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     

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 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     

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     

Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix

Summary. In this paper we propose four algorithms to compute truncated pivoted QR approximations to a sparse matrix. Three are based on the Gram–Schmidt algorithm and the other on Householder triangularization. All four algorithms leave the original matrix unchanged, and the only additional storage requirements are arrays to contain the factorization itself. Thus, the algorithms are particularly suited to determining low-rank approximations to a sparse matrix. Received February 23, 1998 / Revised version received April 16, 1998     

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     

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     

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     

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     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

Accurate computation of the zeros of the generalized Bessel polynomials

Summary. A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper. Received July 25, 1997 / Revised version received May 19, 1999 / Published online June 8, 2000     

Detection of Hopf points by counting sectors in the complex plane

Summary. Let ( real) be a family of real by matrices. A value of is called a Hopf value if has a conjugate pair of purely imaginary eigenvalues , . We describe a technique for detecting Hopf values based on the evolution of the Schur complement of in a bordered extension of where varies along the positive imaginary axis of the complex plane. We compare the efficiency of this method with more obvious methods such as the use of the QR algorithm and of the determinant function of as well as with recent work on the Cayley transform. In particular, we show the advantages of the Schur complement method in the case of large sparse matrices arising in dynamical problems by discretizing boundary value problems. The Hopf values of the Jacobian matrices are important in this setting because they are related to the Hopf bifurcation phenomenon where steady state solutions bifurcate into periodic solutions. Received September 15, 1994 / Revised version received July 7, 1995     

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     

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     

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     

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     

Necessary and sufficient conditions for the convergence of Orthomin(k) on singular and inconsistent linear systems

Summary. We consider the convergence of Orthomin(k) on singular and inconsistent linear systems. Criteria for the breakdown of Orthomin(k) are discussed and analyzed. Moreover, necessary and sufficient conditions for the convergence of Orthomin(k) for any right hand side are given, and a rate of convergence is provided as well. Finally, numerical experiments are shown to confirm the convergence theorem. Received April 1, 1995 / Revised version received October 1, 1997 / Published online July 12, 2000     

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