| Abstract: | In this paper, we study an operator s which maps every n-by-n symmetric matrix A_(n) to a matrix s(A_(n)) that minimizes || B_(n)-A_(n) || F over the set of all matrices B_(n) that can be diagonalized by the sine transform. The matrix s(A_(n)) , called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices An. The cost of constructing s(A_(n)) is the same as that of optimal circulant preconditioner c(A_(n)) which is defined in 8]. The s(A_(n)) has been proved in 6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper , we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given. |
