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  ??1.2.2??

       ::::math::::lnGamma 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.

COMANDS
       ::::math::::lnGamma 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( lnGamma( 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 IE 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::::lnGamma 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.



math                                  4.2                     combinatorics(n)