We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigurous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of "strong block minimal bases pencils", which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. We hope that this robustness property will allow us to extend the results in this paper to other contexts.
Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidade de Lisboa（里斯本大学）；Faculdade de Economia do Porto（波尔图经济与管理学院）；Boston University（波士顿大学）
Fast and backward stable computation of the eigenvalues and eigenvectors of matrix polynomials
Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid（马德里自治大学数学科学研究所）；Nazarbayev University（纳扎尔巴耶夫大学）；ISTI, Area della ricerca CNR（意大利国家研究委员会ISTI研究所）；KU Leuven（荷语鲁汶大学）；Washington State University（华盛顿州立大学）
A unified approach to Fiedler-like pencils via strong block minimal bases pencils
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；KU Leuven（荷语鲁汶大学）；Dos Pueblos High School（多斯普埃洛高中）；University of Virginia（弗吉尼亚大学）
