| H-matrix is a special class of matrices that arises in the mathematical sci-ences and applications of engineering. It plays an important role in computationalmathematics, theory of dynamical systems, mathematical physics and mathemat-ics of economics etc. In recent years, many domestic and overseas scholars havebeen working on its properties and criterion, and have obtained lots of importantresults.In this paper, based on some recent methods, we present several direct cri-terion for H-matrices, which improve and generalize the corresponding results;we also provide several iterative algorithms for identifying H-matrices, which canhave fewer number of iterations and less running time than the earlier ones.In chapter one, we introduce some background knowledge of H-matrices,including the meaning and recent works of the topic, we also present the summaryof this paper, and several basic symbols, definitions, lemmas as well.In chapter two, by constructing di?erent positive diagonal matrices andapplying some techniques of inequalities, we give some simple criterion for H-matrices, which are theoretical basis for several iterative algorithms in chapterfour.In chapter three, by using the properties ofα-(chain) diagonally dominantmatrices, applying some techniques of inequalities, and partitioning the subscriptset of the elements with di?erent row types, we obtain several new methods forH-matrices, which improve and generalize some recent results. We also give thecorresponding criterion when the given matrix is irreducible. Finally, numericalexamples for the e?ectiveness of the methods are presented.In chapter four, based on the early results, we provide several iterative algo-rithms for distinguishing H-matrices by constructing di?erent iterative matrices,we also prove the convergence and superiority of the proposed algorithms. Nu-merical examples show that the new methods can have fewer number of iterations,less running time and larger identifying range than the earlier ones.
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