Standard method. Currently there is only one method for these functions.

uniform_method::accurate

checks for additional

s

and

d

data types. For

integer

data types, it uses

d

as a

BRNG

data type (

s

BRNG

data type is used in

uniform_method::standard

method on GPU).

Generates normally distributed random number x thru the pair of uniformly distributed numbers u1 and u2 according to the formula:

Generates normally distributed random numbers x1 and x2 thru the pair of uniformly distributed numbers u1 and u2 according to the formulas: Lognormal distribution: generated normally distributed random numbers x1 and x2 are converted to lognormal distribution.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

Normally distributed random numbers x1 and x2 are produced through the pair of uniformly distributed numbers u1 and u2 according to the formulas: Then x1 and x2 are converted to lognormal distribution.

Inverse cumulative distribution function (ICDF) method.

Inverse cumulative distribution function (ICDF) method.

For

α > 1

, a gamma distributed random number is generated as a cube of properly scaled normal random number; for

0.6 ≤α < 1

, a gamma distributed random number is generated using rejection from Weibull distribution; for

α < 0.6

, a gamma distributed random number is obtained using transformation of exponential power distribution; for

α = 1

, gamma distribution is reduced to exponential distribution.

Cheng-Johnk-Atkinson method. For

min(p, q) > 1

, Cheng method is used; for

min(p, q) < 1

, Johnk method is used, if

q + K·p2+ C≤ 0 (K = 0.852..., C=-0.956...)

otherwise, Atkinson switching algorithm is used; for

max(p, q) < 1

, method of Johnk is used; for

min(p, q) < 1, max(p, q)> 1

, Atkinson switching algorithm is used (CJA stands for Cheng, Johnk, Atkinson); for

p = 1or q = 1

, inverse cumulative distribution function method is used; for

p = 1

and

q = 1

, beta distribution is reduced to uniform distribution.

(most common): If

ν ≥ 17

or ν is odd and

5 ≤ ν ≤ 15

, a chi-square distribution is reduced to a Gamma distribution with these parameters: Shape

α = ν / 2

Offset a = 0 Scale factor

β = 2

. The random numbers of the Gamma distribution are generated.

BoxMuller method for multivariate Gaussian distribution. BoxMuller_2 method for multivariate Gaussian distribution. Inverse cumulative distribution function (ICDF) method.

Acceptance/rejection method for

ntrial·min(p, 1p) ≥ 30

with decomposition into four regions:

Two parallelograms

Triangle

Left exponential tail

Right exponenetial tail

Acceptance/rejection method for

λ≥ 27

with decomposition into four regions:

Two parallelograms

Triangle

Left exponential tail

Right exponenetial tail

for

λ≥ 1

, method based on Poisson inverse CDF approximation by Gaussian inverse CDF; for

λ < 1

, table lookup method is used.

Acceptance/rejection method for large mode of distribution with decomposition into three regions:

Rectangular

Left exponential tail

Right exponential tail

Acceptance/rejection method for: with decomposition into five regions:

Rectangular

(2) trapezoid

Left exponential tail

Right exponential tail

Multinomial distribution with parameters

m

,

k

, and a probability vector

p

. Random numbers of the multinomial distribution are generated by Poisson Approximation method.