


math::combinatorics(n)   Tcl Math Library  math::combinatorics(n)


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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

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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.




math                          1.2.3                             1





math::combinatorics(n)   Tcl Math Library  math::combinatorics(n)


              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                             2


