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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| Vydané v: | SIAM journal on matrix analysis and applications Ročník 27; číslo 3; s. 638 - 653 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2005
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| ISSN: | 0895-4798, 1095-7162 |
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| Abstract | 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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| AbstractList | 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 $O(n\log^3(n))$. |
| Author | Lu, Yi Hurvich, Clifford M. |
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| Cites_doi | 10.1137/0610039 10.1137/S0036144594276474 10.1109/78.134471 10.1137/0910009 10.1063/1.526133 10.2307/2373789 10.1111/1467-9892.00099 10.1137/1.9780898719253 10.1016/S0024-3795(98)00015-9 10.1007/978-1-4612-1426-7 10.1137/0609005 10.1007/BF02575833 10.1002/0471722235 10.1017/CBO9780511810817 10.1016/S0024-3795(97)00067-0 10.1016/S0024-3795(98)80014-1 10.1002/sapm1986742171 10.1007/s002110050413 10.1007/978-1-4419-0320-4 10.1111/1467-9892.00075 10.1137/0729066 10.1090/S0025-5718-1992-1106960-1 10.1137/0909051 10.1007/BF01389448 10.1016/S0024-3795(98)80002-5 10.1137/S0036142999354083 10.1214/aos/1018031103 |
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| Keywords | 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 |
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| References | Fisher M. E. (R20) 1968; 32 Koopmans L. (R27) 1974 R25 R24 Beran Jan (R4) 1994 R29 R28 Jolley L. (R26) 1961 R1 Golub Gene (R21) 1996 R2 R3 R5 R6 R7 R8 R9 R10 R31 R12 R34 R11 R33 R14 R36 R13 R35 R16 R38 R15 R37 R18 Grenander Ulf (R22) 1984 |
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| Snippet | 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... |
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| SubjectTerms | 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 |
| Title | On the Complexity of the Preconditioned Conjugate Gradient Algorithm for Solving Toeplitz Systems with a Fisher--Hartwig Singularity |
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