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math::combinatorics - Combinatorial functions in the Tcl Math Library

- package require
**Tcl 8.2** - package require
**math ?1.2.3?**

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 whenexp( 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 SF Trackers.
Please also report any ideas for enhancements you may have for either
package and/or documentation.

Mathematics