2126 ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2008, Vol. 48, No. 12, pp. 2126–2139. © Pleiades Publishing, Ltd., 2008. Original Russian Text © M. Van Barel, Kh.D. Ikramov, A.A. Chesnokov, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 12, pp. 2092–2106. 1. INTRODUCTION Consider the system of linear equations Tx = b (1) with a nonsingular symmetric Toeplitz coefficient matrix T of order n × n and a vector b ∈  n (or  n ) as the right-hand side. In this paper, we present a new iterative algorithm for solving systems of this type. The com- plexity of this algorithm is O( ), where κ(T) is the condition number of T , and its memory requirements are O(n). A matrix T ∈  n × n is said to be Toeplitz if T = (t ij ) = (a j – i ); that is, (2) Toeplitz matrices arise in various problems of applied mathematics such as the numerical solution of differ- ential equations (see [1, 2]), Padé approximation (see [3–5]), and polynomial zero localization (see [6–8]). The fact that a Toeplitz matrix is determined by only 2n – 1 parameters underlies the development of numerous fast and superfast algorithms for solving systems of type (1). Algorithms of this type use the struc- ture of the matrix. Two classes of fast direct methods having a complexity of O(n 2 ) arithmetic operations are the methods of the Levinson and Schur type, respectively. The references and other information con- cerning these methods can be found in [9]. In general, the above-mentioned methods require that the leading principal submatrices of T be nonsin- gular. Furthermore, an implementation of these methods in floating-point arithmetic may be numerically unstable if those submatrices, albeit nonsingular, are ill-conditioned. κ T ( ) n n log log T a 0 a 1 a 2 … a n 1 – a 1 – a 0 a 1 a 2 – a 1 – a 0 a 2 a 1 a n 1 – ( ) – … a 2 – a 1 – a 0 ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ . = … … … … … … … A Continuation Method for Solving Symmetric Toeplitz Systems M. Van Barel a , Kh. D. Ikramov b , and A. A. Chesnokov a, b a Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200 A, B-3001 Leuven, Belgium e-mail: marc.vanbarel@cs.kuleuven.be b Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992 Russia e-mail: ikramov@cs.msu.su, ache@mccme.ru Received December 29, 2007; in final form, May 22, 2008 Abstract—A fast algorithm is proposed for solving symmetric Toeplitz systems. This algorithm contin- uously transforms the identity matrix into the inverse of a given Toeplitz matrix T . The memory require- ments for the algorithm are O(n), and its complexity is O( ), where κ(T) is the condition number of T . Numerical results are presented that confirm the efficiency of the proposed algorithm. DOI: 10.1134/S0965542508120026 Keywords: Toeplitz matrices, circulants, superfast algorithm, continuation method, iterative refinement, eigenvalues. κ T ( ) n n log log