void	nag_quasi_init_scrambled (Nag_QuasiRandom_Sequence genid, Nag_QuasiRandom_Scrambling stype, Integer idim, Integer iref[], Integer liref, Integer iskip, Integer nsdigi, Integer state[], NagError *fail)

3 Description

nag_quasi_init_scrambled (g05ync) 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 functions nag_quasi_rand_normal (g05yjc), nag_quasi_rand_lognormal (g05ykc) or nag_quasi_rand_uniform (g05ymc).

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 = Nag_QuasiRandom_Sobol

. 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 = Nag_QuasiRandom_SobolA659

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 = Nag_QuasiRandom_Nied

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 Arguments

1: $genid$ – Nag_QuasiRandom_SequenceInput

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

$genid = Nag_QuasiRandom_Sobol$: Sobol generator.
$genid = Nag_QuasiRandom_SobolA659$: Sobol (A659) generator.
$genid = Nag_QuasiRandom_Nied$: Niederreiter generator.

Constraint:

genid = Nag_QuasiRandom_Sobol

Nag_QuasiRandom_SobolA659

Nag_QuasiRandom_Nied

2: $stype$ – Nag_QuasiRandom_ScramblingInput

On entry: must identify the scrambling method to use.

$stype = Nag_NoScrambling$: No scrambling. This is equivalent to calling nag_quasi_init (g05ylc).
$stype = Nag_OwenLike$: Owen like scrambling.
$stype = Nag_FaureTezuka$: Faure–Tezuka scrambling.
$stype = Nag_OwenFaureTezuka$: Owen and Faure–Tezuka scrambling.

Constraint:

stype = Nag_NoScrambling

Nag_OwenLike

Nag_FaureTezuka

Nag_OwenFaureTezuka

3: $idim$ – IntegerInput

On entry: the number of dimensions required.

Constraints:

if $genid = Nag_QuasiRandom_Sobol$ , $1 \leq idim \leq 50000$ ;
if $genid = Nag_QuasiRandom_SobolA659$ , $1 \leq idim \leq 50000$ ;
if $genid = Nag_QuasiRandom_Nied$ , $1 \leq idim \leq 318$ .

4: $iref [liref]$ – IntegerCommunication Array

On exit: contains initialization information for use by the generator functions nag_quasi_rand_normal (g05yjc), nag_quasi_rand_lognormal (g05ykc) and nag_quasi_rand_uniform (g05ymc). iref must not be altered in any way between initialization and calls of the generator functions.

5: $liref$ – IntegerInput

On entry: the dimension of the array iref.

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 = Nag_QuasiRandom_Sobol

Nag_QuasiRandom_SobolA659

, otherwise nsdigi is ignored. If

nsdigi < 1

nsdigi > 30

then all the digits are scrambled.

8: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

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: $fail$ – 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, $idim = 〈value〉$ .
Constraint: $1 \leq idim \leq 〈value〉$ .

On entry, $iskip = 〈value〉$ .
Constraint: $0 \leq iskip \leq 2^{30}$ .

On entry, $liref = 〈value〉$ .
Constraint: $liref \geq 32 \times idim + 7$ .
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_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
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

Not applicable.

8 Parallelism and Performance

nag_quasi_init_scrambled (g05ync) 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

The additional computational cost in using a scrambled quasi-random sequence over a non-scrambled one comes entirely during the initialization. Once nag_quasi_init_scrambled (g05ync) has been called the computational cost of generating a scrambled sequence and a non-scrambled one is identical.

10 Example

This example calls nag_rand_init_repeatable (g05kfc), nag_quasi_rand_uniform (g05ymc) and nag_quasi_init_scrambled (g05ync) to estimate the value of the integral