244 research outputs found
Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding
This paper is concerned with the reduction of a unitary matrix U to CMV-like
shape. A Lanczos--type algorithm is presented which carries out the reduction
by computing the block tridiagonal form of the Hermitian part of U, i.e., of
the matrix U+U^H. By elaborating on the Lanczos approach we also propose an
alternative algorithm using elementary matrices which is numerically stable. If
U is rank--structured then the same property holds for its Hermitian part and,
therefore, the block tridiagonalization process can be performed using the
rank--structured matrix technology with reduced complexity. Our interest in the
CMV-like reduction is motivated by the unitary and almost unitary eigenvalue
problem. In this respect, finally, we discuss the application of the CMV-like
reduction for the design of fast companion eigensolvers based on the customary
QR iteration
Differential qd algorithm with shifts for rank-structured matrices
Although QR iterations dominate in eigenvalue computations, there are several
important cases when alternative LR-type algorithms may be preferable. In
particular, in the symmetric tridiagonal case where differential qd algorithm
with shifts (dqds) proposed by Fernando and Parlett enjoys often faster
convergence while preserving high relative accuracy (that is not guaranteed in
QR algorithm). In eigenvalue computations for rank-structured matrices QR
algorithm is also a popular choice since, in the symmetric case, the rank
structure is preserved. In the unsymmetric case, however, QR algorithm destroys
the rank structure and, hence, LR-type algorithms come to play once again. In
the current paper we discover several variants of qd algorithms for
quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg
matrices becomes a direct generalization of dqds algorithm for tridiagonal
matrices. Therefore, it can be applied to such important matrices as companion
and confederate, and provides an alternative algorithm for finding roots of a
polynomial represented in the basis of orthogonal polynomials. Results of
preliminary numerical experiments are presented
Global Convergence of Hessenberg Shifted QR II: Numerical Stability
We develop a framework for proving rapid convergence of shifted QR algorithms
which use Ritz values as shifts, in finite arithmetic. Our key contribution is
a dichotomy result which addresses the known forward-instability issues
surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure
which provably either computes a set of approximate Ritz values of a Hessenberg
matrix with good forward stability properties, or leads to early decoupling of
the matrix via a small number of QR steps.
Using this framework, we show that the shifting strategy introduced in Part I
of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in
finite arithmetic with a polylogarithmic bound on the number of bits of
precision required, when invoked on matrices of controlled eigenvector
condition number and minimum eigenvalue gap
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
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
Finding Eigenvalues of Unitary Matrices
The study introduces methods of finding eigenvalues for unitary matrices and pencils. Bunse-Gerstner and Elsner ([2]) proposed an algorithm of using the Schur parameter pencil to solve eigenproblems for unitary matrices and pencils. This thesis reviews the Schur parameter pencil algorithm. The method is divided into two phases: Reducing a unitary pencil to a Schur parameter form and QR-type shifted iteration. The algorithm is proved to be backward stable and more efficient than the standard QR/QZ algorithm. However, during the process of reduction, norms of vectors are frequently compared for numerical stability, which causes a lot of extra work for computations. Based on the idea in [8], we introduce a modified Schur parameter algorithm to avoid such frequent comparison. The modified algorithm is still divided into two phases similar to the one in [2]. A detailed reduction process and shifted iteration are described in this thesis
Spectral computation with third-order tensors using the t-product
The tensor t-product, introduced by Kilmer and Martin [26], is a powerful
tool for the analysis of and computation with third-order tensors. This paper
introduces eigentubes and eigenslices of third-order tensors under the
t-product. The eigentubes and eigenslices are analogues of eigenvalues and
eigenvectors for matrices. Properties of eigentubes and eigenslices are
investigated and numerical methods for their computation are described. The
methods include the tensor power method, tensor subspace iteration, and the
tensor QR algorithm. Computed examples illustrate the performance of these
methods
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