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## 1Purpose

g05kjc 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

 #include
 void g05kjc (Integer n, Integer state[], NagError *fail)
The function may be called by the names: g05kjc, nag_rand_init_skipahead or nag_rand_skip_ahead.

## 3Description

g05kjc 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 g05kjc the base generator defined by state would produce random numbers ${x}_{1},{x}_{2},{x}_{3},\dots$, then after calling g05kjc the generator will produce random numbers ${x}_{n+1},{x}_{n+2},{x}_{n+3},\dots$.
One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05kjc.
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: $\mathbf{n}$Integer Input
On entry: $n$, the number of places to skip ahead.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication 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.
3: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
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 function g05kfc or g05kgc.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_ARRAY
On entry, cannot use skip-ahead 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.

Not applicable.

## 8Parallelism and Performance

g05kjc is not threaded in any implementation.

Calling g05kjc and then generating a series of uniform values using g05sac is more efficient than, but equivalent to, calling g05sac 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
1. (a)call g05kjc once with ${\mathbf{n}}=k×m$, or
2. (b)call g05kjc $k$ 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
1. (a)spawning the $K$ threads and calling g05kjc 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
2. (b)calling g05kjc on a single thread with ${\mathbf{n}}=m$, spawning the $K$ threads and then calling g05kjc 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 g05kfc and then uses g05kjc to advance the sequence 50 places before generating five variates from a uniform distribution using g05sac.

### 10.1Program Text

Program Text (g05kjce.c)

None.

### 10.3Program Results

Program Results (g05kjce.r)