Heike Faßbender

Heike Faßbender, Philip Saltenberger

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential linearizations for P are analyzed. All pencils in M1(P) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M1(P) is a (strong) linearization of P are given. The equivalence of some of them to the Z-rank-condition (see Mackey et al. 2006) is pointed out. Results on the vector space dimensions, the genericity of linearizations in and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Throughout the paper, structural resemblances between the matrix pencils in L1 , i.e. the results obtained in Mackey et al. 2006, and their generalized versions are pointed out.

Christian Bertram, Heike Faßbender

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration. As the general approach suggested here leads to complex valued computation even for real problems, we present a general realification approach based on similarity transformation.

Heike Faßbender, Javier Perez, Nikta Shayanfar

Many applications give rise to structured matrix polynomials. The problem of constructing structure-preserving strong linearizations of structured matrix polynomials is revisited in this work and in the forthcoming ones \cite{PartII,PartIII}. With the purpose of providing a much simpler framework for structure-preserving linearizations for symmetric and skew-symmetric matrix polynomial than the one based on Fiedler pencils with repetition, we introduce in this work the families of (modified) symmetric and skew-symmetric block Kronecker pencils. These families provide a large arena of structure-preserving strong linearizations of symmetric and skew-symmetric matrix polynomials. When the matrix polynomial has degree odd, these linearizations are strong regardless of whether the matrix polynomial is regular or singular, and many of them give rise to structure-preserving companion forms. When some generic nonsingularity conditions are satisfied, they are also strong linearizations for even-degree regular matrix polynomials. Many examples of structure-preserving linearizations obtained from Fiedler pencils with repetitions found in the literature are shown to belong (modulo permutations) to these families of linearizations. In particular, this is shown to be true for the well-known block-tridiagonal symmetric and skew-symmetric companion forms. Since the families of symmetric and skew-symmetric block Kronecker pencils belong to the recently introduced set of minimal bases pencils \cite{Fiedler-like}, they inherit all its desirable properties for numerical applications. In particular, it is shown that eigenvectors, minimal indices, and minimal bases of matrix polynomials are easily recovered from those of any of the linearizations constructed in this work.