Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblem

Recently an efficient method (DACG) for the partial solution of the symmetric generalized eigenproblem Ax = lambda Bx has been developed, based on the conjugate gradient (CG) minimization of the Rayleigh quotient over successive deflated subspaces of decreasing size. The present paper provides a numerical analysis of the asymptotic convergence rate rho(j) of DACG in the calculation of the eigenpair lambda(j), u(j), when the scheme is preconditioned with A(-1). It is shown that, when the search direction are A-conjugate, rho(j) is well approximated by 4/xi(j), where xi(j) is the Hessian condition number of a Rayleigh quotient defined in appropriate oblique complements of the space spanned by the leftmost eigenvectors u(1), u(2),..., u(j-1) already calculated. It is also shown that 1/xi(j) is equal to the relative separation between the eigenvalue lambda(j) currently sought and the next higher one lambda(j+1). A modification of DACG (MDACG) is studied, which involves a new set of CG search directions which are made M-conjugate, with M a matrix approximating the Hessian. By distinction, MDACG has an asymptotic rate of convergence which appears to be inversely proportional to the square root of xi(j), in complete agreement with the theoretical results known for the CG solution to linear systems. (C) 1997 by John Wiley & Sons, Ltd.