The Study Of Diagonal Dominance And Its Nonlinear Generalization

The coefficient matrices of systems of linear equations generated in the study ofequilibrium theory, input-output analysis, bearing ink oscillation are usually M-matrices;while in control theory, theory of neural networks large-scale systems, as well as lineartime-delay systems, the stability of the corresponding systems is often manifested in thecondition that whether the coefficient matrices of the underlying linear equations are H?matrices, which are expected to have diagonal dominance to some extent; in scientific andengineering computing, such as computational electromagnetics, computational ?uid dy-namics, optimization, oil and gas exploration, the basic principles in relevant disciplinesare expressed as partial differential equations or integral equations. After discretizationby finite difference method, finite element method, boundary element method, domaindecomposition algorithms, these partial differential equations are often treated into large-scale sparse linear equations, of whose solution, the existence, uniqueness, and conver-gence properties and stability of related methods are all related with certain diagonaldominance of the coefficient matrix. Thus, it is seen that diagonal dominance of matricesis vital both in objective cognition and research.It is well known that diagonally dominance of matrices plays a significant role ininvestigation of linear problems. It is natural that nonlinear diagonal dominance, which isa nonlinear generalization of diagonal dominance of matrices, is of corresponding impor-tance when dealing with nonlinear process.This will constitute the two themes of this study: one is detailed and comprehensiveinvestigation of practical criteria for the quite important and extensive diagonal domi-nance of matrices, while the other is in-depth study of relevant nonlinear diagonal domi-nance. The dissertation is consisted of six chapters with four parts:The first part has an investigation of practical criteria of nonsingular H? matrices.A large number of practical criteria of nonsingular H? matrices are characterized bymeans of matrix entries with different models to partition the matrix row index sets. Theadvantages of the presented criteria are demonstrated by numerical examples and thepractical programs for determining nonsingular H? matrices are given within Matlabplatform by making use of the obtained research results obtained. The second part studies the nonlinear generalization of diagonal dominance of ma-trices. First of all, relevant concepts and existing results of nonlinear diagonal dominanceis brie?y introduced. Secondly, an answer is given to an open problem in this area, whichis to make use of the diagonal dominance of the Jacobian matrices of multi-vector-valuedfunctions to investigate the diagonal dominance of the nonlinear mapping. Finally, somekinds of nonlinear diagonal dominance are proposed and properties of these nonlineardiagonal dominance are studied.The nonlinear Schur complement is studied in the following part. The concept of theschur complement has existed in linear theory for around 100 years of history, which hasimportant applications in matrix theory and numerical analysis, etc., and has become animportant tool for theory. It still remains blank that whether there is a corresponding intro-duction in nonlinear theory which can be used as a research tool for nonlinear problems.In this part, the concept of the nonlinear Schur complement is introduced and propertiesof such nonlinear Schur complement are studied. Moreover, this concept is applied to thesolution of nonlinear eigenvalue problems.The concept of the Perron complement was introduced by American mathemati-cian Carl D. Meyer in the study of steady-state distribution vectors in finite-state Markovchains. This tool can not only help to handle practical computational problems of thesteady-state distribution vectors in large-scale Markov chains, but also is a mathematicalhot issue in the study of properties of Perron complement itself. In the final part, theconcept of generalized Perron complement is proposed and extended to the widest range.Properties of the generalized Perron complement are investigated.