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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
Source PublicationSIAM Journal on Scientific Computing

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.

KeywordGohberg-semencul Formula Preconditioned Conjugate Gradient Method Preconditioners Recursive-based Method Toeplitz Matrices
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Indexed BySCIE
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000183166600003
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Cited Times [WOS]:11   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionFaculty of Science and Technology
Personal research not belonging to the institution
Affiliation1.Department of Mathematics The University of Hong Kong,Hong Kong
2.Department of Applied MathematicsGuandong University of Technology,China
3.Faculty of Science and Technology University of Macau,China
Recommended Citation
GB/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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