2016/11/22 by Maribel Bueno Cachadina, Froilán M. Dopico, Cachadina, Maribel Bueno +9
Computer Science · Mathematics · Physics and Astronomy · #15A18 #15A22 #15A54 #65F15 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1611.07170
openalex publication_date 2016/11/22 · openalex created_date 2022/09/02 · openalex updated_date 2026/08/01
The standard way of solving the polynomial eigenvalue problem associated with\na matrix polynomial is to embed the matrix polynomial into a matrix pencil,\ntransforming the problem into an equivalent generalized eigenvalue problem.\nSuch pencils are known as linearizations. Many of the families of\nlinearizations for matrix polynomials available in the literature are\nextensions of the so-called family of Fiedler pencils. These families are known\nas generalized Fiedler pencils, Fiedler pencils with repetition and generalized\nFiedler pencils with repetition, or Fiedler-like pencils for simplicity. The\ngoal of this work is to unify the Fiedler-like pencils approach with the more\nrecent one based on strong block minimal bases pencils introduced in\n citecanonical. To this end, we introduce a family of pencils that we have\nnamed extended block Kronecker pencils, whose members are, under some generic\nnonsingularity conditions, strong block minimal bases pencils, and show that,\nwith the exception of the non proper generalized Fiedler pencils, all\nFiedler-like pencils belong to this family modulo permutations. As a\nconsequence of this result, we obtain a much simpler theory for Fiedler-like\npencils than the one available so far. Moreover, we expect this unification to\nallow for further developments in the theory of Fiedler-like pencils such as\nglobal or local backward error analyses and eigenvalue conditioning analyses of\npolynomial eigenvalue problems solved via Fiedler-like linearizations.\n