math::combinatorics(n) Tcl Math Library math::combinatorics(n) ______________________________________________________________________________
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
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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 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 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 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.
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 SF Trackers [http://source-
forge.net/tracker/?group_id=12883]. Please also report any ideas for
enhancements you may have for either package and/or documentation.
CATEGORY
Mathematics
math 1.2.3 math::combinatorics(n)
Mac OS X 10.8 - Generated Wed Sep 5 16:28:27 CDT 2012