G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05YJF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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($\mathit{liref}$), IFAIL REAL (KIND=nag_wp) XMEAN($\mathit{idim}$), STD($\mathit{idim}$), QUAS(N,$\mathit{idim}$)

## 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:
• $\mathit{idim}={\mathbf{IDIM}}$, the number of dimensions required, see G05YLF or G05YNF;
• $\mathit{liref}={\mathbf{LIREF}}$, the length of IREF as supplied to the initialization routines G05YLF or G05YNF.
1:     $\mathrm{XMEAN}\left(\mathit{idim}\right)$ – REAL (KIND=nag_wp) arrayInput
On entry: specifies, for each dimension, the mean of the Normal distribution.
2:     $\mathrm{STD}\left(\mathit{idim}\right)$ – REAL (KIND=nag_wp) arrayInput
On entry: specifies, for each dimension, the standard deviation of the Normal distribution.
Constraint: ${\mathbf{STD}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,\mathit{idim}$.
3:     $\mathrm{N}$ – INTEGERInput
On entry: the number of quasi-random numbers required.
Constraint: ${\mathbf{N}}\ge 0$ and ${\mathbf{N}}+\text{previous number of generated values}\le {2}^{31}-1$.
4:     $\mathrm{QUAS}\left({\mathbf{N}},\mathit{idim}\right)$ – REAL (KIND=nag_wp) arrayOutput
On exit: contains the N quasi-random numbers of dimension idim
5:     $\mathrm{IREF}\left(\mathit{liref}\right)$ – 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:     $\mathrm{IFAIL}$ – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, incorrect initialization has been detected.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=3$
A standard deviation is negative.
${\mathbf{IFAIL}}=4$
There have been too many calls to the generator.
${\mathbf{IFAIL}}=-99$
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

Not applicable.

## 8  Parallelism and Performance

G05YJF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G05YJF 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 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 Sobol, Sobol (A659) and Niederreiter quasi-random number generators in G05YJF have been parallelized, but require quite large problem sizes to see any significant performance gain. The Faure generator is serial.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (g05yjfe.f90)

### 10.2  Program Data

Program Data (g05yjfe.d)

### 10.3  Program Results

Program Results (g05yjfe.r)