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combinatorics - Combinatorial functions in the Tcl Math Library
package require Tcl 8.2
package require math 1.2
::math::ln_Gamma z
::math::factorial n
::math::choose n k
::math::Beta z w
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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, SIAM 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.