beta

beta function

Calling Sequence

z = beta(x,y)

Arguments

:x, y 2 positive reals or 2 matrices (or vectors) of positive reals of

same size.

: :z a real or a matrix of the same size than x with `z(i,j) =

beta(x(i,j),y(i,j))`.

:

Description

Computes the complete beta function :

For small x and y the algorithm uses the expression in function of the gamma function, else it applies the exponential function onto the result of the betaln function provided with the DCDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameter (see cdfbet for more information about DCDFLIB).

Examples

// example 1 :
beta(5,2) - beta(2,5)   // symetry (must be exactly 0)
beta(0.5,0.5)           // exact value is pi

// example 2 : an error study based on the relation  B(1,x) = 1/x
// (computing 1/x must lead to only a relative error of eps_m, so
//  it may be used near as a reference to evaluate the error in B(1,x))
x = `logspace`_(-8,8,20000)';
e = beta(`ones`_(x),x) - (1)./x;
er = `abs`_(e) .* x;
ind = `find`_(er ~= 0);
eps = `ones`_(x(ind))*`number_properties`_("eps");
`clf`_()
`plot2d`_(x(ind),[er(ind) eps 2*eps],style=[1 2 3],logflag="ll",leg="er@eps_m@2 eps_m")
`xtitle`_("approximate relative error in computing beta(1,x)")
`show_window`_()

// example 3 : plotting the beta function
t = `linspace`_(0.2,10,60);
X = t'*`ones`_(t); Y = `ones`_(t')*t;
Z = beta(X,Y);
`clf`_()
`plot3d`_(t, t, Z, flag=[2 4 4], leg="x@y@z", alpha=75, theta=30)
`xtitle`_("The beta function on [0.2,10]x[0.2,10]")
`show_window`_()

See Also

• gamma The gamma function.
• cdfbet cumulative distribution function Beta distribution