NAG Library Routine Document

G05YNF

+− 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

G05YNF initializes a scrambled quasi-random generator prior to calling G05YJF, G05YKF or G05YMF. It must be preceded by a call to one of the pseudorandom initialization routines G05KFF or G05KGF.

2 Specification

SUBROUTINE G05YNF (

GENID, STYPE, IDIM, IREF, LIREF, ISKIP, NSDIGI, STATE, IFAIL)

INTEGER

GENID, STYPE, IDIM, IREF(LIREF), LIREF, ISKIP, NSDIGI, STATE(*), IFAIL

3 Description

G05YNF selects a quasi-random number generator through the input value of GENID, a method of scrambling through the input value of STYPE and initializes the IREF communication array for use in the routines G05YJF, G05YKF or G05YMF.

Scrambled quasi-random sequences are an extension of standard quasi-random sequences that attempt to eliminate the bias inherent in a quasi-random sequence whilst retaining the low-discrepancy properties. The use of a scrambled sequence allows error estimation of Monte–Carlo results by performing a number of iterates and computing the variance of the results.

This implementation of scrambled quasi-random sequences is based on TOMS Algorithm 823 and details can be found in the accompanying paper, Hong and Hickernell (2003). Three methods of scrambling are supplied; the first a restricted form of Owen's scrambling (Owen (1995)), the second based on the method of Faure and Tezuka (2000) and the last method combines the first two.

Scrambled versions of the Niederreiter sequence and two sets of Sobol sequences are provided. The first Sobol sequence is obtained using

GENID = 1

. The first 10000 direction numbers for this sequence are based on the work of Joe and Kuo (2008). For dimensions greater than 10000 the direction numbers are randomly generated using the pseudorandom generator specified in STATE (see Jäckel (2002) for details). The second Sobol sequence is obtained using

GENID = 2

and referred to in the documentation as ‘Sobol (A659)’. The first 1111 direction numbers for this sequence are based on Algorithm 659 of Bratley and Fox (1988) with the extension proposed by Joe and Kuo (2003). For dimensions greater than 1111 the direction numbers are once again randomly generated. The Niederreiter sequence is obtained by setting

GENID = 3

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

Faure H and Tezuka S (2000) Another random scrambling of digital (t,s)-sequences Monte Carlo and Quasi-Monte Carlo Methods Springer-Verlag, Berlin, Germany (eds K T Fang, F J Hickernell and H Niederreiter)

Hong H S and Hickernell F J (2003) Algorithm 823: implementing scrambled digital sequences ACM Trans. Math. Software 29:2 95–109

Jäckel P (2002) Monte Carlo Methods in Finance Wiley Finance Series, John Wiley and Sons, England

Joe S and Kuo F Y (2003) Remark on Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software (TOMS) 29 49–57

Joe S and Kuo F Y (2008) Constructing Sobol sequences with better two-dimensional projections SIAM J. Sci. Comput. 30 2635–2654

Niederreiter H (1988) Low-discrepancy and low dispersion sequences Journal of Number Theory 30 51–70

Owen A B (1995) Randomly permuted (t,m,s)-nets and (t,s)-sequences Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics 106 Springer-Verlag, New York, NY 299–317 (eds H Niederreiter and P J-S Shiue)

5 Parameters

1: GENID – INTEGERInput

On entry: must identify the quasi-random generator to use.

$GENID = 1$: Sobol generator.
$GENID = 2$: Sobol (A659) generator.
$GENID = 3$: Niederreiter generator.

Constraint:

GENID = 1

2

3

2: STYPE – INTEGERInput

On entry: must identify the scrambling method to use.

$STYPE = 0$: No scrambling. This is equivalent to calling G05YLF.
$STYPE = 1$: Owen like scrambling.
$STYPE = 2$: Faure–Tezuka scrambling.
$STYPE = 3$: Owen and Faure–Tezuka scrambling.

Constraint:

STYPE = 0

1

2

3

3: IDIM – INTEGERInput

On entry: the number of dimensions required.

Constraints:

if $GENID = 1$ , $1 \leq IDIM \leq 50000$ ;
if $GENID = 2$ , $1 \leq IDIM \leq 50000$ ;
if $GENID = 3$ , $1 \leq IDIM \leq 318$ .

4: IREF(LIREF) – INTEGER arrayCommunication Array

On exit: contains initialization information for use by the generator routines G05YJF, G05YKF and G05YMF. IREF must not be altered in any way between initialization and calls of the generator routines.

5: LIREF – INTEGERInput

On entry: the dimension of the array IREF as declared in the (sub)program from which G05YNF is called.

Constraint:

LIREF \geq 32 \times IDIM + 7

6: ISKIP – INTEGERInput

On entry: the number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators.

Constraint:

0 \leq ISKIP \leq 2^{30}

7: NSDIGI – INTEGERInput

On entry: controls the number of digits (bits) to scramble when

GENID = 1

2

, otherwise NSDIGI is ignored. If

NSDIGI < 1

NSDIGI > 30

then all the digits are scrambled.

8: STATE( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

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

9: 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

$IFAIL = 1$: On entry, $GENID \neq 1$ , $2$ or $3$ .

$IFAIL = 2$: $STYPE \neq 0$ , $1$ , $2$ or $3$ .

$IFAIL = 3$

On entry,	$IDIM < 1$ ,
or	IDIM is too large.

$IFAIL = 5$: On entry, LIREF is too small.

$IFAIL = 6$: The value of $ISKIP < 0$ or ISKIP is too large.

$IFAIL = 8$

On entry,

STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

The additional computational cost in using a scrambled quasi-random sequence over a non-scrambled one comes entirely during the initialization. Once G05YNF has been called the computational cost of generating a scrambled sequence and a non-scrambled one is identical.

9 Example

This example calls G05KFF, G05YMF and G05YNF 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,

where

s

, the number of dimensions, is set to

8

NAG Library Routine DocumentG05YNF

+− 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

NAG Library Routine Document

G05YNF