The author investigates preconditioning methods for linear algebraic systems \(Ax=f\) with a dense positive definite matrix \(A\). He calls a conditioning matrix \(C\) optimal if it minimizes \(\| C-A\|\) and superoptimal if it minimizes \(\| I-C^{-1} A\|\), both in the Frobenius norm. The conditioners are taken from the class of circulant, doubly circulant or generally multilevel circulant matrices. The author shows that the construction of superoptimal preconditioners by a fast Fourier transform algorithms has the complexity \(O(n^ 2\log_ 2 n)\) in the general (positive definite) case and \(O(n\log_ 2 n)\) in the case of coefficient matrices of Toeplitz or double Toeplitz type.