The Research Of Divide And Conquer Algorithms For Skew-symmetric Tridiagonal Eigenvalue Problems

This thesis investigates divide-and-conquer method for eigenvalue problems of skew-symmetric tridiagonal matrices, which turns the eigenvalue problems of skew-symmetric tridiagonal matrix to the eigenvalue problems of symmetric tridiagonal matrix, which avoids the complex operation and reduces the computation amount Considering the diverse structures of the symmetric tridiagonal matrices, one presents various divide-and-conquer algorithms based on different divide-and-conquer strategies. The article includes three main parts:The first part introduces various divide-and-conquer strategies, properties of Laguerre iteration as well as how to deal with the clusters of pathological eigenvalues by Laguerre iteration. At last, one gives the stopping criteria.Based on the theorem foundation of skew-symmetric tridiagonal matrix which given by professor Lian Qingrong et al, the second part presents a divide-and-conquer algorithm that is on the basis of the rank-one perturbation of matrix and Laguerre iteration. The algorithm is suitable for calculate those lager eigenvalues by modulo. But it may lose the significant figures in case of those smaller eigenvalues by modulo.The third part discusses the eigenvalues relationship between skew-symmetric tridiagonal matrix and its concomitant matrix (special symmetric diagonal matrix with diagonal elements are zero). One gives a split-merge algorithm for the concomitant matrix based on the rank-two perturbation of matrix and Laguerre iteration. Its advantages are to keep the matrices structure and positive-negative eigenvalues in pair. And practical computation process needs only computer nonnegative eigenvalues, and that cuts down the computation amountDivide-and-conquer method possesses the good characteristic which is speed-fast and flexible bringing out, which can not only search for all the eigenvalues, but also get some fixed eigenvalues or eigenvalues in given interval. It's worth noticing that out algorithm does well in its natural parallelism that is suitable for solving the eigenvalue problems of large scale matrices. Above all, the method is an eigenvalue-searching method with widespread prospect.