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Numerische Mathematik

, Volume 67, Issue 3, pp 303–313 | Cite as

Positive definiteness in the numerical solution of Riccati differential equations

  • Luca Dieci
  • Timo Eirola

Summary.

In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization.

Mathematics Subject Classification (1991): 65L07 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Luca Dieci
    • 1
  • Timo Eirola
    • 2
  1. 1.Department of Mathematics, Georgia Tech, Atlanta, GA 30332 USA E-mail address: dieci@math.gatech.edu DE
  2. 2.Institute of Mathematics, Helsinki University of Technology, SF-02150 Espoo, Finland E-mail address: Timo.Eirola@hut.fi DE

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