Randomized preconditioning of the MBA algorithm

Authors:
Victor Y. Pan;Guoliang Qian;Ai-Long Zheng
Affiliations:
Lehman College of CUNY, Bronx, NY, USA;The Graduate Center of CUNY, New York, NY, USA;The Graduate Center of CUNY, New York, NY, USA
Venue:
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Year:
2011

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Preface

Theoretical Computer Science

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Abstract

MBA algorithm inverts a structured matrix in nearly linear arithmetic time but requires a serious restriction on the input class. We remove this restriction by means of randomization and extend the progress to some fundamental computations with polynomials, e.g., computing their GCDs and AGCDs, where most effective known algorithms rely on computations with matrices having Toeplitz-like structure. Furthermore, our randomized algorithms fix rank deficiency and ill conditioning of general and structured matrices. At the end we comment on a wide range of other natural extensions of our progress and underlying ideas.