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lightning-bolt - Euclidean algorithm in Lightning-Bolt form...

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Euclidean algorithm in Lightning-Bolt form Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 8 January, 2011 (at 20:49 ) The Euclidean algorithm, EU , is often presented by a series of equations. I have found the following table-form convenient, both because it organises the computation, and gives a name to each number in the table. Because the update-rule follows the shape of a lightning-bolt, I call it the LBolt algorithm. Henceforth, all variables are integers unless explic- itly stated otherwise. Given integers r 0 and r 1 ( for the time being, assume each is positive ) we will compute G := Gcd( r 0 , r 1 ) as well as a pair S,T of ezout multipliers satisfying G = S · r 0 + T · r 1 . 1 : [ There is a one-parameter family of B´ ezout-pairs; the alg will compute a particular pair. ] I’ll explain via an example. Suppose we want the Gcd of r 0 := 114 and r 1 := 33. Then initialize the table as: n r n q n s n t n 0 114 1 0 1 33 0 1 In order to compute a BP ( ezout-Pair ), we’ll need
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