When does the Lanczos algorithm compute exactly?. ETNA - Electronic Transactions on Numerical Analysis
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9 juin 2022
- handle: 10670/1.iqzajj
- issn: 1068-9613
- doi: 10.1553/etna_vol55s547
info:eu-repo/semantics/openAccess
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Science and space Space research Natural philosophy Philosophy, Natural Papers Exploration of space Space research Solar system--Exploration Space exploration (Astronautics) Matrix algebra Algebra, Matrix Matrixes (Algebra) Cracovians (Mathematics) Systems, Linear Independence Self-governmentCiter ce document
Dorota Šimonová et al., « When does the Lanczos algorithm compute exactly?. ETNA - Electronic Transactions on Numerical Analysis », Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschafte, ID : 10.1553/etna_vol55s547
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In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for an implementation of the conjugate gradient method called cgLanczos. This implementation then computes approximations that agree with their exact counterparts to a relative accuracy given by the machine precision and the condition number of the system matrix. The results are extended to the Arnoldi algorithm, the nonsymmetric Lanczos algorithm, the Golub-Kahan bidiagonalization, the block-Lanczos algorithm, and their counterparts for solving linear systems.