Analysis Of Special Matrices And Iteration Solutions Of Saddle Point Problems

Since the early 20th century, applications of nonnegative matrices, H-matrices, M-matricesand other closely related the special matrices are increasingly widespread. Specialmatrix analysis plays a key role in numerical algebra, possessing wide applications indomains of computational mathematics, mathematical physics, economics, biology andphysics etc. Some special matrices and numerical characteristics are deeply studied, anditerative solutions of saddle point problems are also researched in this doctoral dissertation.The thesis consists of the main results and innovations as follows:1. Two classes of special matrices are studied: H-matrices and doubly diagonallydominant matrices. The problem of the k-subdirect sum of two H-matrices is studied.Sufficient conditions when k-subdirect sum of two H-matrices is an H-matrix are presented,which generalize related results by numerical experiments. Sufficient conditionsare provided that the k-subdirect sum of two doubly diagonally dominant (DDD) matricesstill belongs to the same class. Examples are given to illustrate the conditions presented.Furthermore, the problem of the k-subdirect sum of two S- strictly diagonally dominantmatrices is investigated which supplements that by Bru, Pedroche and Szyld.2. Two special matrices classes of M-matrices and inverse M-matrices are investigated.Firstly, the upper infinity norm bounds for the inverse of weakly chained diagonallydominant M-matrices are obtained. Making use of the special structures ofweakly chained diagonally dominant M-matrices and the relationships between entriesof M-matrices and inverse of M-matrices, estimate of the upper infinity norm boundsfor inverse of M-matrices is derived. Furthermore, estimate of the smallest eigenvalue ofM-matrices is also presented.Moreover, relations between inverse M-matrices and SPP (Strict Path Product)matrices are studied. We give a new value with which any SPP matrix may be made intoan inverse M-matrix by adding the value to the diagonal of a SPP matrix. The questionwhether a 4×4 SPP matrix is a P-matrix or not is settled. All results presented in thispart are superior to the corresponding existing ones.3. Estimates of numerical characteristics of matrices are presented. New criteriafor nonsingularity are derived, which generalize strictly diagonally dominance and prop- erties of B-matrices. These criteria are used to obtain new inclusion intervals for realeigenvalues of real matrices. And based on C-matrices and (?)-matrices, new criteria fornonsingularity are derived. Exclusion intervals of real eigenvalues of real matrices arealso analyzed according to the criteria of nonsingularity.Necessary and sufficient conditions for A to be nonsingular/singular are obtained,where A is a (generalized) block diagonally dominant matrix. In addition, several criticaltheorems for the nonsingularity/singularity of block diagonally equipotent matrices arederived.4. According to splittings of matrices, some new iterative methods are employed tosolve saddle point problems. By selecting different precondition matrices, correspondingiterative algorithms are obtained. Numerical results show that our algorithms are efficient.