The routine may be called by the names g05ynf or nagf_rand_quasi_init_scrambled.
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,g05ykforg05ymf.
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. Software14(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. Software29:2 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 Theory30 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 Statistics106 Springer-Verlag, New York, NY 299–317 (eds H Niederreiter and P J-S Shiue)
1: – IntegerInput
On entry: must identify the quasi-random generator to use.
Sobol (A659) generator.
, or .
2: – IntegerInput
On entry: must identify the scrambling method to use.
No scrambling. This is equivalent to calling g05ylf.
Owen like scrambling.
Owen and Faure–Tezuka scrambling.
, , or .
3: – IntegerInput
On entry: the number of dimensions required.
if , ;
if , ;
if , .
4: – Integer arrayCommunication Array
On exit: contains initialization information for use by the generator routines g05yjf,g05ykfandg05ymf. iref must not be altered in any way between initialization and calls of the generator routines.
5: – IntegerInput
On entry: the dimension of the array iref as declared in the (sub)program from which g05ynf is called.
6: – IntegerInput
On entry: the number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators.
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: – 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: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, .
On entry, .
On entry, .
On entry, .
On entry, state vector has been corrupted or not initialized.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g05ynf 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 routine. 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 g05ynf has been called the computational cost of generating a scrambled sequence and a non-scrambled one is identical.
This example calls g05kff,g05ymfandg05ynf to estimate the value of the integral