generalized_continued_fraction.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 18 January 2018
# https://github.com/trizen

# Numerical evaluation of a generalized continued fraction.

# See also:
#   https://en.wikipedia.org/wiki/Generalized_continued_fraction

#
## Compute the continued fraction numerically
#

func contfrac2num (from, to, num, den) {
    return 0 if (from > to)
    num.run(from) / (den.run(from) + __FUNC__(from + 1, to, num, den))
}

#
## Compute only the numerator of the continued fraction
#

func cfrac_num(from, to, num, den) is cached {

    return 1 if (to == from-1)
    return 0 if (to == from)

    den.run(to-1)*__FUNC__(from, to-1, num, den) + num.run(to-1)*__FUNC__(from, to-2, num, den)
}

#
## Compute only the denominator of the continued fraction
#

func cfrac_den(from, to, num, den) is cached {

    return 1 if (to == from)
    return 0 if (to == from-1)

    den.run(to-1)*__FUNC__(from, to-1, num, den) + num.run(to-1)*__FUNC__(from, to-2, num, den)
}

var from = float(1)
var to   = float(100)

#
## Example for [n=1..Infinity, n^2 / (2n+1)] = 4/Pi - 1
#

var num = {|n| n**2  }
var den = {|n| 2*n + 1 }

say contfrac2num(from, to,      num, den)
say contfrac2num(from, to+100,  num, den)
say contfrac2num(from, to+1000, num, den)
say contfrac2num(from, to+2000, num, den)

var a = cfrac_num(from, 30, num, den)
var b = cfrac_den(from, 30, num, den)

say "Numerator   = #{a}"
say "Denominator = #{b}"
say "Expansion   = #{a/b}"