A SPECTRAL ALGORITHM FOR ENVELOPE REDUCTION OF SPARSE MATRICES Dedicated to William Kahan and Beresford Parlett on the occasion of their th birthdays

We descibe a new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a speci ed eigenvec tor of the Laplacian This Laplacian eigenvector solves a continuous relaxation of a related discrete problem called the minimum sum problem The permutation vector computed by the spectral algorithm is a closest permutation vector to the speci ed Laplacian eigenvector Numerical results show that the new reordering algorithm usu ally computes smaller envelope sizes than those obtained from current algorithms such as the Gibbs Poole Stockmeyer GPS algorithm or the reverse Cuthill McKee RCM algorithm in SPARSPAK in some cases reducing the envelope size by more than a fac tor of two The work involved in an envelope factorization scheme is reduced by the square of the savings in storage