G115: Approximate Vavilov Distribution and its Inverse

Author(s): A. Rotondi, P. Montagna, K.S. Kölbig	Library: MATHLIB
Submitter: K.S. Kölbig	Submitted: 10.12.1993
Language: Fortran	Revised:

The VAVLOV package contains subprograms for fast approximate calculation of functions related to the Vavilov distribution.

For and , the Vavilov density function is mathematically defined by

where c is an arbitrary real constant and

is the exponential integral, , and is Euler's constant.

The Vavilov distribution function is defined by

and its inverse by .

The function can be used to generate Vavilov random numbers (see Usage).

Structure:

SUBROUTINE and FUNCTION subprograms
User Entry Names: VAVSET, VAVDEN, VAVDIS, VAVRND, VAVRAN
External References: LOCATR, DENLAN, DISLAN
COMMON Block Names and Lenghts: /G115C1/ 226

Usage:

    CALL VAVSET(RKAPPA,BETA2,MODE)

sets auxiliary quantities used in VAVDEN, VAVDIS and VAVRND; this call has to precede a reference to any of these entries.

RKAPPA

The variable (the straggling parameter); ( ).

BETA2

The variable (the square of velocity in unit c); ( ).

MODE

in the particular case that VAVDEN only is referenced after the call to VAVSET.

In any arithmetic expression,

VAVDEN(X) has an approximate value of

VAVDIS(X) has an approximate value of

VAVRND(X) has an approximate value of

RKAPPA and BETA2 are defined by the last call to VAVSET prior to a reference to VAVDEN, VAVDIS, or VAVRND.

To generate a set of Vavilov random numbers with identical and , VAVSET should be called once and then VAVRND be referenced repeatedly, using as argument X a random number from a uniform distribution over the interval (0,1).

In any arithmetic expression,

VAVRAN(RKAPPA,BETA2,X) has an approximate value of .

To generate one Vavilov random number for given values of and , VAVRAN should be used, using as argument X a random number from a uniform distribution over the interval (0,1).

VAVDEN, VAVDIS, VAVRND, VAVRAN and X, RKAPPA, BETA2 are of type REAL, and MODE is of type INTEGER.

Method:

The method is discribed in Ref. 1.

Accuracy:

The accuracy depends on the parameters. Although often rather poor from a mathematical point of view, it is usually sufficient for the intended application in physics (see Notes).

Restrictions:

No test is made whether the parameters and are in the specified ranges.

Notes:

1. Representing the Vavilov functions by approximations which are both fast and accurate is a difficult task. In view of the requirements in physics, speed is much more important than accuracy. This is taken into account for the present routines.
2. For a more accurate, but much slower, calculation of the Vavilov density and distribution functions, use VVILOV (G116).
3. For , the Vavilov distribution can be replaced by the Landau distribution (LANDAU (G110)), taking into account that .
4. For , the Vavilov distribution can be replaced by the Gaussian distribution with mean
   and variance .

References:

1. A. Rotondi and P. Montagna, Fast calculation of Vavilov distribution, Nucl. Instr. and Meth. B47 (1990) 215-224.