scholarly journals The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs

2019 ◽  
Vol 40 (1) ◽  
pp. 175-193 ◽  
Author(s):  
Diego Armentano ◽  
Carlos Beltrán
Keyword(s):  
Eigenvalue Problem ◽  
Random Inputs ◽  
Polynomial Eigenvalue Problem

Polynomial Eigenvalue Problem

2020 ◽  
pp. 195-210
Author(s):  
Federico Milano ◽  
Ioannis Dassios ◽  
Muyang Liu ◽  
Georgios Tzounas
Keyword(s):  
Eigenvalue Problem ◽  
Polynomial Eigenvalue Problem

scholarly journals Linearization schemes for Hermite matrix polynomials

2015 ◽  
Author(s):  
Nikta Shayanfar ◽  
Heike Fassbender
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Matrix Polynomial ◽  
Multiple Input Multiple Output ◽  
Matrix Pencil ◽  
Least Square ◽  
Coupled Systems ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
The Matrix

The polynomial eigenvalue problem is to find the eigenpair of $(\lambda,x) \in \mathbb{C}\bigcup \{\infty\} \times \mathbb{C}^n \backslash \{0\}$ that satisfies $P(\lambda)x=0$, where $P(\lambda)=\sum_{i=0}^s P_i \lambda ^i$ is an $n\times n$ so-called matrix polynomial of degree $s$, where the coefficients $P_i, i=0,\cdots,s$, are $n\times n$ constant matrices, and $P_s$ is supposed to be nonzero. These eigenvalue problems arise from a variety of physical applications including acoustic structural coupled systems, fluid mechanics, multiple input multiple output systems in control theory, signal processing, and constrained least square problems. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Such methods convert the eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploit and preserve the structure and properties of the original eigenvalue problem. The linearizations have been extensively studied with respect to the basis that the matrix polynomial is expressed in. If the matrix polynomial is expressed in a special basis, then it is desirable that its linearization be also expressed in the same basis. The reason is due to the fact that changing the given basis ought to be avoided \cite{H1}. The authors in \cite{ACL} have constructed linearization for different bases such as degree-graded ones (including monomial, Newton and Pochhammer basis), Bernstein and Lagrange basis. This contribution is concerned with polynomial eigenvalue problems in which the matrix polynomial is expressed in Hermite basis. In fact, Hermite basis is used for presenting matrix polynomials designed for matching a series of points and function derivatives at the prescribed nodes. In the literature, the linearizations of matrix polynomials of degree $s$, expressed in Hermite basis, consist of matrix pencils with $s+2$ blocks of size $n \times n$. In other words, additional eigenvalues at infinity had to be introduced, see e.g. \cite{CSAG}. In this research, we try to overcome this difficulty by reducing the size of linearization. The reduction scheme presented will gradually reduce the linearization to its minimal size making use of ideas from \cite{VMM1}. More precisely, for $n \times n$ matrix polynomials of degree $s$, we present linearizations of smaller size, consisting of $s+1$ and $s$ blocks of $n \times n$ matrices. The structure of the eigenvectors is also discussed.

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

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