We propose a new type of preconditioners for symmetric Toeplitz system Tx = b. When applying iterative methods to solve linear system with matrix T, we often use some preconditioner C by the preconditioned conjugate gradient (PCG) method[3]. If T is a symmetric positive definite Toeplitz matrix, two kinds of preconditioners are investigated: the "optimal" one , which minimizes ∥C - T∥F, and the "superoptimal" one , which minimize ∥I - C-1T∥F[8]. In this paper, we present a general approach to the design of Toeplitz preconditioners based on the optimal investigating and also preconditioners C with preserving the characteristic of the given matrix T. Fast all resulting preconditioners can be inverted via fast transform algorithms with O(NlogN) operations. For a wide class of problems, PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(NlogN)[2]. © Springer-Verlag 2004.
CITATION STYLE
Baik, R., & Baik, S. W. (2004). A Design and Analysis of Circulant Preconditioners. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 3314, 245–251. https://doi.org/10.1007/978-3-540-30497-5_39
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