Research On Several Problems In Matrix And Tensor Calculation

In this paper, first, we investigate the convergence of the general nonstation-ary iterative method for singular linear system, and propose sufficient conditions for the convergence of it. Furthermore, we apply our results to singular Hermi-tian positive semidefinite linear systems. Some quotient convergence results in [11] are improved such that the convergence are obtained. Also, several classical convergence results of the stationary iteration method [11,12,40,66,129,132] are extended to the nonstationary case, and some new convergence results for the nonstationary iterative scheme are established. We also utilize our results to prove the convergence of nonstationary iterated Tikhonov regularization [7,48], multisplitting algorithms [12,15,23] and nonstationary two-stage iterative algo-rithms [11,14] for solving singular Hermitian positive semidefinite linear systems.Second we investigate the effective condition numbers for the generalized Sylvester equation(AX-YB,DX-YE)=(C,F) where A,D∈Rm×m and B,E∈Rn×n.We also apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires O(m2n+mn2) flops. Numerical examples illustrate the sharpness of our pertur-bation bounds.Third, we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE). The bounds of the backward error and three types of upper bounds for the perturbed STE with or without dropping the second order terms are presented. The classic perturbation results for the Sylvester equation are extended to the high order case.At last, we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes, called weakly nondangling nodes, to another single node, and the PageRank could be obtained by applying the power method to the reduced matrix. Further more, the reduced matrix is also stochastic with the same nonzero eigenvalues as the Google matrix. Numerical examples illustrate that the new algorithm could’save a large amount of operations for computing the full PageRank vector.