arXiv subjects
Publications and source records attributed to David S. Watkins.
Pole-swapping algorithms, generalizations of bulge-chasing algorithms, have been shown to be a viable alternative to the bulge-chasing QZ algorithm for solving the generalized eigenvalue problem for a matrix pencil A - λB. It is natural to try to devise a pole-swapping algorithm that solves the standard eigenvalue problem for a single matrix A. This paper introduces such an algorithm and shows that it is competitive with Francis's bulge-chasing QR algorithm.
Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.
Pole-swapping algorithms are generalizations of bulge-chasing algorithms for the generalized eigenvalue problem. Structure-preserving pole-swapping algorithms for the palindromic and alternating eigenvalue problems, which arise in control theory, are derived. A refinement step that guarantees backward stability of the algorithms is included. This refinement can also be applied to bulge-chasing algorithms that had been introduced previously, thereby guaranteeing their backward stability in all cases.
This work is a continuation of "Fast and backward stable computation of roots of polynomials" by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015. In that paper we introduced a companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the companion matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a companion QZ algorithm that solves a generalized eigenvalue problem for a companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus the companion QR algorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's \texttt{roots} command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The companion QZ algorithm has the same favorable backward error as companion QR, provided that the polynomial coefficients are properly scaled.
In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of $k$ matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods.
Because of the potentially large number of important applications of nonlinear optics, researchers have expended a great deal of effort to optimize the second-order molecular nonlinear-optical response, called the hyperpolarizability. The focus of our present studies is the {\em intrinsic} hyperpolarizability, which is a scale-invariant quantity that removes the effects of simple scaling, thus being the relevant quantity for comparing molecules of varying sizes. Past theoretical studies have focused on structural properties that optimize the intrinsic hyperpolarizability, which have characterized the structure of the quantum system based on the potential energy function, placement of nuclei, geometry, and the effects of external electric and magnetic fields. Those previous studies focused on single-electron models under the influence of an average potential. In the present studies, we generalize our calculations to two-electron systems and include electron interactions. As with the single-electron studies, universal properties are found that are common to all systems -- be they molecules, nanoparticles, or quantum gases -- when the hyperpolarizability is near the fundamental limit.
Previous studies have used numerical methods to optimize the hyperpolarizability of a one-dimensional quantum system. These studies were used to suggest properties of one-dimensional organic molecules, such as the degree of modulation of conjugation, that could potentially be adjusted to improve the nonlinear-optical response. However, there were no conditions set on the optimized potential energy function to ensure that the resulting energies were consistent with what is observed in real molecules. Furthermore, the system was placed into a one-dimensional box with infinite walls, forcing the wavefunctions to vanish at the ends of the molecule. In the present work, the walls are separated by a distance much larger than the molecule's length; and, the variations of the potential energy function are restricted to levels that are more typical of a real molecule. In addition to being a more physically-reasonable model, our present approach better approximates the bound states and approximates the continuum states - which are usually ignored. We find that the same universal properties continue to be important for optimizing the nonlinear-optical response, though the details of the wavefunctions differ from previous result.
We use numerical optimization to study the properties of (1) the class of one-dimensional potential energy functions and (2) systems of point charges in two-dimensions that yield the largest hyperpolarizabilities, which we find to be within 30% of the fundamental limit. We investigate the character of the potential energy functions and resulting wavefunctions and find that a broad range of potentials yield the same intrinsic hyperpolarizability ceiling of 0.709.
We use numerical optimization to find a one-dimensional potential energy function that yields the largest hyperpolarizability, which we find is within 30% of the fundamental limit. Our results reveal insights into the character of the potential energy functions and wavefunctions that lead to the largest hyperpolarizability. We suggest that donor-acceptor molecules with a conjugated bridge with many sites of reduced conjugation to impart conjugation modulation may be the best paradigm for making materials with huge hyperpolarizabilities that approach the fundamental limit.
Extensive studies in the past have focused on precise calculations of the nonlinear-optical susceptibility of thousands of molecules. In this work, we use the broader approach of considering how geometry and symmetry alone play a role. We investigate the nonlinear optical response of potential energy functions that are given by a superposition of force centers (representing the nuclear charges) that lie in various planar geometrical arrangements. We find that for certain specific geometries, such as an octupolar-like molecule with donors and acceptors of varying strengths at the branches, the hyperpolarizability is near the {\em fundamental limit}. In these cases, the molecule is observed to be well approximated by a three-level model - consistent with the three-level ansatz previously used to calculate the {\em fundamental limits}. However, when the hyperpolarizability is below the {\em apparent limit} (about a factor of thirty below the {\em fundamental limit}) the system is no longer representable by a three-level model; where both two-level and a many-state models are found to be appropriate, depending on the symmetry.