# NAG Library Routine Document

## 1Purpose

g05khf allows for the generation of multiple, independent, sequences of pseudorandom numbers using the leap-frog method.

## 2Specification

Fortran Interface
 Subroutine g05khf ( n, k,
 Integer, Intent (In) :: n, k Integer, Intent (Inout) :: state(*), ifail
#include nagmk26.h
 void g05khf_ (const Integer *n, const Integer *k, Integer state[], Integer *ifail)

## 3Description

g05khf adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the leap-frog method (see the G05 Chapter Introduction for details).
If, prior to calling g05khf the base generator defined by state would produce random numbers ${x}_{1},{x}_{2},{x}_{3},\dots$, then after calling g05khf the generator will produce random numbers ${x}_{k},{x}_{k+n},{x}_{k+2n},{x}_{k+3n},\dots$.
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 g05khf.
The leap-frog algorithm can be used in conjunction with the NAG basic generator, both the Wichmann–Hill I and Wichmann–Hill II generators, the Mersenne Twister and L'Ecuyer.

## 4References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the total number of sequences required.
Constraint: ${\mathbf{n}}>0$.
2:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of the current sequence.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
3:     $\mathbf{state}\left(*\right)$ – 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.
4:     $\mathbf{ifail}$ – 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: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=4$
On entry, cannot use leap-frog with the base generator defined by state.
${\mathbf{ifail}}=-99$
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

g05khf is not threaded in any implementation.

The leap-frog method tends to be less efficient than other methods of producing multiple, independent sequences. See the G05 Chapter Introduction for alternative choices.

## 10Example

This example creates three independent sequences using g05khf, after initialization by g05kff. Five variates from a uniform distribution are then generated from each sequence using g05saf.

### 10.1Program Text

Program Text (g05khfe.f90)

### 10.2Program Data

Program Data (g05khfe.d)

### 10.3Program Results

Program Results (g05khfe.r)

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