For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitia...For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. A priori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.展开更多
基金Project supported by the China State Major Key Projects for Basic Researchesthe National Natural Science Foundation of China (Grant No. 19571014)the Doctoral Program (97014113), the Foundation of Excellent Young Scholors of Ministry of Education
文摘For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. A priori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.
