Etude statistique des erreurs dans l'arithmétique des ordinateurs; application au contrôle des résultats d'algorithmes numériques.
Evaluation de l'incertitude sur la solution d'un système Linéaire.
Fehleranalyse für die Gauß-Elimination zur Berechnung der Lösung minimaler Länge.
Formally certified floating-point filters for homogeneous geometric predicates
Floating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this...
Forward Error Analysis of Gaussian Elimination. Part I: Error and Residual Estimates.
Forward Error Analysis of Gaussian Elimination. Part II: Stability Theorems.
Intervallarithmetik - mit Zirkel und Lineal.
Lie groups and error analysis.
Measuring and maintaining consistency: a hybrid FTF algorithm
Due to the versatility as well as its ease of implementation, the Fast Transversal Filters algorithm is attractive for many adaptive filtering applications. However, it is not widely used because of its undesirable tendency to diverge when operating in finite precision arithmetic. To compensate, modifications to the algorithm have been introduced that are either occasional (performed when a predefined condition(s) is violated) or structured as part of the normal update iteration. However, in neither...
Méthode numérique de détection de la singularité d'une matrice.
Mixed precision GMRES-based iterative refinement with recycling
With the emergence of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes for solving linear systems have recently been developed. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than the alternative of recomputing the LU factors in a higher precision. In this work, we incorporate the idea of Krylov subspace recycling, a well-known technique for reusing information across sequential invocations,...
Multiple-Precision Correctly rounded Newton-Cotes quadrature
Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics. We study the Newton-Cotes quadrature scheme in the context of multiple-precision arithmetic and give enough details on the algorithms and the error bounds to enable software...
Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding
For a real square matrix and an integer , let denote the matrix formed from by rounding off all its coefficients to decimal places. The main problem handled in this paper is the following: assuming that has some property, under what additional condition(s) can we be sure that the original matrix possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number...
Note sur la suppression des étiquettes en programmation
Numerical Stability for Solving Nonlinear Equations.
Numerical Stability of Decent Methods for Solving Linear Equations.
Numerical Stability of the Chebyshev Method for the Solution of Large Linear Systems.
Numerické metody a počítače
On a direct method for the solution of nearly uncoupled Markov chains.
On error estimation in the conjugate gradient method and why it works in finite precision computations.