G110 Landau Distribution

The LANDAU function subprogram package contains six independent subprograms for the calculation of the following functions related to the Landau distribution:

$The\; density$	$\phi (\lambda )$	$\{12\pi i\}\int$c-i&inf;c+i&inf;exp(λs + s lns) ds,
$the\; distribution$	$\Phi (\lambda )$	$\int$-&inf;λφ(λ) d λ,
$the\; derivative$	$\phi \text{'}(\lambda )$	$\{d\phi (\lambda )d\lambda \},$
$the\; first\; moment$	$\Phi$1(x)	$\{1\Phi (x)\}\; \int$-&inf;xλφ(λ) d λ,
$the\; second\; moment$	$\Phi$2(x)	$\{1\Phi (x)\}\; \int$-&inf;xλ2φ(λ) d λ,
$the\; inverse\; of\Phi (x)$	$\Psi (x)$	$\Phi -1(x).$

The function $\Psi (x)$ can be used to generate Landau random numbers (see Usage).

Structure:

FUNCTION subprograms
User Entry Names: DENLAN, DISLAN, DIFLAN, XM1LAN, XM2LAN, RANLAN
Obsolete User Entry Names: DSTLAN $\equiv$ DISLAN

Usage:

In any arithmetic expression,

DENLAN(X)	$\phi (X)$ ,
DISLAN(X)	$\Phi (X)$ ,
DIFLAN(X)	$\phi \text{'}(X)$ ,
XM1LAN(X)	$\Phi$1(X) ,
XM2LAN(X)	$\Phi$2(X) ,
RANLAN(X)	$\Psi (X)$ ,

where DENLAN, DISLAN, DIFLAN, XM1LAN, XM2LAN, RANLAN and X are of type REAL.

To generate a set of Landau random numbers, RANLAN should be referenced repeatedly, using as argument a random number from a uniform distribution over the interval (0,1).

Method:

Approximation by rational functions. For reason of speed, RANLAN proceeds mainly by table look-up and quadratic interpolation.

Accuracy:

At least six significant digits (five for RANLAN) are correct.

Restrictions:

1. Underflow may occur for DENLAN, DISLAN and DIFLAN if X is negative and (moderately) large.
2. No test is made whether X for RANLAN lies outside the interval (0,1), and hence no error message is printed.

Notes:

This program package is a version of the CPC Program Library package LANDAU (Ref. 1).

References:

1. K.S. Kölbig and B. Schorr, A program package for the Landau distribution, Computer Phys. Comm. 31 (1984) 97--111.

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