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 g05khc the base generator defined by state would produce random numbers

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

, then after calling g05khc the generator will produce random numbers

x_{ℓ}, x_{ℓ + n}, x_{ℓ + 2 n}, x_{ℓ + 3 n}, \dots

, where

ℓ = k + n - 1

. Note that the first

n - 1

random numbers are discarded when using leap-frog.

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 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.

4 References

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

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the total number of sequences required.

Constraint: $n > 0$ .
2: $k$ – Integer Input: On entry: $k$ , the number of the current sequence.

Constraint: $0 < k \leq n$ .
3: $state [\dim]$ – Integer Communication 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.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INT_2: On entry, $k = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $0 < k \leq n$ .
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 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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05khc is not threaded in any implementation.

9 Further Comments

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.

10 Example

This example creates three independent sequences using g05khc, after initialization by g05kfc. Five variates from a uniform distribution are then generated from each sequence using g05sac.