Introduction

A Matlab code to generate random numbers following multi-dimensional Cauchy distributions is provided here.

Citation

The following publication contains mathematical details of the method and the code, which should be cited whenever relevant.

• J.-S. Lee, C. H. Park, and T. Ebrahimi, "Theory and applications of hybrid simulated annealing," in Handbook of Optimization, I. Zelinka, V. Snasel, and A. Abraham (eds.), Springer, pp. 395-422, 2013

Code

cauchyn.m

function [x, r] = cauchyn(t, y, T, m, n) % x - n-dimensional cauchy random numbers % r - norm of x % [t, y] - lookup table % T - Cauchy parameter % m - number of random numbers % n - random number dimension u = rand(m,1); c = find(y < 0); if isempty(c) y1 = y; t1 = t; else y1 = y(c(size(c,1))+1:size(y,1),:); t1 = t(c(size(c,1))+1:size(t,1),:); end theta = interp1(y1, t1, u); r = T * tan(theta); x = randn(m,n); x = repmat(r, [1,size(x,2)]) .* x ./ repmat(sqrt(sum(x .* x, 2)), [1,size(x, 2)]);

cauchyn_gen.m

function [t,y] = cauchyn_gen(n, s) % n - dimension % s - size of lookup table [t,y] = ode45(@cauchyn_dydt, [pi/(2*s):pi/(2*s):pi/2], 0, 'AbsTol', n);

cauchyn_dydt.m

function dydt = cauchyn_dydt(t, y, n) dydt = sin(t)^(n-1);

Usage

1. Determine the number of random numbers (m), their dimension (n), and the Cauchy parameter (T).
2. Get the n-dimensional CDF lookup table having a size of s. s=20000 is a reasonable choice.
   [t, y] = cauchyn_gen(n, s)
3. Generate n-dimensional Cauchy random numbers x.
   [x, r] = cauchyn(t, y, T, m, n)