389 research outputs found
A framework for structured linearizations of matrix polynomials in various bases
We present a framework for the construction of linearizations for scalar and
matrix polynomials based on dual bases which, in the case of orthogonal
polynomials, can be described by the associated recurrence relations. The
framework provides an extension of the classical linearization theory for
polynomials expressed in non-monomial bases and allows to represent polynomials
expressed in product families, that is as a linear combination of elements of
the form , where and
can either be polynomial bases or polynomial families
which satisfy some mild assumptions. We show that this general construction can
be used for many different purposes. Among them, we show how to linearize sums
of polynomials and rational functions expressed in different bases. As an
example, this allows to look for intersections of functions interpolated on
different nodes without converting them to the same basis. We then provide some
constructions for structured linearizations for -even and
-palindromic matrix polynomials. The extensions of these constructions
to -odd and -antipalindromic of odd degree is discussed and
follows immediately from the previous results
Set-theoretic defining equations of the tangential variety of the Segre variety
We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the
defining equations of the tangential variety of a Segre product of projective
spaces. We introduce and study the concept of exclusive rank. For the proof of
this conjecture we use a connection to the author's previous work
\cite{oeding_pm_paper, oeding_thesis} and re-express the tangential variety as
the variety of principal minors of symmetric matrices that have exclusive rank
no more than one.Comment: 13 page
On pole-swapping algorithms for the eigenvalue problem
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
Annual Report of the Municipal Officers for the Town of Palmyra, Maine Municipal Year 2015
Original scanned reports courtesy of Palmyra Historical Societ
On deflations in extended QR algorithms
In this paper we discuss the deflation criterion used in the extended QR algorithm based on the chasing of rotations. We provide absolute and relative perturbation bounds for this deflation criterion. Further, we present a generalization of aggressive early deflation to the extended QR algorithms. Aggressive early deflation is the key technique for the identification and deflation of already converged, but hidden, eigenvalues. Often these possibilities for deflation are not detected by the standard technique. We present numerical results underpinning the power of aggressive early deflation also in the context of extended QR algorithms. We further generalize these ideas by the transcription of middle deflations. © 2014 Society for Industrial and Applied Mathematics
A new deflation criterion for the QZ algorithm
The QZ algorithm computes the Schur form of a matrix pencil. It is an
iterative algorithm and at some point, it must decide that an eigenvalue has
converged and move on with another one. Choosing a criterion that makes this
decision is nontrivial. If it is too strict, the algorithm might waste
iterations on already converged eigenvalues. If it is not strict enough, the
computed eigenvalues might be inaccurate. Additionally, the criterion should
not be computationally expensive to evaluate. This paper introduces a new
criterion based on the size of and the gap between the eigenvalues. This is
similar to the work of Ahues and Tissuer for the QR algorithm. Theoretical
arguments and numerical experiments suggest that it outperforms the most
popular criteria in terms of accuracy. Additionally, this paper evaluates some
commonly used criteria for infinite eigenvalues.Comment: 10 pages, 6 figure
A rational QZ method
We propose a rational QZ method for the solution of the dense, unsymmetric
generalized eigenvalue problem. This generalization of the classical QZ method
operates implicitly on a Hessenberg, Hessenberg pencil instead of on a
Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace
iteration driven by a polynomial, the rational QZ method allows for nested
subspace iteration driven by a rational function, this creates the additional
freedom of selecting poles. In this article we study Hessenberg, Hessenberg
pencils, link them to rational Krylov subspaces, propose a direct reduction
method to such a pencil, and introduce the implicit rational QZ step. The link
with rational Krylov subspaces allows us to prove essential uniqueness
(implicit Q theorem) of the rational QZ iterates as well as convergence of the
proposed method. In the proofs, we operate directly on the pencil instead of
rephrasing it all in terms of a single matrix. Numerical experiments are
included to illustrate competitiveness in terms of speed and accuracy with the
classical approach. Two other types of experiments exemplify new possibilities.
First we illustrate that good pole selection can be used to deflate the
original problem during the reduction phase, and second we use the rational QZ
method to implicitly filter a rational Krylov subspace in an iterative method
Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices
In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved. First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info. In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm. © 2014 Elsevier B.V. All rights reserved
Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices
It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which aside from poles at infinity and zero, also contain finite non-zero poles. Furthermore, the algorithms are generalized to deal with block rational Krylov subspaces and techniques to exploit the symmetry when working with Hermitian matrices are also presented. For each variant of the algorithm numerical experiments illustrate the power of the new approach. The experiments involve matrix functions, Ritz-value computations, and the solutions of matrix equations
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