A generalized global Arnoldi method for ill-posed matrix equations

This paper discusses the solution of large-scale linear discrete ill-posed problems with a noise-contaminated right-hand side. Tikhonov regularization is used to reduce the influence of the noise on the computed approximate solution. We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects the structure of the equation, for the solution of the regularized problem. Theoretical properties of the method are shown and applications to image deblurring are described.     

A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

The inverse-free preconditioned Krylov subspace method of Golub and Ye [G.H. Golub, Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comp. 24 (2002) 312-334] is an efficient algorithm for computing a few extreme eigenvalues of the symmetric generalized eigenvalue problem. In this paper, we first present an analysis of the preconditioning strategy based on incomplete factorizations. We then extend the method by developing a block generalization for computing multiple or severely clustered eigenvalues and develop a robust black-box implementation. Numerical examples are given to illustrate the analysis and the efficiency of the block algorithm.     

A block incomplete orthogonalization method for large nonsymmetric eigenproblems

The incomplete orthogonalization method (IOM) proposed by Saad for computing a few eigenpairs of large nonsymmetric matrices is generalized into a block incomplete orthogonalization method (BIOM). It is studied how the departure from symmetry A – A ^H affects the conditioning of the block basis vectors generated by BIOM, and some relationships are established between the approximate eigenpairs obtained by BIOM and Ritz pairs. It is proved that BIOM behaves much like generalized block Lanczos methods if the basis vectors of the block Krylov subspace generated by it are strongly linearly independent. However, it is shown that BIOM may generate a nearly linearly dependent basis for a general nonsymmetric matrix. Numerical experiments illustrate the convergence behavior of BIOM.This work was supported in part by the Graduiertenkolleg at the University of Bielefeld, Germany.     

Implementation of a restarted Krylov subspace method for the evaluation of matrix functions

A new implementation of restarted Krylov subspace methods for evaluating f(A)b for a function f, a matrix A and a vector b is proposed. In contrast to an implementation proposed previously, it requires constant work and constant storage space per restart cycle. The convergence behavior of this scheme is discussed and a new stopping criterion based on an error indicator is given. The performance of the implementation is illustrated for three parabolic initial value problems, requiring the evaluation of exp(A)b.     

Eigenvalue inclusion regions from inverses of shifted matrices

We consider eigenvalue inclusion regions based on the field of values, pseudospectra, Gershgorin region, and Brauer region of the inverse of a shifted matrix. A family of these inclusion regions is derived by varying the shift. We study several properties, one of which is that the intersection of a family is exactly the spectrum. The numerical approximation of the inclusion sets for large matrices is also examined.     

Rational Krylov for Nonlinear Eigenproblems,an Iterative Projection Method

In recent papers Ruhe suggested a rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similarly to the Arnoldi method the search space is expanded by the direction from residual inverse iteration. Numerical methods demonstrate that the rational Krylov method can be accelerated considerably by replacing an inner iteration by an explicit solver of projected problems.     

Extracting partial canonical structure for large scale eigenvalue problems

We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

调和Arnoldi方法的一种变形

讨论求解大规模非对称矩阵内部特征问题的一种方法,与标准的调和A rnold i方法相比,该方法仍用调和R itz值作为特征值的近似,而在近似特征向量选取方面,我们充分利用A rnold i过程所提供的最末一个基向量的信息,在多1维K ry lov子空间中选取一个向量-称之为改进的调和R itz向量-作为所求的特征向量的近似.理论分析和数值试验均表明这种变形的调和A rnold i方法的可行性和有效性.     

解大型非对称特征问题的精化块不完全正交化算法

0引言 块Arnoldi方法~[5]是解大型非对称特征值问题的正交投影方法，然而Jia~[3]的分析表