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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

Quadratic eigenvalue problems.

Description: In this report we will describe some nonlinear eigenvalue problems that arise in the areas of solid mechanics, acoustics, and coupled structural acoustics. We will focus mostly on quadratic eigenvalue problems, which are a special case of nonlinear eigenvalue problems. Algorithms for solving the quadratic eigenvalue problem will be presented, along with some example calculations.
Date: April 1, 2007
Creator: Walsh, Timothy Francis & Day, David Minot
Partner: UNT Libraries Government Documents Department

Time-periodic solutions of the Benjamin-Ono equation

Description: We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.
Date: April 1, 2008
Creator: Ambrose , D.M. & Wilkening, Jon
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

Comparative Test Results for Two ODE Solvers: EPISODE and GEAR

Description: This is a sequel to the paper ''A comparison of two ODE codes: GEAR and EPISODE,'' and is concerned with the testing of two superficially similar ODE packages, GEAR and EPISODE. Fourteen basic test problems, some with several cases, are the basis for the testing. These problems represent several types-nonlinear systems with real and complex eigenvalues, linear systems with varied diagonal dominance, linear scalar problems, stiff and non-stiff problems, chemical kinetics with and without diurnal effect, and systems arising from the use of the numerical method of lines. Some problems are included in order to examine the options and error returns. The test results are presented in two forms: raw output and a comparative display of operation counts and of timings for the best method in the GEAR package and the best method in the EPISODE package. This approach allows a comparison of the consequences of the fixed-step interpolate strategy (GEAR) for changing step size against the truly variable step size strategy (EPISODE). It is concluded that EPISODE is generally faster than GEAR for problems involving wave fronts or transients on the interior of the interval of integration. For linear or simply decaying problems, these roles are usually reversed.
Date: March 1977
Creator: Byrne, G. D.; Hindmarsh, A. C.; Jackson, Kenneth R. & Brown, H. Gordon
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

Bound states and the Bekenstein bound

Description: We explore the validity of the generalized Bekenstein bound, S<= pi M a. We define the entropy S as the logarithm of the number of states which have energy eigenvalue below M and are localized to a flat space region of width alpha. If boundary conditions that localize field modes are imposed by fiat, then the bound encounters well-known difficulties with negative Casimir energy and large species number, as well as novel problems arising only in the generalized form. In realistic systems, however, finite-size effects contribute additional energy. We study two different models for estimating such contributions. Our analysis suggests that the bound is both valid and nontrivial if interactions are properly included, so that the entropy S counts the bound states of interacting fields.
Date: October 16, 2003
Creator: Bousso, Raphael
Partner: UNT Libraries Government Documents Department

The Use of the Power Method to Find Dominant Eigenvalues of Matrices

Description: This paper is the result of a study of the power method to find dominant eigenvalues of square matrices. It introduces ideas basic to the study and shows the development of the power method for the most well-behaved matrices possible, and it explores exactly which other types of matrices yield to the power method. The paper also discusses a type of matrix typically considered impossible for the power method, along with a modification of the power method which works for this type of matrix. It gives an overview of common extensions of the power method. The appendices contain BASIC versions of the power method and its modification.
Date: July 1992
Creator: Cavender, Terri A.
Partner: UNT Libraries

Homologous Recombination in Q-Beta Rna Bacteriophage

Description: Q-Beta phage RNAs with inactivating insertion (8 base) or deletion (17 base) mutations within their replicase genes were transfected into Escherichia coli spheroplasts containing QB replicase provided in trans by a resident plasmid. Replicase-defective (Rep~) Q3 phage produced by these spheroplasts were unable to form plaques on cells lacking this plasmid. When individual Rep~ phage were isolated and grown to high titer in cells containing plasmid derived Q3 replicase, revertant Q3 phage (Rep'), with the original mutation (insertion or deletion) repaired, were obtained at a frequency of ca. 1 x 108. RNA recombination via a "template switching" mechanism involving Q3 replicase, the mutant phage genome, and the plasmid-derived replicase mRNA was shown to be the primary means by which these mutant phages reverted to wild type.
Date: May 1992
Creator: Palasingam, Kampan
Partner: UNT Libraries

Spatially-discretized high-temperature approximations and theirO(N) implementation on a grid

Description: We consider the problem of performing imaginary-time propagation of wavefunctions on a grid. We demonstrate that spatially-continuous high-temperature approximations can be discretized in such a way that their convergence order is preserved. Requirements of minimal computational work and reutilization of data then uniquely determine the optimal grid, quadrature technique, and propagation method. It is shown that the optimal propagation technique is O(N) with respect to the grid size. The grid technique is utilized to compare the Monte Carlo efficiency of the Trotter-Suzuki approximation against a recently introduced fourth-order high-temperature approximation, while circumventing the issue of statistical noise, which usually prevents such comparisons from being carried out. We document the appearance of a systematic bias in the Monte Carlo estimators that involve temperature differentiation of the density matrix, bias that is due to the dependence of the eigenvalues on the inverse temperature. This bias is negotiated more successfully by the short-time approximations having higher convergence order, leading to non-trivial computational savings.
Date: October 1, 2006
Creator: Predescu, Cristian
Partner: UNT Libraries Government Documents Department

On Ideal Stability of Cylindrical Localized Interchange Modes

Description: Stability of cylindrical localized ideal pressure-driven interchange plasma modes is revisited. Converting the underlying eigenvalue problem into the form of the Schroedinger equation gives a new simple way of deriving the Suydam stability criterion and calculating the growth rates of unstable modes. Near the marginal stability limit the growth rate is exponentially small and the mode has a double-peak structure.
Date: May 15, 2007
Creator: Umansky, M V
Partner: UNT Libraries Government Documents Department

Multidimensional spacial eigenmode analysis.

Description: Several recent papers have examined higher mode eigenvalues and eigenfunctions fo r multiplying systems. The general application focus of these papers is related to determining the dominance ratio, which is of great interest to people analyzing loosely coupled fissile systems . For large systems, we derived some simple approximations to the dominance ratio, and we continue this analysis in this paper. In the previous papers, we were able to utilize semianalytical techriiques because we mainly examined one-dimensional Cartesian systems. In this paper we analyze the effectiveness of using reflective boundary conditions for multi-dimensional system s and expand past work by examining two- and three-dimensional eigenfunctions.
Date: January 1, 2003
Creator: Parsons, Donald Kent & Kornreich, D. E. (Drew E.)
Partner: UNT Libraries Government Documents Department

Oracle inequalities for SVMs that are based on random entropy numbers

Description: In this paper we present a new technique for bounding local Rademacher averages of function classes induced by a loss function and a reproducing kernel Hilbert space (RKHS). At the heart of this technique lies the observation that certain expectations of random entropy numbers can be bounded by the eigenvalues of the integral operator associated to the RKHS. We then work out the details of the new technique by establishing two new oracle inequalities for SVMs, which complement and generalize orevious results.
Date: January 1, 2009
Creator: Steinwart, Ingo
Partner: UNT Libraries Government Documents Department

Krylov subspace iterations for the calculation of K-Eigenvalues with sn transport codes

Description: We apply the Implicitly Restarted Arnoldi Method (IRAM), a Krylov subspace iterative method, to the calculation of k-eigenvalues for criticality problems. We show that the method can be implemented with only modest changes to existing power iteration schemes in an SN transport code. Numerical results on three dimensional unstructured tetrahedral meshes are shown. Although we only compare the IRAM to unaccelerated power iteration, the results indicate that the IRAM is a potentially efficient and powerful technique, especially for problems with dominance ratios approaching unity. Key Words: criticality eigenvalues, Implicitly Restarted Arnoldi Method (IRAM), deterministic transport methods
Date: January 1, 2002
Creator: Warsa, J. S. (James S.); Wareing, T. A. (Todd A.); Morel, J. E.; McGhee, J. M. (John M.) & Lehoucq, R. B. (Richard B.)
Partner: UNT Libraries Government Documents Department

Benefits of IEEE-754 features in modern symmetric tridiagonaleigensolvers

Description: Bisection is one of the most common methods used to compute the eigenvalues of symmetric tridiagonal matrices. Bisection relies on the Sturm count: For a given shift a, the number of negative pivots in the factorization T - {sigma}I = LDL{sup T} equals the number of eigenvalues of T that are smaller than a. In IEEE-754 arithmetic, the value oo permits the computation to continue past a zero pivot, producing a correct Sturm count when T is unreduced. Demmel and Li showed that using oo rather than testing for zero pivots within the loop could significantly improve performance on certain architectures. When eigenvalues are to be computed to high relative accuracy, it is often preferable to work with LDL{sup T} factorizations instead of the original tridiagonal T. One important example is the MRRR algorithm. When bisection is applied to the factored matrix, the Sturm count is computed from LDL{sup T} which makes differential stationary and progressive qds algorithms the methods of choice. While it seems trivial to replace T by LDL{sup T}, in reality these algorithms are more complicated: In IEEE-754 arithmetic, a zero pivot produces an overflow followed by an invalid exception (NaN, or 'Not a Number') that renders the Sturm count incorrect. We present alternative, safe formulations that are guaranteed to produce the correct result. Benchmarking these algorithms on a variety of platforms shows that the original formulation without tests is always faster provided that no exception occurs. The transforms see speed-ups of up to 2.6x over the careful formulations. Tests on industrial matrices show that encountering exceptions in practice is rare. This leads to the following design: First, compute the Sturm count by the fast but unsafe algorithm. Then, if an exception occurs, recompute the count by a safe, slower alternative. The new Sturm count algorithms improve ...
Date: March 12, 2006
Creator: Marques, Osni; Riedy, Jason E. & Vomel, Christof
Partner: UNT Libraries Government Documents Department

A Convergence of LDPC Based on Eigenvalues

Description: Low-density parity check (LDPC) codes are very popular among error correction codes because of their high-performance capacity. Numerous investigations have been carried out to analyze the performance and simplify the implementation of LDPC codes. Relatively slow convergence of iterative decoding algorithm affects the performance of LDPC codes. Faster convergence can be achieved by reducing the number of iterations during the decoding process. In this thesis, a new approach for faster convergence is suggested by choosing a systematic parity check matrix that yields lowest Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. MATLAB simulations are used to study the impact of eigenvalues on the number of iterations of the LDPC decoder. It is found that for a given (n, k) LDPC code, a parity check matrix with lowest SSEM converges quickly as compared to the parity check matrix with high SSEM. In other words, a densely connected graph that represents the parity check matrix takes more iterations to converge than a sparsely connected graph.
Date: August 2017
Creator: Kharate, Neha Ashok
Partner: UNT Libraries

Asymptotically Optimal High-Order Accurate Algorithms for the Solution of Certain Elliptic PDEs

Description: The main goal of the project, "Asymptotically Optimal, High-Order Accurate Algorithms for the Solution of Certain Elliptic PDE's" (DE-FG02-03ER25577) was to develop fast, high-order algorithms for the solution of scattering problems and spectral problems of photonic crystals theory. The results we obtained lie in three areas: (1) asymptotically fast, high-order algorithms for the solution of eigenvalue problems of photonics, (2) fast, high-order algorithms for the solution of acoustic and electromagnetic scattering problems in the inhomogeneous media, and (3) inversion formulas and fast algorithms for the inverse source problem for the acoustic wave equation, with applications to thermo- and opto- acoustic tomography.
Date: November 26, 2008
Creator: Kunyansky, Leonid
Partner: UNT Libraries Government Documents Department

Navier-Stokes Solvers and Generalizations for Reacting Flow Problems

Description: This is an overview of our accomplishments during the final term of this grant (1 September 2008 -- 30 June 2012). These fall mainly into three categories: fast algorithms for linear eigenvalue problems; solution algorithms and modeling methods for partial differential equations with uncertain coefficients; and preconditioning methods and solvers for models of computational fluid dynamics (CFD).
Date: January 27, 2013
Creator: Elman, Howard C
Partner: UNT Libraries Government Documents Department

Resolving the sign ambiguity in the singular value decomposition.

Description: Many modern data analysis methods involve computing a matrix singular value decomposition (SVD) or eigenvalue decomposition (EVD). Principal components analysis is the time-honored example, but more recent applications include latent semantic indexing, hypertext induced topic selection (HITS), clustering, classification, etc. Though the SVD and EVD are well-established and can be computed via state-of-the-art algorithms, it is not commonly mentioned that there is an intrinsic sign indeterminacy that can significantly impact the conclusions and interpretations drawn from their results. Here we provide a solution to the sign ambiguity problem and show how it leads to more sensible solutions.
Date: October 1, 2007
Creator: Bro, Rasmus (University of Copenhagen, Frederiksberg C, Denmark); Acar, Evrim (Rensselaer Polytechnic Institute, Troy, NY) & Kolda, Tamara Gibson
Partner: UNT Libraries Government Documents Department