Showing papers on "Bidiagonalization published in 2005"

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

Abstract: New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the new methods to be competitive with available schemes.

Core Problems in Linear Algebraic Systems

Abstract: For any linear system $A x \approx b$ we define a set of core problems and show that the orthogonal upper bidiagonalization of $[b, A]$ gives such a core problem. In particular we show that these core problems have desirable properties such as minimal dimensions. When a total least squares problem is solved by first finding a core problem, we show the resulting theory is consistent with earlier generalizations, but much simpler and clearer. The approach is important for other related solutions and leads, for example, to an elegant solution to the data least squares problem. The ideas could be useful for solving ill-posed problems.

Tikhonov Regularization with a Solution Constraint

Abstract: Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization This paper presents a modification of a numerical method proposed by Golub and von Matt for quadratically constrained least-squares problems and applies it to Tikhonov regularization of large-scale linear discrete ill-posed problems The method is based on partial Lanczos bidiagonalization and Gauss quadrature Computed examples illustrating its performance are presented

Tikhonov Regularization of Large Symmetric Problems

Abstract: SUMMARY Many popular solution methods for large discrete ill-posed problems are based on Tikhonov regularization and compute a partial Lanczos bidiagonalization of the matrix. The computational eort required by these methods is not reduced significantly when the matrix of the discrete ill-posed problem, rather than being a general nonsymmetric matrix, is symmetric and possibly indefinite. This paper describes new methods, based on partial Lanczos tridiagonalization of the matrix, that exploit symmetry. Computed examples illustrate that one of these methods can require significantly less computational work than available structure-ignoring schemes. Copyright c ∞ 2000 John Wiley & Sons, Ltd.

On Accuracy Properties of One-Sided Bidiagonalization Algorithm and Its Applications

Abstract: The singular value decomposition (SVD) of a general matrix is the fundamental theoretical and computational tool in numerical linear algebra. The most efficient way to compute the SVD is to reduce the matrix to bidiagonal form in a finite number of orthogonal (unitary) transformations, and then to compute the bidiagonal SVD. This paper gives detailed error analysis and proposes modifications of recently proposed one-sided bidiagonalization procedure, suitable for parallel computing. It also demonstrates its application in solving two common problems in linear algebra.

Theory and boundary element methods for near-field acoustic holography

Abstract: We consider the problem of recovering surface vibrations from acoustic pressure measurements taken in the interior or the exterior of a region. We give two formulations of the problem. One is based on a representation of the pressure as layer potentials and the other is based on approximation by a class of specific solutions to the Helmholtz equation. Boundary element methods are developed to approximate the integral operators. A conjugate gradient algorithm based on Lanczos bidiagonalization is applied to compute regularized solutions to the ill-conditioned equations in the presence of measurement errors. Numerical examples which compare the two formulations are presented.

Using Core Formulations for Ill‐posed Linear Systems

Abstract: We investigate the application of core problem formulations to the context of ill-posed linear algebraic systems. We note that in order to impose regularization on such systems by using core formulations, one is lead to a technique similar to the Truncated (Total) Least Squares method. Choosing an appropriate truncation level from given data can be made efficient by incorporating the truncation level decision as a stopping criterion into a partial bidiagonalization algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)