Extended Euclid's Method


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In "The Art of Computer Programming, Volume 1", Donald Knuth describes the algorithm for Euclid's Method.

Euclid's method find the greatest common divisor for two numbers. The greatest common divisor of two integers \(m\) and \(n\) is the largest integer \(d\) such that both \(m\) and \(n\) are divisible by \(d\) .

The extended algorithm goes farther than just finding the gcd. The extended method finds integers \(a\) and \(b\) such that

\begin{equation*} am + bn = \gcd(m,n) \end{equation*}

This OCaml code is a straightforward translation from the algorithm in Knuth's book, and it is written longer than it needs to be to make the names clearer. Also, this is an example of replacing a loop with recursion.

let euclid_extended m n =
  let rec loop a b a' b' c d =
    let q = c / d in
    let r = c - q * d in
    Printf.printf "| %d | %d | %d | %d | %d | %d|\n" a a' b b' c d;
    if r = 0 then
      a,b,d
    else
       let c_new = d in
       let d_new = r in
       let a_new = a' - q * a in
       let a'_new = a in
       let b_new = b' - q * b in
       let b'_new = b in
       loop a_new b_new a'_new b'_new c_new d_new
  in
  loop 0 1 1 0 m n

let test() =
  let m, n = 1769, 551 in
  Printf.printf "| a| a' | b | b' | c| d|\n|---\n";
  let a,b,d = euclid_extended m n in
  Printf.printf "a %d:  b: %d\n" a b;
  Printf.printf "test result: %d\n" (a * m + b * n);

Running the code produces the following table.

a a' b b' c d
0 1 1 0 1769 551
1 0 -3 1 551 116
-4 1 13 -3 116 87
5 -4 -16 13 87 29

Here we can easily verify that \(5 * 1769 - 16 * 551 = 29\) . The table gives us a nice way to see that \(a m + b n = d\) during all of the steps of the algorithm. Of course, Knuth already has the same table in his book, but some people, like me, don't really learn something until they have done it themselves.