Efficient computation with structured matrices and arithmetic expressions
type: Dataset
description: Designing efficient code in practice for a given computation is a hard task. In this thesis, we tackle this issue in two different situations. The first part of the thesis introduces some algorithmic improvements in structured linear algebra. We first show how to extend an algorithm by Cardinal for inverting Cauchy-like matrices to the other common structures. This approach, which mainly relies on products of the type "structured matrix × matrix", leads to a theoretical speed-up of a factor up to 7 that we also observe in practice. Then, we extend some works on Toeplitz-like matrices and prove that, for any of the common structures, the product of an n×n structured matrix of displacement rank α by an n×α matrix can be computed in Õ(α^(ω-1)n). This leads to direct inversion algorithms in Õ(α^(ω-1)n) , that do not rely on a reduction to the Toeplitz-like case. The second part of the thesis deals with the implementation of arithmetic expressions. This topic raises several issues like finding the minimum number of operations, and maximizing the speed or the accuracy when using some finite-precision arithmetic. Making use of the inductive nature of arithmetic expressions enables the design of algorithms that help to answer such questions. We thus present a set of algorithms for generating evaluation schemes, counting them, and optimizing them according to one or several criteria. These algorithms are part of a library that we have developed and used, among other things, in order to decrease the running time of a code generator for a mathematical library, and to study optimality issues about the evaluation of a small degree polynomial with scalar coefficients at a matrix point.
landing page: https://theses.hal.science/tel-00688388
identifier: NNT: 2011ENSL0652
keyword: algebre-lineaire-structuree
keyword: arithmetic-expressions
keyword: code-generation
keyword: combinatoire-et-optimisation-des-schemas-devaluation
keyword: combinatorics-and-optimization-of-evaluation-schemes
keyword: expressions-arithmetiques
keyword: generation-de-code
keyword: infoeu-reposemanticsdoctoralthesis
keyword: infoinfo-ohcomputer-science-csother-csoh
keyword: inversion-de-matrices
keyword: matrix-inversion
keyword: matrix-product
keyword: produit-de-matrices
keyword: structured-linear-algebra
keyword: theses
issued: 2026-05-21T17:29:29.099896
modified: 2026-05-21T17:29:29.099902
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Efficient computation with structured matrices and arithmetic expressions
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00688388
issued: 2026-05-21T17:29:29.136768
modified: 2026-05-21T17:29:29.067047