The Lanczos method is often used to solve a large scale symmetric matrix eigenvalue problem. It is well-known that the single-vector Lanczos method can only find one copy of any multiple eigenvalue and encounters slow convergence towards clustered eigenvalues. On the other hand, the block Lanczos method can compute all or some of the copies of a multiple eigenvalue and, with a suitable block size, also compute clustered eigenvalues much faster. The existing convergence theory due to Saad for the block Lanczos method, however, does not fully reflect this phenomenon since the theory was established to bound approximation errors in each individual approximate eigenpairs. Here, it is argued that in the presence of an eigenvalue cluster,
the entire approximate eigenspace associated with the cluster should be considered as a whole, instead of each individual approximate eigenvectors, and likewise for approximating clusters of eigenvalues. In this paper, we obtain error bounds on approximating eigenspaces and eigenvalue clusters. Our bounds are much sharper than the existing ones and expose true rates of convergence of the block Lanczos method towards eigenvalue clusters. Furthermore, their sharpness is independent of
the closeness of eigenvalues within a cluster. Numerical examples are presented to support our claims.
We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative
third order tensor.
The method involves the selection of a
parameter $\theta_k$ in the $k$th iteration.
For every positive starting vector, the method converges quadratically
and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration.
It is also shown that $\theta_k=1$ near convergence.
The computational work for each iteration of the proposed method
is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be $1$ directly)
that for each iteration of the Ng--Qi--Zhou algorithm, which is linearly convergent
for essentially positive tensors.
We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.