# Least squares residuals and minimal residual methods

 dc.contributor.author Liesen, Jörg dc.contributor.author Rozlozník, Miroslav dc.contributor.author Strakoš, Zdeněk dc.date.accessioned 2017-12-20T12:05:49Z dc.date.available 2017-12-20T12:05:49Z dc.date.issued 2012 dc.description.abstract We study Krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step (minimal residual (MR) methods). MR methods are often formulated in terms of a sequence of least squares (LS) problems of increasing dimension. We present several basic identities and bounds for the LS residual. These results are interesting in the general context of solving LS problems. When applied to MR methods, they show that the size of the MR residual is strongly related to the conditioning of different bases of the same Krylov subspace. Using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of MR methods. Different bases also lead to different implementations which are mathematically equivalent but can differ numerically. Our theoretical results are used for a finite precision analysis of implementations of the GMRES method [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869]. We explain that the choice of the basis is fundamental for the numerical stability of the implementation. As demonstrated in the case of Simpler GMRES [H. F. Walker and L. Zhou, Numer. Linear Algebra Appl., 1 (1994), pp. 571--581], the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis. In particular, we prove that Simpler GMRES is inherently less numerically stable than the Classical GMRES implementation due to Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869]. en dc.identifier.eissn 1095-7197 dc.identifier.issn 1064-8275 dc.identifier.uri https://depositonce.tu-berlin.de/handle/11303/7298 dc.identifier.uri http://dx.doi.org/10.14279/depositonce-6571 dc.language.iso en en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject.ddc 518 Numerische Analysis de dc.subject.ddc 512 Algebra de dc.subject.other linear systems en dc.subject.other least squares problems en dc.subject.other Krylov subspace methods en dc.subject.other minimal residual methods en dc.subject.other GMRES en dc.subject.other convergence en dc.subject.other rounding errors en dc.title Least squares residuals and minimal residual methods en dc.type Article en dc.type.version publishedVersion en dcterms.bibliographicCitation.doi 10.1137/S1064827500377988 en dcterms.bibliographicCitation.issue 5 en dcterms.bibliographicCitation.journaltitle SIAM Journal on Scientific Computing en dcterms.bibliographicCitation.originalpublishername Society for Industrial and Applied Mathematics en dcterms.bibliographicCitation.originalpublisherplace Philadelphia, Pa en dcterms.bibliographicCitation.pageend 1525 en dcterms.bibliographicCitation.pagestart 1503 en dcterms.bibliographicCitation.volume 23 en tub.accessrights.dnb domain en tub.affiliation Fak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Numerische Lineare Algebra de tub.affiliation.faculty Fak. 2 Mathematik und Naturwissenschaften de tub.affiliation.group FG Numerische Lineare Algebra de tub.affiliation.institute Inst. Mathematik de tub.publisher.universityorinstitution Technische Universität Berlin en

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