# NAG FL Interfaceg05yjf (quasi_​normal)

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## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine g05yjf ( std, n, quas, iref,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: iref(*), ifail Real (Kind=nag_wp), Intent (In) :: xmean(*), std(*) Real (Kind=nag_wp), Intent (Inout) :: quas(n,*)
#include <nag.h>
 void g05yjf_ (const double xmean[], const double std[], const Integer *n, double quas[], Integer iref[], Integer *ifail)
The routine may be called by the names g05yjf or nagf_rand_quasi_normal.

## 3Description

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).

## 4References

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

## 5Arguments

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: $\mathbf{xmean}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array xmean must be at least $\mathit{idim}$.
On entry: specifies, for each dimension, the mean of the Normal distribution.
2: $\mathbf{std}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array std must be at least $\mathit{idim}$.
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: $\mathbf{n}$Integer Input
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: $\mathbf{quas}\left({\mathbf{n}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array quas must be at least $\mathit{idim}$.
On exit: contains the n quasi-random numbers of dimension idim
5: $\mathbf{iref}\left(*\right)$Integer array Communication Array
Note: the dimension of the array iref must be at least $\mathit{liref}$.
On entry: contains information on the current state of the sequence.
On exit: contains updated information on the state of the sequence.
6: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, iref has either not been initialized or has been corrupted.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, $i=⟨\mathit{\text{value}}⟩$ and ${\mathbf{std}}\left(i\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{std}}\left(i\right)\ge 0.0$.
${\mathbf{ifail}}=4$
There have been too many calls to the generator.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism 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.

## 10Example

This example calls g05ylf to initialize the generator and then g05yjf to generate a sequence of five four-dimensional variates.

### 10.1Program Text

Program Text (g05yjfe.f90)

### 10.2Program Data

Program Data (g05yjfe.d)

### 10.3Program Results

Program Results (g05yjfe.r)