Block Methods

Implicit restarting can also be used in conjunction with a block Arnoldi   method. A block method attempts to use the block Krylov subspace 

where has b columns, to approximate eigenvalues and eigenvectors. A block Arnoldi reduction uses a subspace drawn from a sequence of subspace iterates. The details associated with implicitly restarting a block Arnoldi reduction are laid out in [4,24].

Block methods are used for two major reasons. The first one is to aid in reliably determining multiple and/or clustered eigenvalues.   Although [26] indicates that an unblocked Arnoldi method coupled with an appropriate deflation strategy may be used to compute multiple and/or clustered eigenvalues, a relatively small convergence tolerance is required to reliably compute clustered eigenvalues. Many problems do not require this much accuracy, and such a criterion can result in unnecessary computation. The second reason for using a block formulation is related to computational efficiency. Often, when a matrix-vector product with ${\bf A}$ is very costly, it is possible to compute the action of ${\bf A}$ on several vectors at once with roughly the same cost as computing a single matrix-vector product. This can happen, for example, when the matrix is so large that it must be retrieved from disk each time a matrix-vector product is performed. In this situation, a block method may have considerable advantages.

The performance tradeoffs of block methods and potential improvements to deflation techniques are under investigation.