Linearization is a widely used method for solving polynomial eigenvalue problems (PEPs) and rational eigenvalue problem (REPs) in which the PEP/REP is transformed to a generalized eigenproblem and then solve this generalized eigenproblem with algorithms available in the literature. Fiedler-like pencils (Fiedler pencils (FPs), generalized Fiedler pencils (GFPs), Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs)) are well known classes of strong linearizations. GFPs are an intriguing family of linearizations, and GF pencils are the fundamental building blocks of FPRs and GFPRs. As a result, FPRs and GFPRs have distinctive features and they provide structure-preserving linearizations for structured matrix polynomials. But GFPRs do not use the full potential of GF pencils. Indeed, not all the GFPs are FPRs or GFPRs, and vice versa. The main aim of this paper is two-fold. First, to build a unified framework for all the Fiedler-like pencils FPs, GFPs, FPRs and GFPRs. To that end, we construct a new family of strong linearizations (named as EGFPs) of a matrix polynomial $P(\lam)$ that subsumes all the Fiedler-like linearizations. A salient feature of the EGFPs family is that it allows the construction of structured preserving banded linearizations with low bandwidth for structured (symmetric, Hermitian, palindromic) matrix polynomial. Low bandwidth structured linearizations may be useful for numerical computations. Second, to utilize EGFPs directly to form a family of Rosenbrock strong linearizations of an $n \times n$ rational matrix $G(\lam)$ associated with a realization. We describe the formulas for the construction of low bandwidth linearizations for $P(\lam)$ and $G(\lam)$. We show that the eigenvectors, minimal bases/indices of $P(\lam)$ and $G(\lam)$ can be easily recovered from those of the linearizations of $P(\lam)$ and $G(\lam)$.

翻译：线性化是求解多项式特征值问题（PEPs）和有理特征值问题（REPs）的广泛使用的方法，其中将PEP/REP转化为广义特征问题，并利用文献中现有的算法求解该广义特征问题。类Fiedler铅笔（Fiedler铅笔（FPs）、广义Fiedler铅笔（GFPs）、带重复的Fiedler铅笔（FPRs）和带重复的广义Fiedler铅笔（GFPRs））是众所周知的强线性化类别。GFPs是一类有趣的线性化族，而GF铅笔是FPRs和GFPRs的基本构建模块。因此，FPRs和GFPRs具有独特特性，并为结构化矩阵多项式提供保结构的线性化。但GFPRs并未充分利用GF铅笔的全部潜力。实际上，并非所有GFPs都是FPRs或GFPRs，反之亦然。本文的主要目标有两个。首先，为所有类Fiedler铅笔FPs、GFPs、FPRs和GFPRs构建一个统一框架。为此，我们构造了一个新的强线性化族（称为EGFPs），用于矩阵多项式$P(\lam)$，该族包含了所有类Fiedler线性化。EGFPs族的一个显著特点是，对于结构化（对称、埃尔米特、回文）矩阵多项式，它允许构建低带宽的保结构带状线性化。低带宽结构化线性化可能对数值计算有用。其次，直接利用EGFPs构建与实现相关联的$n \times n$有理矩阵$G(\lam)$的一族Rosenbrock强线性化。我们描述了为$P(\lam)$和$G(\lam)$构建低带宽线性化的公式。我们证明，$P(\lam)$和$G(\lam)$的特征向量、最小基/指数可以很容易地从$P(\lam)$和$G(\lam)$的线性化中恢复。