NAME

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

SYNOPSIS

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

DESCRIPTION

The sequence of factorials, 1, 2, 6, 24, 120, etc, being the product 1*2*3*...*i.

FUNCTIONS

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::Factorials->new (key=>value,...)

Create and return a new sequence object.

$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 = $seq->value_to_i_floor($value)

Return the index i of $value or of the next factorial below $value.

$i = $seq->value_to_i_estimate($value)

Return an estimate of the i corresponding to $value.

FORMULAS

Value to i Estimate

The current code uses Stirling's approximation

log(n!) ~= n*log(n) - n

by seeking i for which the target factorial "value" has

i*log(i) - i == log(value)

Newton's method is used to solve for i,

target t=log(value)
f(x) = x*log(x) - x - t, seeking f(x)=0
f'(x) = log(x)

iterate new_x = x - f(x)/f'(x)
              = (x+target)/log(x)

Just two iterations is quite close

t = log(value)
i0 = t
i1 = (i0+t)/log(t)
   = 2t/log(t)
i2 = (i1+t)/log(i1)

i ~= int(i2)

SEE ALSO

Math::NumSeq, Math::NumSeq::Primorials

HOME PAGE

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE

Copyright 2010, 2011, 2012 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.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.