source: CGBLisp/trunk/src/xgcd.lisp

;;; -*- Mode: Lisp; Syntax: Common-Lisp; Package: Grobner; Base: 10 -*-
#|
  *--------------------------------------------------------------------------*
  |  Copyright (C) 1994, Marek Rychlik (e-mail: rychlik@math.arizona.edu)    |
  |    Department of Mathematics, University of Arizona, Tucson, AZ 85721    |
  |                                                                          |
  | Everyone is permitted to copy, distribute and modify the code in this    |
  | directory, as long as this copyright note is preserved verbatim.         |
  *--------------------------------------------------------------------------*
|#

(defpackage "XGCD"
  (:use "COMMON-LISP")
  (:export xgcd))
(in-package "XGCD")

(proclaim '(optimize (speed 0) (space 0) (safety 3) (debug 3)))

(defun xgcd (X Y)
  "Extended gcd; the call
      (xgcd X Y)
returns a multiple value list:
- GCD
- U,V  such that they solve the equation
     GCD=U*X+V*Y
- U1,V1 such that
     LCM=U1*X=V1*Y (up to the sign)."
  (labels ((xgcd-aux (x0 x1 u0 v0 u1 v1)
             (cond 
              ((= x1 0) (values x0 u0 v0 u1 v1))
              (t (multiple-value-bind (n x2) (floor x0 x1)
                   (let ((u2 (- u0 (* n u1)))
                         (v2 (- v0 (* n v1))))
                     (xgcd-aux x1 x2 u1 v1 u2 v2)))))))
    (xgcd-aux X Y 1 0 0 1)))