subtraction in bash arrays Post: 302274949

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combinatorics

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

_________________________________________________________________

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  SF	Trackers [http://sourceforge.net/tracker/?group_id=12883].  Please also report any ideas for enhancements you may have for
       either package and/or documentation.

math								       1.2.3						    math::combinatorics(n)