Computation of minimum eigenvalue through minimization of rayleigh's quotient for large sparse matrices using vector computer:

Abstract

Rayleigh Quotient Minimization methods for the calculation of minimal eigenpair achieve great efficiency when ad hoc preconditioners are employed. The performance of different preconditioners on a vector computer is compared and analyzed from a computational standpoint. The numerical experiments are carried out on large symmetric sparse positive definite matrices arising from finite element discretizations of practical problems. Diagonal, polynomial and Kershaw preconditioners are considered for the generalized and the classical eigenvalue problems. Speed-up factors and MFLOPS are calculated. The results show that a good vectorization level of the computational code is achieved. The speed-up factors obtained with the best schemes are generally high and uniformly distributed. Numerical experiments suggest that the highest efficiency for the solution of the classical problem is achieved in vector computers by diagonal preconditioning. This conclusion differs from the results that can be obtained in scalar computers. All the computations were performed on a CRAY X-MP/48 supercomputer.

Keywords:

• Eigenvalue
• eigenvector
• sparse matrices
• Rayleigh quotient
• vector computers