Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

Upper bound and stability of scaled pseudoinverses

Summary. For given matrices and where is positive definite diagonal, a weighed pseudoinverse of is defined by and an oblique projection of is defined by . When is of full column rank, Stewart [3] and O'Leary [2] found sharp upper bound of oblique projections which is independent of , and an upper bound of weighed pseudoinverse by using the bound of . In this paper we discuss the sharp upper bound of over a set of positive diagonal matrices which does not depend on the upper bound of , and the stability of over . Received September 29, 1993 / Revised version received October 31, 1994     

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order may be bounded from below by with , and that this bound may be improved at most by a factor . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations. Received December 1, 1997 / Revised version received February 25, 1999 / Published online 16 March 2000     

Symmetric Gaussian elimination for cauchy-type matrices with application to positive definite Toeplitz matrices

The problem of solving linear equations with a Toeplitz matrix appears in many applications. Often is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix is proposed. The resulting new linear equations can be solved in operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform a Hermitian Toeplitz matrix into a Cauchy-type matrix by applying the Fourier transform. We will prove some useful properties of and formulate a symmetric Gaussian elimination algorithm for positive definite . Using the symmetry and persymmetry of we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show that this new algorithm is fast and reliable. Received March 24, 1995 / Revised version received December 13, 1995     

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     

Quadratic convergence of scaled matrices in Jacobi method

Summary. A quadratic convergence bound for scaled Jacobi iterates is proved provided the initial symmetric positive definite matrix has simple eigenvalues. The bound is expressed in terms of the off-norm of the scaled initial matrix and the minimum relative gap in the spectrum. The obtained result can be used to predict the stopping moment in the two-sided and especially in the one-sided Jacobi method. Received October 31, 1997 / Revised version received March 8, 1999 / Published online July 12, 2000     

Block LU factorizations of M–matrices

Summary. It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular M–matrix can have when L and U are M–matrices. Received November 21, 1994 / Revised version received August 4, 1997     

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     

Componentwise perturbation analyses for the QR factorization

Summary. This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, , R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation are explained and extended. Weaker rigorous bounds are also given. Received April 11, 1999 / Published online October 16, 2000     

Pseudozeros of polynomials and pseudospectra of companion matrices

Summary. It is well known that the zeros of a polynomial are equal to the eigenvalues of the associated companion matrix . In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The is the set of zeros of all polynomials obtained by coefficientwise perturbations of of size ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The is another subset of defined as the set of eigenvalues of matrices with ; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then and are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties. Received June 15, 1993     

Semiconvergence of nonnegative splittings for singular matrices

Summary. In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above. Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000     

Stability analysis and fast algorithms for triangularization of Toeplitz matrices

Summary. A new algorithm for triangularizing an Toeplitz matrix is presented. The algorithm is based on the previously developed recursive algorithms that exploit the Toeplitz structure and compute each row of the triangular factor via updating and downdating steps. A forward error analysis for this existing recursive algorithm is presented, which allows us to monitor the conditioning of the problem, and use the method of corrected semi-normal equations to obtain higher accuracy for certain ill-conditioned matrices. Numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in . Received April 30, 1995 / Revised version received February 12, 1996     

On multisplitting methods for band matrices

Summary. We present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively. These classes are based on overlapping blocks of the underlying coefficient matrix which is assumed to be a band matrix. We show that under suitable conditions the spectral radius of the iteration matrix does not depend on the weights of the method even if these weights are allowed to be negative. For a certain class of splittings we prove an optimality result for with respect to the weights provided that is an M–matrix. This result is based on the fact that the multisplitting method can be represented by a single splitting which in our situation surprisingly turns out to be a regular splitting. Furthermore we show by numerical examples that weighting factors may considerably improve the convergence. Received July 18, 1994 / Revised version received November 20, 1995     

On close eigenvalues of tridiagonal matrices

Summary. A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element and must be a direct sum of two complementary blocks, both of which have the eigenvalue. Yet it is well known that a small spectral gap does not necessarily imply that some is small, as is demonstrated by the Wilkinson matrix. In this note, it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the coupling the two blocks may not be small. In particular, some explanatory bounds are derived and a connection to the Lanczos algorithm is observed. The nonsymmetric problem is also included. Received April 8, 1992 / Revised version received September 21, 1994     

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     

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     

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     

Regions of exact convergence of the USSOR method applied on generalized consistently ordered matrices

Summary. Using the theory of nonnegative matrices and regular splittings, exact convergence and divergence domains of the Unsymmetric Successive Overrelaxation (USSOR) method, as it pertains to the class of Generalized Consistently Ordered (GCO) matrices, are determined. Our recently derived upper bounds, for the convergence of the USSOR method, re also used as effective tools. Received October 17, 1993 / Revised version received December 19, 1994     

Vandermonde matrices on integer nodes

Summary. This paper deals with Vandermonde matrices whose nodes are the first integer numbers. We give an analytic factorization of such matrices and explicit formulas for the entries of their inverses, and explore their computational issues. We also give asymptotic estimates of the Frobenius norm of both and its inverse. Received July 28, 1995 / Revised version received July 4, 1997