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Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices

Clément Pernet 1 Hippolyte Signargout 2, 1 Pierre Karpman 1 Gilles Villard 2 
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+Hankel like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured projections, which have been recently successfully applied for computing the bivariate resultant. A first baby-step/giant step approach - directly derived using known techniques on structured matrices - gives a randomized Monte Carlo algorithm for the minimal polynomial of an n×n Toeplitz or Hankel-like matrix of displacement rank alpha using O˜(n^(w-c(w)) alpha^c(w) ) arithmetic operations, where w is the exponent of matrix multiplication and c(2.373) ≈ 0.523 for the best known value of w. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in O˜(n^(2−1/w)) operations when the displacement rank is considered constant. Previous algorithms required 2 operations while the exponents presented here are respectively less than 1.86 and 1.58 with the best known estimate for w.
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Submitted on : Friday, April 2, 2021 - 5:28:18 PM
Last modification on : Tuesday, October 25, 2022 - 4:19:20 PM
Long-term archiving on: : Saturday, July 3, 2021 - 6:54:05 PM


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Clément Pernet, Hippolyte Signargout, Pierre Karpman, Gilles Villard. Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices. ISSAC 2021 - International Symposium on Symbolic and Algebraic Computation, Jul 2021, Saint Petersburg, Russia. pp.249-256, ⟨10.1145/3452143.3465542⟩. ⟨hal-03189115⟩



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