Spectral Theory Of Tensors And Iterative Methods For Several Numerical Algebra Problems

Tensor plays an important role in theoretical physics, quantum mechanics, magnetic resonance imaging, higher order Markov chains and many other areas. Saddle point problems are widely used in domains of science and engineering such as solutions of PDEs with high orders, computational electromagnetics, fluid mechanics and optimization problems. There are two parts: the first one is about the properties of tensors, the second is concerning iterative methods for several numerical algebra problems, including preconditioning techniques for iteration solutions of saddle problems, SOR alternating variable method for computing the global maxima of the maximal correlation problem,modified Newton method for computing the minimal solution of the quadratic vector equation.Firstly, some contributions to the bounds of the subdominant eigenvalues are made,when is a positive square tensor. The Hopf inequality for the subdominant eigenvalues of the positive tensor is given. When the tensors are no need square or rectangular, some properties of H-singular values of a positive tensor are presented. An algorithm to find the largest singular value of a positive tensor is given, the monotone convergence of the algorithm is proved, some numerical experiments are given to show that our algorithm is efficient. When is an irreducible and nonnegative tensor, we derive some bounds for the largest eigenvalue(Z-eigenvalue, H-eigenvalue, or B-eigenvalue) of nonnegative tensors. We derive upper and lower bounds for the minimum eigenvalue of M-tensors,then, we give the Ky Fan theorem for M-tensors.Secondly, we give a lower bound for the real eigenvalue of a special saddle point matrix. The HSS splitting for saddle point problems, which is introduced by Benzi,etc., is further studied. A modified generalization of the Hermitian and skew-Hermitian method(MGHSS) is given. Theoretical analysis and numerical experiments show that the present preconditioners can be better than the corresponding HSS preconditioners.We consider block diagonal preconditioners for solving generalized saddle point linear systems and show properties of eigenvalues of the preconditioned matrix.Thirdly, based on a core engine in seeking global maxima of the maximal correlation problem(MCP), A SOR alternating variable method(AVM) is proposed. This algorithm is proved to enjoy monotone convergence,it is also shown that for a nonnegative irreducible matrix, the algorithm converges to the global maximizer from any nonnegative starting point.Finally, the existence of the solution of the quadratic vector equation arising in Markovian binary trees is given. A modified Newton method for computing the minimal solution of the equation is presented. The monotone convergence of the modified Newton method is proved. Numerical experiments show the efficiency of our method.