On the Complexity of the Preconditioned Conjugate Gradient Algorithm for Solving Toeplitz Systems with a Fisher--Hartwig Singularity

The Toeplitz matrix $T_n$ with generating function $f ( \omega ) = |1 - e ^{-i \omega}|^{-2d} h( \omega )$, where $d \in (-\frac{1}{2}, \frac{1}{2})\setminus \{0\}$ and $h(\omega)$ is positive, continuous on $[-\pi,\pi]$, and differentiable on $[-\pi,\pi]\setminus\{0\}$, has a Fisher--Hartwig singul...

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Veröffentlicht in:	SIAM journal on matrix analysis and applications Jg. 27; H. 3; S. 638 - 653
Hauptverfasser:	Lu, Yi, Hurvich, Clifford M.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Schlagworte:	Algebra Algorithms Eigenvalues Exact sciences and technology Fourier transforms Inference from stochastic processes; time series analysis Linear and multilinear algebra, matrix theory Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Probability and statistics Sciences and techniques of general use Statistics Time series Conjugate gradient method 62M10 Generating function 62M15 Toeplitz matrix Matrix function Periodogram Fisher-Hartwig singularity spectral density; 15A12 preconditioned conjugate gradient algorithm 65F10 expected periodogram Algorithm complexity Circulant matrix Fisher Hartwig singularity 65F15 Spectral density Preconditioning conjugate gradient algorithm 62M20
ISSN:	0895-4798, 1095-7162
Online-Zugang:	Volltext
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Zusammenfassung:	The Toeplitz matrix $T_n$ with generating function $f ( \omega ) = \|1 - e ^{-i \omega}\|^{-2d} h( \omega )$, where $d \in (-\frac{1}{2}, \frac{1}{2})\setminus \{0\}$ and $h(\omega)$ is positive, continuous on $[-\pi,\pi]$, and differentiable on $[-\pi,\pi]\setminus\{0\}$, has a Fisher--Hartwig singularity [M. E. Fisher and R. E. Hartwig (1968), Adv. Chem. Phys., 32, pp. 190--225]. The complexity of the preconditioned conjugate gradient (PCG) algorithm is known [R. H. Chan and M. Ng (1996), SIAM Rev., 38, pp. 427--482] to be $O(n\log n)$ for Toeplitz systems when $d = 0$. However, the effect on the PCG algorithm of the Fisher--Hartwig singularity in $T_n$ has not been explored in the literature. We show that the complexity of the conjugate gradient (CG) algorithm for solving $T_n x=b$ without any preconditioning grows asymptotically as $n^{1+\|d\|}\log (n)$. With T. Chan's optimal circulant preconditioner $C_n$ [T. Chan (1988), SIAM J. Sci. Statist. Comput., 9, pp. 766--771], the complexity of the PCG algorithm is $O(n\log^3(n))$.
Bibliographie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14
ISSN:	0895-4798 1095-7162
DOI:	10.1137/040612117