Numerische Mathematik

, Volume 60, Issue 1, pp 115–131 | Cite as

Smooth factorizations of matrix valued functions and their derivatives

  • Peter Kunkel
  • Volker Mehrmann


In this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.

Mathematics Subject Classification (1991)

65L02 65F25 93C15 93B40 


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  1. [BCP] Brenan, K., Campbell, S.L., Petzold, L.R. (1989): Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Elsevier, North HollandGoogle Scholar
  2. [BBMN] Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N. (1990): Numerical computation of an analytic singular value decomposition of a matrix valued function. FSP Mathematisierung, Universität Bielefeld, Materialien LI XGoogle Scholar
  3. [BMN1] Bunse-Gerstner, A., Mehrmann, V., Nichols, N. (1989): Derivative feedback for descriptor systems. FSP Mathematisierung, Universität Bielefeld, Materialien LV IIIGoogle Scholar
  4. [BMN2] Bunse-Gerstner, A., Mehrmann, V., Nichols, N. (1992): Numerical methods for the regularization of descriptor systems. SIAM J. Matrix Anal. Appl. (to appear)Google Scholar
  5. [CPS] Chapman, M.J., Pearson, D.W., Shields, D.N. (1988): Partial singular value assignment in the design of robust observers for discrete time descriptor systems. IMA J. Control Inf. 203–213Google Scholar
  6. [DHZ] Deuflhard P., Hairer, E., Zugck, J. (1987): One step and extrapolation methods for differential algebraic systems. Numer. Math.51, 501–516Google Scholar
  7. [G] Gantmacher, F.R. (1959): Theory of Matrices, Vol. II. Chelsea, New YorkGoogle Scholar
  8. [GM] Griepentrop, E., März, R. (1986): Differential-algebraic equations and their numerical treatment. Teubner-Texte, LeipzigGoogle Scholar
  9. [GP] Gear, C.W., Petzold, L.R. (1983): Differential-algebraic systems and matrix pencils. In: B. Kågström, A. Ruhe, eds., Matrix Pencils, Lecture Notes in Mathematics 943. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. [GV] Golub, G.H., Van Loan, C.F. (1983): Matrix Computations, The Johns Hopkins University Press, Baltimore, Md.Google Scholar
  11. [H] Hansen, B. (1990): Comparing different concepts to treat differential algebraic equations. Preprint 220, Humboldt-Universität Berlin, Sektion MathematikGoogle Scholar
  12. [KM] Kunkel, P., Mehrmann, V. (1990): Numerical solution of differential algebraic Riccati equations. Linear Algebra Appl.137/138, 39–66Google Scholar
  13. [M1] Mehrmann, V. (1987): The Linear Quadratic Control Problem, Theory and Numerical Algorithms. Habilitationsschrift, Universität BielefeldGoogle Scholar
  14. [M2] Mehrmann, V. (1989): Existence, uniqueness and stability of solutions to singular linear quadratic optimal control problems. Linear Algebra Appl.121, 291–331Google Scholar
  15. [R1] Rheinboldt, W.C. (1988): On the computation of multidimensional solution manifolds of parametrized equations. Numer. Math.53, 165–181Google Scholar
  16. [R2] Rheinboldt, W.C. (1990): On the theory and numerics of differential-algebraic equations. In: Proceedings of the SERC Summer School at Lancaster (UK). Oxford University Press, OxfordGoogle Scholar
  17. [VLK] Verghese, G.C., Lévy, B.C., Kailath, T. (1981): A general state space for singular systems. IEEE Trans. Autom. ControlAC-26, 811–831Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Peter Kunkel
    • 1
  • Volker Mehrmann
    • 2
  1. 1.Fachbereich MathematikUniversität OldenburgOldenburgFederal Republic of Germany
  2. 2.Institut für Geometrie und praktische MathematikRWTH AachenAachenFederal Republic of Germany

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