On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces

The main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. We provide a new stability characterization of oblique projection generalized inverses of closed linear operators under T-bounded perturbations, which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces

In the present paper, we propose a computer-assisted procedure to prove the invertibility of a linear operator in a Hilbert space and to compute a verified norm bound of its inverse. A number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates. The approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications. Several verification examples that confirm the actual effectiveness of the proposed procedure are reported.     

An optimum iterative method for solving any linear system with a square matrix

A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.     

Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems

Two classes of incomplete factorization preconditioners are considered for nonsymmetric linear systems arising from second order finite difference discretizations of non-self-adjoint elliptic partial differential equations. Analytic and experimental results show that relaxed incomplete factorization methods exhibit numerical instabilities of the type observed with other incomplete factorizations, and the effects of instability are characterized in terms of the relaxation parameter. Several stabilized incomplete factorizations are introduced that are designed to avoid numerically unstable computations. In experiments with two-dimensional problems with variable coefficients and on nonuniform meshes, the stabilized methods are shown to be much more robust than standard incomplete factorizations.The work presented in this paper was supported by the National Science Foundation under grants DMS-8607478, CCR-8818340, and ASC-8958544, and by the U.S. Army Research Office under grant DAAL-0389-K-0016.     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators

Summary. The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of finding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to the particular case of stochastic matrices. Moreover, an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled Markov chains, as well as for Markov chains with other zero-nonzero structures. Received May 25, 1991/Revised version received February 23, 1994     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

Polynomial least squares fitting in the Bernstein basis

The problem of polynomial least squares fitting in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.     

On linear preservers of normal maps

We study group induced cone (GIC) orderings generating normal maps. Examples of normal maps cover, among others, the eigenvalue map on the space of n × n Hermitian matrices as well as the singular value map on n × n complex matrices. In this paper, given two linear spaces equipped with GIC orderings induced by groups of orthogonal operators, we investigate linear operators preserving normal maps of the orderings. A characterization of the preservers is obtained in terms of the groups. The result is applied to show that the normal structure of the spaces is preserved under the action of the operators. In addition, examples are given.     

Perturbation analysis of generalized singular subspaces

This paper, as a continuation of the paper [20] in Numerische Mathematik, studies the subspaces associated with the generalized singular value decomposition. Second order perturbation expansions, Fréchet derivatives and condition numbers, and perturbation bounds for the subspaces are derived. Received January 26, 1996 / Revised version received May 14, 1997     

Perturbation bounds for polynomials

Using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of Ostrowski (Acta Math 72:99–257, 1940), Elsner et al. (Linear Algebra Appl 142:195–209, 1990). A comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of Beauzamy (Can Math Bull 42(1):3–12, 1999). Using the matrix theoretical approach we also improve the backward stability result of Edelman and Murakami (Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, SIAM, Philapdelphia, 1994; Math Comput 64:210–763, 1995).     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

Block elimination with one refinement solves bordered linear systems accurately

Linear systems with a fairly well-conditioned matrixM of the form , for which a black box solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the black box has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the black box is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.Dedicated to the memory of J. H. Wilkinson     

Perturbation of systems of linear algebraic equations

A classical perturbation result for nonsingular systems of linear algebraic equations is extended to general consistent systems under any norm. An optimal perturbation result is also obtained for general linear least squares problems under a Euclidean norm.     

Group actions on Banach spaces and a geometric characterization of a-T-menability

We prove a geometric characterization of a-T-menability through proper, affine, isometric actions on the Banach spaces Lp[0,1] for 1<p<2. This answers a question of A. Valette.