A note on best conditioned preconditioners

Abstract

We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition numberκ(M −1/2 AM −1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM Q = {Q*Λ N Q | Λ n is ann-by-n complex diagonal matrix} andM + ={Q*Λ n Q | Λ n is ann-by-n positive definite diagonal matrix}. The preconditionerM b that minimizesκ(M −1/2 AM −1/2 ) overM + is called the best conditioned preconditioner for the matrixA overM + . We prove that ifQAQ* has Young's Property A, thenM b is nothing new but the minimizer of ‖M −A‖ F overM Q . Here ‖ · ‖ F denotes the Frobenius norm. Some applications are also given here.