87 research outputs found
A study of matrix equations
Matrix equations have been studied by Mathematicians for many
years. Interest in them has grown due to the fact that these
equations arise in many different fields such as vibration analysis,
optimal control, stability theory etc.
This thesis is concerned with methods of solution of various
matrix equations with particular emphasis on quadratic matrix
equations. Large scale numerical techniques are not investigated
but algebraic aspects of matrix equations are considered.
Many established methods are described and the solution of a
matrix equation by consideration of an equivalent system of
multivariable polynomial equations is investigated. Matrix equations
are also solved by a method which combines the given equation with
the characteristic equation of the unknown matrix.
Several iterative processes used for the solution of scalar
equations are applied directly to the matrix equation. A new
iterative process based on elimination methods is also described
and examples given.
The solutions of the equation x2 = P are obtained by a method
which derives a set of polynomial equations connecting the
characteristic coefficients of X and P. It is also shown that
the equation X2 = P has an infinite number of solutions if P is a
derogatory matrix.
Acknowledgement
From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
The problem of reducing an algebraic Riccati equation to a unilateral quadratic matrix equation (UQME) of the
kind is analyzed. New reductions are introduced
which enable one to prove some theoretical and computational properties.
In particular we show that the structure preserving doubling algorithm
of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the
cyclic reduction algorithm applied to a suitable UQME. A new algorithm
obtained by complementing our reductions with the shrink-and-shift tech-
nique of Ramaswami is presented. Finally, faster algorithms which require
some non-singularity conditions, are designed. The non-singularity re-
striction is relaxed by introducing a suitable similarity transformation of
the Hamiltonian
Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction
We review a family of algorithms for Lyapunov- and Riccati-type equations
which are all related to each other by the idea of \emph{doubling}: they
construct the iterate of another naturally-arising fixed-point
iteration via a sort of repeated squaring.
The equations we consider are Stein equations , Lyapunov
equations , discrete-time algebraic Riccati equations
, continuous-time algebraic Riccati equations
, palindromic quadratic matrix equations , and
nonlinear matrix equations . We draw comparisons among these
algorithms, highlight the connections between them and to other algorithms such
as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge
Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms
We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton\u27s iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence
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