BEST-CONDITIONED CIRCULANT PRECONDITIONERS

We discuss the solutions to a class of Hermitian positive definite systems Ax = b by the preconditioned conjugate gradient method with circulant preconditioner C. In general, the smaller the condition number kappa(C-1/2 AC-1/2) is, the faster the convergence of the method will be. The circulant matrix C(b) that minimizes kappa(C-1/2 AC-1/2) is called the best-conditioned circulant preconditioner for the matrix A. We prove that if F AF* has Property A, where F is the Fourier matrix, then C(b) minimizes parallel-to C - A parallel-to F over all circulant matrices C. Here parallel-to . parallel-to F denotes the Frobenius norm. We also show that there exists a noncirculant Toeplitz matrix A such that F AF* has Property A.