T. F. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comp., 9(1988), 766-771. Home/Search Document Not in Database Summary Related Articles Check |
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Circulant Preconditioners for Toeplitz Matrices with Positive.. - Chan, Yeung (1991) (4 citations) Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
Optimal Trigonometric Preconditioners for Elliptic and.. - Chan, Ching, Wong (1994) (1 citation) Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems , SIAM J. Sci. Statist. Comput., V9 (1988), 766--771.
Fast Iterative Solvers for Toeplitz-plus-Band - Systems Raymond Chan Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766-771.
Circulant Preconditioned Toeplitz Least - Squares Iterations Raymond Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771. 20
Multigrid Method for Ill-Conditioned Symmetric - Toeplitz Systems Raymond Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., 9 (1988), 766--771.
Toeplitz Preconditioners for Toeplitz Systems with Nonnegative.. - Chan (1990) (3 citations) Self-citation (Chan) (Correct)
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Chan, T., An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comp., V9 (1988), pp. 766-771.
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comp., 9 (1988), pp. 766-771.
Circulant Preconditioners for Elliptic Problems - Raymond Chan Univeristy (1992) (7 citations) Self-citation (Chan) (Correct)
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T.F. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comp., Vol. 9, pp. 766-771, Jul. 1988.
Circulant Preconditioners For Toeplitz Matrices With Piecewise.. - Yeung, Chan (1992) (4 citations) Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comp., 9 (1988), pp. 766--771.
Scientific Applications of Iterative Toeplitz Solvers - Ng, Chan Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comput. 9, (1988) 766--771.
Constrained Minimax Approximation and Optimal Preconditioners.. - Chan, Tang Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comp., V9 (1988), pp. 766--771.
Spectral Equivalence and Proper Clusters for Matrices from.. - Tyrtyshnikov, Chan Self-citation (Chan) (Correct)
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T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput. 9: 766--771 (1988).
Circulant Integral Operators as Preconditioners for.. - Chan, Jin, Ng (1995) Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
FFT-Based Preconditioners for Toeplitz-Block Least Squares.. - Chan, Nagy, Plemmons (1992) (9 citations) Self-citation (Chan) (Correct)
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T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
Toeplitz-circulant Preconditioners for Toeplitz Systems and.. - Chan, Ching (1994) (1 citation) Self-citation (Chan) (Correct)
....of these direct methods for symmetric positive definite matrices are discussed in Bunch [5] Here we will consider solving Toeplitz systems by the preconditioned conjugate gradient squared (PCGS) method. There are many circulant preconditioning strategies for Toeplitz systems, see for instance [23, 11, 16, 15, 26]. The convergence results for these circulant preconditioners are all based on the regularity of the function g( whose Fourier coefficients give the diagonals of T n . The function g( with 2 [ Gamma; is called the generating function of the sequence of Toeplitz matrices T n . A general ....
....i.e. if T n is constant along its diagonals. It is said to be circulant if its diagonals t k further satisfies t n Gammak = t Gammak for 0 k n Gamma 1. The idea of using circulant matrices as preconditioners for Toeplitz matrices has been studied extensively in recent years, see for instance [23, 11, 16, 26, 15]. In this paper, we will concentrate ourselves in the T. Chan circulant preconditioners. The results for the other circulant preconditioners can be obtained similarly, see x5. For a given Toeplitz matrix T n with diagonals ft j g j= Gamma(n Gamma1) the T. Chan circulant preconditioner to T n ....
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., 9 (1988), 767--771.
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....systems with generating functions that are positive functions in the Wiener class, the method has a super linear convergence rate due to the clustering of the eigenvalues of the preconditioned matrices. Several circulant preconditioners have been proposed since then, see for example T. Chan [4] and Tyrtyshinkov [7] For any n by n matrix A n , the circulant preconditioner proposed in T. Chan [4] called the optimal circulant preconditioner, is defined to be the minimizer of kC n Gamma A n k F over the space of all n by n circulant matrices C n . Here k Delta k F denotes the Frobenius ....
....a super linear convergence rate due to the clustering of the eigenvalues of the preconditioned matrices. Several circulant preconditioners have been proposed since then, see for example T. Chan [4] and Tyrtyshinkov [7] For any n by n matrix A n , the circulant preconditioner proposed in T. Chan [4], called the optimal circulant preconditioner, is defined to be the minimizer of kC n Gamma A n k F over the space of all n by n circulant matrices C n . Here k Delta k F denotes the Frobenius norm. The circulant preconditioner given in Tyrtyshinkov [7] is defined to be the minimizer of kI ....
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 766--771.
Unknown - Self-citation (Chan) (Correct)
....function f , the Fourier coefficients of which give the entries of A n , is a positive function in the Wiener class, then for n sufficiently large, S n and S n are uniformly bounded in the 2 norm and the eigenvalues of the preconditioned matrix S are clustered around one. T. Chan in [5] proposed another circulant matrix C n that is obtained by averaging the corresponding diagonals of A n with the diagonals of A n being extended to length n by a wrap around. He proved that such C n minimizes kC n Gamma A n k F over all circulant matrices, where k Delta k F means the Frobenius ....
....that the coefficients of Q in (1) can also be written as a pq = tr (Q Gammaj A) 3) where tr ( Delta) means the trace. From (3) it can be checked easily that if A n = a i Gammaj ) is a Toeplitz matrix in M n Thetan , then c(A n ) is the circulant preconditioner given in T. Chan [5]. More precisely, the entries c ij = c i Gammaj of c(A n ) are given by c k = fka k Gamman (n Gamma k)a k g ; k = 0; Delta Delta Delta n Gamma 1: 4) We will call c the circulant operator. In the following, we will investigate some of its properties that will be needed later on. Theorem ....
T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 766--771.
The Best Circulant Preconditioners for Hermitian Toeplitz Systems - Chan, Yip, Ng (1999) (1 citation) Self-citation (Chan) (Correct)
....matrices as preconditioners for solving Toeplitz systems. The number of operations per iteration is of order O(n log n) as circulant systems can be solved efficiently by fast Fourier transforms. Several successful circulant preconditioners have been introduced and analyzed; see for instance [11, 5]. In these papers, the given Toeplitz matrix A n is assumed to be generated by a generating function f , i.e. the diagonals a j of A n are given by the Fourier coefficients of f . It was shown that if f is a positive function in the Wiener class (i.e. the Fourier coefficients of f are absolutely ....
....for the Toeplitz matrices generated by a function with a zero of order 2p, their condition numbers grow like O(n ) see [19] Hence the number of iterations required for convergence will increase like O(n ) see [2, p.24] Tyrtyshnikov [23] has proved that the Strang [21] and the T. Chan [11] preconditioners both fail in this case. To tackle this problem, non circulant type preconditioners have been proposed, see [6, 4, 18, 16] The basic idea behind these preconditioners is to find a function g that matches the zeros of f . Then the preconditioners are constructed based on the ....
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 766--771.
Copyright information to be inserted by the Publishers.. - Hung-Cheong Chan Raymond Self-citation (Chan) (Correct)
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
Toeplitz Preconditioners For Hermitian Toeplitz - Systems Raymond Chan Self-citation (Chan) (Correct)
....O(n log n) The iterative approach is to use preconditioned conjugate gradient method with circulant matrices as preconditioners for the solution of Toeplitz systems, see Strang [21] Several successful circulant preconditioners have been proposed and analyzed, see for instance Chan [4] T. Chan [8], Huckle [15] Ku and Kuo [18] Tismenetsky [22] and Tyrtyshnikov [24] In these papers, the Toeplitz matrix A n is assumed to be generated by a generating function f , i.e. the diagonals of A n are given by the Fourier coefficients of f . It has been shown that if f is a positive function in the ....
....s = 1, our Toeplitz preconditioners T n reduce to the well known circulant preconditioners mentioned above, depending on the kernel function we used. As an example, if the kernel function is the Fej er function, then T n is just the inverse of the T. Chan circulant preconditioner proposed in [8]. For integers s 1, we will show that the Toeplitz preconditioner T n thus constructed can be written as a sum of so called f g circulant matrices, see Davis [11, p.84] or x4 for definition) More precisely, we have V t where V t are f t g circulant matrices with t = e . As a ....
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
Sine Transform Based Preconditioners for Symmetric Toeplitz.. - Chan, Ng, Wong (1994) (9 citations) Self-citation (Chan) (Correct)
....C that one can define to be preconditioners for the system Tx = b. Since the convergence rate of the PCG method depends on how good the preconditioner C approximates T , much attention has been focused on searching a circulant matrix C which is close to the matrix T in certain norms, see T. Chan [6], Tyrtyshnikov [22] and Huckle [18] T. Chan in [6] proposed a circulant preconditioner c(T ) which is the minimizer of jjC Gamma T jj F over all circulant matrices C. Here jj Delta jj F denotes the Frobenius norm. He called c(T ) the optimal circulant preconditioner and showed that the first ....
....system Tx = b. Since the convergence rate of the PCG method depends on how good the preconditioner C approximates T , much attention has been focused on searching a circulant matrix C which is close to the matrix T in certain norms, see T. Chan [6] Tyrtyshnikov [22] and Huckle [18] T. Chan in [6] proposed a circulant preconditioner c(T ) which is the minimizer of jjC Gamma T jj F over all circulant matrices C. Here jj Delta jj F denotes the Frobenius norm. He called c(T ) the optimal circulant preconditioner and showed that the first column entries c j of c(T ) are given by c j = jt ....
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T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., V9 (1988), pp. 766--771.
The Spectrum of a Family of Circulant Preconditioned Toeplitz.. - Chan (1988) (3 citations) Self-citation (Chan) (Correct)
....Kong Hong Kong July 1987 Revised February 88 Abstract. We study the solutions of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm jjC Gamma Ajj F , see T. Chan [5]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of C A. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, we establish the invertiblity of C, find the asymptotic behaviour of ....
....a positive function in the Wiener class, then for n sufficiently large, S n and S n are uniformly bounded in the l 2 norm and the eigenvalues of the preconditioned matrix S n A n are clustered around 1. We remark that the assumptions on f also imply that A n are positive definite. T. Chan [5] recently proposed another circulant matrix C n that is obtained by averaging the corresponding diagonals of A n with the diagonals of A n being extended to length n by a wrap around. More precisely, the entries c ij = c ji Gammajj of C n are given by c k = ka n Gammak (n Gamma k)a k ; 0 k ....
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Chan, T., An Optimal Circulant Preconditioner for Toeplitz Systems, UCLA Math. Dept., CAM Report 87-06, June 1987.
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T. F. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comp., 9(1988), 766-771.
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T. F. Chan, An optimal circulant preconditioner for Toeplitz systems,SIAMJ.Sci. Statist. Comput., 9 (1988), pp. 766--771.
A Nearly Optimal Preconditioner for the Navier-Stokes.. - Hemmingsson-Fränden.. (Correct)
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T. F. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 766-771.
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