Krešimir Veselić

Jacobi's method is more accurage than QR

It is shown that Jacobi’s method (with a proper stopping criterion) computes small eigenvalues of symmetric positive definite matrices with a uniformly better relative accuracy bound than QR, divide and conquer, traditional bisection, or any algorithm which first involves tridiagonalizing the matrix. Modulo an assumption based on extensive numerical tests, Jacobi’s method is optimally accurate in the following sense: if the matrix is such that small relative errors in its entries cause small relative errors in its eigenvalues, Jacobi will compute them with nearly this accuracy. In other words, as long as the initial matrix has small relative errors in each component, even using infinite precision will not improve on Jacobi (modulo factors of dimensionality). It is also shown that the eigenvectors are computed more accurately by Jacobi than previously thought possible. Similar results are proved for using one-sided Jacobi for the singular value decomposition of a general matrix.

New Fast and Accurate Jacobi SVD Algorithm. I

This paper is the result of concerted efforts to break the barrier between numerical accuracy and run-time efficiency in computing the fundamental decomposition of numerical linear algebra—the singular value decomposition (SVD) of general dense matrices. It is an unfortunate fact that the numerically most accurate one-sided Jacobi SVD algorithm is several times slower than generally less accurate bidiagonalization-based methods such as the QR or the divide-and-conquer algorithm. Our quest for a highly accurate and efficient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm. The new algorithm has inherited all good high accuracy properties of the Jacobi algorithm, and it can outperform the QR algorithm.

A note on one-sided Jacobi algorithm

SummaryWe propose a “one-sided” or “implicit” variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.

On linear vibrational systems with one dimensional damping II

We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linearn-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n3 operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.

Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix

Abstract Some new estimates for the eigenvalue decay rate of the Lyapunov equation AX + XA T = B with a low rank right-hand side B are derived. The new bounds show that the right-hand side B can greatly influence the eigenvalue decay rate of the solution. This suggests a new choice of the ADI-parameters for the iterative solution. The advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix.

PERTURBATION OF PSEUDORESOLVENTS AND ANALYTICITY IN 1/c IN RELATIVISTIC QUANTUM MECHANICS.

The analytic functional calculus, relatively bounded and analytic perturbations of pseudoresolvents have been studied. As an application, the nonrelativistic limit of the Dirac and Klein-Gordon operator in the presence of an external static field has been considered. It has been proved that the resolvents of these operators have only a removable singularity atc=∞. This implies the analyticity atc=∞ of the eigenvalues and eigenvectors corresponding to the bound states of the mentioned operators.

On the Perturbation of the Cholesky Factorization

The perturbation of the Cholesky factor of a perturbed positive definite matrix is considered. Estimates are included for small perturbations in the spectral norm as well as for large perturbations in the Euclidean norm. The results can be applied to floating point perturbations as well.

Wielandt and Ky-Fan theorem for matrix pairs

The generalization of Wielandt and Ky-Fan theorem is given for Hermitian matrix pairs, and some new eigenvalue perturbation estimates are obtained. An application is made on a class of quadratic matrix pencils.

Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-inde finite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensur ing the second representation the- orem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

Trace minimization and definiteness of symmetric pencils

Abstract A symmetric matrix pencil A - λB of order n is called positive definite if there is a μ such that the matrix A − μB is positive definite. We consider the case with B nonsingular and show that the definiteness is closely related to the existence of min Tr X T AX under the condition X T BX = J 1 where J 1 is a given diagonal matrix of order ≤ n and J 2 1 = I . We also prove an analog of the Cauchy interlacing theorem for some such pencils.