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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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 27; no. 3; pp. 638 - 653
Main Authors: Lu, Yi, Hurvich, Clifford M.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Subjects:
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 Access:Get full text
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Summary: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 $O(n\log^3(n))$.
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ISSN:0895-4798
1095-7162
DOI:10.1137/040612117