naginterfaces.library.stat.pdf_​vavilov

naginterfaces.library.stat.pdf_vavilov(x, comm)

pdf_vavilov returns the value of the Vavilov density function ${\varphi }_V(\lambda \text{;}\kappa ,{\beta }^2)$.

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

For full information please refer to the NAG Library document for g01mu

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01muf.html

Parameters

xfloat

The argument $\lambda$ of the function.

commdict, communication object

Communication structure.

This argument must have been initialized by a prior call to init_vavilov().

Returns

pdffloat

The value of the Vavilov density function ${\varphi }_V(\lambda \text{;}\kappa ,{\beta }^2)$.

Raises

NagValueError

(errno $1$)

Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.

Notes

pdf_vavilov evaluates an approximation to the Vavilov density function ${\varphi }_V(\lambda \text{;}\kappa ,{\beta }^2)$ given by

$${\varphi }_V(\lambda \text{;}\kappa ,{\beta }^2)=\frac{1}{2\pi i}{\int }_{c-i\infty }^{c+i\infty }{e}^{\lambda s}f(s\text{;}\kappa ,{\beta }^2)ds\text{,}$$

where $\kappa >0$ and $0\le {\beta }^2\le 1$, $c$ is an arbitrary real constant and

$$f(s\text{;}\kappa ,{\beta }^2)=C(\kappa ,{\beta }^2)exp\{sln\left(\kappa \right)+(s+\kappa {\beta }^2)[ln\left(\frac{s}{\kappa }\right)+E_1\left(\frac{s}{\kappa }\right)]-\kappa exp(-\frac{s}{\kappa })\}\text{.}$$

$E_1\left(x\right)={\int }_0^x{t}^{-1}(1-e^{-t})dt$ is the exponential integral, $C(\kappa ,{\beta }^2)=exp\left\{\kappa (1+\gamma {\beta }^2)\right\}$ and $\gamma$ is Euler’s constant.

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

For values of $\kappa \le 0.01$, the Vavilov distribution can be replaced by the Landau distribution since ${\lambda }_V=({\lambda }_L-ln\left(\kappa \right))/\kappa$. For values of $\kappa \ge 10$, the Vavilov distribution can be replaced by a Gaussian distribution with mean $\mu =\gamma -1-{\beta }^2-ln\left(\kappa \right)$ and variance ${\sigma }^2=(2-{\beta }^2)/2\kappa$.

References

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