# NAG Library Function Document

## 1Purpose

nag_rand_leap_frog (g05khc) allows for the generation of multiple, independent, sequences of pseudorandom numbers using the leap-frog method.

## 2Specification

 #include #include
 void nag_rand_leap_frog (Integer n, Integer k, Integer state[], NagError *fail)

## 3Description

nag_rand_leap_frog (g05khc) 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 nag_rand_leap_frog (g05khc) the base generator defined by state would produce random numbers ${x}_{1},{x}_{2},{x}_{3},\dots$, then after calling nag_rand_leap_frog (g05khc) the generator will produce random numbers ${x}_{k},{x}_{k+n},{x}_{k+2n},{x}_{k+3n},\dots$.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_leap_frog (g05khc).
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:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{k}}\le {\mathbf{n}}$.
NE_INT_ARRAY
On entry, cannot use leap-frog with the base generator defined by state.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

nag_rand_leap_frog (g05khc) 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 nag_rand_leap_frog (g05khc), after initialization by nag_rand_init_repeatable (g05kfc). Five variates from a uniform distribution are then generated from each sequence using nag_rand_basic (g05sac).

### 10.1Program Text

Program Text (g05khce.c)

None.

### 10.3Program Results

Program Results (g05khce.r)

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