A unified framework for the numerical solution and analysis of generalized algebraic quadratic matrix equations with engineering and scientific applications : theory and software design and implementation - Volume 1

A unified framework for the numerical solution and analysis of all algebraic quadratic matrix equations is presented in this thesis. In a global approach, it is shown how all systems of algebraic quadratic equations can be equivalently considered as special cases of a single generalized algebraic quadratic matrix equation. Hence, the thesis is devoted to the theoretical development of the numerical solution and analysis of this general matrix equation and to the design and implementation of relevant computer software. The essential tools for the developments of the numerical algorithms are the probability-1 homotopy methods. In addition, perturbation theory is used in order to conduct numerical analysis studies of the designed algorithms and to derive error estimates for the computed solutions. All the above are then implemented in software in the form of a MATLAB toolbox.;It is shown how the numerical solution and analysis of many design and analysis problems in engineering and science can be formulated as special cases within the proposed framework. Several problems are considered: classical and generalized algebraic Riccati equations, modern robust control system designs, second order matrix polynomial equations for the solution of the quadratic eigenvalue problem, computation of equilibrium points in noncooperative Nash games in economic systems, chemical kinetics modelling, computation of equilibrium points of quadratic dynamical systems (which often appear in bifurcation and in chaos theory). For the above cases, relevant numerical examples from the literature are used to illustrate the efficiency and success of the proposed framework.;Both theoretical and application issues are analysed and discussed in detail. Because the presentation of the thesis is highly mathematical and several issues are new, all necessary background information is provided in order to make the thesis an independent self-study.