Positive definiteness in the numerical solution of Riccati
differential equations |
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Authors: | Luca Dieci Timo Eirola |
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Institution: | (1) Department of Mathematics, Georgia Tech, Atlanta, GA 30332 USA E-mail address: dieci@math.gatech.edu , DE;(2) Institute of Mathematics, Helsinki University of Technology, SF-02150 Espoo, Finland E-mail address: Timo.Eirola@hut.fi , DE |
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Abstract: | 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.
Received April 21, 1993 |
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Keywords: | Mathematics Subject Classification (1991): 65L07 |
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