Title:

The numerical solution of quadratic matrix equations

Methods for computing an efficient and accurate numerical solution of the real monic unilateral quadratic matrix equation, are few. They are not guaranteed to work on all problems. One of the methods performs a sequence of Newton iterations until convergence occurs whilst another is a matrix analogy of the scalar polynomial algorithm. The former fails from a poor starting point and the latter fails if no dominant solution exists. A recent approach, the Elimination method, is analysed and shown to work on problems for which other methods fail. . The method requires the coefficients of the characteristic polynomial of a matrix to be computed and to this end a comparative numerical analysis of a number of methods for computing the coefficients is performed. A new minimisation approach for solving the quadratic matrix equation is proposed and shown to compare very favourably with existing methods . . A special case of the quadratic matrix equation is the matrix square root problem, where P = o. There have been a number of method proposed for it's solution, the more successful ones being based upon Newton iterations or the Schur factorisation. The Elimination method is used as a basis for generating three methods for solving the matrix square root problem. By means of a numerical analysis and results it is shown that for small order problems the Elimination methods compare favourably with the existing methods. The algebraic Riccati equation of stochastic and optimal control is, where the solution of interest is the symmetric nonnegative definite one. The current methods are based on Newton iterations or the determination of the invariant subspace of the associated Hamiltonian matrix. A new method based on a reformulation of Newton's method is presented. The method reduces the work involved at each iteration by introducing a Schur factorisation and a sparse linear system solver. Numerical results suggest that it may compare favourably with wellestablished methods. Central to the numerical issues are the discussions on conditioning, stability and accuracy. For a method to yield accurate results, the problem must be wellconditioned and the method that solves the problem must be stableconsequently discussions on conditioning and stability feature heavily in this thesis. The units of measure we use to compare the speed of the methods are the operations count and the Central Processor Unit (CPU) time. We show how the CPU time accurately reflects the amount of work done by an algorithm and that the operations counts of the algorithms correspond with the respective CPU times.
