Generates pseudorandom numbers from a Cauchy distribution.

Routine RNCHY generates pseudorandom numbers from a standard Cauchy distribution. The probability density function is

Use of the inverse CDF technique would yield a Cauchy deviate from a uniform (0, 1) deviate, u, as tan[π(u ‑ 5)]. Rather than evaluating a tangent directly, however, RNCHY generates two uniform (‑1, 1) deviates, x1 and x2. These values can be thought of as sine and cosine values. If

is less than or equal to 1, then x1/x2 is delivered as the Cauchy deviate; otherwise, x1 and x2 are rejected and two new uniform (–1, 1) deviates are generated. This method is also equivalent to taking the ratio of two independent normal deviates.

Deviates from the Cauchy distribution with median T and first quartile T ‑ S, that is, with density

can be obtained by scaling the output from RNCHY. The following statements (in single precision) would yield random deviates from this Cauchy distribution.

The Cauchy distribution is a member of the symmetric stable family of distributions. The routine RNSTA can be used to generate deviates from this more general family of distributions or even from the stable family not requiring symmetry.

In this example, RNCHY is used to generate five pseudorandom deviates from a Cauchy distribution.