Claude-Pierre Jeannerod

Learn More
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)
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 (̃n3d) 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)
Linear systems with structures such as Toeplitz, Vandermonde or Cauchy-likeness can be solved in O (̃α2n) 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 (̃αω−1n), where ω is a feasible exponent for matrix multiplication(More)
The asymptotically fastest known divide-and-conquer methods for inverting dense structured matrices are essentially variations or extensions of the Morf/Bitmead-Anderson algorithm. Most of them must deal with the growth in length of intermediate generators, and this is done by incorporating various generator compression techniques into the algorithms. One(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 r̂ such that |r̂ − xT y| 6 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)
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 rowrank matrices. We present three different ways of computing an LSPdecomposition, that are both rank-sensitive and based on matrix multiplication. In each case,(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)
We provide a detailed analysis of Kahan’s algorithm for the accurate computation of the determinant of a 2 × 2 matrix. This algorithm requires the availability of a fused multiply-add instruction. Assuming radix-β, precision-p floating-point arithmetic with β even, p ≥ 2, and barring overflow or underflow we show that the absolute error of Kahan’s algorithm(More)