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 Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions Ng,Michael K.1; Sun,Hai Wei2; Jin,Xiao Qing3 2003-10-20 Source Publication SIAM Journal on Scientific Computing ISSN 10648275 Volume 24Issue:5Pages:1507-1529 Abstract In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems Ax = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of A (the principal submatrix of A) with the Gohberg-Semencul formula as a preconditioner for A. The inverse of A can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of A and does not require the explicit knowledge of the generating function f of A. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach. Keyword Gohberg-semencul Formula Preconditioned Conjugate Gradient Method Preconditioners Recursive-based Method Toeplitz Matrices DOI 10.1137/S1064827500378155 URL View the original Indexed By SCI Language 英语 WOS Research Area Mathematics WOS Subject Mathematics, Applied WOS ID WOS:000183166600003 Fulltext Access Citation statistics Cited Times [WOS]:10   [WOS Record]     [Related Records in WOS] Document Type Journal article Collection Faculty of Science and TechnologyPersonal research not belonging to the institution Affiliation 1.Department of Mathematics The University of Hong Kong,Hong Kong2.Department of Applied MathematicsGuandong University of Technology,China3.Faculty of Science and Technology University of Macau,China Recommended CitationGB/T 7714 Ng,Michael K.,Sun,Hai Wei,Jin,Xiao Qing. Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions[J]. SIAM Journal on Scientific Computing,2003,24(5):1507-1529. APA Ng,Michael K.,Sun,Hai Wei,&Jin,Xiao Qing.(2003).Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions.SIAM Journal on Scientific Computing,24(5),1507-1529. MLA Ng,Michael K.,et al."Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions".SIAM Journal on Scientific Computing 24.5(2003):1507-1529.
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