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

SUMMARY The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is the first part paper used to investigate sensitivity spectrum. Explicit expressions for structured distance closest normal matrix, departure from normality, ϵ ‐pseudospectrum derived. second discusses applications theory inverse eigenvalue problems, construction Chebyshev polynomial‐based Krylov subspace bases, Tikhonov regularization. Copyright © 2012 John Wiley & Sons, Ltd.

The need to evaluate expressions of the form $f(A)$ or $f(A)b$, where f is a nonlinear function, A large sparse $n\times n$ matrix, and b an n-vector, arises in many applications. This paper describes how Faber transform applied field values can be used determine improved error bounds for popular polynomial approximation methods based on Arnoldi process. Applications rational and, particular, process also are discussed.

The restoration of two-dimensional images in the presence noise by Wiener's minimum mean square error filter requires solution large linear systems equations. When is white and Gaussian, under suitable assumptions on image, these equations can be written as a Sylvester's equation \[ T_1^{ - 1} \hat F + FT_2 = C \] for matrix $\hat F$ representing restored image. matrices $T_1 $ $T_2 are symmetric positive definite Toeplitz matrices. We show that ADI iterative method well suited equations,...

The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] is one of the most popular iterative methods for solution large linear systems equations Ax=b with a nonsymmetric sparse matrix. This particularly attractive when good preconditioner available. present paper describes two new determining preconditioners from spectral information gathered Arnoldi process during iterations algorithm. These seek to determine an invariant subspace...

A new hybrid iterative algorithm is proposed for solving large nonsymmetric systems of linear equations. Unlike other algorithms, which first estimate eigenvalues and then apply this knowledge in further iterations, avoids eigenvalue estimates. Instead, it runs GMRES until the residual norm drops by a certain factor, re-applies polynomial implicitly constructed via Richardson iteration with Leja ordering. Preliminary experiments suggest that frequently outperforms restarted algorithm.

Tikhonov regularization is one of the most popular methods for solving linear systems equations or least-squares problems with a severely ill-conditioned matrix A. This method replaces given problem by penalized problem. The present paper discusses measuring residual error (discrepancy) in seminorm that uses fractional power Moore-Penrose pseudoinverse AA T as weighting matrix. Properties this are discussed. Numerical examples illustrate proposed scheme suitable may give approximate...

Recently Laurie presented a new algorithm for the computation of $(2n+1)$-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This first determines symmetric tridiagonal matrix order $2n+1$ from certain mixed moments, then computes partial spectral factorization. We describe that does not require entries to be determined, thereby avoids computations can sensitive perturbations. Our uses consolidation phase divide-and-conquer eigenproblem. also discuss how applied...

The GMRES method by Saad and Schultz is one of the most popular iterative methods for solution large sparse non-symmetric linear systems equations. implementation proposed uses Arnoldi process modified Gram-Schmidt (MGS) to compute orthonormal bases certain Krylov subspaces. MGS requires many vector-vector operations, which can be difficult implement efficiently on vector parallel computers due low granularity these operations. We present a new in which, each subspace used, we first...

This paper presents a new efficient approach for the solution of $\ell_p$-$\ell_q$ minimization problem based on application successive orthogonal projections onto generalized Krylov subspaces increasing dimension. The are generated according to iteratively reweighted least-squares strategy approximation $\ell_p$/$\ell_q$-norms by weighted $\ell_2$-norms. Computed image restoration examples illustrate that it suffices carry out only few iterations achieve high-quality restorations....

Summary Generalized cross validation is a popular approach to determining the regularization parameter in Tikhonov regularization. The chosen by minimizing an expression, which easy evaluate for small‐scale problems, but prohibitively expensive compute large‐scale ones. This paper describes novel method, based on Gauss‐type quadrature, upper and lower bounds desired expression. These are used determine problems. Computed examples illustrate performance of proposed method demonstrate its...

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice matrix may significantly affect quality computed solution. When identity, iterated can yield approximate solutions higher than (standard) regularization. This paper provides an analysis with a different from identity. Computed examples illustrate performance this method.