# NAG Library Routine Document

## 1Purpose

g05ykf generates a quasi-random sequence from a log-normal distribution. It must be preceded by a call to one of the initialization routines g05ylf or g05ynf.

## 2Specification

Fortran Interface
 Subroutine g05ykf ( 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(ldquas,*)
#include <nagmk26.h>
 void g05ykf_ (const double xmean[], const double std[], const Integer *n, double quas[], Integer iref[], Integer *ifail)

## 3Description

g05ykf generates a quasi-random sequence from a log-normal distribution by first generating a uniform quasi-random sequence which is then transformed into a log-normal sequence using the exponential of 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) arrayInput
Note: the dimension of the array xmean must be at least $\mathit{idim}$.
On entry: specifies, for each dimension, the mean of the underlying Normal distribution.
Constraint: $\left|{\mathbf{xmean}}\left(\mathit{i}\right)\right|\le \left|-\mathrm{log}\left({\mathbf{x02amf}}\right)-10.0×{\mathbf{std}}\left(\mathit{i}\right)\right|$, for $\mathit{i}=1,2,\dots ,\mathit{idim}$.
2:     $\mathbf{std}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array std must be at least $\mathit{idim}$.
On entry: specifies, for each dimension, the standard deviation of the underlying Normal distribution.
Constraint: ${\mathbf{std}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,\mathit{idim}$.
3:     $\mathbf{n}$ – 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:     $\mathbf{quas}\left({\mathbf{n}},*\right)$ – Real (Kind=nag_wp) arrayOutput
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 arrayCommunication 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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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: ${\mathbf{std}}\left(i\right)\ge 0$.
${\mathbf{ifail}}=4$
There have been too many calls to the generator.
${\mathbf{ifail}}=5$
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{xmean}}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{xmean}}\left(i\right)\right|\le 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

g05ykf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05ykf 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 g05ykc 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 g05ykf to produce a sequence of five four-dimensional quasi-random numbers variates.

### 10.1Program Text

Program Text (g05ykfe.f90)

### 10.2Program Data

Program Data (g05ykfe.d)

### 10.3Program Results

Program Results (g05ykfe.r)