combinatorics (3)

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

CATEGORY

Mathematics