Math::NumSeq::Factorials -- factorials i! = 1*2*...*i
use Math::NumSeq::Factorials; my $seq = Math::NumSeq::Factorials->new; my ($i, $value) = $seq->next;
The factorials being product 1*2*3*...*i, 1 to i inclusive.
1, 2, 6, 24, 120, 720, ... starting i=1
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::Factorials->new ()
Create and return a new sequence object.
$seq->seek_to_i($i)
Move the current sequence position to $i
. The next call to next()
will return $i
and corresponding value.
$value = $seq->ith($i)
Return 1*2*...*$i
. For $i==0
this is considered an empty product and the return is 1.
$bool = $seq->pred($value)
Return true if $value
is a factorial, ie. equal to 1*2*...*i
for some i.
$i = $seq->value_to_i($value)
$i = $seq->value_to_i_floor($value)
Return the index i of $value
. If $value
is not a factorial then value_to_i()
returns undef
, or value_to_i_floor()
the i of the next lower value which is or undef
if $value < 1
.
$i = $seq->value_to_i_estimate($value)
Return an estimate of the i corresponding to $value
.
The current code uses Stirling's approximation
log(n!) ~= n*log(n) - n
by seeking an i for which the target factorial "value" has
i*log(i) - i == log(value)
Newton's method is applied to solve for i,
target=log(value) f(x) = x*log(x) - x - target wanting f(x)=0 f'(x) = log(x) iterate next_x = x - f(x)/f'(x) = (x+target)/log(x)
Just two iterations is quite close
target = log(value) i0 = target i1 = (i0+target)/log(target) = 2*target/log(target) i2 = (i1+target)/log(i1) i ~= int(i2)
Math::NumSeq, Math::NumSeq::Primorials
Math::BigInt (bfac()
), Math::Combinatorics (factorial()
, Math::NumberCruncher (Factorial()
Math::BigApprox (Fact()
http://user42.tuxfamily.org/math-numseq/index.html
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
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