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 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.

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   Trackers
       [http://core.tcl.tk/tcllib/reportlist].  Please also report any ideas for enhancements you
       may have for either package and/or documentation.

       When proposing code changes, please provide unified diffs, i.e the output of diff -u.

       Note further that attachments are strongly preferred over inlined patches. Attachments can
       be  made  by going to the Edit form of the ticket immediately after its creation, and then
       using the left-most button in the secondary navigation bar.

CATEGORY

       Mathematics