In [1]:
using Symata;
Compute the gamma function for a=3
. The result and estimated error are returned.
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a = 3.0
NIntegrate(Exp(-t)*t^(a-1), [t,0,Infinity])
Out[2]:
Define a function gamma
, where gamma(a)
is the gamma function at a
and gamma(a,x)
is the lower incomplete gamma function. The default value of the second argument is Infinity
.
In [3]:
gamma(a_, x_:Infinity) := NIntegrate(Exp(-t)*t^(a-1), [t,0,x])[1]
Test the function on a List
of numbers.
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Map(gamma, [3.0, 4.0, 5.0])
Out[4]:
The integrand in the function gamma
is a Symata expression that is re-evaluated every time the integration routine asks for a value. Using a compiled function in the integrand is more efficient.
Create the compiled function like this.
In [5]:
gammaint = Compile([t] , Exp(-t)*t^(a-1));
gammaint
refers to a Julia function of one variable, x
. The variable a
is free. To use gammaint
, we have to export the Symata variable to Julia using ExportJ
(or SetJ
).
The alternative gamma function is
In [6]:
gamma1(a1_, x_:Infinity) := (SetJ(a,a1), NIntegrate(gammaint, [0,x]))[1]
In [7]:
gamma1(3.0)
Out[7]:
Timing shows that gamma1
is about 50 times faster than gamma
.
In [8]:
Timing(gamma(3.0))
Out[8]:
In [9]:
Timing(gamma1(3.0))
Out[9]:
In [10]:
VersionInfo()
In [11]:
InputForm(Now())
Out[11]: