Numerical aspects of gram-schmidt orthogonalization of vectors
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TLDR
Several variants of Gram-Schmidt orthogonalization are reviewed from a numerical point of view in this paper, and it is shown that the classical and modified variants correspond to the Gauss-Jacobi and Gauss -Seidel iterations for linear systems.About:
This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 73 citations till now. The article focuses on the topics: Orthogonalization & Gram–Schmidt process.read more
Citations
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A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
TL;DR: In this article, a new method for the iterative computation of a few extremal eigenvalues of a symmetric matrix and their associated eigenvectors is proposed, based on an old and almost unknown method of Jacobi.
Journal ArticleDOI
A Jacobi--Davidson Iteration Method for Linear Eigenvalue Problems
TL;DR: A new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors is proposed that has improved convergence properties and that may be used for general matrices.
Journal ArticleDOI
Numerics of Gram-Schmidt orthogonalization
TL;DR: The classical and modified Gram-Schmidt (CGS) orthogonalization is one of the fundamental procedures in linear algebra as mentioned in this paper, and it is equivalent to the factorization AQ1R, where Q1∈Rm×n with orthonormal columns and R upper triangular.
Journal ArticleDOI
An iterative solution method for solving f ( A ) x = b , using Krylov subspace information obtained for the symmetric positive definite matrix A
TL;DR: It will be shown how intermediate information obtained by the conjugate gradients (cg) algorithm can be used to solve f(A)x = b iteratively in an efficient way, for suitable functions f.
References
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Journal ArticleDOI
Iterative refinement of linear least squares solutions II
TL;DR: In this paper, an iterative procedure is developed for reducing the rounding errors in the computed least squares solution to an overdetermined system of equations, where the method relies on computing accurate residuals to a certain augmented system of linear equations, by using double precision accumulation of inner products.
Book ChapterDOI
Projection methods for solving large sparse eigenvalue problems
TL;DR: A unified approach to several methods for computing eigenvalues and eigenvectors of large sparse matrices, and includes the most commonly used algorithms for solving large sparse eigenproblems like the Lanczos algorithm, Arnoldi's method and the subspace iteration.
Book ChapterDOI
The two-sided arnoldi algorithm for nonsymmetric eigenvalue problems
TL;DR: Algorithms for computing a few eigenvalues of a large nonsymmetric matrix and an algorithm which computes both left and right eigenvector approximations, by applying the Arnoldi algorithm both to the matrix and its transpose are described.