Recursion Formulae for the Characteristic Polynomial of Symmetric Banded Matrices Werner Kratz und Markus Tentler Preprint Series: 2007-15 Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften UNIVERSITAT¨ ULM Recursion Formulae for the Characteristic Polynomial of Symmetric Banded Matrices Werne Kratz und Markus Tentler Preprint Series: 2007-15 Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften UNIVERSITAT¨ ULM Recursion formulae for the characteristic polynomial of symmetric banded matrices Werner Kratz and Markus Tentler Abstract . In this article we treat the algebraic eigenvalue problem for real, symmetric, and banded matrices of size N × N , say. For symmetric, tridi- agonal matrices, there is a well-known two-term recursion to evaluate the characteristic polynomials of its principal submatrices. This recursion is of complexity O(N) and it requires additions and multiplications only. More- over, it is used as the basis for a numerical algorithm to compute particular eigenvalues of the matrix via bisection. We derive similar recursion formu- lae with the same complexity O(N) for symmetric matrices with arbitrary bandwidth, containing divisions. The main results are divisionfree recursions for penta- and heptadiagonal symmetric matrices. These recursions yield sim- ilarly as in the tridiagonal case effective (with complexity O(N) ), fast, and stable algorithms to compute their eigenvalues. Running head: Recursion formulae for banded matrices Key words: Banded matrix, eigenvalue problem, Sturm-Liouville equation, pentadiagonal matrix, heptadiagonal matrix, bisection method AMS subject classi¯cation: 15A18; 65F15, 15A15, 39A10, 15A24 W. Kratz and M. Tentler Universit¨atUlm Institut f¨urAngewandte Analysis D - 89069 Ulm, Germany e-mail:
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[email protected] Recursion formulae for the characteristic polynomial of symmetric banded matrices Werner Kratz and Markus Tentler 1 Introduction In this article we consider the algebraic eigenvalue problem for real, sym- metric, and banded matrices of size N × N , say.