Best-Conditioned Circulant Preconditioners
Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review
|Journal / Publication||Linear Algebra and Its Applications|
|Publication status||Published - 15 Mar 1995|
|Link to Scopus||https://www.scopus.com/record/display.uri?eid=2-s2.0-21844491101&origin=recordpage|
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.