Nekrasov Tensor And Its Criteria

H-matrices and Nekrasov matrices are classes of matrices which play impor-tant roles in matrix theory. They have numerous applications in many areas such as numerical algebra, control theory. Recently, H-matrices have been extended to H-tensors. In this paper, we define the Nekrasov tensors, and prove that they are equivalent to nonsingular H-tensors. Besides, we give serial practical conditions for the Nekrasov tensors.In chapter one, we introduce background knowledge and recent research for tensors and introduce some basic symbols and definitions.In chapter two, we give the definition of Nekrasov tensors, and discuss some properties, such as the principal subtensors of Nekrasov tensors are Nekrasov ten-sors, the Hadamard product of Nekrasov tensors is Nekrasov tensor.In chapter three, we investigate the relationship between nonsingular H-tensors and generalized Nekrasov tensors. We present that strictly diagonally dominant tensors are Nekrasov tensors, Nekrasov tensors and irreducible weak Nekrasov tensors are nonsingular H-tensors. Generalized Nekrasov tensor are non-singular H-tensors, and vice versa.In chapter four, we obtain several direct criteria and iterative algorithms for generalized Nekrasov tensors by analysising the structure of the tensors, construct-ing special positive diagonal matrices,combing inequality transformation, the ef-fectiveness of the results obtained are illustrated by numerical examples.