generalized_continued_fraction_parts_iter.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 23 January 2018
# https://github.com/trizen

# Numerical evaluation of a generalized continued fraction, computing
# the numerator and the denominator separately (iterative algorithm).

#
## Compute the numerator of the continued fraction
#

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

    var (f1, f2) = (1, 0)

    for ((); to > from; ++from) {
        (f1, f2) = (f2, den.run(from)*f2 + num.run(from)*f1)
    }

    return f2
}

#
## Compute the denominator of the continued fraction
#

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

    var (f1, f2) = (0, 1)

    for ((); to > from; ++from) {
        (f1, f2) = (f2, den.run(from)*f2 + num.run(from)*f1)
    }

    return f2
}

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

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

var a = cfrac_num(1, 30, num, den)
var b = cfrac_den(1, 30, num, den)

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