1Math::NumSeq::FactorialUss(e3r)Contributed Perl DocumentMaattiho:n:NumSeq::Factorials(3)
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NAME

6       Math::NumSeq::Factorials -- factorials i! = 1*2*...*i
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SYNOPSIS

9        use Math::NumSeq::Factorials;
10        my $seq = Math::NumSeq::Factorials->new;
11        my ($i, $value) = $seq->next;
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DESCRIPTION

14       The factorials being product 1*2*3*...*i, 1 to i inclusive.
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16           1, 2, 6, 24, 120, 720, ...
17           starting i=1
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FUNCTIONS

20       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
21       classes.
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23       "$seq = Math::NumSeq::Factorials->new ()"
24           Create and return a new sequence object.
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26   Iterating
27       "$seq->seek_to_i($i)"
28           Move the current sequence position to $i.  The next call to
29           "next()" will return $i and corresponding value.
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31   Random Access
32       "$value = $seq->ith($i)"
33           Return "1*2*...*$i".  For "$i==0" this is considered an empty
34           product and the return is 1.
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36       "$bool = $seq->pred($value)"
37           Return true if $value is a factorial, ie. equal to "1*2*...*i" for
38           some i.
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40       "$i = $seq->value_to_i($value)"
41       "$i = $seq->value_to_i_floor($value)"
42           Return the index i of $value.  If $value is not a factorial then
43           "value_to_i()" returns "undef", or "value_to_i_floor()" the i of
44           the next lower value which is or "undef" if "$value < 1".
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46       "$i = $seq->value_to_i_estimate($value)"
47           Return an estimate of the i corresponding to $value.
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FORMULAS

50   Value to i Estimate
51       The current code uses Stirling's approximation
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53           log(n!) ~= n*log(n) - n
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55       by seeking an i for which the target factorial "value" has
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57           i*log(i) - i == log(value)
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59       Newton's method is applied to solve for i,
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61           target=log(value)
62           f(x) = x*log(x) - x - target      wanting f(x)=0
63           f'(x) = log(x)
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65           iterate next_x = x - f(x)/f'(x)
66                          = (x+target)/log(x)
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68       Just two iterations is quite close
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70           target = log(value)
71           i0 = target
72           i1 = (i0+target)/log(target)
73              = 2*target/log(target)
74           i2 = (i1+target)/log(i1)
75
76           i ~= int(i2)
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SEE ALSO

79       Math::NumSeq, Math::NumSeq::Primorials
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81       Math::BigInt ("bfac()"), Math::Combinatorics ("factorial()",
82       Math::NumberCruncher ("Factorial()" Math::BigApprox ("Fact()"
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HOME PAGE

85       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

88       Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019 Kevin Ryde
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90       Math-NumSeq is free software; you can redistribute it and/or modify it
91       under the terms of the GNU General Public License as published by the
92       Free Software Foundation; either version 3, or (at your option) any
93       later version.
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95       Math-NumSeq is distributed in the hope that it will be useful, but
96       WITHOUT ANY WARRANTY; without even the implied warranty of
97       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
98       General Public License for more details.
99
100       You should have received a copy of the GNU General Public License along
101       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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105perl v5.34.0                      2021-07-22       Math::NumSeq::Factorials(3)
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