Journal ArticleDOI
On Jacobi methods for singular value decompositions
Vjeran Hari,Krevšimir Veselić +1 more
TLDR
An improvement of the Jacobi singular value decomposition algorithm is proposed in this article, where the matrix is first reduced to a triangular form and the row-cyclic strategy preserves the triangularity.Abstract:
An improvement of the Jacobi singular value decomposition algorithm is proposed The matrix is first reduced to a triangular form It is shown that the row-cyclic strategy preserves the triangularity Further improvements lie in the convergence properties It is shown that the method converges globally and a proof of the quadratic convergence is indicated as well The numerical experiments confirm these theoretical predictions Our method is about 2-3 times slower than the standard QR method but it almost reaches the latter if the matrix is diagonally dominant or of low rankread more
Citations
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Journal ArticleDOI
Parallel Numerical Linear Algebra
TL;DR: This work discusses basic principles of paralled processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms, and presents direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigene value problem, and the singular value decomposition.
Journal ArticleDOI
New Fast and Accurate Jacobi SVD Algorithm. I
Zlatko Drmač,Krešimir Veselić +1 more
TL;DR: The quest for a highly accurate and efficient SVD algorithm has led to a new, superior variant of the Jacobi algorithm, which has inherited all good high accuracy properties and can outperform the QR algorithm.
Journal ArticleDOI
On Rank-Revealing Factorisations
TL;DR: A systematic treatment of algorithms for determining RRQR factorisations and presents "hybrid" algorithms that solve the optimisation problems almost exactly (up to a factor proportional to the size of the matrix).
Journal ArticleDOI
Computing the generalized singular value decomposition
Zhaojun Bai,James Demmel +1 more
TL;DR: A new numerical method for computing the GSVD of two matrices A and B is presented, a variation on Paige''s method, which differs from previous algorithms in guaranteeing both backward stability and convergence.
Journal ArticleDOI
A Jacobi eigenreduction algorithm for definite matrix pairs
TL;DR: In this article, a Jacobi eigen reduction algorithm for symmetric definite matrix pairs was proposed, which works only on one matrix and uses J-orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process.
References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
Book
The Symmetric Eigenvalue Problem
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
Journal ArticleDOI
The Solution of Singular-Value and Symmetric Eigenvalue Problems on Multiprocessor Arrays
Richard P. Brent,Franklin T. Luk +1 more
TL;DR: Parallel Jacobi-like algorithms are presented for computing a singular-value decomposition of an $(m \geqq n) matrix and an eigenvalue decompositions of an $n \times n$ symmetric matrix.