# NAG Library Routine Document

## 1Purpose

g05kff initializes the selected base generator, as used by the group of pseudorandom number routines (see g05khfg05kjf, g05ncf, g05ndf, g05pdfg05pjf, g05pxfg05pzf, g05rcf, g05rdf, g05ryf, g05rzf and g05safg05tlf) and the quasi-random scrambled sequence initialization routine, g05ynf.

## 2Specification

Fortran Interface
 Subroutine g05kff ( seed,
 Integer, Intent (In) :: genid, subid, seed(lseed), lseed Integer, Intent (Inout) :: state(lstate), lstate, ifail
#include nagmk26.h
 void g05kff_ (const Integer *genid, const Integer *subid, const Integer seed[], const Integer *lseed, Integer state[], Integer *lstate, Integer *ifail)

## 3Description

g05kff selects a base generator through the input value of the arguments genid and subid, and then initializes it based on the values given in the array seed.
A given base generator will yield different sequences of random numbers if initialized with different values of seed. Alternatively, the same sequence of random numbers will be generated if the same value of seed is used. It should be noted that there is no guarantee of statistical properties between sequences, only within sequences.
A definition of some of the terms used in this description, along with details of the various base generators can be found in the G05 Chapter Introduction.

## 4References

L'Ecuyer P and Simard R (2002) TestU01: a software library in ANSI C for empirical testing of random number generators Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal http://www.iro.umontreal.ca/~lecuyer
Maclaren N M (1989) The generation of multiple independent sequences of pseudorandom numbers Appl. Statist. 38 351–359
Matsumoto M and Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator ACM Transactions on Modelling and Computer Simulations
Wichmann B A and Hill I D (2006) Generating good pseudo-random numbers Computational Statistics and Data Analysis 51 1614–1622
Wikramaratna R S (1989) ACORN - a new method for generating sequences of uniformly distributed pseudo-random numbers Journal of Computational Physics 83 16–31

## 5Arguments

1:     $\mathbf{genid}$ – IntegerInput
On entry: must contain the type of base generator to use.
${\mathbf{genid}}=1$
NAG basic generator.
${\mathbf{genid}}=2$
Wichmann Hill I generator.
${\mathbf{genid}}=3$
Mersenne Twister.
${\mathbf{genid}}=4$
Wichmann Hill II generator.
${\mathbf{genid}}=5$
ACORN generator.
${\mathbf{genid}}=6$
L'Ecuyer MRG32k3a generator.
See the G05 Chapter Introduction for details of each of the base generators.
Constraint: ${\mathbf{genid}}=1$, $2$, $3$, $4$, $5$ or $6$.
2:     $\mathbf{subid}$ – IntegerInput
On entry: if ${\mathbf{genid}}=2$, subid indicates which of the $273$ sub-generators to use. In this case, the  sub-generator is used.
If ${\mathbf{genid}}=5$, subid indicates the values of $k$ and $p$ to use, where $k$ is the order of the generator, and $p$ controls the size of the modulus, $M$, with $M={2}^{\left(p×30\right)}$. If ${\mathbf{subid}}<1$, the default values of $k=10$ and $p=2$ are used, otherwise values for $k$ and $p$ are calculated from the formula, ${\mathbf{subid}}=k+1000\left(p-1\right)$.
If ${\mathbf{genid}}=6$ and  the range of the generator is set to $\left(0,1\right]$, otherwise the range is set to $\left(0,1\right)$; in this case the sequence is identical to the implementation of MRG32k3a in TestU01 (see L'Ecuyer and Simard (2002)) for identical seeds.
For all other values of genid, subid is not referenced.
3:     $\mathbf{seed}\left({\mathbf{lseed}}\right)$ – Integer arrayInput
On entry: the initial (seed) values for the selected base generator. The number of initial values required varies with each of the base generators.
If ${\mathbf{genid}}=1$, one seed is required.
If ${\mathbf{genid}}=2$, four seeds are required.
If ${\mathbf{genid}}=3$, $624$ seeds are required.
If ${\mathbf{genid}}=4$, four seeds are required.
If ${\mathbf{genid}}=5$, $\left(k+1\right)p$ seeds are required, where $k$ and $p$ are defined by subid. For the ACORN generator it is recommended that an odd value is used for ${\mathbf{seed}}\left(1\right)$.
If ${\mathbf{genid}}=6$, six seeds are required.
If insufficient seeds are provided then the first ${\mathbf{lseed}}-1$ values supplied in seed are used and the remaining values are randomly generated using the NAG basic generator. In such cases the NAG basic generator is initialized using the value supplied in ${\mathbf{seed}}\left({\mathbf{lseed}}\right)$.
Constraint: ${\mathbf{seed}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{lseed}}$.
4:     $\mathbf{lseed}$ – IntegerInput
On entry: the size of the seed array.
Constraint: ${\mathbf{lseed}}\ge 1$.
5:     $\mathbf{state}\left({\mathbf{lstate}}\right)$ – Integer arrayCommunication Array
On exit: contains information on the selected base generator and its current state.
6:     $\mathbf{lstate}$ – IntegerInput/Output
On entry: the dimension of the state array, or a value $<1$. If the Mersenne Twister (${\mathbf{genid}}=3$) is being used and the skip ahead routine g05kjf or g05kkf will be called subsequently, then you must ensure that ${\mathbf{lstate}}\ge 1260$.
On exit: if ${\mathbf{lstate}}<1$ on entry, then the required length of the state array for the chosen base generator, otherwise lstate is unchanged. When ${\mathbf{genid}}=3$ (Mersenne Twister) a value of $1260$ is returned, allowing for the skip ahead routine to be subsequently called. In all other cases the minimum length, as documented in the constraints below, is returned.
Constraints:
• if ${\mathbf{genid}}=1$, ${\mathbf{lstate}}\ge 17$;
• if ${\mathbf{genid}}=2$, ${\mathbf{lstate}}\ge 21$;
• if ${\mathbf{genid}}=3$, ${\mathbf{lstate}}\ge 633$;
• if ${\mathbf{genid}}=4$, ${\mathbf{lstate}}\ge 29$;
• if ${\mathbf{genid}}=5$, ${\mathbf{lstate}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left(k+1\right)×p+9,14\right)+3$, where $k$ and $p$ are defined by subid;
• if ${\mathbf{genid}}=6$, ${\mathbf{lstate}}\ge 61$;
• otherwise ${\mathbf{lstate}}<1$.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ or $-1$ unless the routine detects an error or a warning has been flagged (see Section 6).
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 argument, 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.

## 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, ${\mathbf{genid}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{genid}}=1$, $2$, $3$, $4$, $5$ or $6$.
${\mathbf{ifail}}=3$
On entry, invalid seed.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{lseed}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lseed}}\ge 1$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{lstate}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lstate}}\le 0$ or ${\mathbf{lstate}}\ge 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-1$
Required length of state array returned in lstate but state array not initialized.
${\mathbf{ifail}}=-99$
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

g05kff is not threaded in any implementation.

None.

## 10Example

This example prints the first five pseudorandom real numbers from a uniform distribution between $0$ and $1$, generated by g05saf after initialization by g05kff.

### 10.1Program Text

Program Text (g05kffe.f90)

### 10.2Program Data

Program Data (g05kffe.d)

### 10.3Program Results

Program Results (g05kffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017