berle-4-5-10

berle-4-5-10 - 202: Lecture Notes GARSIA April 6, 2010 1 ....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 202: Lecture Notes GARSIA April 6, 2010 1 . Computing GCDs 1. The Berlekamp Algorithm The Euclidean algorithm is the process which yields the greatest common divisor d of two given integers A and B or more generally the GCD of any two elements of a Euclidean ring. It is shown by Berlekamp ( * ) that d may be computed by the following modified version of the Euclidean algorithm. We start by setting r- 2 = A , r- 1 = B , p- 2 = 0 , p- 1 = 1 , q- 2 = 1 , q- 1 = 0 1 . 1 then compute a k ,r k ,p k ,q k according to the recursions (**) ( i ) r k- 2 = a k r k- 1 + r k ( integer division ) ( ii ) p k = a k p k- 1 + p k- 2 ( iii ) q k = a k q k- 1 + q k- 2 . ( k = 0 , 1 , 2 ,... ) 1 . 2 since r k decreases at least by one at each step after n < degree ( B ) steps we shall have r n = 0. we will show that these recursions force the following basic identities ( a ) q n p n- 1- p n q n- 1 = (- 1) n ( b ) A = r n- 1 p n ( c ) B = r n- 1 q n . 1 . 3 Multiplying the first of these equations by r n- 1 and using 1.3 (b) and 1.3 (c) we derive that B p n- 1- A q n- 1 = (- 1) n r n- 1 1...
View Full Document

This note was uploaded on 09/22/2010 for the course MATH MATH187 taught by Professor Math187 during the Spring '10 term at UCSD.

Page1 / 3

berle-4-5-10 - 202: Lecture Notes GARSIA April 6, 2010 1 ....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online