We derive computable expressions of structured backward errors of approximate eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We also characterize minimal structured perturbations such that approximate eigenelements are exact eigenelements of the perturbed polynomials. We detect structure preserving linearizations which have almost no adverse effect on the structured backward errors of approximate eigenelements of the *-palindromic and *-anti-palindromic polynomials.
We identify a relationship between the solutions of a nonsymmetric algebraic T-Riccati equation (T-NARE) and the deflating subspaces of a palindromic matrix pencil, obtained by arranging the coefficients of the T-NARE. The interplay between T-NARE and palindromic pencils allows one to derive both theoretical properties of the solutions of the equation, and new methods for its numerical solution. In particular, we propose methods based on the (palindromic) QZ algorithm and the doubling algorithm, whose effectiveness is demonstrated by several numerical tests
In the framework of Polynomial Eigenvalue Problems, most of the matrix polynomials arising in applications are structured polynomials (namely (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve Polynomial Eigenvalue Problems is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as {\em companion linearizations}. It is well known, however, that is not possible to construct companion linearizations that preserve any of the previous structures …
A standard approach to compute the roots of a univariate polynomial is to compute the eigenvalues of an associated \emph{confederate} matrix instead, such as, for instance the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure exploiting QR algorithm. In [J. Aurentz et al., Fast and backward stable comp…
We study the backward stability of running a backward stable eigenstructure solver on a pencil $S(λ)$ that is a strong linearization of a rational matrix $R(λ)$ expressed in the form $R(λ)=D(λ)+ C(λI_\ell-A)^{-1}B$, where $D(λ)$ is a polynomial matrix and $C(λI_\ell-A)^{-1}B$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $R(λ)$, which are highly structured pencils. Backward stable eigenstructure solvers applied to $S(λ)$ will compute the exact eigenstructure of a perturbed pencil $\widehat S(λ):=S(λ)+Δ_S(λ)$ and the special structure of $S(λ)…
Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidade de Lisboa（里斯本大学）；Faculdade de Economia do Porto（波尔图经济与管理学院）；Boston University（波士顿大学）
A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials $p(z)$. Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of $p(z)$ that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of $p(z)$. It uses a companion linearization $C(z) = A-zB$ of $p(z)$ to which we added an extra zero …
Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials
Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid（马德里自治大学数学科学研究所）；Nazarbayev University（纳扎尔巴耶夫大学）；ISTI, Area della ricerca CNR（意大利国家研究委员会ISTI研究所）；KU Leuven（荷语鲁汶大学）；Washington State University（华盛顿州立大学）
For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-an…
We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [Eidelman, Y., Gemignani, L., and Gohberg, I., Numer. Algorithms, 47.3 (2008): pp. 253-273] introducing an aggressive early deflation strategy, and showing that the rank-structure allows to parallelize the algorithm avoiding data dependencies which would be present in the unstructured QR. We exploit the particular structure of the colleague linearization to achieve quadratic complexity and linear storage requirements. The (unbalanced) QR iteration used for Chebyshev roo…
A unified approach to Fiedler-like pencils via strong block minimal bases pencils
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；KU Leuven（荷语鲁汶大学）；Dos Pueblos High School（多斯普埃洛高中）；University of Virginia（弗吉尼亚大学）
