module Alea::Core::SpecFun::Gamma
Overview
Implementation of the Gamma Special functions (incomplete, incomplete regularized).
Defined in:
Class Method Summary
-
.incomplete(a : Float64, x : Float64, uorl : Symbol) : Float64
Estimates the Incomplete Gamma Function
-
.incomplete(a : Float32, x : Float32, uorl : Symbol) : Float32
Estimates the Incomplete Gamma Function
-
.incomplete_reg(a : Float64, x : Float64, uorl : Symbol) : Float64
Estimates the Incomplete Regularized Gamma function in double precision.
-
.incomplete_reg(a : Float32, x : Float32, uorl : Symbol) : Float32
Estimates the Incomplete Regularized Gamma function in single precision.
Class Method Detail
Estimates 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 withinMAX_ITER
iterations.
Estimates 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 withinMAX_ITER
iterations.
Estimates the Incomplete Regularized Gamma function in double precision.
@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 withinMAX_ITER
iterations.
Estimates the Incomplete Regularized Gamma function in single precision.
@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 withinMAX_ITER
iterations.