Displacement structure approach to discrete-trigonometric-transform based preconditioners of G.Strang type and of T.Chan type

In this paper adisplacement structure technique is used to design a class of newpreconditioners for theconjugate gradient method applied to the solution of large Toeplitz linear equations. Explicit formulas are suggested for the G.Strang-type and for the T.Chan-type preconditioners belonging to any of 8 classes of matrices diagonalized by the correspondingdiscrete cosine or sine transforms. Under the standard Wiener class assumption theclustering property is established for all of these preconditioners, guaranteeing a rapid convergence of the preconditioned conjugate gradient method. The formulas for the G.Strang-type preconditioners have another important application: they suggest a wide variety of newO(m logm) algorithms for multiplication of a Toeplitz matrix by a vector, based on any of the 8 DCT’s and DST’s. Recentlytransformations of Toeplitz matrices to Vandermonde-like or Cauchy-like matrices have been found to be useful in developing accuratedirect methods for Toeplitz linear equations. Here it is suggested to further extend the range of the transformation approach by exploring it foriterative methods; this technique allowed us to reduce the complexity of each iteration of the preconditioned conjugate gradient method to 4 discrete transforms per iteration.