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Register a new userOrthogonal iterations on Structured Pencils
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Added by
Gianna Maria Del Corso
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse factorization -- named LFR factorization -- using orthogonal Hessenberg matrices. These new subspace trackers reach a complexity of only $O(nk^2)$ operations per time update, where $n$ and $k$ are the size of the matrix and of the small rank perturbation, respectively.
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We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $Ntimes N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.
In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${mathbb R}^{ntimes n}$, smoothly depending on parameters $xin Omegasubset {mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.
Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $Uin mathbb C^{ntimes n}$ is unitary block circulant and $X, Y inmathbb{C}^{n times k}$, have recently appeared in the literature. Most of these algorithms rely on the decomposition of $A$ as product of scalar companion matrices which turns into a factored representation of the Hessenberg reduction of $A$. In this paper we generalize the approach to encompass Hessenberg matrices of the form $A=U + XY^H$ where $U$ is a general unitary matrix. A remarkable case is $U$ unitary diagonal which makes possible to deal with interpolation techniques for rootfinding problems and nonlinear eigenvalue problems. Our extension exploits the properties of a larger matrix $hat A$ obtained by a certain embedding of the Hessenberg reduction of $A$ suitable to maintain its structural properties. We show that $hat A$ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first $k$ rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR/QZ iteration. The resulting algorithm is fast and backward stable.
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.
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