nag_rand_skip_ahead (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 nag_rand_skip_ahead (g05kjc) the base generator defined by state would produce random numbers

x_{1}, x_{2}, x_{3}, \dots

, then after calling nag_rand_skip_ahead (g05kjc) the generator will produce random numbers

x_{n + 1}, x_{n + 2}, x_{n + 3}, \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_skip_ahead (g05kjc).

The skip-ahead algorithm can be used in conjunction with any of the six base generators discussed in Chapter g05.

4

References

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

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the number of places to skip ahead.

Constraint: $n \geq 0$ .
2: $state [\dim]$ – IntegerCommunication Array: Note: the dimension, $\dim$ , 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: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error 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.
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 nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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 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.

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_rand_skip_ahead (g05kjc) is not threaded in any implementation.

9

Further Comments

Calling nag_rand_skip_ahead (g05kjc) and then generating a series of uniform values using nag_rand_basic (g05sac) is more efficient than, but equivalent to, calling nag_rand_basic (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 \times m

places you can either

(a)	call nag_rand_skip_ahead (g05kjc) once with $n = k \times m$ , or
(b)	call nag_rand_skip_ahead (g05kjc) $k$ times with $n = m$ , 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 nag_rand_skip_ahead (g05kjc) once on each thread with $n = (k - 1) \times m$ , where $k$ is a thread ID, taking a value between $1$ and $K$ , or
(b)	calling nag_rand_skip_ahead (g05kjc) on a single thread with $n = m$ , spawning the $K$ threads and then calling nag_rand_skip_ahead (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.

10

Example

This example initializes a base generator using nag_rand_init_repeatable (g05kfc) and then uses nag_rand_skip_ahead (g05kjc) to advance the sequence 50 places before generating five variates from a uniform distribution using nag_rand_basic (g05sac).