428 research outputs found
Approximated structured pseudospectra
No full textPseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra
Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz-type matrices
No full textThe sensitivity of eigenvalues of structured matrices under general or structured perturbations of the matrix entries has been thoroughly studied in the literature. Error bounds are available, and the pseudospectrum can be computed to gain insight. Few investigations have focused on analyzing the sensitivity of eigenvectors under general or structured perturbations. This paper discusses this sensitivity for tridiagonal Toeplitz and Toeplitz-type matrices
Software for limited memory restarted - minimization methods using generalized Krylov subspaces
No full textThis paper describes software for the solution of finite-dimensional minimization problems with two terms, a fidelity term and a regularization term. The sum of the p-norm of the former and the q-norm of the latter is minimized, where 0 < p, q ≤ 2. We note that the “p-norm” is not a norm when 0 < p < 1, and similarly for the “q-norm”. This kind of minimization problems arises when solving linear discrete ill-posed problems, such as certain problems in image restoration. They also find applications in statistics. Recently, limited-memory restarted numerical methods that are well suited for the solution of large-scale minimization problems of this kind were described by the authors in [Adv. Comput. Math., 49 (2023), Art. 26]. These methods are based on the application of restarted generalized Krylov subspaces. This paper presents software for these solution methods
Inverse problems for regularization matrices
No full textDiscrete ill-posed problems are difficult to solve, because their solution is very sensitive to errors in the data and to round-off errors introduced during the solution process. Tikhonov regularization replaces the given discrete ill-posed problem by a nearby penalized least-squares problem whose solution is less sensitive to perturbations. The penalization term is defined by a regularization matrix, whose choice may affect the quality of the computed solution significantly. We describe several inverse matrix problems whose solution yields regularization matrices adapted to the desired solution. Numerical examples illustrate the performance of the regularization matrices determined
A modified truncated singular value decomposition method for discrete ill-posed problems
No full textTruncated singular value decomposition is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank-k approximant, which we denote by Ak. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L-curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as Ak. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by Ak. © 2014 John Wiley & Sons, Ltd
The structured distance to normality of banded Toeplitz matrices
No full textSpectral properties of normal (2k + 1)- banded Toeplitz matrices of order n, with k <= left perpendicularn/ 2right perpendicular, are described. Formulas for the distance of (2k + 1)-banded Toeplitz matrices to the algebraic variety of similarly structured normal matrices are presented
Computing unstructured and structured polynomial pseudospectrum approximations
No full textIn many applications it is important to sensitivity of eigenvalues of a matrix polynomial polynomial. The sensitivity commonly is described pseudospectra. However, the determination of pseudospectra of matrix polynomials is very demanding computationally. This paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank-one or projected rank-one perturbations. These perturbations are inspired by Wilkinson's analysis of eigenvalue sensitivity. This approach allows the approximation of both structured and unstructured pseudospectra. Computed examples show the method to perform much better than a method based on random rank-one perturbations both for the approximation of structured and unstructured (i.e., standard) polynomial pseudospectra
On the banded Toeplitz structured distance to symmetric positive semidefiniteness
No full textThis paper is concerned with the determination of a close real banded
positive definite Toeplitz matrix in the Frobenius norm to a given square real
banded matrix. While it is straightforward to determine the closest banded
Toeplitz matrix to a given square matrix, the additional requirement of
positive definiteness makes the problem difficult. We review available
theoretical results and provide a simple approach to determine a banded
positive definite Toeplitz matrix.Comment: 14 pages,2 figure
Generalized circulant Strang-type preconditioners
No full textStrang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {eif}-circulant Strang-type preconditioner. Copyright (C) 2011 John Wiley & Sons, Ltd
The structured distance to normality of Toeplitz matrices with application to preconditioning
No full textA formula for the distance of a Toeplitz matrix to the subspace of {e(i phi)}-circulant matrices is presented, and applications of {e(i phi)}-circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright (C) 2010 John Wiley & Sons, Ltd
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