Denis Trystram

Parallel Gaussian elimination on an MIMD computer

This paper introduces a graph-theoretic approach to analyse the performances of several parallel Gaussian-like triangularization algorithms on an MIMD computer. We show that the SAXPY, GAXPY and DOT algorithms of Dongarra, Gustavson and Karp, as well as parallel versions of the LDMt, LDLt, Doolittle and Cholesky algorithms, can be classified into four task graph models. We derive new complexity results and compare the asymptotic performances of these parallel versions.

An orthogonal systolic array for the algebraic path problem

This paper is devoted to the design of an orthogonal systolic array ofn(n+1) elementary processors which can solve any instance of the Algebraic Path Problem within only 5n−2 time steps, and is compared with the 7n−2 time steps of the hexagonal systolic array of Rote [8].ZusammenfassungEs wird ein orthogonales systolisches Feld (systolic array) mitn(n+1) einfachen Prozessoren entworfen, das das algebraische Wegproblem in nur 5n−2 Schritten lösen kann, im Vergleich zu 7n−2 Schritten beim hexagonalen systolischen Feld von Rote [8].

Parallel Implementation of the Algebraic Path Problem

The Algebraic Path Problem is a general framework which unifies several algorithms arising from various fields of computer science. Rote [11] introduces a general algorithm to solve any instance of the APP, as well as a hexagonal systolic array of (n+1)2 elementary processors which can solve the problem in 7n-2 time steps. We propose a new algorithm to solve the APP, and demonstrate its equivalence with Rotes algorithm. The new algorithm is more suitable to parallelization: we propose an orthogonal systolic array of n(n+1) processors which solves the APP within only 5n-2 steps. Finally, we give some experiments on the implementation of our new algorithm in the parallel environment developped by IBM at ECSEC in Roma.

Parallel solution of dense linear systems using diagonalization methods

We study the parallel implementation of two diagonalization methods for solving dense linear systems: the well known Gauss-Jordan method and a new one introduced by Huard. The number of arithmetic operations performed by the Huard method is the same as for Gaussian elimination, namely 2n 3/3, less than for the Jordan method, namely n 3. We introduce parallel versions of these methods, compare their performances and study their complexity. We assume a shared memory computer with a number of processors p of the order of n, the size of the problem to be solved, We show that the best parallel version for Jordans method is by rows whereas the best one for Huards method is by columns. Our main result states that for a small number of processors the parallel Huard method is faster than the parallel Jordan method and slower otherwise. The separation is obtained for p = 0.44n.

Gass Elimination Algorithms for MIMD Computers

This paper uses a graph-theoretic approach to analyse the performances of several parallel variations of the Gaussian triangularization algorithm on an MIMD computer. Dongarra et al. [DGK] have studied various parallel implementations of this method for a vector pipeline machine. We obtain complexity results permitting to select among these parallel algorithms.

An incomplete factorization algorithm for adaptive filtering

Abstract Many applications in signal processing lead to the solution of consecutive linear systems with very similar matrices. Here we present an efficient algorithm for solving this basic problem and compare it to the well-known Cholesky factorization.

Systolic implementation of the adaptive solution to normal equations

We are interested in the systolic computation of projection operators entering digital signal processing, or more precisely, solution of the so-called normal equations involved in adaptive systems. The systolic array proposed achieves a real-time adaptive solution, i.e., updates the left- and right-hand sides of the linear equation and computes its solution at each time step.