Research On M - Matrix (tensor) Least Eigenvalue Estimation And Its Related Problems

The theory of M-matrices is one of the important research topics in numeri-cal algebras and matrix theories and it has important applications in many fields. The researches of M-matrices contain the following problems as the hotspot:Es-timations of bounds on the smallest eigenvalue of the Hadamard product of an M-matrix and its inverse; Estimations (Computions) of the smallest eigenvalue of an M-matrix (tensor); The numerical solution of linear equations which the coefficient matrix is a Z-matrix.In this dissertation, we research the above issues and obtain the following results:First, several sequences of the upper and lower bounds of the minimum eigenvalue of the Hadamard product of a nonsingular M-matrix and its inverse, and a corresponding algorithm is given. It is proved that the upper bound se-quences are monotone decreasing and the lower bound sequences are monotone increasing, and improved the related results of [H.B. Li, T.Z. Huang, S.Q. Shen and H. Li. Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse. Linear Algebra Appl,2007,(420):235-247] and [Y.T. Li, F.B. Chen and D.F. Wang. New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse. Linear Algebra Appl,2009,(430):1423-1431]. Numerical examples are given to illustrate the availability and practicability of the algorithm and show that the lower bound sequences could ap-proach the true value of the minimum eigenvalue in some cases. Second, a lower bound sequence with monotone increasing of the minimum eigenvalue of a strictly diagonally dominant M-matrix, and its related algorithm are establish. Next, the estimations (computations) of the minimum eigenvalue of M-tensors, which is a generalization of the M-matrix on the higher order, is studied. The estimates and algorithm of the minimum eigenvalue are given and numerical examples show that the estimates can achieve true value in some cases and it is proved that the algorithm can converge to the true value theoretically.Last, as an application, we give two types of new preconditioned GAOR methods to solve linear equations which the coefficient matrices is Z-matrices and it is proved that in some cases the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods in [H.L. Shen, X.H. Shao and T. Zhang. Preconditioned iterative methods for solving weighted linear least squares problems. Appl. Math. Mech.-Engl. Ed.,2012,(33):375-384] and [J.H. Yun. Comparison results on the preconditioned GAOR method for generalized least squares problems. Intern. J. Comput. Math.,2012,(89):2094-2105] whenever these methods are convergent, and a coefficient matrix of the linear equation after preliminary conditioning is an M-matrix.