The Pad e - Rayleigh - Ritz Method for SolvingLarge

We make use of the Pad e approximants and Krylov's sequence x; Ax; ; : : :; A m?1 x in the projection methods to compute a few Ritz values of a hermitian matrix A of order n. This process consists of approaching the poles of R x () = ((I ?A) ?1 x; x), the mean value of the resolvant of A, by those of m ? 1=m] Rx (), where m ? 1=m] Rx () is the Pad e approximant of order m of the function R x (). This is equivalent to approach some eigenvalues of A by the roots of the polynomial of degree m of the denominator of m?1=m] Rx (). This projection method, called the Pad e-Rayleigh-Ritz (PRR) method, provides a simple way to determine the minimumpolynomial of x in the Krylov's method for the symmetrical case. The numerical stability of the PRR method can be ensured if there is not \considerable" variation in the matrix elements of A and if the projection subspace m is \suuciently" 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 comparison between them with regard to stability and parallelism. We then try to justify the use of this method under some assumptions.