Thomas Ericsson

The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems

A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.

How to implement the spectral transformation

The general, linear eigenvalue equations (H X M)z = 0, where H and M are real symmetric matrices with M positive semidefinite, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs (X, z). When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues X near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction. The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration. A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when M is ill-conditioned or singular. We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well. 1. Overview. This contribution is an addendum to the paper by Ericsson and Ruhe [1] and also [7]. The value of the spectral transformation is reiterated in a later section. Here we outline our implementation of this transformation. The equation to be solved, for an eigenvalue A and eigenvector z, is (1) (H XM)z = 0, H and M are real symmetric n X n matrices, and M is positive semidefinite. A practical instance of (1) occurs in dynamic analysis of structures, where H and M are the stiffness and mass matrices, respectively. We assume that a linear combination of H and M is positive definite. It then follows that all eigenvalues X are real. In addition, one has a real scalar a, distinct from any eigenvalue, and we seek a few eigenvalues A close to a, together with their eigenvectors z. Ericsson and Ruhe replace (1) by a standard eigenvalue equation (2) [C(H aM) -CT v]y = 0, where C is the Choleski factor of M; M = CTC and y = Cz. If M is singular then so is C, but fortunately the eigenvector z can be recovered from y via z = (H aM) -ICTy. Of course, there is no intention to invert (H a M) explicitly. The Received May 14, 1984; revised December 20, 1985. 1980 Mathematics Subject Classification. Primary 65F15. * This research was supported in part by the AFOSR contract F49620-84-C-0090. The third author was also supported in part by the Swedish Natural Science Research Council. * *The paper was written while this author was visiting the Center for Pure and Applied Mathematics, University of California, Berkeley, California 94720. ?1987 American Mathematical Society 0025-5718/87

Biomechanical effects of environmental and engineered particles on human airway smooth muscle cells

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Dosimetric characterization of radionuclides for systemic tumor therapy: influence of particle range, photon emission, and subcellular distribution.

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Galacturonic acid groups in birch xylan

The past decade has seen significant increases in combustion-generated ambient particles, which contain a nanosized fraction (less than 100 nm), and even greater increases have occurred in engineered nanoparticles (NPs) propelled by the booming nanotechnology industry. Although inhalation of these particulates has become a public health concern, human health effects and mechanisms of action for NPs are not well understood. Focusing on the human airway smooth muscle cell, here we show that the cellular mechanical function is altered by particulate exposure in a manner that is dependent upon particle material, size and dose. We used Alamar Blue assay to measure cell viability and optical magnetic twisting cytometry to measure cell stiffness and agonist-induced contractility. The eight particle species fell into four categories, based on their respective effect on cell viability and on mechanical function. Cell viability was impaired and cell contractility was decreased by (i) zinc oxide (40–100 nm and less than 44 μm) and copper(II) oxide (less than 50 nm); cell contractility was decreased by (ii) fluorescent polystyrene spheres (40 nm), increased by (iii) welding fumes and unchanged by (iv) diesel exhaust particles, titanium dioxide (25 nm) and copper(II) oxide (less than 5 μm), although in none of these cases was cell viability impaired. Treatment with hydrogen peroxide up to 500 μM did not alter viability or cell mechanics, suggesting that the particle effects are unlikely to be mediated by particle-generated reactive oxygen species. Our results highlight the susceptibility of cellular mechanical function to particulate exposures and suggest that direct exposure of the airway smooth muscle cells to particulates may initiate or aggravate respiratory diseases.

Lanczos algorithms and field of value rotations for symmetric matrix pencils

Various radionuclides have been proposed for systemic tumor therapy. However, in most dosimetric analysis of proposed radionuclides the charged particles are taken into consideration while the potential photons are ignored. The photons will cause undesirable irradiation of normal tissue, and increase the probability of toxicity in, e.g., the bone marrow. The aim of this study was to investigate the dosimetric properties according to particle range, photon emission, and subcellular radionuclide distribution, of a selection of radionuclides used or proposed for radionuclide therapy, and to investigate the possibility of dividing radionuclides into groups according to their dosimetric properties. The absorbed dose rate to the tumors divided by the absorbed dose rate to the normal tissue (TND) was estimated for different tumor sizes in a mathematical model of the human body. The body was simulated as a 70-kg ellipsoid and the tumors as spheres of different sizes (1 ng-100 g). The radionuclides were either assumed to be uniformly distributed throughout the entire tumor and normal tissue, or located in the nucleus or the cytoplasm of the tumor cells and on the cell membrane of the normal cells. Fifty-nine radionuclides were studied together with monoenergetic electrons, positrons, and alpha particles. The tumor and normal tissue were assumed to be of water density. The activity concentration ratio between the tumor and normal tissue was assumed to be 25. The radionuclides emitting low-energy electrons combined with a low photon contribution, and the alpha emitters showed high TND values for most tumor sizes. Electrons with higher energy gave reduced TND values for small tumors, while a higher photon contribution reduced the TND values for large tumors. Radionuclides with high photon contributions showed low TND value for all tumor sizes studied. The radionuclides studied could be divided into four main groups according to their TND values: beta emitters, Auger electron emitters, photon emitters, and alpha emitters. The TND values of the beta emitters were not affected by the subcellular distribution of the radionuclide. The TND values of the Auger electron emitters were affected by the subcellular radionuclide distribution. The photon emitters showed low TND values that were only slightly affected by the subcellular radionuclide distribution. The alpha emitters showed high TND values that were only slightly affected by the subcellular radionuclide distribution. This dosimetric characterization of radionuclides may be valuable in choosing the appropriate radionuclides for specific therapeutic applications.

Anion-exchange chromatography of dicarboxylic hydroxy acids

SummaryA large number of reducing xylose end groups are present in birch wood (Betula verrucosa) while all galacturonic acid moieties are non-reducing. A large portion of the reducing xylose end groups are removed during treatments with alkali used for the islation of xylan, while reducing galacturonic acid moieties are formed by endwise degradation during this procedure.

Treatment of Birch Xylan with Chlorine in Aqueous Solution.

Abstract Iterative algorithms for the eigensolution of symmetric pencils of matrices are considered. It is shown that the symmetric Lanczos algorithm, the nonsymmetric Lanczos algorithm, and the Arnoldi algorithm are closely related in this case. The applicability of this class of algorithms to indefinite pencils is discussed. A new field of values concept is used to describe the symmetric pencil problem. Then spectral transformation corresponds to a rotation in the complex plane, and the inertia count gives the number of eigenvalues corresponding to points in one of two half planes.

Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

Abstract In the anion-exchange chromatography of dicarboxylic acids with one uronic acid and one aldonic acid moiety, the distribution coefficients (Dv) decrease markedly with increasing molar volume when sodium acetate is used as the eluent. This decrease in an oligomeric series can be predicted from the Gibbs-Donnan equation. For isomers, Dv increases with decreasing distance between the carboxyl groups. The shorter this distance, the higher is the acid strength, which strongly affects Dv in acetic acid.