The approximation of generalized turning points\newline by Krylov subspace methods

Summary. Certain types of singular solutions of nonlinear parameter-dependent operator equations were characterized by Griewank and Reddien [5, 6] as regular solutions of suitable augmented systems. For their numerical approximation an approach based on the use of Krylov subspaces is here presented. The application to boundary value problems is illustrated by numerical examples. Received March 8, 1993 / Revised version received December 13, 1993     

An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices

Summary. In this paper we propose an algorithm based on Laguerre's iteration, rank two divide-and-conquer technique and a hybrid strategy for computing singular values of bidiagonal matrices. The algorithm is fully parallel in nature and evaluates singular values to tiny relative error if necessary. It is competitive with QR algorithm in serial mode in speed and advantageous in computing partial singular values. Error analysis and numerical results are presented. Received March 15, 1993 / Revised version received June 7, 1994     

Iteration schemes for the divide-and-conquer eigenvalue solver

Summary. We propose globally convergent iteration schemes for updating the eigenvalues of a symmetric matrix after a rank-1 modification. Such calculations are the core of the divide-and-conquer technique for the symmetric tridiagonal eigenvalue problem. We prove the superlinear convergence right from the start of our schemes which allows us to improve the complexity bounds of [3]. The effectiveness of our algorithms is confirmed by numerical results which are reported and discussed. Received September 22, 1993     

A note on the pivoted symmetric LR iteration

Summary. We prove that the diagonally pivoted symmetric LR algorithm on a positive definite matrix is globally convergent. Received December 23, 1997 / Revised version received August 3, 1998 / Published online August 19, 1999     

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order , where is the machine precision, the new method computes the eigenvalues to full possible accuracy. Received April 8, 1996 / Revised version received December 20, 1996     

Bounds on the error of an approximate invariant subspace for non-self-adjoint matrices

Summary. Suppose one approximates an invariant subspace of an matrix in which in not necessarily self--adjoint. Suppose that one also has an approximation for the corresponding eigenvalues. We consider the question of how good the approximations are. Specifically, we develop bounds on the angle between the approximating subspace and the invariant subspace itself. These bounds are functions of the following three terms: (1) the residual of the approximations; (2) singular--value separation in an associated matrix; and (3) the goodness of the approximations to the eigenvalues. Received December 1, 1992 / Revised version received October 20, 1993     

Computing the generalized singular values/vectors of large sparse or structured matrix pairs

Summary. We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair . The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair . Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm. Received April 1, 1994 / Revised version received December 15, 1994     

Backward errors for the inverse eigenvalue problem

Backward errors for the symmetric matrix inverse eigenvalue problem with respect to an approximate solution are defined, and explicit expressions of the backward errors are derived. The expressions may be useful for testing the stability of practical algorithms. Received August 4, 1997 / Revised version received May 11, 1998     

A matrix analysis of Arnoldi and Lanczos methods

Summary. In this paper we propose a matrix analysis of the Arnoldi and Lanczos procedures when used for approximating the eigenpairs of a non-normal matrix. By means of a new relation between the respective representation matrices, we relate the corresponding eigenvalues and eigenvectors. Moreover, backward error analysis is used to theoretically justify some unexpected experimental behaviors of non-normal matrices and in particular of banded Toeplitz matrices. Received June 19, 1996 / Revised version received November 3, 1997     

Riccati-based preconditioner for computing invariant subspaces of large matrices

Summary. This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported. Received June 15, 2000 / Revised version received January 22, 2001 / Published online October 17, 2001     

On worst-case condition numbers of a nondefective multiple eigenvalue

Summary. This paper is a continuation of the author [6] in Numerische Mathematik. Let be a nondefective multiple eigenvalue of multiplicity of an complex matrix , and let be the secants of the canonical angles between the left and right invariant subspaces of corresponding to the multiple eigenvalue . The analysis of this paper shows that the quantities are the worst-case condition numbers of the multiple eigenvalue . Received September 28, 1992 / Revised version received January 18, 1994     

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     

The Lidskii-Mirsky-Wielandt theorem – additive and multiplicative versions

Summary. We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices. Our results strengthen and generalize those in the literature. Received November 20, 1996 / Revised version received January 27, 1998     

Accurate singular values and differential qd algorithms

Summary. We have discovered a new implementation of the qd algorithm that has a far wider domain of stability than Rutishauser's version. Our algorithm was developed from an examination of the {Cholesky~LR} transformation and can be adapted to parallel computation in stark contrast to traditional qd. Our algorithm also yields useful a posteriori upper and lower bounds on the smallest singular value of a bidiagonal matrix. The zero-shift bidiagonal QR of Demmel and Kahan computes the smallest singular values to maximal relative accuracy and the others to maximal absolute accuracy with little or no degradation in efficiency when compared with the LINPACK code. Our algorithm obtains maximal relative accuracy for all the singular values and runs at least four times faster than the LINPACK code. Received August 8, 1993/Revised version received May 26, 1993     

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix , or a pair of left and right deflating subspaces of a regular matrix pencil . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Received September 20, 1994 / Revised version received February 5, 1996     

A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues

Summary. We prove that the 2-norm distance from an matrix A to the matrices that have a multiple eigenvalue is equal to where the singular values are ordered nonincreasingly. Therefore, the 2-norm distance from A to the set of matrices with multiple eigenvalues is Received February 19, 1998 / Revised version received July 15, 1998 / Published online: July 7, 1999     

A note on properties of splittings of singular symmetric positive semidefinite matrices

Summary. Recently, Benzi and Szyld have published an important paper [1] concerning the existence and uniqueness of splittings for singular matrices. However, the assertion in Theorem 3.9 on the inheriting property of P-regular splitting for singular symmetric positive semidefinite matrices seems to be incorrect. As a complement of paper [1], in this short note we point out that if a matrix T is resulted from a P-regular splitting of a symmetric positive semidefinite matrix A, then splittings induced by T are not all P-regular. Received January 7, 1999 / Published online December 19, 2000     

Numerical methods for finding multiple eigenvalues of matrices depending on parameters

Summary. Let be a square matrix dependent on parameters and , of which we choose as the eigenvalue parameter. Many computational problems are equivalent to finding a point such that has a multiple eigenvalue at . An incomplete decomposition of a matrix dependent on several parameters is proposed. Based on the developed theory two new algorithms are presented for computing multiple eigenvalues of with geometric multiplicity . A third algorithm is designed for the computation of multiple eigenvalues with geometric multiplicity but which also appears to have local quadratic convergence to semi-simple eigenvalues. Convergence analyses of these methods are given. Several numerical examples are presented which illustrate the behaviour and applications of our methods. Received December 19, 1994 / Revised version received January 18, 1996     

Convergence of block two-stage iterative methods for symmetric positive definite systems

Summary. We study the convergence of two-stage iterative methods for solving symmetric positive definite (spd) systems. The main tool we used to derive the iterative methods and to analyze their convergence is the diagonally compensated reduction (cf. [1]). Received December 11, 1997 / Revised version received March 25, 1999 / Published online May 30, 2001     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995