The preconditioned inverse iteration for hierarchical matrices

The preconditioned inverse iteration [Ney01a] is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix M . Here we use this method to find the smallest eigenvalues of a hierarchical matrix [Hac99]. The storage complexity of the datasparse H-matrices is almost linear. We use H-arithmetic to precondition with an approximate inverse of M or an approximate Cholesky decomposition of M . In general H-arithmetic is of linear-polylogarithmic complexity, so the computation of one eigenvalue is cheap. We extend the ideas to the computation of inner eigenvalues by computing an invariant subspaces S of (M − μI) by subspace preconditioned inverse iteration. The eigenvalues of the generalized matrix Rayleigh quotient μM (S) are the wanted inner eigenvalues of M . The idea of using (M − μI) instead of M is known as folded spectrum method [WZ94]. Numerical results substantiate the convergence properties and show that the computation of the eigenvalues is superior to existing algorithms for non-sparse matrices.