Block LU factors of generalized companion matrix pencils
نویسندگان
چکیده
We present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems. In particular, we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange, and Bernstein/Bézier bases. Additionally, we give a pair of explicit LU factors associated with each pencil and a prescription for block pivoting when required.
منابع مشابه
Explicit Block-structures for Block-symmetric Fiedler-like Pencils∗
In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial P (λ), regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular P (λ) in an easy way, allowing the computation of the minimal indices of a singular P (λ) in an easy way, etc. As a consequence of this resea...
متن کاملA Unified Approach to Fiedler-like Pencils via Strong Block Minimal Bases Pencils
The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such pencils are known as linearizations. Many of the families of linearizations for matrix polynomials available in the literature are extensions of the so-called fam...
متن کاملA note on companion pencils
Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding polynomial. In the classical companion pencil case (A,B) only the coefficient of the highest degree appears in B’s lower right corner. We will show, however, that all coefficients of the polynomial ...
متن کاملHigher numerical ranges of matrix polynomials
Let $P(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. In this paper, some algebraic and geometrical properties of the $k$-numerical range of $P(lambda)$ are investigated. In particular, the relationship between the $k$-numerical range of $P(lambda)$ and the $k$-numerical range of its companion linearization is stated. Moreover, the $k$-numerical...
متن کاملLarge Vector Spaces of Block-symmetric Strong Linearizations of Matrix Polynomials
M. I. BUENO∗, F. M. DOPICO †, S. FURTADO ‡, AND M. RYCHNOVSKY § Abstract. Given a matrix polynomial P (λ) = Pk i=0 λ Ai of degree k, where Ai are n × n matrices with entries in a field F, the development of linearizations of P (λ) that preserve whatever structure P (λ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P (...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Theor. Comput. Sci.
دوره 381 شماره
صفحات -
تاریخ انتشار 2007