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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_int_log (g05tf)

## Purpose

nag_rand_int_log (g05tf) generates a vector of pseudorandom integers from the discrete logarithmic distribution with parameter a$a$.

## Syntax

[r, state, x, ifail] = g05tf(mode, n, a, r, state)
[r, state, x, ifail] = nag_rand_int_log(mode, n, a, r, state)

## Description

nag_rand_int_log (g05tf) generates n$n$ integers xi${x}_{i}$ from a discrete logarithmic distribution, where the probability of xi = I${x}_{i}=I$ is
 P (xi = I) = − (aI)/( I × log(1 − a) ) ,   I = 1,2, … , $P (xi=I) = - aI I × log(1-a) , I=1,2,… ,$
where 0 < a < 1.$0
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 nag_rand_int_log (g05tf) with the same parameter value can then use this reference vector to generate further variates.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_int_log (g05tf).

## References

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

## Parameters

### Compulsory Input Parameters

1:     mode – int64int32nag_int scalar
A code for selecting the operation to be performed by the function.
mode = 0${\mathbf{mode}}=0$
Set up reference vector only.
mode = 1${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rand_int_log (g05tf).
mode = 2${\mathbf{mode}}=2$
Set up reference vector and generate variates.
mode = 3${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: mode = 0${\mathbf{mode}}=0$, 1$1$, 2$2$ or 3$3$.
2:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     a – double scalar
a$a$, the parameter of the logarithmic distribution.
Constraint: 0.0 < a < 1.0$0.0<{\mathbf{a}}<1.0$.
4:     r(lr) – double array
lr, the dimension of the array, must satisfy the constraint
• if mode = 0${\mathbf{mode}}=0$ or 2$2$, lr must not be too small, but the lower limit is too complicated to specify;
• if mode = 1${\mathbf{mode}}=1$, lr must remain unchanged from the previous call to nag_rand_int_log (g05tf).
If mode = 1${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rand_int_log (g05tf).
If mode = 3${\mathbf{mode}}=3$, r is not referenced by nag_rand_int_log (g05tf).
5:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

lr

### Output Parameters

1:     r(lr) – double array
The reference vector.
2:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
3:     x(n) – int64int32nag_int array
The n$n$ pseudorandom numbers from the specified logarithmic distribution.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, mode0${\mathbf{mode}}\ne 0$, 1$1$, 2$2$ or 3$3$.
ifail = 2${\mathbf{ifail}}=2$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, a ≤ 0.0${\mathbf{a}}\le 0.0$, or a ≥ 1.0${\mathbf{a}}\ge 1.0$.
ifail = 4${\mathbf{ifail}}=4$
On entry, a is not the same as when r was set up in a previous call to nag_rand_int_log (g05tf) with mode = 0${\mathbf{mode}}=0$ or 2$2$.
On entry, the r vector was not initialized correctly, or has been corrupted.
ifail = 5${\mathbf{ifail}}=5$
On entry, lr is too small when mode = 0${\mathbf{mode}}=0$ or 2$2$.
ifail = 6${\mathbf{ifail}}=6$
 On entry, state vector was not initialized or has been corrupted.

Not applicable.

None.

## Example

```function nag_rand_int_log_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
mode = int64(3);
n = int64(10);
a = 0.9999;
r = zeros(1, 1);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[r, state, x, ifail] = nag_rand_int_log(mode, n, a, r, state)
```
```

r =

0

state =

17
1234
1
0
6694
27818
10435
15383
17917
13895
19930
8
0
1234
1
1
1234

x =

6
23
2765
30
3
1
299
968
166
4

ifail =

0

```
```function g05tf_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
mode = int64(3);
n = int64(10);
a = 0.9999;
r = zeros(1, 1);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[r, state, x, ifail] = g05tf(mode, n, a, r, state)
```
```

r =

0

state =

17
1234
1
0
6694
27818
10435
15383
17917
13895
19930
8
0
1234
1
1
1234

x =

6
23
2765
30
3
1
299
968
166
4

ifail =

0

```