G115 Approximate Vavilov Distribution and its Inverse

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

For $\kappa >0$ and $0\; \le \beta 2\le 1$ , the Vavilov density function is mathematically defined by

$\phi$_V(λ;κ,β²) = {12πi} ∫_c-i&inf;^c+i&inf;e^λs f(s;κ,β²) ds,

where c is an arbitrary real constant and

$f(s;\kappa ,\beta 2)\; =\; C(\kappa ,\beta 2)\hspace{0.17em}exps\; ln\kappa +\; (s+\kappa \beta 2)\hspace{0.17em}[\; ln(\{s\kappa \})+E$₁({sκ}) ]-κ exp(-{sκ}) .

$E$₁(x)=∫₀^xt^-1 (1-e^-t) dt is the exponential integral, $C(\kappa ,\beta 2)=exp\kappa (1+\beta 2\gamma )$ , and $\gamma =0.57721...\hspace{0.17em}$ is Euler's constant.

The Vavilov distribution function is defined by

$\Phi$_V(λ;κ,β²) = ∫_-&inf;^λφ_V(λ;κ,β²) dλ

and its inverse by $\Psi$_V(x;κ,β²)=Φ_V^-1(x;κ,β²) .

The function $\Psi$_V(x;κ,β²) 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: LOCATF (E106), DENLAN, DISLAN (G110)
COMMON Block Names and Lenghts: /G115C1/ 226

Usage:

    CALL VAVSET(RKAPPA,BETA2,MODE)

RKAPPA

The variable $\kappa$ (the straggling parameter); ($0.01\; \le \kappa \le 12$ ).

BETA2

The variable $\beta 2$ (the square of velocity in unit c); ($0\; \le \beta 2\le 1$ ).

MODE

$=\; 1;$
$=\; 0$ in the particular case that VAVDEN only is referenced after the call to VAVSET.

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 $\kappa$

and $\beta 2$ , 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 $\Psi$_V(X;RKAPPA,BETA2) .

To generate one Vavilov random number for given values of $\kappa$

and $\beta 2$ , 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 $\kappa$ and $\beta 2$

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 $\kappa \le 0.01$ , the Vavilov distribution can be replaced by the Landau distribution ( LANDAU (G110)), taking into account that $\lambda$_V=(λ_L-lnκ)/κ .
4. For $\kappa \ge 10$ , the Vavilov distribution can be replaced by the Gaussian distribution with mean
   $\mu =\gamma -1-\beta 2-ln\kappa$ and variance $\sigma 2=(2-\beta 2)/(2\kappa )$ .

References:

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

G116