The Padé-Rayleigh-Ritz method for solving large hermitian eigenproblems

Abstract

We make use of the Pade approximants and the Krylov sequencex, Ax,,...,A m−1 x in the projection methods to compute a few Ritz values of a large hermitian matrixA of ordern. This process consists in approaching the poles ofR x(λ)=((I−λA)−1 x,x), the mean value of the resolvant ofA, by those of [m−1/m]Rx(λ), where [m−1/m]Rx(λ) is the Pade approximant of orderm of the functionR x(λ). This is equivalent to approaching some eigenvalues ofA by the roots of the polynomial of degreem of the denominator of [m−1/m]Rx(λ). This projection method, called the Pade-Rayleigh-Ritz (PRR) method, provides a simple way to determine the minimum polynomial ofx in the Krylov subspace methods for the symmetrical case. The numerical stability of the PRR method can be ensured if the projection subspacem is “sufficiently” small. The mainly expensive portion of this method is its projection phase, which is composed of the matrix-vector multiplications and, consequently, is well suited for parallel computing. This is also true when the matrices are sparse, as recently demonstrated, especially on massively parallel machines. This paper points out a relationship between the PRR and Lanczos methods and presents a theoretical comparison between them with regard to stability and parallelism. We then try to justify the use of this method under some assumptions.