The Properties And Applications Of Some Classes Of H-matrices And H-tensors

The H-matrices are defined as the very important class of special matrices in the matrix theory,which have a widely range of applications in many scientific com-putation and engineering applications fields,such as computational mathematics,numerical algebra,economical mathematics,electric power system theory,control theory.The H-tensor is an extension of the H-matrix,which has a widely range of applications in higher order Markov chains,quantum entanglement problem,hypergraph theory,magnetic resonance imaging,automatic control,blind source separation,simulation,polynomial optimization,etc..The Schur complement of Matrix plays a very important role in order reduction processing of large matrix computation and precondition method for solving linear equations,and becomes one of our research hot topic in the past ten years.The paper will mainly has a research on the properties,the decisive methods and the applications of the H-matrices and the H-tensors,and the key results are as follows.(1)The structure and properties of the Schur complement of ?(product ?)-diagonally dominant matrices and its application in solving large linear equations are studied.Firstly,the paper present the ?(product ?)-diagonally dominant de-gree on the Schur complement of ?(product ?)-diagonally dominant matrices.The disc theorem for the Schur complement of ?(product ?)-diagonally dominant ma-trices is proved and suggested,which improves and extends some related results.Additionally,because the iterative method is closely related to spectral radius es-timation,the paper gives the spectral radius estimates for the inverse matrices of?(product ?)-diagonally dominant matrices and their Schur complements.Then,a new iterative method is designed for the linear equations with the coefficient matrix which is ?(product ?)-diagonally dominant matrix.Further,by combining the iterative method with the Schur-based iterative method,the paper suggested to establish a new iterative method called the Schur complement-based iterative algorithm and proved its convergence.Finally,some numcrical examples arc given to verify that the Schur complement-based iterative algorithm can not only reduce the order,but also have a good effect in terms of convergence.(2)The paper has got the application of the Schur complement of Nekrasov matrices in solving linear equations.Firstly,the spectral radius estimation of the inverse of the Schur complement of Nekrasov matrices is gained;and then,some new methods called Schur-based super relaxation iteration(SSSOR)method and Schur-based conjugate gradient(SCG)method to solve the linear equations by reducing order are introduced respectively,finally the validity of the method is illustrated by the numerical examples given.(3)The paper has introduced two generalized irreducible Nekrasov matrices,which are irreducible ?-Nekrasov matrices and irreducible ?-S-Nekrasov matrices,and it has analyzed the relationships among the involved matrices and irreducible H-matrices.(4)The paper has studied on the closure property of H-tensors under Hadamard product,and has demonstrated that the Hadamard products of Hadamard pow-ers of strong H-tensors are still strong H-tensors.Then the paper has bounded the minimal real eigenvalues of the comparison tensors of the Hadamard products involving strong H-tensors.Finally,the paper has attained the bounds of these eigenvalues through the properties of these H-tensors.