A Uniied Superfast Divide-and-conquer Algorithm for Structured Matrices over Abstract Elds

We propose a superfast divide-and-conquer algorithm that uses 2n ? 2 random parameters , O(n) memory space and O((n log 2 n) log log n) operations in a xed arbitrary eld in order to compute the rank and a basis for the null space of a structured n n matrix X given with its short generator, as well as to solve a linear system Xy = b or to determine its inconsistency. If rank X = n, the algorithm also computes det X and a short generator of X ?1. The cost bounds cover correctness veriication for the output but not the cost of the generation of random parameters. The algorithm gives a uniied treatment of various classes of structured matrices including ones of Toeplitz, Hankel, Vandermonde and Cauchy types.

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