NAG Library Function Document
nag_quasi_init_scrambled (g05ync)
1
Purpose
nag_quasi_init_scrambled (g05ync) initializes a scrambled quasi-random generator prior to calling
nag_quasi_rand_normal (g05yjc),
nag_quasi_rand_lognormal (g05ykc) or
nag_quasi_rand_uniform (g05ymc). It must be preceded by a call to one of the pseudorandom initialization functions
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc).
2
Specification
#include <nag.h> |
#include <nagg05.h> |
|
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
. 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
.
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:
– Nag_QuasiRandom_SequenceInput
-
On entry: must identify the quasi-random generator to use.
- Sobol generator.
- Sobol (A659) generator.
- Niederreiter generator.
Constraint:
, or .
- 2:
– Nag_QuasiRandom_ScramblingInput
-
On entry: must identify the scrambling method to use.
- No scrambling. This is equivalent to calling nag_quasi_init (g05ylc).
- Owen like scrambling.
- Faure–Tezuka scrambling.
- Owen and Faure–Tezuka scrambling.
Constraint:
, , or .
- 3:
– IntegerInput
-
On entry: the number of dimensions required.
Constraints:
- if , ;
- if , ;
- if , .
- 4:
– 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:
– IntegerInput
-
On entry: the dimension of the array
iref.
Constraint:
.
- 6:
– IntegerInput
-
On entry: the number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators.
Constraint:
.
- 7:
– IntegerInput
-
On entry: controls the number of digits (bits) to scramble when
or
, otherwise
nsdigi is ignored. If
or
then all the digits are scrambled.
- 8:
– IntegerCommunication Array
-
Note: the dimension,
, 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:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 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 had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- 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.
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.
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
where
, the number of dimensions, is set to
.
10.1
Program Text
Program Text (g05ynce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (g05ynce.r)