(i)if $a < 1$ , a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are $f_{1} (x) = c a x^{a - 1} / t^{a}$ and $f_{2} (x) = (1 - c) e^{- (x - t)}$ , where $c = t / (t + a e^{- t})$ , and the switching parameter, $t$ , is taken as $1 - a$ . This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which $t = 1$ ;
(ii)if $a = 1$ , the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used;
(iii)if $a > 1$ , the algorithm given by Best (1978) is used. This is based on using a Student's $t$ -distribution with two degrees of freedom as the target distribution in an envelope rejection method.

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 g05sjc.

4 References

Ahrens J H and Dieter U (1974) Computer methods for sampling from gamma, beta, Poisson and binomial distributions Computing 12 223–46

Best D J (1978) Letter to the Editor Appl. Statist. 27 181

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of pseudorandom numbers to be generated.

Constraint: $n \geq 0$ .
2: $a$ – double Input: On entry: $a$ , the parameter of the gamma distribution.

Constraint: $a > 0.0$ .
3: $b$ – double Input: On entry: $b$ , the parameter of the gamma distribution.

Constraint: $b > 0.0$ .
4: $state [\dim]$ – 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.
5: $x [n]$ – double Output: On exit: the $n$ pseudorandom numbers from the specified gamma distribution.
6: $fail$ – 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: $n \geq 0$ .
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.
NE_REAL: On entry, $a = ⟨ value ⟩$ .
Constraint: $a > 0.0$ .

On entry, $b = ⟨ value ⟩$ .
Constraint: $b > 0.0$ .
NW_REAL: On entry, $a = ⟨ value ⟩$ .
For very small shape parameter values, variates are approximate.

7 Accuracy

For very small values of the shape parameter,

a

, the G6 algorithm is not appropriate due to loss of accuracy in the transformations used; for example,

t = 1 - a

. An alternative algorithm based on approximations to the density functions has been added at MK30.2 for the case

a < \sqrt{ε}

, where

ε

is the machine precision. A warning exit has also been added for this case to indicate that approximations have been used.

8 Parallelism and Performance

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

g05sjc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.