1math::combinatorics(n)         Tcl Math Library         math::combinatorics(n)
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NAME

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

11       package require Tcl  8.2
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13       package require math  ?1.2.3?
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15       ::math::ln_Gamma z
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17       ::math::factorial x
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19       ::math::choose n k
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21       ::math::Beta z w
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DESCRIPTION

26       The  math  package contains implementations of several functions useful
27       in combinatorial problems.
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COMMANDS

30       ::math::ln_Gamma z
31              Returns the natural logarithm of  the  Gamma  function  for  the
32              argument z.
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34              The Gamma function is defined as the improper integral from zero
35              to positive infinity of
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38                t**(x-1)*exp(-t) dt
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41       The approximation used in the Tcl Math Library is from  Lanczos,  ISIAM
42       J.  Numerical  Analysis,  series  B, volume 1, p. 86.  For "x > 1", the
43       absolute error of the result is claimed to be smaller than  5.5*10**-10
44       -- that is, the resulting value of Gamma when
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47                exp( ln_Gamma( x) )
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49
50              is  computed  is expected to be precise to better than nine sig‐
51              nificant figures.
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53       ::math::factorial x
54              Returns the factorial of the argument x.
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56              For integer x, 0  <=  x  <=  12,  an  exact  integer  result  is
57              returned.
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59              For  integer x, 13 <= x <= 21, an exact floating-point result is
60              returned on machines with IEEE floating point.
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62              For integer x, 22 <= x <= 170, the result is exact to 1 ULP.
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64              For real x, x >= 0, the  result  is  approximated  by  computing
65              Gamma(x+1)  using  the ::math::ln_Gamma function, and the result
66              is expected to be precise to better than nine  significant  fig‐
67              ures.
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69              It  is  an  error to present x <= -1 or x > 170, or a value of x
70              that is not numeric.
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72       ::math::choose n k
73              Returns the binomial coefficient C(n, k)
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76                 C(n,k) = n! / k! (n-k)!
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79              If both parameters are integers and the result fits in 32  bits,
80              the result is rounded to an integer.
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82              Integer results are exact up to at least n = 34.  Floating point
83              results are precise to better than nine significant figures.
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85       ::math::Beta z w
86              Returns the Beta function of the parameters z and w.
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89                 Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)
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92              Results are returned as a floating point number precise to  bet‐
93              ter  than nine significant digits provided that w and z are both
94              at least 1.
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BUGS, IDEAS, FEEDBACK

97       This document, and the package it describes, will  undoubtedly  contain
98       bugs  and  other  problems.  Please report such in the category math of
99       the  Tcllib  Trackers  [http://core.tcl.tk/tcllib/reportlist].   Please
100       also  report any ideas for enhancements you may have for either package
101       and/or documentation.
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103       When proposing code changes, please provide unified diffs, i.e the out‐
104       put of diff -u.
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106       Note  further  that  attachments  are  strongly  preferred over inlined
107       patches. Attachments can be made by going  to  the  Edit  form  of  the
108       ticket  immediately  after  its  creation, and then using the left-most
109       button in the secondary navigation bar.
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CATEGORY

112       Mathematics
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116tcllib                               1.2.3              math::combinatorics(n)
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