NAG Library Routine Document
G05YNF initializes a scrambled quasi-random generator prior to calling G05YJF
. It must be preceded by a call to one of the pseudorandom initialization routines G05KFF
||GENID, STYPE, IDIM, IREF(LIREF), LIREF, ISKIP, NSDIGI, STATE(*), IFAIL
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
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
. 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
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
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)
- 1: GENID – INTEGERInput
: must identify the quasi-random generator to use.
- Sobol generator.
- Sobol (A659) generator.
- Niederreiter generator.
, or .
- 2: STYPE – INTEGERInput
: must identify the scrambling method to use.
- No scrambling. This is equivalent to calling G05YLF.
- Owen like scrambling.
- Faure–Tezuka scrambling.
- Owen and Faure–Tezuka scrambling.
, , or .
- 3: IDIM – INTEGERInput
On entry: the number of dimensions required.
- if , ;
- if , ;
- if , .
- 4: IREF(LIREF) – INTEGER arrayCommunication Array
: contains initialization information for use by the generator routines G05YJF
must not be altered in any way between initialization and calls of the generator routines.
- 5: LIREF – INTEGERInput
: the dimension of the array IREF
as declared in the (sub)program from which G05YNF is called.
- 6: ISKIP – INTEGERInput
On entry: the number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators.
- 7: NSDIGI – INTEGERInput
: controls the number of digits (bits) to scramble when
, otherwise NSDIGI
is ignored. If
then all the digits are scrambled.
- 8: STATE() – INTEGER arrayCommunication Array
the actual argument supplied must be the array STATE
supplied to the initialization routines G05KFF
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
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
On entry, , or .
, , or .
|or||IDIM is too large.|
On entry, LIREF
is too small.
The value of
is too large.
|On entry,||STATE vector was not initialized or has been corrupted.|
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.
This example calls G05KFF
and G05YNF to estimate the value of the integral
, the number of dimensions, is set to
9.1 Program Text
Program Text (g05ynfe.f90)
9.2 Program Data
Program Data (g05ynfe.d)
9.3 Program Results
Program Results (g05ynfe.r)