void	g05tgc (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer m, Integer k, const double p[], double r[], Integer lr, Integer state[], Integer x[], Integer pdx, NagError *fail)

The function may be called by the names: g05tgc, nag_rand_int_multinomial or nag_rand_gen_multinomial.

3 Description

g05tgc generates a sequence of

n

groups of

k

integers

x_{i, j}

, for

j = 1, 2, \dots, k

and

i = 1, 2, \dots, n

, from a multinomial distribution with

m

trials and

k

outcomes, where the probability of

x_{i, j} = I_{j}

for each

j = 1, 2, \dots, k

P (i_{1} = I_{1}, \dots, i_{k} = I_{k}) = \frac{m!}{\prod_{j = 1}^{k} I_{j}!} \prod_{j = 1}^{k} p_{j}^{I_{j}} = \frac{m!}{I_{1}! I_{2}! \dots I_{k}!} p_{1}^{I_{1}} p_{2}^{I_{2}} \dots p_{k}^{I_{k}},

where

\sum_{j = 1}^{k} p_{j} = 1 and \sum_{j = 1}^{k} I_{j} = m .

A single trial can have several outcomes (

k

) and the probability of achieving each outcome is known (

p_{j}

). After

m

trials each outcome will have occurred a certain number of times. The

k

numbers representing the numbers of occurrences for each outcome after

m

trials is then a single sample from the multinomial distribution defined by the parameters

k

m

and

p_{j}

, for

j = 1, 2, \dots, k

. This function returns

n

such samples.

When

k = 2

this distribution is equivalent to the binomial distribution with parameters

m

and

p = p_{1}

(see g05tac).

The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to g05tgc with the same parameter values can then use this reference vector to generate further variates. The reference array is generated only for the outcome with greatest probability. The number of successes for the outcome with greatest probability is calculated first as for the binomial distribution (see g05tac); the number of successes for other outcomes are calculated in turn for the remaining reduced multinomial distribution; the number of successes for the final outcome is simply calculated to ensure that the total number of successes is

m

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 g05tgc.

4 References

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

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mode$ – Nag_ModeRNG Input

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to g05tgc.
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.
$mode = Nag_GenerateWithoutReference$: Generate variates without using the reference vector.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

Nag_GenerateWithoutReference

3: $n$ – Integer Input

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

4: $m$ – Integer Input

On entry:

m

, the number of trials of the multinomial distribution.

Constraint:

m \geq 0

5: $k$ – Integer Input

On entry:

k

, the number of possible outcomes of the multinomial distribution.

Constraint:

k \geq 2

6: $p [k]$ – const double Input

On entry: contains the probabilities

p_{j}

, for

j = 1, 2, \dots, k

, of the

k

possible outcomes of the multinomial distribution.

Constraint:

0.0 \leq p [j - 1] \leq 1.0

and

\sum_{j = 1}^{k} p [j - 1] = 1.0

7: $r [lr]$ – double Communication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to g05tgc.

mode = Nag_GenerateWithoutReference

, r is not referenced and may be NULL.

On exit: if

mode \neq Nag_GenerateWithoutReference

, the reference vector.

8: $lr$ – Integer Input

Note: for convenience p_max will be used here to denote the expression

p_max = \max_{j} (p [j])

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 30 + 20 \times \sqrt{m \times p_max \times (1 - p_max)}$ ;
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ ,
$\begin{array}{l} lr & > & \min (m, INT [m \times p_max + 7.25 \times \sqrt{m \times p_max \times (1 - p_max)} + 8.5]) \\ - \max (0, INT [m \times p_max - 7.25 \times \sqrt{m \times p_max \times (1 - p_max)}]) + 9 \end{array}$ ;
if $mode = Nag_GenerateFromReference$ , lr must remain unchanged from the previous call to g05tgc.

9: $state [\dim]$ – 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.

10: $x [\dim]$ – Integer Output

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times k)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the first

n

rows of

X (i, j)

each contain

k

pseudorandom numbers representing a

k

-dimensional variate from the specified multinomial distribution.

11: $pdx$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq k$ .

12: $fail$ – 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, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .

On entry, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = ⟨ value ⟩$ , minimum length required $= ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdx = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $pdx \geq k$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq n$ .
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_PREV_CALL: The value of m or k is not the same as when r was set up in a previous call.
Previous value of $m = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Previous value of $k = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
NE_REAL_ARRAY: On entry, at least one element of the vector p is less than $0.0$ or greater than $1.0$ .

On entry, the sum of the elements of p do not equal one.
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05tgc 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.

9 Further Comments

The reference vector for only one outcome can be set up because the conditional distributions cannot be known in advance of the generation of variates. The outcome with greatest probability of success is chosen for the reference vector because it will have the greatest spread of likely values.