combinatorics(n) 4.2 "Tcl Math Library"

NAME

combinatorics - Combinatorial functions in the Tcl Math Library

SYNOPSIS

package require Tcl 8.2
package require math ?1.2.2?

::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.