combinatorics(n)						 Tcl Math Library						  combinatorics(n)

NAME

       combinatorics - Combinatorial functions in the Tcl Math Library

SYNOPSIS

       package require Tcl 8.2

       package require math ?1.2?

       ::math::ln_Gamma z

       ::math::factorial x

       ::math::choose n k

       ::math::Beta z w

DESCRIPTION

       The math package contains implementations of several functions useful in combinatorial problems.

COMMANDS

       ::math::ln_Gamma z
	      Returns the natural logarithm of the Gamma function for the argument z.

	      The Gamma function is defined as the improper integral from zero to positive infinity of
		t**(x-1)*exp(-t) dt

	      The  approximation  used	in the Tcl Math Library is from Lanczos, ISIAM J. Numerical Analysis, series B, volume 1, p. 86.  For "x >
	      1", the absolute error of the result is claimed to be smaller than 5.5*10**-10 -- that is, the resulting value of Gamma when
		exp( ln_Gamma( x) )
	      is computed is expected to be precise to better than nine significant figures.

       ::math::factorial x
	      Returns the factorial of the argument x.

	      For integer x, 0 <= x <= 12, an exact integer result is returned.

	      For integer x, 13 <= x <= 21, an exact floating-point result is returned on machines with IEEE floating point.

	      For integer x, 22 <= x <= 170, the result is exact to 1 ULP.

	      For real x, x >= 0, the result is approximated by computing Gamma(x+1) using  the  ::math::ln_Gamma  function,  and  the	result	is
	      expected to be precise to better than nine significant figures.

	      It is an error to present x <= -1 or x > 170, or a value of x that is not numeric.

       ::math::choose n k
	      Returns the binomial coefficient C(n, k)
		 C(n,k) = n! / k! (n-k)!
	      If both parameters are integers and the result fits in 32 bits, the result is rounded to an integer.

	      Integer results are exact up to at least n = 34.	Floating point results are precise to better than nine significant figures.

       ::math::Beta z w
	      Returns the Beta function of the parameters z and w.
		 Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)
	      Results  are  returned  as  a floating point number precise to better than nine significant digits provided that w and z are both at
	      least 1.

math									4.2							  combinatorics(n)