Using fast matrix multiplication to solve structured linear systems

Authors Eric Schost, Alin Bostan, Claude-Pierre Jeannerod

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Eric Schost
Alin Bostan
Claude-Pierre Jeannerod

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Eric Schost, Alin Bostan, and Claude-Pierre Jeannerod. Using fast matrix multiplication to solve structured linear systems. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)


Structured linear algebra techniques are a versatile set of tools; they enable one to deal at once with various types of matrices, with features such as Toeplitz-, Hankel-, Vandermonde- or Cauchy-likeness. Following Kailath, Kung and Morf (1979), the usual way of measuring to what extent a matrix possesses one such structure is through its displacement rank, that is, the rank of its image through a suitable displacement operator. Then, for the families of matrices given above, the results of Bitmead-Anderson, Morf, Kaltofen, Gohberg-Olshevsky, Pan (among others) provide algorithm of complexity $O(alpha^2 n)$, up to logarithmic factors, where $n$ is the matrix size and $alpha$ its displacement rank. We show that for Toeplitz- Vandermonde-like matrices, this cost can be reduced to $O(alpha^(omega-1) n)$, where $omega$ is an exponent for linear algebra. We present consequences for Hermite-Pad'e approximation and bivariate interpolation.
  • Structured matrices
  • matrix multiplication
  • Hermite-Pade
  • bivariate interpolation


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