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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_triangular (g05sp)

## Purpose

nag_rand_dist_triangular (g05sp) generates a vector of pseudorandom numbers from a triangular distribution with parameters xmin${x}_{\mathrm{min}}$, xmed${x}_{\mathrm{med}}$ and xmax${x}_{\mathrm{max}}$.

## Syntax

[state, x, ifail] = g05sp(n, xmin, xmed, xmax, state)
[state, x, ifail] = nag_rand_dist_triangular(n, xmin, xmed, xmax, state)

## Description

The triangular distribution has a PDF (probability density function) that is triangular in profile. The base of the triangle ranges from x = xmin$x={x}_{\mathrm{min}}$ to x = xmax$x={x}_{\mathrm{max}}$ and the PDF has a maximum value of 2/(xmaxxmin) $\frac{2}{{x}_{\mathrm{max}}-{x}_{\mathrm{min}}}$ at x = xmed$x={x}_{\mathrm{med}}$. If xmin = xmed = xmax${x}_{\mathrm{min}}={x}_{\mathrm{med}}={x}_{\mathrm{max}}$ then x = xmed$x={x}_{\mathrm{med}}$ with probability 1; otherwise the triangular distribution has PDF:
 f(x) = (x − xmin)/(xmed − xmin) × 2/(xmax − xmin) ​ if ​xmin ≤ x ≤ xmed, f(x) = (xmax − x)/(xmax − xmed) × 2/(xmax − xmin) ​ if ​xmed < x ≤ xmax, f(x) = 0 ​ otherwise.
$f(x) = x-xmin xmed-xmin × 2 xmax-xmin ​ if ​xmin≤x≤xmed, f(x)= xmax-x xmax-xmed ×2xmax-xmin ​ if ​xmed
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_dist_triangular (g05sp).

## References

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

## Parameters

### Compulsory Input Parameters

1:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     xmin – double scalar
The end point xmin${x}_{\mathrm{min}}$ of the triangular distribution.
3:     xmed – double scalar
The median of the distribution xmed${x}_{\mathrm{med}}$ (also the location of the vertex of the triangular distribution at which the PDF reaches a maximum).
Constraint: ${\mathbf{xmed}}\ge {\mathbf{xmin}}$.
4:     xmax – double scalar
The end point xmax${x}_{\mathrm{max}}$ of the triangular distribution.
Constraint: ${\mathbf{xmax}}\ge {\mathbf{xmed}}$.
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.

None.

### Output Parameters

1:     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.
2:     x(n) – double array
The n$n$ pseudorandom numbers from the specified triangular distribution.
3:     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, n < 0${\mathbf{n}}<0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, ${\mathbf{xmed}}<{\mathbf{xmin}}$.
ifail = 4${\mathbf{ifail}}=4$
On entry, ${\mathbf{xmax}}<{\mathbf{xmed}}$.
ifail = 5${\mathbf{ifail}}=5$
 On entry, state vector was not initialized or has been corrupted.

Not applicable.

None.

## Example

```function nag_rand_dist_triangular_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmin = -1;
xmed = 0.5;
xmax = 1;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_triangular(n, xmin, xmed, xmax, state)
```
```

state =

17
1234
1
0
4110
11820
23399
29340
17917
13895
19930
8
0
1234
1
1
1234

x =

0.3817
-0.4348
0.4960
0.5509
-0.4398

ifail =

0

```
```function g05sp_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmin = -1;
xmed = 0.5;
xmax = 1;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sp(n, xmin, xmed, xmax, state)
```
```

state =

17
1234
1
0
4110
11820
23399
29340
17917
13895
19930
8
0
1234
1
1
1234

x =

0.3817
-0.4348
0.4960
0.5509
-0.4398

ifail =

0

```