NAG FL Interface
g01muf (pdf_​vavilov)

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1 Purpose

g01muf returns the value of the Vavilov density function ϕV(λ;κ,β2).
It is intended to be used after a call to g01zuf.

2 Specification

Fortran Interface
Function g01muf ( x, rcomm, ifail)
Real (Kind=nag_wp) :: g01muf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x, rcomm(322)
C Header Interface
#include <nag.h>
double  g01muf_ (const double *x, const double rcomm[], Integer *ifail)
The routine may be called by the names g01muf or nagf_stat_pdf_vavilov.

3 Description

g01muf evaluates an approximation to the Vavilov density function ϕV(λ;κ,β2) given by
ϕV(λ;κ,β2)=12πi c-i c+ieλsf(s;κ,β2)ds,  
where κ>0 and 0β21, c is an arbitrary real constant and
f(s;κ,β2)=C(κ,β2)exp{slnκ+(s+κβ2)[ln(sκ)+E1(sκ)]-κexp(-sκ)} .  
E1(x)=0xt-1(1-e-t)dt is the exponential integral, C(κ,β2)=exp{κ(1+γβ2)} 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 κ0.01, the Vavilov distribution can be replaced by the Landau distribution since λV=(λL-lnκ)/κ. For values of κ10, the Vavilov distribution can be replaced by a Gaussian distribution with mean μ=γ-1-β2-lnκ and variance σ2=(2-β2)/2κ.

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 Real (Kind=nag_wp) Input
On entry: the argument λ of the function.
2: rcomm(322) Real (Kind=nag_wp) array Communication Array
On entry: this must be the same argument rcomm as returned by a previous call to g01zuf.
3: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Either the initialization routine has not been called prior to the first call of this routine or a communication array has become corrupted.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

At least five significant digits are usually correct.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01muf is not threaded in any implementation.

9 Further Comments

g01muf can be called repeatedly with different values of λ provided that the values of κ and β2 remain unchanged between calls. Otherwise, g01zuf must be called again.

10 Example

This example evaluates ϕV(λ;κ,β2) at λ=2.5, κ=0.4 and β2=0.1, and prints the results.

10.1 Program Text

Program Text (g01mufe.f90)

10.2 Program Data

Program Data (g01mufe.d)

10.3 Program Results

Program Results (g01mufe.r)