Sine transform based preconditioners for symmetric Toeplitz systems
The authors construct an optimal sine transform based preconditioner which is defined to be the minimizer of \(|B- A|_F\) over the set of matrices \(B\) that can be diagonalized by sine transforms. They prove that for general \(n\)-by-\(n\) matrices \(A\), these optimal preconditioners can be constructed in \(O(n^2)\) real operations and in \(O(n)\) real operations if \(A\) is Toeplitz. They also show that the convergence properties of these optimal sine transform preconditioners are the same as that of the optimal circulant ones when they are employed to solve Toeplitz systems. Numerical examples are given to illustrate the convergence analysis.
scientific article; zbMATH DE number 841763
scientific article