An engineered algorithm for the Smith form of an integer matrix

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【标题】An engineered algorithm for the Smith form of an integer matrix

【作者】 B. David Saunders  Zhendong Wan 

【摘要】A variety of algorithms for computing Smith normal forms of integer matrices are known. Their worst case asymptotic complexities have steadily improved over the past four decades. However; in practice; an asymptotically inferior algorithm often outperforms an asymptotically better one. We offer an “engineered” algorithm for Smith forms of integer matrices; which is designed to combine the best aspects of previous algorithms. The fundamental base of our method is the Smith form algorithm of Eberly; Giesbrecht; and Villard. Our algorithm shares its worst case complexity. We provide several improvements such as a “bonus” idea which allows two adjacent invariant factors to be found at the price one; and a more practically efficient perturbation method. We “engineer in” judicious use of other algorithms in situations where they are more efficient. We discover the importance of seqarating rough and smooth parts of invariant factors and find an adaptive algorithm for computing each part. A number of experimental measurements suggest substantial benefit from our engineered Smith form algorithm.