Numerical Linear Algebra with Applications, Vol. 0(0), 0–0 (2000) Approximate Inverse Preconditioning in the Parallel Solution of Sparse Eigenproblems Luca Bergamaschi, Giorgio Pini Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Universit` a di Padova, via Belzoni 7, 35131 Padova, Italy Flavio Sartoretto Dipartimento di Informatica Universit` a di Venezia, Via Torino 155, 30171 Mestre VE, Italy Abstract A preconditioned scheme for solving sparse symmetric eigenproblems is proposed. The solution strategy relies upon the DACG algorithm, which is a Preconditioned Conjugate Gra- dient algorithm for minimizing the Rayleigh Quotient. A comparison with the well established ARPACK code, shows that when a small number of the leftmost eigenpairs is to be computed, DACG is more efficient than ARPACK. Effective convergence acceleration of DACG is shown to be performed by a suitable approximate inverse preconditioner (AINV). The performance of such a preconditioner is shown to be safe, i.e. not highly dependent on a drop tolerance parameter. On sequential machines, AINV preconditioning proves a practicable alternative to the effective incomplete Cholesky factorization, and is more efficient than Block Jacobi. Due to its parallelizability, the AINV preconditioner is exploited for a parallel implementation of the DACG algorithm. Numerical tests account for the high degree of parallelization attainable on a Cray T3E machine and confirm the satisfactory scalability properties of the algorithm. A final comparison with PARPACK shows the (relative) higher efficiency of AINV-DACG. KEY WORDS generalized eigenproblem, sparse approximate inverse, parallel algorithm 1. Introduction The computation of a small number of the leftmost eigenpairs (the smallest eigenval- ues and corresponding eigenvectors) of the problem where and are large, sparse, symmetric positive definite matrices, is an important task in many scientific applications. Typical examples are the vibrational analysis of mechanical structures [1], 1070–5325/00/000000–01$5.50 Received c 2000 by John Wiley & Sons, Ltd. Revised