Best-conditioned circulant preconditioners Original Research Article

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 κ(C−1/2 AC−1/2) is, the faster the convergence of the method will be. The circulant matrix Cb that minimizes κ(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 Cb minimizes short parallelC − Ashort parallelF over all circulant matrices C. Here short parallel · short parallelF denotes the Frobenius norm. We also show that there exists a noncirculant Toeplitz matrix A such that F AF* has Property A.