2021/06/03 by Dorota Šimonová, Šimonová, Dorota, Petr Tichý +1
Computer Science · Physics and Astronomy · #65F10 #65F15 #Electromagnetic Scattering and Analysis #FOS: Mathematics #G.1.3 #Matrix Theory and Algorithms #Neural Networks and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2106.02068
openalex publication_date 2021/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 a variant of the conjugate gradient method that produces then almost exact results. 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.