Numerical Methods For Judging Generalized Diagonally Dominant Matrices

The generalized diagonally dominant matrix (H-matrix) is a relatively activeresearch field in computational mathematics, cybernetics and matrix theory. It iswidely used in many subjects such as computational mathematics, mathemati-cal physics, economics, biology, dynamical system theory and intelligence scienceetc.. Many practical problems can be summarized to determine the H-matrix.However, it is rather difficult in practice to determine whether a matrix is anH-matrix or not. Therefore, it has great theoretical and practical value to studythe numerical methods for judging the H-matrix, to provide the concise and prac-tical criteria and to construct fast and e-cient iterative identification algorithms.Up to now, within the scope of the field, many researchers have done a lot ofin-depth study and acquired some very valuable results in many respects, such asH-matrix properties, criteria and the iterative identification algorithm and so on.In this paper,we mainly discuss numerical criteria of the H-matrix, the it-eration identification algorithm and the parallel processing iteration algorithm.Further we extend these results to generalized block strictly diagonally dominantmatrices(block H-matrix). The main results and innovations are as follows:Firstly,we introduce the research background of the problems of judging gen-eralized diagonally dominant matrices,as well as several definitions, notations andthe main work of this paper.Secondly,by using properties of the M-matrices and H-matrices, we obtaina series of new concise and practical criteria for nonsingular H-matrices by meanof constructing new positively diagonal transformation matrix factors and someinequality techniques. Some relate results are improved. Several numerical exam-ples are presented to illustrate the e-ectiveness of the proposed results. Moreover,by applying block matrix skills and the properties of matrix norm, we extend theresults of H-matrix to the generalized block strictly diagonally dominant matrixand obtain some su-cient conditions for judging such matrices.Thirdly, we study the iterative criterion of the H-matrix and provide a newtype non-parameter interleaved iterative identification algorithm which can deter- mine whether an irreducible matrix is an H-matrix or not only by fewer iteration.Furthermore, the convergence of the algorithm is proved and the superiority of theiteration speed as well as the computation amount compared with the earlier onesare shown by several numerical examples.Lastly, further researches are focused on the parallel computing method forthe iterative criterion of the H-matrix. A parallel interleaved iterative identifi-cation algorithm is proposed to improve the corresponding algorithm for generalmatrices. The convergence of the new algorithm is proved and the in-uences ofthe parallel computing to the convergence and computation e-ectiveness of theiterative algorithm are discussed. It is proved both by theoretical and numericalexperiments that the new parallel algorithm only requires less iteration and com-putation time. Furthermore, We study the iterative criterion for judging the blockH-matrix and provide a parallel interleaved iterative identification algorithm forit. Therefore, it will help to the generation of new quick identification algorithmsand the corresponding development of the theory by studying the parallel com-puting method.