## Superlinear PCG methods for symmetric Toeplitz systems

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## Abstract:

In this paper we deal with the solution, by means of preconditioned conjugate gradient (PCG) methods, of $n\times n$ symmetric Toeplitz systems $A_n(f) \mathbf { x}= \mathbf { b}$ with nonnegative generating function $f$. Here the function $f$ is assumed to be continuous and strictly positive, or is assumed to have isolated zeros of even order. In the first case we use as preconditioner the*natural*and the

*optimal*$\tau$ approximation of $A_n(f)$ proposed by Bini and Di Benedetto, and we prove that the related PCG method has a superlinear rate of convergence and a total arithmetic cost of $O(n\log n)$ ops. Under the second hypothesis we cannot guarantee that the

*natural*$\tau$ matrix is positive definite, while for the

*optimal*we show that, in the ill-conditioned case, this can be really a bad choice. Consequently, we define a new $\tau$ matrix for preconditioning the given system; then, by applying the ShermanâMorrisonâWoodbury inversion formula to the preconditioned system, we introduce a small, constant number of subsidiary systems which can be solved again by means of the previous PCG method. Finally, we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis.

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## Additional Information

**Stefano Serra**- Affiliation: Dipartimento di Informatica, Corso Italia 40, 56100 Pisa (ITALY)
- MR Author ID: 332436
- Email: serra@mail.dm.unipi.it
- Received by editor(s): February 7, 1996
- Received by editor(s) in revised form: October 15, 1996, and July 18, 1997
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 793-803 - MSC (1991): Primary 65F10, 65F15
- DOI: https://doi.org/10.1090/S0025-5718-99-01045-5
- MathSciNet review: 1620251