++ed by:
Kevin Ryde
and 1 contributors

# 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 factorials being product 1*2*3*...*i, 1 to i inclusive.

1, 2, 6, 24, 120, 720, ...
starting i=1

# FUNCTIONS

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

\$seq = Math::NumSeq::Factorials->new ()

Create and return a new sequence object.

## Iterating

\$seq->seek_to_i(\$i)

Move the current sequence position to \$i. The next call to next() will return \$i and corresponding value.

## Random Access

\$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.

# FORMULAS

## Value to i Estimate

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::BigInt (bfac()), Math::Combinatorics (factorial(), Math::NumberCruncher (Factorial() Math::BigApprox (Fact()