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We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean <i>n</i>times <i>n</i> matrices in <b>K</b>[<i>x</i>] of degree bounded by <i>d</i>, with <b>K</b> a commutative field. Under the straight-line program model we(More)
Linear systems with structures such as Toeplitz, Vandermonde or Cauchy-likeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O˜denotes the omission of logarithmic factors. We show that for such matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication(More)
We present an inversion algorithm for nonsingular n × n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and, when n is a power of two, requires O˜(n 3 d) field operations for a generic input; the soft-O notation O˜indicates some missing log(nd) factors. Up to such logarithmic factors, this asymptotic complexity(More)
In this paper, we study the problem of computing an LSP-decomposition of a matrix over a field. This decomposition is an extension to arbitrary matrices of the well-known LUP-decomposition of full row-rank matrices. We present three different ways of computing an LSP-decomposition, that are both rank-sensitive and based on matrix multiplication. In each(More)
For matrices with displacement structure, basic operations like multiplication, inversion , and linear system solving can all be expressed in terms of the following task: evaluate the product AB, where A is a structured n × n matrix of displacement rank α, and B is an arbitrary n × α matrix. Given B and a so-called generator of A, this product is(More)
Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the well known variations of such decompositions and transformations that have been proposed in the literature. We present an(More)
Given two floating-point vectors x, y of dimension n and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number r such that | r − x T y| nu|x| T |y| with u the unit roundoff. This result, which holds for any radix and with no restriction on n, can be seen as a(More)
Linear systems with structures such as Toeplitz-, Vandermonde-or Cauchy-likeness can be solved in <i>O</i>~(&#945;<sup>2</sup><i>n</i>) operations, where <i>n</i> is the matrix size, &#945; is its displacement rank, and <i>O</i>~denotes the omission of logarithmic factors. We show that for Toeplitz-like and Vandermonde-like trices, this cost can be reduced(More)
In this articlc, \ve study square matrices pcrturbcd by a pa-rarncter E. An efficient algorithm conlputing the z-espansiou of the eigcnvalucs in forinal Laurent-Puiseux series is provided , for \vhicli the computation of the characteristic polynomial is not rccluired. 15;e show 110~ to reduce the init,ial mat.ris so t.hat. the Lidskii-Edelman-Yla(More)