Iterative Algorithms Based On A Sufficient And Necessary Condition Of H-Matrices And Some Special Properties And Applications Of Its Subclasses

As a common and important special matrix,non-singular H-matrix is widely used in many fields,such as matrix theory,computational mathematics,cybernetics,power system theory,neural networks and intelligent science and engineering and so on.In recent years,many important results have emerged on the algorithms of solving some equations based on M-matrix(H-matrix),the bounds of constrained solutions.For many theoretical and practical problems,especially for large-scaled problems,one of the first problems people will face is to determine whether some matrices are H-matrices.On the other hand,in view of the extensive application backgrounds of H-matrix,all kinds of special properties of H-matrix also need to be explored,especially the properties and methods that can help solve large-scaled problems.Therefore,iterative algorithms for H-matrices and some properties with applications are discussed in this paper,including an iterative algorithm based on necessary and sufficient conditions for H-matrices;The closure property for Nekrasov matrices with its application in solving large linear equations by Schur-based method;New upper bounds for strictly diagonally dominant M-matrices and its applications in linear complementarity problems(LCPs).· At first,we propose a necessary and sufficient condition for Hmatrix,and then an iterative algorithm for H-matrices is designed.Through theoretical analysis of the algorithm,our algorithm is not affected by the rounding errors of computer in some cases,which provides theoretical support for the implementation of iterative algorithm of H-matrix on computer,and can judge H-matrix more accurately.Further more,we also propose an accelerate algorithm.In the end,some numerical experiments are presented to show the effectiveness and superiority of our algorithms;· We give a sufficient condition that Nekrasov matrices satisfy the closed property.This result generalizes the existing results and makes Nekrasov matrices also have closed property via their nonleading principal submatrices in some cases.Therefore,the closed property of Nekrasov matrix provides a theoretical basis for solving large linear equations by using Schur-based method,and some numerical experiments are presented to show the effectiveness and superiority in some cases;· By partition and summation,we obtain two new upper bounds for the inverse of strictly diagonally dominant(SDD)M-matrices by reduction method.Various numerical experiments with random matrices show that the upper bound(which is expressed by means of some determinants of third order matrices)is much sharper than existing ones.In addition,the new upper bounds for the matrix inverse can be utilized for the error analysis in LCPs for error bounds,and we propose two sharper error bounds for B-matrices by computing the maximum value of multivariate functions.In addition,some other numerical experiments for error bounds of LCPs are presented to show the efficiency and superiority of our results.