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PUBLICATION INTERNE No 961
PARALLEL COMPUTATION OF SPECTRAL PORTRAITS OF MATRICES BY BIDIAGONALIZATION
ISSN 1166-8687
A.N. MALYSHEV
IRISA CAMPUS UNIVERSITAIRE DE BEAULIEU - 35042 RENNES CEDEX - FRANCE
INSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTE`MES ALE´ATOIRES Campus de Beaulieu – 35042 Rennes Cedex – France Te´l. : (33) 99 84 71 00 – Fax : (33) 99 84 71 71
Parallel computation of spectral portraits of matrices by bidiagonalization A.N. Malyshev Programme 6 | Calcul scienti que, modelisation et logiciel numerique Projet ALADIN Publication interne n961 | octobre 1995 | 16 pages
Abstract: We describe parallel programs for computation of spectral portraits
of matrices on Paragon and Connection Machine 5. The method used consists of bidiagonal reduction of a complex square matrix by unitary Householder transformations and computation of the minimal singular value of the resulting real bidiagonal matrix by the bisection procedure employing Sturm sequences. The computation of bidiagonal reduction uses the block-cyclic distribution of matrices on a rectangular processor grid in order to get good load balancing. Since the computation of spectral portraits needs to calculate the minimal singular values in many points, the bisection procedure is performed independently on several parallel processors. Timing results are provided. Key-words: spectral portrait, bidiagonal reduction, bisection method, parallelism, Paragon, Connection Machine 5 (Resume : tsvp)
Institute of Mathematics, Novosibirsk, 630090, Russia (
[email protected]). This work was carried out while the author was visiting IRISA. The work was supported by the European Commission (Copernicus project PORTRAIT CP940682)
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
Centre National de la Recherche Scientifique (URA 227) Universite´ de Rennes 1 – Insa de Rennes
Institut National de Recherche en Informatique et en Automatique – unite´ de recherche de Rennes
Calcul parallele des portraits spectraux de matrices par bidiagonalisation
Resume : Ce rapport decrit di erents programmes paralleles sur les calculateurs Paragon (Intel) et CM 5 (Thinking Machine Corp.) pour le calcul de portraits spectraux de matrices. La methode employee consiste a reduire une matrice complexe sous forme bidiagonale par des transformations unitaires de Householder puis a calculer la plus petite valeur singuliere de la matrice bidiagonale reelle obtenue par une procedure de bissection fondee sur des suites de Sturm. La reduction sous forme bidiagonale utilise une distribution cyclique par blocs des matrices sur une grille rectangulaire de processeurs a n d'assurer un bon equilibre de charge. Comme le calcul de portraits spectraux necessite le calcul de la plus petite valeur singuliere d'un grand nombre de matrices, la procedure de bissection est executee en parallele sur plusieurs processeurs. On rapporte en n les resultats des essais numeriques. Mots-cle : portrait spectral, reduction bidiagonale, methode de bisection, parallelisme, Paragon, Connection Machine 5
Calcul parallele des portraits spectraux
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1 Theoretical discussion
1.1 De nition of the problem
Last years it was demonstrated that many properties of the spectrum of non-normal matrices can be accounted for by means of the pseudospectrum or -spectrum of a matrix [13, 18]. Let us consider this notion in more detail. Given an N -by-N matrix A with real or complex elements, the -spectrum of A is de ned by the following formula: (A) = f 2 C j9; kk ; det(A + ? I ) = 0g:
(1)
In other words, (A) is the union of the spectra of all possible perturbed matrices A + , where the perturbation matrix satis es kk . There are two di erent types of the -spectrum, complex or real, depending on which types of perturbations are admissible. For instance, if the matrix A is real and only real perturbations are admitted, we come to the notion of a real -spectrum R(A). If the matrix A is complex, then with complex perturbations we obtain a complex -spectrum C (A). It was shown in [16] that ! ) ( A ? < I ?
= I R (2) (A) = 2 C j max 2N ?1 ?1= I A ?
