@article{e337c715c6aa48ada0bc5c49c6d0a28e, title = "Best-Conditioned Circulant Preconditioners", abstract = "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 ∥C − A∥F over all circulant matrices C. Here ∥·∥F denotes the Frobenius norm. We also show that there exists a noncirculant Toeplitz matrix A such that F AF∗ has Property A. ", author = "Chan, {Raymond H.} and Wong, {C. K.}", year = "1995", month = mar, day = "15", doi = "10.1016/0024-3795(93)00171-U", language = "English", volume = "218", pages = "205--211", journal = "Linear Algebra and Its Applications", issn = "0024-3795", publisher = "Elsevier Inc.", }