  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 $G(\lam)$.
