NAG Library Routine Document

G01MUF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01MUF returns the value of the Vavilov density function

ϕ_{V} (λ; κ, β^{2})

, via the routine name.

It is intended to be used after a call to G01ZUF.

2 Specification

FUNCTION G01MUF (

X, RCOMM, IFAIL)

REAL (KIND=nag_wp) G01MUF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X, RCOMM(322)

3 Description

G01MUF 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 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $λ$ of the function.
2: RCOMM( $322$ ) – REAL (KIND=nag_wp) arrayCommunication Array: On entry: this must be the same parameter RCOMM as returned by a previous call to G01ZUF.
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

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

7 Accuracy

At least five significant digits are usually correct.

8 Further Comments

G01MUF can be called repeatedly with different values of

λ

provided that the values of

κ

and

β^{2}

remain unchanged between calls. Otherwise, G01ZUF must be called again. This is illustrated in Section 9.

9 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 Routine DocumentG01MUF