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W. Auzinger:
"Stability estimates via orthogonal polynomials";
Vortrag: SDIDE - Stability and Discretization Issues in Differential Equations, Wien (eingeladen); 17.09.2008 - 21.09.2008.

Our main focus is on the stability problem for multistep methods applied to linear ODE systems, e.g. depending on a stiffness parameter. This can be analyzed by means of formulating the scheme as a one-step recursion in a higher dimension and considering the powers of the associated companion matrices. For the purpose of quantitative
stability estimates for such families of matrices, uniform information about the location of their eigenvalues (the characteristic roots of the underlying scheme) must be available. Moreover, in order to derive satisfactory estimates, an appropriately chosen similarity transformation is required.

For BDF schemes applied to stiff model problems, a transformation to bidiagonal form has been successfully used for this purpose. We give a review of this type of stability estimates, including the system case, and we interpret bidiagonalization as a special orthogonalization process. Moreover, we propose and discuss a general framework for deriving stability estimates, especially for the case of companion matrices, which is based on systems of orthogonal polynomials. Scaling of the basis polynomials is an important issue and is discussed by means of examples.

Numerical analysis, ODEs, stability