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Acceleration of Euclidean algorithm and extensions V. Y. Pan, X. Wang, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation , pp. 207-213, 2002. 田錦燕 95/9/15. Outline. 摘要 Smith Normal Form Smith normal form 與 Extended Euclidean algorithm
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Acceleration of Euclidean algorithm and extensionsV. Y. Pan, X. Wang, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pp. 207-213, 2002 田錦燕 95/9/15
Outline • 摘要 • Smith Normal Form • Smith normal form與Extended Euclidean algorithm • Accelerated Extended Euclidean Algorithm
摘要 • 目的：Solve linear systems • 原來方法：利用Extended Euclidean algorithm求得Smith normal form係數 • 本篇論文提出Accelerated Extended Euclidean Algorithm
Smith Normal Form • where a1(x) ,a2(x), ...,am(x) are monic nonzero elements of with degrees at least one and satisfying a1(x) |a2(x)| ...|am(x) , where f | g | h |… means f divides g , which in turn divides , and so on (Dummit and Foote 1998, pp. 390-391 and 414). This form is known as Smith normal form, and the elements are called the invariant factors of .
Smith normal form與Extended Euclidean algorithm rj = sj m + tj f ≡ tj f mod m , Complexity : • Smith normal form係數為η/δ
