NAG Library Function Document

nag_prob_density_vavilov (g01muc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_prob_density_vavilov (g01muc) returns the value of the Vavilov density function

ϕ_{V} (λ; κ, β^{2})

It is intended to be used after a call to nag_init_vavilov (g01zuc).

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_density_vavilov (double x, const double comm_arr[])

3 Description

nag_prob_density_vavilov (g01muc) 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

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $λ$ of the function.
2: comm_arr[ $322$ ] – const doubleCommunication Array: On entry: this must be the same argument comm_arr as returned by a previous call to nag_init_vavilov (g01zuc).

6 Error Indicators and Warnings

None.

7 Accuracy

At least five significant digits are usually correct.

8 Parallelism and Performance

Not applicable.

9 Further Comments

nag_prob_density_vavilov (g01muc) can be called repeatedly with different values of

λ

provided that the values of

κ

and

β^{2}

remain unchanged between calls. Otherwise, nag_init_vavilov (g01zuc) must be called again. This is illustrated in Section 10.

10 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.

NAG Library Function Documentnag_prob_density_vavilov (g01muc)