NDLI: Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems

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BIT Numerical Mathematics : Volume 44, Issue 4, December 2004

Preface

Introduction to the Contents of Issue 44:4

Incomplete Orthogonal Distance Regression

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

Stability and Sensitivity of Tridiagonal LU Factorization without Pivoting

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Generalized Gauss–Radau and Gauss–Lobatto Formulae

Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

Stability of the High Frequency Fast Multipole Method for Helmholtz’ Equation in Three Dimensions

Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers

Analysis of a Class of Parallel Multigrid Smoothers

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

BIT Numerical Mathematics : Volume 44, Issue 3, August 2004

BIT Numerical Mathematics : Volume 44, Issue 2, May 2004

BIT Numerical Mathematics : Volume 44, Issue 1, March 2004

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Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems

Content Provider	Springer Nature Link
Author	Hochstenbach, M. E.
Copyright Year	2004
Abstract	For the accurate approximation of the minimal singular triple (singular value and left and right singular vector) of a large sparse matrix, we may use two separate search spaces, one for the left, and one for the right singular vector. In Lanczos bidiagonalization, for example, such search spaces are constructed. In SIAM J. Sci. Comput., 23(2) (2002), pp. 606–628, the author proposes a Jacobi–Davidson type method for the singular value problem, where solutions to certain correction equations are used to expand the search spaces. As noted in the mentioned paper, the standard Galerkin subspace extraction works well for the computation of large singular triples, but may lead to unsatisfactory approximations to small and interior triples. To overcome this problem for the smallest triples, we propose three harmonic and a refined approach. All methods are derived in a number of different ways. Some of these methods can also be applied when we are interested in the largest or interior singular triples. Theoretical results as well as numerical experiments indicate that the results of the alternative extraction processes are often better than the standard approach. We show that when Lanczos bidiagonalization is used for the subspace expansion, the standard, harmonic, and refined extraction methods are all essentially equivalent. This gives more insight in the success of Lanczos bidiagonalization to find the smallest singular triples. Finally, we show that the extraction processes for the smallest singular values may give an approximation to a least squares problem at low additional costs. The truncated SVD is also discussed in this context.
Starting Page	721
Ending Page	754
Page Count	34
File Format	PDF
ISSN	00063835
Journal	BIT Numerical Mathematics
Volume Number	44
Issue Number	4
e-ISSN	15729125
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2004-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	SVD singular value problem large sparse matrix subspace method subspace extraction two-sided approach harmonic extraction refined extraction Rayleigh quotient Lanczos bidiagonalization Saad’s theorem least squares problem truncated SVD Numeric Computing Mathematics Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

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Probabilistic Upper Bounds for the Matrix Two-Norm

On the computation of a truncated SVD of a large linear discrete ill-posed problem

Using cross-product matrices to compute the SVD

Some properties of LSQR for large sparse linear least squares problems

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

Lanczos bidiagonalization-based inverse solution methods applied to electrical imaging of the heart by using reduced lead-sets: A simulation study

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

A contribution to perturbation analysis for total least squares problems

Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems

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Probabilistic Upper Bounds for the Matrix Two-Norm

On the computation of a truncated SVD of a large linear discrete ill-posed problem

Using cross-product matrices to compute the SVD

Some properties of LSQR for large sparse linear least squares problems

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

Lanczos bidiagonalization-based inverse solution methods applied to electrical imaging of the heart by using reduced lead-sets: A simulation study

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

A contribution to perturbation analysis for total least squares problems

Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems