Filtered conjugate residual-type algorithms with applications

It is often necessary to filter out an eigenspace of a given matrix A before performing certain computations with it. The eigenspace usually corresponds to undesired eigenvalues in the underlying application. One such application is in information retrieval, where the method of latent semantic indexing replaces the original matrix with a lower-rank one using tools based on the singular value decomposition. Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector. Filtering has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies in the underlying problem. Another common application is to compute an invariant subspace of a symmetric matrix associated with eigenvalues in a given interval. In this case, it is necessary to filter out eigenvalues that are not in the interval of the wanted eigenvalues. This paper presents a few conjugate gradient-like methods to provide solutions to these types of problems by iterative procedures which utilize only matrix-vector products.