combinatorics man page on Ubuntu

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math::combinatorics(3tcl)      Tcl Math Library	     math::combinatorics(3tcl)

______________________________________________________________________________

NAME
       math::combinatorics - Combinatorial functions in the Tcl Math Library

SYNOPSIS
       package require Tcl  8.2

       package require math  ?1.2.3?

       ::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 sig‐
	      nificant 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  fig‐
	      ures.

	      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  bet‐
	      ter  than nine significant digits provided that w and z are both
	      at least 1.

BUGS, IDEAS, FEEDBACK
       This document, and the package it describes, will  undoubtedly  contain
       bugs  and  other	 problems.  Please report such in the category math of
       the	   Tcllib	  SF	     Trackers	       [http://source‐
       forge.net/tracker/?group_id=12883].   Please  also report any ideas for
       enhancements you may have for either package and/or documentation.

CATEGORY
       Mathematics

math				     1.2.3	     math::combinatorics(3tcl)
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