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W. Auzinger:
"Uniform bounds for powers of companion matrices, with application to difference schemes";
Vortrag: Mathematisches Seminar, Universität Györ, Györ; 23.05.2008.

Linear multistep methods applied to stiff ODEs can be analyzed by means of formulating the scheme as a one-step recursion in a higher dimension and considering the associated iteration matrix (companion matrix) depending on the stiffness parameter. For the purpose of uniform stability estimates for such a family of matrices, quantitative information about the location of eigenvalues (the characteristic roots of the underlying scheme) is required. Moreover, the matrices has to transformed in an appropriate way.

For BDF schemes, a similarity transformation to bidiagonal form has been successfully used. In this talk we interpret this transformation as a spezial orthogonalization process. Moreover, we discuss the general question how systems of orthogonal polynomials may be used to derive power bounds for families of stable companion matrices.

companion matrix, uniform power boundedness, orthogonal polynomials