Sine transform based preconditioners for symmetric Toeplitz systems

Chan, Raymond H.; Ng, Michael K.; Wong, Charles

doi:10.1016/0024-3795(94)00049-2

Cited by 42 publications

(32 citation statements)

References 19 publications

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“…We remark that in those two test problems, both A x and B x in (15) are symmetric tridiagonal Toeplitz matrices, which can be diagonalized by sine transform (see, e.g., [17]). Thus, for solving (17), we can use fast sine transform (FST) to diagonalize A x and B x of the matrix M in (18) first.…”

Section: Numerical Resultsmentioning

confidence: 99%

A high‐order compact boundary value method for solving one‐dimensional heat equations

Sun

Zhang

2003

Numerical Methods Partial

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We combine fourth-order boundary value methods (BVMs) for discretizing the temporal variable with fourth-order compact difference scheme for discretizing the spatial variable to solve one-dimensional heat equations. This class of new compact difference schemes achieve fourth-order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second-order Crank-Nicolson scheme.

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Section: Numerical Resultsmentioning

confidence: 99%

A high‐order compact boundary value method for solving one‐dimensional heat equations

Sun

Zhang

2003

Numerical Methods Partial

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“…Then it is easy to prove D −1 Ψ n S n e 1 = ∆ n 1 n ; see [3]. Hence, S n is determined by its first column and ∆ n can be constructed in O(n log n) flops.…”

Section: Lemma 32 (Chan Ng and Wong [3]) Let A N = [A Jk ] Be An mentioning

confidence: 97%

“…From Lemma 6 in Chan [3] we know that for all > 0 there exist two positive integers N 1 , N 2 such that for ∀n > N 1 and k ≥ 0, at most N 2 eigenvalues of [A n − ρ (k) n B n − S (k) n ] have absolute values large than . In addition, from Lemma 5.1, S (k)−1 n 2 is uniformly bounded, which leads to the result that the spectra of S…”

Section: Convergence Analysismentioning

confidence: 98%

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Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

Wang

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tWe employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix A n − ρB n involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem A n x = λB n x, where A n and B n are given matrices of suitable sizes. The sine transform-based preconditioner can improve the spectral distribution of the shifted Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method. The sine transformbased preconditioner can be implemented efficiently by the fast transform algorithm. A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for a large matrix.

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“…This computation can be performed in O(n log n) operations using the DST. The following algorithm makes this computation [36].…”

Section: Triangular Decomposition Of Symmetric Cauchy-like Matricesmentioning

confidence: 99%

A multilevel parallel algorithm to solve symmetric Toeplitz linear systems

2007

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This paper presents a parallel algorithm to solve a structured linear system with a symmetric-Toeplitz matrix. Our main result concerns the use of a combination of shared and distributed memory programming tools to obtain a multilevel algorithm that exploits the actual different hierarchical levels of memory and computational units present in parallel architectures. This gives, as a result, a so-called parallel hybrid algorithm that is able to exploit each of these different configurations. Our approach has been done not only by means of combining standard implementation tools like OpenMP and MPI, but performing the appropriate mathematical derivation that allows this derivation. The experimental results over different representations of available parallel hardware configurations show the usefulness of our proposal.

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Sine transform based preconditioners for symmetric Toeplitz systems

Cited by 42 publications

References 19 publications

A high‐order compact boundary value method for solving one‐dimensional heat equations

A high‐order compact boundary value method for solving one‐dimensional heat equations

Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

A multilevel parallel algorithm to solve symmetric Toeplitz linear systems

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