module Alea::Core
Included Modules
Defined in:
Constant Summary
-
SPECFUN_EPS32 =
1.0e-06
-
Single precision tolerance
-
SPECFUN_EPS64 =
1.0e-15
-
Double precision tolerance
-
SPECFUN_ITMAX =
1000
-
Iteration limit within which
Alea::NoConvergeError
is not raised
Class Method Summary
-
.inc_gamma(a, x, uorl)
Estimate the Incomplete Gamma Function
-
.inc_gamma_regular(a, x, uorl)
Estimate the Incomplete Regularized Gamma function.
-
.incg_regular_lower(a, x)
Estimate the Incomplete Regular Lower Gamma function.
-
.incg_regular_upper(a, x)
Estimate the Incomplete Regular Upper Gamma function.
-
.poisson_mult(lam : Float | Int, prng : Alea::PRNG)
Ok here to pass the prng: it's a reference and it will not affect repeatability.
-
.poisson_ptrs(lam : Float | Int, prng : Alea::PRNG)
Ok here to pass the prng: it's a reference and it will not affect repeatability.
Class Method Detail
Estimate the Incomplete Gamma Function
When x > (a+1)
, the upper gamma function can be evaluated as
-x a
e * x
G(a,x) ~= --------------
cf(a,x)
where cf(a,x)
is the continued fraction defined above, its coefficients
a(i)
and b(i)
are implemented in #inc_gamma_ctdfr_proc
.
When x < (a+1)
, it is more convenient to apply the following Taylor series
that evaluates the lower incomplete gamma function:
inf
-----
-x a \ G(a) i
g(a,x) ~= e * x * > ---------- * x
/ G(a+1+i)
-----
i=0
Applying the following property of the gamma function:
G(a+1) = a * G(a)
The Taylor series above can be further simplified to:
inf
----- i
-x a \ x
g(a,x) ~= e * x * > -------------------------
/ a * (a+1) * ... * (a+i)
-----
i=0
Once either a lower or an upper incomplete gamma function is evaluated, the other value may be quickly obtained by applying the following property of the incomplete gamma function:
G(a,x) + g(a,x) = G(a)
@parameters:
a
: parameter of the Incomplete Gamma function.x
: the upper integration limit.uorl
: symbol to request the:upper
or:lower
inc. gamma function.
@exceptions:
Alea::NoConvergeError
if no convergence occurs withinSPECFUN_ITMAX
iterations.
Estimate the Incomplete Regularized Gamma function.
@parameters:
a
: parameter of the Incomplete Gamma function.x
: the upper integration limit.uorl
: symbol to request the:upper
or:lower
inc. reg. gamma function.
@references:
@exceptions:
Alea::NoConvergeError
if no convergence occurs withinSPECFUN_ITMAX
iterations.
Estimate the Incomplete Regular Lower Gamma function.
@parameters:
a
: parameter of the Incomplete Gamma function.x
: the upper integration limit.
@references:
@exceptions:
Alea::NoConvergeError
if no convergence occurs withinSPECFUN_ITMAX
iterations.
Estimate the Incomplete Regular Upper Gamma function.
@parameters:
a
: parameter of the Incomplete Gamma function.x
: the upper integration limit.
@references:
@exceptions:
Alea::NoConvergeError
if no convergence occurs withinSPECFUN_ITMAX
iterations.
Ok here to pass the prng: it's a reference and it will not affect repeatability.
Ok here to pass the prng: it's a reference and it will not affect repeatability.