18 research outputs found
Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization
We present a randomized, inverse-free algorithm for producing an approximate
diagonalization of any matrix pencil . The bulk of the
algorithm rests on a randomized divide-and-conquer eigensolver for the
generalized eigenvalue problem originally proposed by Ballard, Demmel, and
Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer
approach can be formulated to succeed with high probability as long as the
input pencil is sufficiently well-behaved, which is accomplished by
generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas,
Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In
particular, we show that perturbing and scaling regularizes its
pseudospectra, allowing divide-and-conquer to run over a simple random grid and
in turn producing an accurate diagonalization of in the backward error
sense. The main result of the paper states the existence of a randomized
algorithm that with high probability (and in exact arithmetic) produces
invertible and diagonal such that and in at most
operations, where is the asymptotic complexity of matrix
multiplication. This not only provides a new set of guarantees for highly
parallel generalized eigenvalue solvers but also establishes nearly matrix
multiplication time as an upper bound on the complexity of exact arithmetic
matrix pencil diagonalization.Comment: 58 pages, 8 figures, 2 table
Pseudospectra, critical points and multiple eigenvalues of matrix polynomials
AbstractWe develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space Lm(Cn×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in Lm(Cn×n). We present a family of natural norms on the space Lm(Cn×n) and show that the norms on the spaces Cm+1 and Cn×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space Lm(Cn×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space Lm(Cn×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials
Nonnormality in Lyapunov Equations
The singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low- rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. We will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically as the numerical range of the coefficient matrix extends farther into the right half-plane. We also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role
Structured Eigenvalue Problems
Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices
A block Newton method for nonlinear eigenvalue problems
We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viabilit
Computing the density of states for optical spectra by low-rank and QTT tensor approximation
In this paper, we introduce a new interpolation scheme to approximate the
density of states (DOS) for a class of rank-structured matrices with
application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter
equation (BSE). The presented approach for approximating the DOS is based on
two main techniques. First, we propose an economical method for calculating the
traces of parametric matrix resolvents at interpolation points by taking
advantage of the block-diagonal plus low-rank matrix structure described in [6,
3] for the BSE/TDA problem. Second, we show that a regularized or smoothed DOS
discretized on a fine grid of size can be accurately represented by a low
rank quantized tensor train (QTT) tensor that can be determined through a least
squares fitting procedure. The latter provides good approximation properties
for strictly oscillating DOS functions with multiple gaps, and requires
asymptotically much fewer () functional calls compared with the full
grid size . This approach allows us to overcome the computational
difficulties of the traditional schemes by avoiding both the need of stochastic
sampling and interpolation by problem independent functions like polynomials
etc. Numerical tests indicate that the QTT approach yields accurate recovery of
DOS associated with problems that contain relatively large spectral gaps. The
QTT tensor rank only weakly depends on the size of a molecular system which
paves the way for treating large-scale spectral problems.Comment: 26 pages, 25 figure
A block Newton method for nonlinear eigenvalue problems
We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability. © Springer-Verlag 2009