# NAG Library Routine Document

## 1Purpose

g05kjf allows for the generation of multiple, independent, sequences of pseudorandom numbers using the skip-ahead method.
The base pseudorandom number sequence defined by state is advanced $n$ places.

## 2Specification

Fortran Interface
 Subroutine g05kjf ( n,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: state(*), ifail
#include nagmk26.h
 void g05kjf_ (const Integer *n, Integer state[], Integer *ifail)

## 3Description

g05kjf adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the skip-ahead method (see the G05 Chapter Introduction for details).
If, prior to calling g05kjf the base generator defined by state would produce random numbers ${x}_{1},{x}_{2},{x}_{3},\dots$, then after calling g05kjf the generator will produce random numbers ${x}_{n+1},{x}_{n+2},{x}_{n+3},\dots$.
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 g05kjf.
The skip-ahead algorithm can be used in conjunction with any of the six base generators discussed in Chapter G05.

## 4References

Haramoto H, Matsumoto M, Nishimura T, Panneton F and L'Ecuyer P (2008) Efficient jump ahead for F2-linear random number generators INFORMS J. on Computing 20(3) 385–390
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of places to skip ahead.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{state}\left(*\right)$ – 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.
3:     $\mathbf{ifail}$ – 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.
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.
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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=3$
On entry, cannot use skip-ahead with the base generator defined by state.
${\mathbf{ifail}}=4$
On entry, the base generator is Mersenne Twister, but the state vector defined on initialization is not large enough to perform a skip ahead. See the initialization routine g05kff or g05kgf.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

g05kjf is not threaded in any implementation.

Calling g05kjf and then generating a series of uniform values using g05saf is more efficient than, but equivalent to, calling g05saf and discarding the first $n$ values. This may not be the case for distributions other than the uniform, as some distributional generators require more than one uniform variate to generate a single draw from the required distribution.
To skip ahead $k×m$ places you can either
 (a) call g05kjf once with ${\mathbf{n}}=k×m$, or (b) call g05kjf $k$ times with ${\mathbf{n}}=m$, using the state vector output by the previous call as input to the next call
both approaches would result in the same sequence of values. When working in a multithreaded environment, where you want to generate (at most) $m$ values on each of $K$ threads, this would translate into either
 (a) spawning the $K$ threads and calling g05kjf once on each thread with ${\mathbf{n}}=\left(k-1\right)×m$, where $k$ is a thread ID, taking a value between $1$ and $K$, or (b) calling g05kjf on a single thread with ${\mathbf{n}}=m$, spawning the $K$ threads and then calling g05kjf a further $k-1$ times on each of the thread.
Due to the way skip ahead is implemented for the Mersenne Twister, approach (a) will tend to be more efficient if more than 30 threads are being used (i.e., $K>30$), otherwise approach (b) should probably be used. For all other base generators, approach (a) should be used. See the G05 Chapter Introduction for more details.

## 10Example

This example initializes a base generator using g05kff and then uses g05kjf to advance the sequence 50 places before generating five variates from a uniform distribution using g05saf.

### 10.1Program Text

Program Text (g05kjfe.f90)

### 10.2Program Data

Program Data (g05kjfe.d)

### 10.3Program Results

Program Results (g05kjfe.r)

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