nag_quasi_init (g05ylc) selects a quasi-random number generator through the input value of genid and initializes the iref communication array for use by the functions nag_quasi_rand_normal (g05yjc), nag_quasi_rand_lognormal (g05ykc) or nag_quasi_rand_uniform (g05ymc).

One of three types of quasi-random generator may be chosen, allowing the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter to be generated.

Two sets of Sobol sequences are supplied, the first, is based on the work of Joe and Kuo (2008). The second, referred to in the documentation as "Sobol (A659)", is based on Algorithm 659 of Bratley and Fox (1988) with the extension to 1111 dimensions proposed by Joe and Kuo (2003). Both sets of Sobol sequences should satisfy the so-called Property A, up to

1111

dimensions, but the first set should have better two-dimensional projections than those produced using Algorithm 659.

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

Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376

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

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.
$genid = Nag_QuasiRandom_Faure$: Faure generator.

Constraint:

genid = Nag_QuasiRandom_Sobol

Nag_QuasiRandom_SobolA659

Nag_QuasiRandom_Nied

Nag_QuasiRandom_Faure

2: $idim$ – IntegerInput

On entry: the number of dimensions required.

Constraints:

if $genid = Nag_QuasiRandom_Sobol$ , $1 \leq idim \leq 10000$ ;
if $genid = Nag_QuasiRandom_SobolA659$ , $1 \leq idim \leq 1111$ ;
if $genid = Nag_QuasiRandom_Nied$ , $1 \leq idim \leq 318$ ;
if $genid = Nag_QuasiRandom_Faure$ , $1 \leq idim \leq 40$ .

3: $iref [liref]$ – 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.

4: $liref$ – IntegerInput

On entry: the dimension of the array iref.

Constraints:

if $genid = Nag_QuasiRandom_Sobol$ , $Nag_QuasiRandom_SobolA659$ or $Nag_QuasiRandom_Nied$ , $liref \geq 32 \times idim + 7$ ;
if $genid = Nag_QuasiRandom_Faure$ , $liref \geq 407$ .

5: $iskip$ – IntegerInput

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

genid = Nag_QuasiRandom_Faure

, iskip is ignored.

Constraint: if

genid = Nag_QuasiRandom_Sobol

Nag_QuasiRandom_SobolA659

Nag_QuasiRandom_Nied

0 \leq iskip \leq 2^{30}

6: $fail$ – 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, $iskip < 0$ or iskip is too large: $iskip = 〈value〉$ , maximum value is $〈value〉$ .

On entry, liref is too small: $liref = 〈value〉$ , minimum length is $〈value〉$ .
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_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 (g05ylc) 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.