Analysis and implementation of an implicitly restarted Arnoldi iteration

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dc.contributor.advisorSorensen, Danny C.en_US
dc.creatorLehoucq, Richard Brunoen_US
dc.date.accessioned2009-06-04T00:44:23Zen_US
dc.date.available2009-06-04T00:44:23Zen_US
dc.date.issued1995en_US
dc.description.abstractThe Arnoldi algorithm, or iteration, is a computationally attractive technique for computing a few eigenvalues and associated invariant subspace of large, often sparse, matrices. The method is a generalization of the Lanczos process and reduces to that when the underlying matrix is symmetric. This thesis presents an analysis of Sorensen's Implicitly Re-started Arnoldi iteration, (scIRA-iteration), by exploiting its relationship with the scQR algorithm. The goal of this thesis is to present numerical techniques that attempt to make the scIRA-iteration as robust as the implicitly shifted scQR algorithm. The benefit is that the Arnoldi iteration only requires the computation of matrix vector products w = Av at each step. It does to rely on the dense matrix similarity transformations required by the EISPACK and LAPACK software packages. Five topics form the contribution of this dissertation. The first topic analyzes re-starting the Arnoldi iteration in an implicit or explicit manner. The second topic is the numerical stability of an scIRA-iteration. The forward instability of the scQR algorithm and the various schemes used to re-order the Schur form of a matrix are fundamental to this analysis. A sensitivity analysis of the Hessenberg decomposition is presented. The practical issues associated with maintaining numerical orthogonality among the Arnoldi/Lanczos basis vectors is the third topic. The fourth topic is deflation techniques for an scIRA-iteration. The deflation strategies introduced make it possible to compute multiple or clustered eigenvalues with a single vector re-start method. The block Arnoldi/Lanczos methods commonly used are not required. The final topic is the convergence typical of an scIRA-iteration. Both formal theory and heuristics are provided for making choices that will lead to improved convergence of an scIRA-iteration.en_US
dc.format.extent193 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH.SCI. 1995 LEHOUCQen_US
dc.identifier.citationLehoucq, Richard Bruno. "Analysis and implementation of an implicitly restarted Arnoldi iteration." (1995) Diss., Rice University. <a href="https://hdl.handle.net/1911/16844">https://hdl.handle.net/1911/16844</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/16844en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.subjectComputer scienceen_US
dc.titleAnalysis and implementation of an implicitly restarted Arnoldi iterationen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematical Sciencesen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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