Formal Correctness and Stability of Dense Linear Algebra Algorithms

Deriving Correct High-Performance Algorithms

Dijkstra observed that verifying correctness of a program is difficult and conjectured that derivation of a program hand-in-hand with its proof of correctness was the answer. We illustrate this goal-oriented approach by applying it to the domain of dense linear algebra libraries for distributed memory parallel computers. We show that algorithms that underlie the implementation of most functiona...

Étude formelle d'algorithmes efficaces en algèbre linéaire. (Formal study of efficient algorithms in linear algebra)

Formal methods have reached a degree of maturity leading to the design of general-purpose proof systems, enabling both to verify the correctness of complex software systems and to formalize advanced mathematics. However, the ease of reasoning on programs is often emphasized more than their efficient execution. The antagonism between these two aspects is particularly significant for computer alg...

Deriving Linear Algebra Libraries

Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goal-oriented programming and formal derivation of algorithms. The problem was that for loop-based programs, a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. We believe that no practical progress was made in the field...

Systematic Generation of Algorithms for Iterative Methods

The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as the Cholesky decomposition and the solution of Sylvester equations. In this thesis, we present an extension of the FLAME methodology to tackle iterative meth...

Cauchy-Rassias Stability of linear Mappings in Banach Modules Associated with a Generalized Jensen Type Mapping