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_{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 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.

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: $state (*)$ – Integer array Communication 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $k = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $0 < k \leq n$ .

$ifail = 3$: On entry, state vector has been corrupted or not initialized.

$ifail = 4$: On entry, cannot use leap-frog with the base generator defined by state.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

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

g05khf 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 g05khf, after initialization by g05kff. Five variates from a uniform distribution are then generated from each sequence using g05saf.

g05kh: FL CL CPP AD PY MB

NAG FL Interfaceg05khf (init_​leapfrog)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g05khf (init_leapfrog)