Sparse Hessenberg reduction and the eigenvalue problem for large sparse matrices

A four-stage algorithm for the efficient solution of the standard eigenvalue problem for large sparse matrices is presented. The matrix whose eigenvalues are desired is first reduced to a block upper triangular form, if possible, to expose those eigenvalues that are readily identified. The reduced matrix Is then scaled and transformed to a sparse Hessenberg matrix with numerical stability control. Laguerre´s iteration is then used to find the remaining eigenvalues. Examples are given.