g01mu returns the value of the Vavilov density function

ϕ_{V} (λ; κ, β^{2})

It is intended to be used after a call to g01zu.

Syntax

C#
public static double g01mu( double x, double[] rcomm, out int ifail )

Visual Basic
Public Shared Function g01mu ( _ x As Double, _ rcomm As Double(), _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01mu( double x, array<double>^ rcomm, [OutAttribute] int% ifail )

F#
static member g01mu : x : float * rcomm : float[] * ifail : int byref -> float

Parameters

x: Type: System..::..Double
On entry: the argument $λ$ of the function.

rcomm: Type: array<System..::..Double>[]()[][]
An array of size [ $322$ ]
On entry: this must be the same parameter rcomm as returned by a previous call to g01zu.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01mu returns the value of the Vavilov density function

ϕ_{V} (λ; κ, β^{2})

It is intended to be used after a call to g01zu.

Description

g01mu evaluates an approximation to the Vavilov density function

ϕ_{V} (λ; κ, β^{2})

given by

ϕ_{V} (λ; κ, β^{2}) = \frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} e^{λ s} f (s; κ, β^{2}) d s,

where

κ > 0

and

0 \leq β^{2} \leq 1

c

is an arbitrary real constant and

f (s; κ, β^{2}) = C (κ, β^{2}) \exp \{s \ln κ + (s + κ β^{2}) [\ln (\frac{s}{κ}) + E_{1} (\frac{s}{κ})] - κ \exp (- \frac{s}{κ})\} .

E_{1} (x) = \int_{0}^{x} t^{- 1} (1 - e^{- t}) d t

is the exponential integral,

C (κ, β^{2}) = \exp \{κ (1 + γ β^{2})\}

and

γ

is Euler's constant.

The method used is based on Fourier expansions. Further details can be found in Schorr (1974).

For values of

κ \leq 0.01

, the Vavilov distribution can be replaced by the Landau distribution since

λ_{V} = (λ_{L} - \ln κ) / κ

. For values of

κ \geq 10

, the Vavilov distribution can be replaced by a Gaussian distribution with mean

μ = γ - 1 - β^{2} - \ln κ

and variance

σ^{2} = (2 - β^{2}) / 2 κ

References

Schorr B (1974) Programs for the Landau and the Vavilov distributions and the corresponding random numbers Comp. Phys. Comm. 7 215–224

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: Either the initialization method has not been called prior to the first call of this method or a communication array has become corrupted.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

At least five significant digits are usually correct.

Parallelism and Performance

None.

Further Comments

g01mu can be called repeatedly with different values of

λ

provided that the values of

κ

and

β^{2}

remain unchanged between calls. Otherwise, g01zu must be called again. This is illustrated in [Example].

Example

This example evaluates

ϕ_{V} (λ; κ, β^{2})

λ = 2.5

κ = 0.4

and

β^{2} = 0.1

, and prints the results.

Example program (C#): g01mue.cs

Example program data: g01mue.d

Example program results: g01mue.r