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## Theory and Methods in Determining the Eigenvalues and Eigenvectors of a Matrix

Description: In the numerous problems of matrix algebra, one finds the problem of determining the eigenvalues of eigenvectors of a matrix quite frequently. The theory and methods leading to the solution of the eigenvalue and eigenvector problem are of considerable interest. The relation between vector spaces, matrices, eigenvalues, and eigenvectors is to be considered in this chapter, with particular concentration directed toward the eigenvalues and eigenvectors shall be developed in the following chapters with detailed examples of the methods.
Date: August 1965
Creator: Waldon, Jerry Herschel
Partner: UNT Libraries

## Improving the Accuracy of Computed Matrix Eigenvalues

Description: A computational method is described for improving the accuracy of a given eigenvalue and its associated eigenvector, arrived at through a computation in a lower precision. The method to be described will increase the accuracy of the pair and do so at a relatively low cost. The technique used is similar to iterative refinement for the solution of a linear system; that is, through the factorization from the low-precision computation, an iterative algorithm is applied to increase the accuracy of the eigenpair. Extended precision arithmetic is used at critical points in the algorithm. The iterative algorithm requires O(n²) operations for each iteration.
Date: August 1980
Creator: Dongarra, J. J.
Partner: UNT Libraries Government Documents Department

## Computations of Eigenpair Subsets with the MRRR Algorithm

Description: The main advantage of inverse iteration over the QR algorithm and Divide & Conquer for the symmetric tridiagonal eigenproblem is that subsets of eigenpairs can be computed at reduced cost. The MRRR algorithm (MRRR = Multiple Relatively Robust Representations) is a clever variant of inverse iteration without the need for reorthogonalization. STEGR, the current version of MRRR in LAPACK 3.0, does not allow for subset computations. The next release of STEGR is designed to compute a (sub-)set of k eigenpairs with {Omicron}(kn) operations. Because of the special way in which eigenvectors are computed, MRRR subset computations are more complicated than when using inverse iteration. Unlike the latter, MRRR sometimes cannot ignore the unwanted part of the spectrum. We describe the problems with what we call 'false singletons'. These are eigenvalues that appear to be isolated with respect to the wanted eigenvalues but in fact belong to a tight cluster of unwanted eigenvalues. This paper analyzes these complications and ways to deal with them.
Date: June 6, 2006
Creator: Marques, Osni A.; Parlett, Beresford N. & Vomel, Christof
Partner: UNT Libraries Government Documents Department

## nu-TRLan User Guide Version 1.0: A High-Performance Software Package for Large-Scale Harmitian Eigenvalue Problems

Description: The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs, users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 ...
Date: October 27, 2008
Creator: Yamazaki, Ichitaro; Wu, Kesheng & Simon, Horst
Partner: UNT Libraries Government Documents Department

## Improving the Accuracy of Computed Eigenvalues and Eigenvectors

Description: This paper describes a computational method for improving the accuracy of a given eigenvalue and its associated eigenvector. The method is analogous to iterative improvement for the solution of linear systems. An iterative algorithm using working precision arithmetic is applied to increase the accuracy of the eigenpair. The only extended precision computation is the residual calculation. The method is related to inverse iteration and to Newton's method applied to the eigenvalue problem.
Date: July 1981
Creator: Dongarra, J. J.; Moler, Cleve B. & Wilkinson, J. H.
Partner: UNT Libraries Government Documents Department

## A Generalized Eigensolver based on Smoothed Aggregation (GES-SA) for Initializing Smoothed Aggregation Multigrid (SA)

Description: Consider the linear system Ax = b, where A is a large, sparse, real, symmetric, and positive definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's relaxation process. For relaxation processes that are typically used in practice, algebraically smooth error corresponds to the near-nullspace of A. Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. This paper discusses the details of a generalized eigensolver based on smoothed aggregation (GES-SA) that is designed to produce an approximation to a minimal eigenvector of A. GES-SA might be very useful as a standalone eigensolver for applications that desire an approximate minimal eigenvector, but the primary aim here is for GES-SA to produce an initial algebraically smooth component that may be used to either create a black-box SA solver or initiate the adaptive SA ({alpha}SA) process.
Date: May 31, 2007
Creator: Brezina, M; Manteuffel, T; McCormick, S; Ruge, J; Sanders, G & Vassilevski, P S
Partner: UNT Libraries Government Documents Department

## Factoring Algebraic Error for Relative Pose Estimation

Description: We address the problem of estimating the relative pose, i.e. translation and rotation, of two calibrated cameras from image point correspondences. Our approach is to factor the nonlinear algebraic pose error functional into translational and rotational components, and to optimize translation and rotation independently. This factorization admits subproblems that can be solved using direct methods with practical guarantees on global optimality. That is, for a given translation, the corresponding optimal rotation can directly be determined, and vice versa. We show that these subproblems are equivalent to computing the least eigenvector of second- and fourth-order symmetric tensors. When neither translation or rotation is known, alternating translation and rotation optimization leads to a simple, efficient, and robust algorithm for pose estimation that improves on the well-known 5- and 8-point methods.
Date: March 9, 2009
Creator: Lindstrom, P & Duchaineau, M
Partner: UNT Libraries Government Documents Department

## Efficient solution of the relativistic APW secular equation for both eigenvalues and eigenvectors

Description: From international symposium on atomic, molecular, solidstate theory, and quantum statistics: Sanibel Island, Florida, USA (20 Jan 1974). The special problems introduced into the solution of the secular equation by the inclusion of relativistic effects are discussed. A method based on the use of linear combination of RAPW's is then described for solving the secular equation. This method greatly reduces the computer time and core space required to solve the secular problem and is particularly advantageous in large systems. (auth)
Date: January 1, 1974
Creator: Koelling, D.D.
Partner: UNT Libraries Government Documents Department

## On the Optimal Number of Eigenvectors for Orbit Correction

Description: The singular value decomposition method is widely used for orbit correction in the storage rings. It is a powerful tool for inverting of the usually rectangular response matrices, which usually have rectangular form. Another advantage is flexibility to choose number of eigenvectors for calculation of required strengths of orbit correctors. In particular, by reduction in number of eigenvectors one can average over ensemble the noise in the beam position monitors. A theoretical approach as well as experimental results on the NSLS VUV ring is presented.
Date: June 23, 2008
Creator: Pinayev,I.
Partner: UNT Libraries Government Documents Department

## Frequency dependent thermal expansion in binary viscoelasticcomposites

Description: The effective thermal expansion coefficient beta* of abinary viscoelastic composite is shown to be frequency dependent even ifthe thermal expansion coefficients beta A and beta B of both constituentsare themselves frequency independent. Exact calculations for binaryviscoelastic systems show that beta* is related to constituent valuesbeta A, beta B, volume fractions, and bulk moduli KA, KB, as well as tothe overall bulk modulus K* of the composite system. Then, beta* isdetermined for isotropic systems by first bounding (or measuring) K* andtherefore beta*. For anisotropic systems with hexagonal symmetry, theprincipal values of the thermal expansion beta*perp and beta*para can bedetermined exactly when the constituents form a layered system. In allthe examples studied, it is shown explicitly that the eigenvectors of thethermoviscoelastic system possess non-negative dissipation -- despite thecomplicated analytical behavior of the frequency dependent thermalexpansivities themselves. Methods presented have a variety ofapplications from fluid-fluid mixtures to fluid-solid suspensions, andfrom fluid-saturated porous media to viscoelastic solid-solidcomposites.
Date: December 1, 2007
Creator: Berryman, James G.
Partner: UNT Libraries Government Documents Department

## Diffraction grating eigenvector for translational and rotational motion

Description: Future energy scaling of high-energy chirped-pulse amplification systems will benefit from the capability to coherently tile diffraction gratings into larger apertures. Design and operation of a novel, accurate alignment diagnostics for coherently tiled diffraction gratings is required for successful implementation of this technique. An invariant diffraction direction and phase for special moves of a diffraction grating is discussed, allowing simplification in the design of the coherently tiled grating diagnostics. An analytical proof of the existence of a unique diffraction grating eigenvector for translational and rotational motion which conserves the diffraction direction and diffracted wave phase is presented.
Date: July 28, 2005
Creator: Rushford, M C; Molander, W A; Nissen, J D; Jovanovic, I; Britten, J A & Barty, C J
Partner: UNT Libraries Government Documents Department

## Parallel eigensolver for H(curl) problems using H1-auxiliary space AMG preconditioning

Description: This report describes an application of the recently developed H{sup 1}-auxiliary space preconditioner for H(curl) problems to the Maxwell eigenvalue problem. The auxiliary space method based on the new (HX) finite element space decomposition introduced in [7], was implemented in the hypre library, [10, 11] under the name AMS. The eigensolver considered in the present paper, referred to as the AME, is an extension of the AMS. It is based on the locally optimal block eigensolver LOBPCG [9] and the parallel AMG (algebraic multigrid) solver BoomerAMG [2] from the hypre library. AME is designed to compute a block of few minimal nonzero eigenvalues and eigenvectors, for general unstructured finite element discretizations utilizing the lowest order Nedelec elements. The main goal of the current report is to document the usage of AME and to illustrate its parallel scalability.
Date: November 15, 2006
Creator: Kolev, T V & Vassilevski, P S
Partner: UNT Libraries Government Documents Department

## Shifted power method for computing tensor eigenpairs.

Description: Recent work on eigenvalues and eigenvectors for tensors of order m {&gt;=} 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax{sup m-1} = {lambda}x subject to {parallel}x{parallel} = 1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a novel shifted symmetric higher-order power method (SS-HOPM), which we showis guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to fnding complex eigenpairs.
Date: October 1, 2010
Creator: Mayo, Jackson R. & Kolda, Tamara Gibson
Partner: UNT Libraries Government Documents Department

## Resolution depths for some transmitter receiverconfigurations

Description: Away from a conductive body, secondary magnetic fields due to currents induced in the body by a time varying external magnetic field are approximated by (equivalent) magnetic dipole fields. Approximating the external magnetic field by its value at the location of the equivalent magnetic dipoles, the equivalent magnetic dipoles' strengths are linearly proportional to the external magnetic field, for a given time dependence of external magnetic field, and are given by the equivalent dipole polarizability matrix. The polarizability matrix and its associated equivalent dipole location is estimated from magnetic field measurements made with at least three linearly independent polarizations of external magnetic fields at the body. Uncertainties in the polarizability matrix elements and its equivalent dipole location are obtained from analysis of a linearized inversion for polarizability and dipole location. Polarizability matrix uncertainties are independent of the scale of the polarizability matrix. Dipole location uncertainties scale inversely with the scale of the polarizability matrix. Uncertainties in principal polarizabilities and directions are obtained from the sensitivities of eigenvectors and eigenvalues to perturbations of a symmetric matrix. In application to synthetic data from a magnetic conducting sphere and to synthetic data from an axially symmetric elliptic conducting body, the estimated polarizability matrices, equivalent dipole locations and principal polarizabilities and directions are consistent with their estimated uncertainties.
Date: August 28, 2002
Creator: Smith, J. Torquil; Morrison, H. Frank & Becker, Alex
Partner: UNT Libraries Government Documents Department

## The Godunov-Inverse Iteration : A fast and accurate solution to the symmetric tridiagonal eigenvalue problem.

Description: We present a new hybrid algorithm based on Godunov's method for computing eigenvectors of symmetric tridiagonal matrices and Inverse Iteration, which we call the Godunov-Inverse Iteration Algorithm. We use eigenvectors computed according to Godunov's method as starting vectors in the Inverse Iteration, replacing any nonnumeric elements of the Godunov eigenvectors with random uniform numbers. We use the right-hand bounds of the Ritz intervals found by the bisection method as Inverse Iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one step of the iteration, producing error estimates that are as good as or superior to those produced by standard Inverse Iteration implementations.
Date: November 27, 2002
Creator: Matsekh, A. M. (Anna M.)
Partner: UNT Libraries Government Documents Department

## Model-Based Signal Processing: Correlation Detection With Synthetic Seismograms

Description: Recent applications of correlation methods to seismological problems illustrate the power of coherent signal processing applied to seismic waveforms. Examples of these applications include detection of low amplitude signals buried in ambient noise and cross-correlation of sets of waveforms to form event clusters and accurately measure delay times for event relocation and/or earth structure. These methods rely on the exploitation of the similarity of individual waveforms and have been successfully applied to large sets of empirical observations. However, in cases with little or no empirical event data, such as aseismic regions or exotic event types, correlation methods with observed seismograms will not be possible due to the lack of previously observed similar waveforms. This study uses model-based signals computed for three-dimensional (3D) Earth models to form the basis for correlation detection. Synthetic seismograms are computed for fully 3D models estimated from the Markov Chain Monte-Carlo (MCMC) method. MCMC uses stochastic sampling to fit multiple seismological data sets. Rather than estimate a single ''optimal'' model, MCMC results in a suite of models that sample the model space and incorporates uncertainty through variability of the models. The variability reflects our ignorance of Earth structure, due to limited resolution, data and modeling errors, and produces variability in the seismic waveform response. Model-based signals are combined using a subspace method where the synthetic signals are decomposed into an orthogonal basis by singular-value decomposition (SVD) and the observed waveforms are represented with a linear combination of a sub-set of eigenvectors (signals) associated with the most significant eigenvalues. We have demonstrated the method by modeling long-period (80-10 seconds) regional seismograms for a moderate (M{approx}5) earthquake near the China-North Korea border. Synthetic seismograms are computed with the Spectral Element Method for a suite of long-wavelength (2 degree) seismic velocity models based on the MCMC method. We are ...
Date: August 30, 2006
Creator: Rodgers, A; Harris, D; Pasyanos, M; Blair, S & Matt, R
Partner: UNT Libraries Government Documents Department

## From Self-consistency to SOAR: Solving Large Scale NonlinearEigenvalue Problems

Description: What is common among electronic structure calculation, design of MEMS devices, vibrational analysis of high speed railways, and simulation of the electromagnetic field of a particle accelerator? The answer: they all require solving large scale nonlinear eigenvalue problems. In fact, these are just a handful of examples in which solving nonlinear eigenvalue problems accurately and efficiently is becoming increasingly important. Recognizing the importance of this class of problems, an invited minisymposium dedicated to nonlinear eigenvalue problems was held at the 2005 SIAM Annual Meeting. The purpose of the minisymposium was to bring together numerical analysts and application scientists to showcase some of the cutting edge results from both communities and to discuss the challenges they are still facing. The minisymposium consisted of eight talks divided into two sessions. The first three talks focused on a type of nonlinear eigenvalue problem arising from electronic structure calculations. In this type of problem, the matrix Hamiltonian H depends, in a non-trivial way, on the set of eigenvectors X to be computed. The invariant subspace spanned by these eigenvectors also minimizes a total energy function that is highly nonlinear with respect to X on a manifold defined by a set of orthonormality constraints. In other applications, the nonlinearity of the matrix eigenvalue problem is restricted to the dependency of the matrix on the eigenvalues to be computed. These problems are often called polynomial or rational eigenvalue problems In the second session, Christian Mehl from Technical University of Berlin described numerical techniques for solving a special type of polynomial eigenvalue problem arising from vibration analysis of rail tracks excited by high-speed trains.
Date: February 1, 2006
Creator: Bai, Zhaojun & Yang, Chao
Partner: UNT Libraries Government Documents Department

## FORTRAN SUBROUTINES FOR HOUSEHOLDER'S METHOD IN THE COMPLEX CASE AND EIGENSYSTEMS OF HERMITIAN MATRICES.

Description: No Description Available.
Date: January 1, 1966
Creator: Mueller, D.J.
Partner: UNT Libraries Government Documents Department

## SLIM--An Early Work Revisited

Description: An early, but at the time illuminating, piece of work on how to deal with a general, linearly coupled accelerator lattice is revisited. This work is based on the SLIM formalism developed in 1979-1981.
Date: July 25, 2008
Creator: Chao, Alex
Partner: UNT Libraries Government Documents Department

## Some matrix results for stable m-dimensional rotations

Description: No Description Available.
Date: April 15, 1974
Creator: Gear, C.W.
Partner: UNT Libraries Government Documents Department

## PCR+ In Diesel Fuels and Emissions Research

Description: In past work for the U.S. Department of Energy (DOE) and Oak Ridge National Laboratory (ORNL), PCR+ was developed as an alternative methodology for building statistical models. PCR+ is an extension of Principal Components Regression (PCR), in which the eigenvectors resulting from Principal Components Analysis (PCA) are used as predictor variables in regression analysis. The work was motivated by the observation that most heavy-duty diesel (HDD) engine research was conducted with test fuels that had been ''concocted'' in the laboratory to vary selected fuel properties in isolation from each other. This approach departs markedly from the real world, where the reformulation of diesel fuels for almost any purpose leads to changes in a number of interrelated properties. In this work, we present new information regarding the problems encountered in the conventional approach to model-building and how the PCR+ method can be used to improve research on the relationship between fuel characteristics and engine emissions. We also discuss how PCR+ can be applied to a variety of other research problems related to diesel fuels.
Date: April 15, 2002
Partner: UNT Libraries Government Documents Department

## Calculating luminosity for a coupled Tevatron lattice

Description: The traditional formula for calculating luminosity assumes an uncoupled lattice and makes use of one-degree-of-freedom lattice functions, {beta}{sub H} and {beta}{sub v}, for relating transverse beam widths to emittances. Strong coupling requires changing this approach. It is simplest to employ directly the linear normal form coordinates of the one turn map. An equilibrium distribution in phase space is expressed as a function of the Jacobian`s eigenvectors and beam size parameters or emittances. Using the equilibrium distributions an expression for the luminosity was derived and applied to the Tevatron lattice, which was coupled due to a quadrupole roll.
Date: May 1, 1995
Creator: Holt, J.A.; Martens, M.A.; Michelotti, L. & Goderre, G.
Partner: UNT Libraries Government Documents Department

## Matrix-free constructions of circulant and block circulant preconditioners

Description: A framework for constructing circulant and block circulant preconditioners (C) for a symmetric linear system Ax=b arising from certain signal and image processing applications is presented in this paper. The proposed scheme does not make explicit use of matrix elements of A. It is ideal for applications in which A only exists in the form of a matrix vector multiplication routine, and in which the process of extracting matrix elements of A is costly. The proposed algorithm takes advantage of the fact that for many linear systems arising from signal or image processing applications, eigenvectors of A can be well represented by a small number of Fourier modes. Therefore, the construction of C can be carried out in the frequency domain by carefully choosing its eigenvalues so that the condition number of C{sup T} AC can be reduced significantly. We illustrate how to construct the spectrum of C in a way such that the smallest eigenvalues of C{sup T} AC overlaps with those of A extremely well while the largest eigenvalues of C{sup T} AC are smaller than those of A by several orders of magnitude. Numerical examples are provided to demonstrate the effectiveness of the preconditioner on accelerating the solution of linear systems arising from image reconstruction application.
Date: December 1, 2001
Creator: Yang, Chao; Ng, Esmond G. & Penczek, Pawel A.
Partner: UNT Libraries Government Documents Department

## TRLAN User Guide

Description: TRLAN is a program designed to find a small number of extreme eigenvalues and their corresponding eigenvectors of a real symmetric matrix. Denote the matrix as A, the eigenvalue as {lambda}, and the corresponding eigenvector as x, they are defined by the following equation, Ax = {lambda}x. There are a number of different implementations of the Lanczos algorithm available. Why another one? Our main motivation is to develop a specialized version that only target the case where one wants both eigenvalues and eigenvectors of a large real symmetric eigenvalue problems that can not use the shift-and-invert scheme. In this case the standard non-restarted Lanczos algorithm requires one to store a large number of Lanczos vectors which can cause storage problem and make each iteration of the method very expensive. The underlying algorithm of TRLAN is a dynamic thick-restart Lanczos algorithm. Like all restarted methods, the user can choose how many vectors can be generated at once. Typically, th e user chooses a moderate size so that all Lanczos vectors can be stored in core. This allows the restarted methods to execute efficiently. This implementation of the thick-restart Lanczos method also uses the latest restarting technique, it is very effective in reducing the time required to compute a desired solutions compared to similar restarted Lanczos schemes, e.g., ARPACK.
Date: March 9, 1999
Creator: Wu, Kesheng & Simon, H.
Partner: UNT Libraries Government Documents Department