Towards a Reproducible Solution of Linear Systems

Roman Iakymchuk (1) , Stef S Graillat (2) , David Defour (3) , Erwin Laure (1) , Enrique S Quintana-Ortí (4)

1 KTH - KTH Royal Institute of Technology [Stockholm]
2 PEQUAN - Performance et Qualité des Algorithmes Numériques
3 ELIAUS - Electronique, Informatique, Automatique et Systèmes
4 Universitat Jaume I = Jaume I University

Solving a linear system of equations is an important underlying part of numerous scientific applications. In this article, we address the issue of non-deterministic and, therefore, non-reproducible solution of linear systems and propose an approach to ensure its reproducibility. Our approach is based on the hierarchical and modular structure of linear algebra algorithms. Consequently, we divide computations into smaller logical blocks – such as a blocked LU factorization, triangular system solve, and matrix-matrix multiplication – and ensure their reproducible results. In this manner, we also split the blocked LU factorization into the unblocked LU and the BLAS-3 routines; the former is built on top of scaling a vector and outer product of two vectors routines from BLAS-1 and-2, accordingly. In this work, our focus is on constructing these building blocks that eventually lead, as we will prove, to the reproducible solution of linear systems.

Mots clés: Linear system; reproducibility; long accumulator; error-free transformation

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Arithmétique des ordinateurs; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]; Logiciel mathématique [cs.MS]

Roman Iakymchuk, Stef S Graillat, David Defour, Erwin Laure, Enrique S Quintana-Ortí. Towards a Reproducible Solution of Linear Systems. Supercomputing Conference 2017-Computational Reproducibility at Exascale Workshop, Nov 2017, Denver, United States. ⟨hal-01633980⟩