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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

or 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

$\max (1, pdquas \times idim)$ 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 Further Comments

None.

9 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.