# Performance of PeIGS

The numerical method used by PeIGS is bisection for eigenvalues and repeated inverse iteration and orthogonalization for eigenvectors. Accuracy and orthogonality are similar to LAPACK's DSPGV and DSPEV. Orthogonality of eigenvectors is enforced even for arbitrarily large clusters that span multiple processors.

PeIGS is both fast and scalable -- on a single processor of a KSR-2 it takes only 45% longer than LAPACK for a 1000-by-1000 generalized eigenproblem, and parallel efficiency is good even for large processor counts. For example, in one of our computational chemistry applications, the standard eigenproblem Ax = kx was solved for all eigenpairs of a 2053-by-2053 matrix. This computation required only 117 seconds on 150 processors of an Intel Paragon computer, a time-to-solution estimated as 77 times faster than the 1 processor time.

PeIGS performance depends to some extent on how eigenvalues cluster within the spectrum. Generally speaking, systems with many eigenvalues close together take longer to solve due to increased computation and communication costs for reorthogonalization. This is an important issue because actual applications with large matrices often have highly clustered spectra.