Validated Solution of Large Linear Systems

نویسنده

Siegfried M. Rump

چکیده

Some new methods will be presented for computing verified inclusions of the solution of large linear systems. The matrix of the linear system is typically of band or sparse structure. There are no prerequisites to the matrix such as being M-matrix, symmetric, positive definite or diagonally dominant. For general band matrices of lower, upper bandwidth p, q of dimension n the computing time is less than n ·(pq+p2 +q). Examples with up to 1.000.000 unknowns will be presented. Zusammenfassung Es werden neuartige Methoden vorgestellt zur Berechnung sicherer Schranken der Lösung großer linearer Gleichungssysteme. Die Matrix des Gleichungssystems hat typischerweise Bandstruktur oder ist spärlich besetzt. Es werden keinerlei Voraussetzungen an die Matrix gestellt wie etwa M-Matrix, symmetrisch, positiv definit oder diagonal dominant. Für Bandmatrizen von oberer bzw. unterer Bandbreite p bzw. q der Dimension n ist die Rechenzeit kleiner als n · (pq + p + q). Es werden Beispiele bis Dimension 1.000.000 diskutiert. 0 Notation Let IR denote the set of real numbers, IR vectors and IRn×n matrices over those. The letter n is only used for the dimension of vectors and matrices, others then n-vectors and n× n-matrices do not occur in this paper. IPT denotes the power set over T, IIT the interval extension for T ∈ {IR, IR, IRn×n}. Usually hyperrectangulars are used but others are not excluded. It should be stressed that interval operations producing validated bounds are rigorously and very efficiently implementable on digital computers, see [25], [1], [5], [28] for details. published in R. Albrecht et al. (eds.): Validation numerics: theory and applications, vol. 9 of Computing Supplementum , pp. 191–212, Springer 1993

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