:

if $RCORD = 1$ , $LDQUAS \geq idim$ ;
if $RCORD = 2$ , $LDQUAS \geq N$ .

5: IREF( $liref$ ) – INTEGER arrayCommunication Array

On entry: contains information on the current state of the sequence.

On exit: contains updated information on the state of the sequence.

6: 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, either $N < 0$ or there have been too many calls to the generator.

$IFAIL = 2$: On entry, $RCORD \neq 1$ or $2$ .

$IFAIL = 4$: On entry, $RCORD = 1$ and $LDQUAS < idim$ .

On entry, $RCORD = 2$ and $LDQUAS < N$ .

$IFAIL = 5$: Incorrect initialization. G05YLF must be called prior to G05YMF and IREF must remain unaltered after this call.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example calls G05YLF and G05YMF to estimate the value of the integral

\int_{0}^{1} \dots \int_{0}^{1} \prod_{i = 1}^{s} |4 x_{i} - 2| d x_{1}, d x_{2}, \dots, d x_{s} = 1 .

In this example the number of dimensions

S

is set to

8

NAG Library Routine DocumentG05YMF