On preconditioning and penalized matrices

An alternative approach to the preconditioning of a system of linear equati ons with a matrix A + rho (CC)-C-T that is the sum of a positive definite m atrix A and a penalization term is proposed. After showing that there is a gap in the spectrum of A + rho (CC)-C-T provided rho is sufficiently large, a preconditioner for A is applied to A + rho (CC)-C-T in such a way that i t preserves the gap in the spectrum but still improves the convergence of t he conjugate gradient method. A bound on the rate of convergence of the con jugate gradient method with our preconditioning based on the estimates by A xelsson is given that depends neither on rho nor on the rank of C. Numerica l experiments confirm the efficiency of the approach presented. Copyright ( C) 1999 John Wiley & Sons, Ltd.