ASYMPTOTIC CONVERGENCE OF CONJUGATE-GRADIENT METHODS FOR THE PARTIAL SYMMETRICAL EIGENPROBLEM

Recently an efficient method (DACG) for the partial solution of the sy mmetric generalized eigenproblem Ax = lambda Bx has been developed, ba sed on the conjugate gradient (CG) minimization of the Rayleigh quotie nt 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), w hen the scheme is preconditioned with A(-1). It is shown that, when th e search direction are A-conjugate, rho(j) is well approximated by 4/x i(j), where xi(j) is the Hessian condition number of a Rayleigh quotie nt defined in appropriate oblique complements of the space spanned by the leftmost eigenvectors u(1), u(2),..., u(j-1) already calculated. I t is also shown that 1/xi(j) is equal to the relative separation betwe en the eigenvalue lambda(j) currently sought and the next higher one l ambda(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 asympt otic 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.