pell_method_for_square_roots.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 04 February 2019
# https://github.com/trizen

# Approximate square roots, using Pell's method.

# See also:
#   https://rosettacode.org/wiki/Pell%27s_equation
#   https://en.wikipedia.org/wiki/Pell%27s_equation
#   https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

func pell_square_root(n, eps=1e-22) {

    var x = n.isqrt
    var y = x
    var z = 1
    var r = 2*x

    var (e1, e2) = (1, 0)
    var (f1, f2) = (0, 1)

    loop {

        y = (r*z - y)
        z = ((n - y*y) / z)
        r = round((x + y) / z)      # floor() also works

        (e1, e2) = (e2, r*e2 + e1)
        (f1, f2) = (f2, r*f2 + f1)

        var A = (e2 + x*f2)
        var B = f2

        if (abs((A/B)**2 - n) <= eps) {
            return A/B
        }
    }
}

for n in [61, 109, 181, 277] {
    var s = pell_square_root(n)
    say "sqrt(#{'%3d' % n}) =~ #{s.as_rat} =~ #{s}"
}

__END__
sqrt( 61) =~ 5380205503727/688864726095 =~ 7.81024967590665439412972327539046655850673601909
sqrt(109) =~ 5886776254306/563850903159 =~ 10.4403065089105501797577550769986386680615397574
sqrt(181) =~ 2900300962727/215577672795 =~ 13.45362404707371031716308866094140080774035985437
sqrt(277) =~ 4157003026204/249770104837 =~ 16.64331697709323806892821623002200061329832127014