NAG Library Function Document

nag_quasi_rand_normal (g05yjc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_quasi_rand_normal (g05yjc) generates a quasi-random sequence from a Normal (Gaussian) distribution. It must be preceded by a call to one of the initialization functions nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync).

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_quasi_rand_normal (Nag_OrderType order, const double xmean[], const double std[], Integer n, double quas[], Integer pdquas, Integer iref[], NagError *fail)

3 Description

nag_quasi_rand_normal (g05yjc) generates a quasi-random sequence from a Normal distribution by first generating a uniform quasi-random sequence which is then transformed into a Normal sequence using the inverse of the Normal CDF. The type of uniform sequence used depends on the initialization function called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter. If the initialization function nag_quasi_init_scrambled (g05ync) was used then the underlying uniform sequence is first scrambled prior to being transformed (see Section 3 in nag_quasi_init_scrambled (g05ync) for details).

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

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

5 Arguments

Note: the following variables are used in the parameter descriptions:

$idim = idim$ , the number of dimensions required, see nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync);
$liref = liref$ , the length of iref as supplied to the initialization functions nag_quasi_init (g05ylc) or nag_quasi_init_scrambled (g05ync).
$tdquas = n$ if $order = Nag_RowMajor$ ; otherwise $tdquas = idim$

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: xmean[ $idim$ ] – const doubleInput

On entry: specifies, for each dimension, the mean of the Normal distribution.

3: std[ $idim$ ] – const doubleInput

On entry: specifies, for each dimension, the standard deviation of the Normal distribution.

Constraint:

std [i - 1] \geq 0.0

, for

i = 1, 2, \dots, idim

4: n – IntegerInput

On entry: the number of quasi-random numbers required.

Constraint:

n \geq 0

and

n + previous number of generated values \leq 2^{31} - 1

5: quas[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array quas must be at least

pdquas \times idim

The dimension, dim, of the array quas must be at least

$\max (1, pdquas \times idim)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdquas)$ when $order = Nag_RowMajor$ .

Where

QUAS (i, j)

appears in this document, it refers to the array element

$quas [(j - 1) \times pdquas + i - 1]$ when $order = Nag_ColMajor$ ;
$quas [(i - 1) \times pdquas + j - 1]$ when $order = Nag_RowMajor$ .

On exit: contains the n quasi-random numbers of dimension idim,

QUAS (i, j)

holds the

i

th value for the

j

th dimension.

6: pdquas – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array quas.

Constraints:

if $order = Nag_ColMajor$ , $pdquas \geq n$ ;
if $order = Nag_RowMajor$ , $pdquas \geq idim$ .

7: iref[ $liref$ ] – IntegerCommunication Array

On entry: contains information on the current state of the sequence.

On exit: contains updated information on the state of the sequence.

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INITIALIZATION: On entry, iref has either not been initialized or has been corrupted.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdquas = ⟨value⟩$ and $idim = ⟨value⟩$ .
Constraint: $pdquas \geq idim$ .
On entry, $pdquas = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdquas \geq n$ .
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.
NE_REAL_ARRAY: On entry, $std [⟨value⟩] = ⟨value⟩$ .
Constraint: $std [i] \geq 0.0$ .
NE_TOO_MANY_CALLS: There have been too many calls to the generator.

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_quasi_rand_normal (g05yjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_quasi_rand_normal (g05yjc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

The Sobol, Sobol (A659) and Niederreiter quasi-random number generators in nag_quasi_rand_normal (g05yjc) have been parallelized, but require quite large problem sizes to see any significant performance gain. Parallelism is only enabled when

order = Nag_ColMajor

. The Faure generator is serial.

9 Further Comments

None.

10 Example

This example calls nag_quasi_init (g05ylc) to initialize the generator and then nag_quasi_rand_normal (g05yjc) to generate a sequence of five four-dimensional variates.