Best Conditioned Circulant Preconditioners
نویسنده
چکیده
In this paper, we discuss the solutions to a class of Hermitian positive deenite system Ax = b by the preconditioned conjugate gradient method with circulant preconditioner C. In general, the smaller the condition number (C ?1=2 AC ?1=2) is, the faster the convergence of the method will be. The circulant matrix C b that minimizes (C ?1=2 AC ?1=2) is called the best conditioned circulant preconditioner for the matrix A. We prove that if FAF has Property A where F is the Fourier matrix, then C b minimizes jjC ? Ajj F over all circulant matrices C. Here jj jj F denotes the Frobenius norm. We also show that there exists non-circulant Toeplitz matrix A such that FAF has Property A.
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