Math::Combinatorics(3) Combinatorial functions in the Tcl Math Library


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


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


::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.


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 []. Please also report any ideas for enhancements you may have for either package and/or documentation.