continued_fractions_for_nth_roots.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 03 February 2019
# https://github.com/trizen

# Approximate n-th roots, using continued fractions.

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

func cfrac_nth_root (z, m, n, y, r, k=1) {

    return 0 if (r <= 0)

    (k² * n² - m²) * y² / ((2*k + 1)*n*(2*z - y) - __FUNC__(z, m, n, y, r - 1, k+1))
}

func nth_root(z, n, r = 100) {
    var (x, y) = z.irootrem(n)

    var m = 1
    var t = cfrac_nth_root(z, m, n, y, r)

    x**m + ((2 * x * m * y) / (n*(2*z - y) - m*y - t))
}

say nth_root(1234, 2)       #=> 35.12833614050059160587031162535630676454048547878
say nth_root(12345, 3)      #=> 23.11161874980726868087197332958829017451713700262
say nth_root(123456, 5)     #=> 10.43043546409766483375317008568663847055013893731