TY - JOUR
T1 - Best-Conditioned Circulant Preconditioners
AU - Chan, Raymond H.
AU - Wong, C. K.
PY - 1995/3/15
Y1 - 1995/3/15
N2 - 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.
AB - 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.
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U2 - 10.1016/0024-3795(93)00171-U
DO - 10.1016/0024-3795(93)00171-U
M3 - 21_Publication in refereed journal
VL - 218
SP - 205
EP - 211
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -