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NAG Toolbox

NAG Toolbox: nag_rand_dist_cauchy (g05sc)

Purpose

nag_rand_dist_cauchy (g05sc) generates a vector of pseudorandom numbers from a Cauchy distribution with median aa and semi-interquartile range bb.

Syntax

[state, x, ifail] = g05sc(n, xmed, semiqr, state)
[state, x, ifail] = nag_rand_dist_cauchy(n, xmed, semiqr, state)

Description

The distribution has PDF (probability density function)
f(x) = 1/(πb (1 + ((xa)/b)2) ).
f(x)=1πb (1+ (x-ab) 2) .
nag_rand_dist_cauchy (g05sc) returns the value
a + b(2y1 1)/(y2),
a+b2y1- 1y2,
where y1y1 and y2y2 are a pair of consecutive pseudorandom numbers from a uniform distribution over (0,1)(0,1), such that
(2y11)2 + y221.
(2y1-1) 2+y221.
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_cauchy (g05sc).

References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of pseudorandom numbers to be generated.
Constraint: n0n0.
2:     xmed – double scalar
aa, the median of the distribution.
3:     semiqr – double scalar
bb, the semi-interquartile range of the distribution.
Constraint: semiqr0.0semiqr0.0.
4:     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.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

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 nn pseudorandom numbers from the specified Cauchy distribution.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, n < 0n<0.
  ifail = 3ifail=3
On entry, semiqr < 0.0semiqr<0.0.
  ifail = 4ifail=4
On entry,state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_dist_cauchy_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmed = 1;
semiqr = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_cauchy(n, xmed, semiqr, state)
 

state =

                   17
                 1234
                    1
                    0
                 9910
                16740
                20386
                10757
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    6.1229
    2.2328
   -2.2118
    0.4118
    0.9892


ifail =

                    0


function g05sc_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmed = 1;
semiqr = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sc(n, xmed, semiqr, state)
 

state =

                   17
                 1234
                    1
                    0
                 9910
                16740
                20386
                10757
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    6.1229
    2.2328
   -2.2118
    0.4118
    0.9892


ifail =

                    0



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