G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05SMF

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

G05SMF generates a vector of pseudorandom numbers from a log-normal distribution with parameters $\mu$ and ${\sigma }^{2}$.

## 2  Specification

 SUBROUTINE G05SMF ( N, XMU, VAR, STATE, X, IFAIL)
 INTEGER N, STATE(*), IFAIL REAL (KIND=nag_wp) XMU, VAR, X(N)

## 3  Description

The distribution has PDF (probability density function)
 $fx = 1 xσ⁢2π exp - ln⁡x-μ 2 2σ2 if ​ x>0 , fx=0 otherwise,$
i.e., $\mathrm{ln}x$ is normally distributed with mean $\mu$ and variance ${\sigma }^{2}$. G05SMF evaluates $\mathrm{exp}{y}_{i}$, where the ${y}_{i}$ are generated by G05SKF from a Normal distribution with mean $\mu$ and variance ${\sigma }^{2}$, for $\mathit{i}=1,2,\dots ,n$.
One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SMF.
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
2:     XMU – REAL (KIND=nag_wp)Input
On entry: $\mu$, the mean of the distribution of $\mathrm{ln}x$.
3:     VAR – REAL (KIND=nag_wp)Input
On entry: ${\sigma }^{2}$, the variance of the distribution of $\mathrm{ln}x$.
Constraint: ${\mathbf{VAR}}\ge 0.0$.
4:     STATE($*$) – INTEGER arrayCommunication Array
Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:     X(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified log-normal distribution.
6:     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, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=2$
On entry, unable to calculate $\mathrm{exp}\left({\mathbf{XMU}}\right)$ as XMU is too large.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{VAR}}<0.0$.
${\mathbf{IFAIL}}=4$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

None.

## 9  Example

This example prints five pseudorandom numbers from a log-normal distribution with mean $1.0$ and variance $2.0$, generated by a single call to G05SMF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05smfe.f90)

### 9.2  Program Data

Program Data (g05smfe.d)

### 9.3  Program Results

Program Results (g05smfe.r)