NAG Library Routine Document

G05YJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G05YJF generates a quasi-random sequence from a Normal (Gaussian) distribution. It must be preceded by a call to one of the initialization routines G05YLF or G05YNF.

2 Specification

SUBROUTINE G05YJF (

XMEAN, STD, N, QUAS, IREF, IFAIL)

INTEGER	N, IREF( $liref$ ), IFAIL
REAL (KIND=nag_wp)	XMEAN( $idim$ ), STD( $idim$ ), QUAS(N, $idim$ )

3 Description

G05YJF 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 routine called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter. If the initialization routine G05YNF was used then the underlying uniform sequence is first scrambled prior to being transformed (see Section 3 in G05YNF 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 Parameters

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

$idim = IDIM$ , the number of dimensions required, see G05YLF or G05YNF;
$liref = LIREF$ , the length of IREF as supplied to the initialization routines G05YLF or G05YNF.

1: XMEAN( $idim$ ) – REAL (KIND=nag_wp) arrayInput: On entry: specifies, for each dimension, the mean of the Normal distribution.
2: STD( $idim$ ) – REAL (KIND=nag_wp) arrayInput: On entry: specifies, for each dimension, the standard deviation of the Normal distribution.
Constraint: $STD (i) \geq 0.0$ , for $i = 1, 2, \dots, idim$ .
3: N – INTEGERInput: On entry: the number of quasi-random numbers required.
Constraint: $N \geq 0$ and $N + previous number of generated values \leq 2^{31} - 1$ .
4: QUAS(N, $idim$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: contains the N quasi-random numbers of dimension idim
5: IREF( $liref$ ) – INTEGER arrayCommunication Array: On entry: contains information on the current state of the sequence.

On exit: contains updated information on the state of the sequence.
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, incorrect initialization has been detected.

$IFAIL = 2$

On entry,

N < 1

$IFAIL = 3$: A standard deviation is negative.

$IFAIL = 4$: There have been too many calls to the generator.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example calls G05YLF to initialize the generator and then G05YJF to generate a sequence of five four-dimensional variates.

NAG Library Routine DocumentG05YJF