Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems

Serra, Stefano

doi:10.1007/bf01934269

Cited by 85 publications

(28 citation statements)

References 15 publications

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“…The associated band-Toeplitz matrix A n (g) results to be the desired preconditioner in the sense that the spectrum of A −1 n (g)A n (f ) lies in (r, R) for any dimension n. The quoted idea resulted to be very flexible and, actually, has been successfully applied to the case of nondefinite Toeplitz problems [29], block Toeplitz problems [27] and, joint with circulant structures, to the case of non-Hermitian Toeplitz problems [8].…”

Section: Introductionmentioning

confidence: 99%

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Serra

1997

Math. Comp.

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Abstract. In this paper we are concerned with the solution of n × n Hermitian Toeplitz systems with nonnegative generating functions f . The preconditioned conjugate gradient (PCG) method with the well-known circulant preconditioners fails in the case where f has zeros. In this paper we consider as preconditioners band-Toeplitz matrices generated by trigonometric polynomials g of fixed degree l. We use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f , is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n. For each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy. We obtain different estimates of l for which the total cost of the proposed PCG methods is optimal and the related rates of convergence are superlinear. Finally, for the most economical strategy, we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems.

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Section: Introductionmentioning

confidence: 99%

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Serra

1997

Math. Comp.

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“…In the literature, there are some papers [25,28,29,34,35,36,38] discussing about iterative block-Toeplitz solvers. In [28,35,36], they considered n-by-n block Toeplitz matrices with m-by-m blocks generated by a Hermitian matrix-valued generating function and analyzed the associated problem of preconditioning using preconditioners generated a nonnegative definite, not essentially singular, matrix-valued functions.…”

mentioning

confidence: 99%

“…In [28,35,36], they considered n-by-n block Toeplitz matrices with m-by-m blocks generated by a Hermitian matrix-valued generating function and analyzed the associated problem of preconditioning using preconditioners generated a nonnegative definite, not essentially singular, matrix-valued functions. In [25,29,34], they considered block-Toeplitz-Toeplitz-block matrices and studied block band-Toeplitz preconditioners. In [38], multigrid methods were applied to solving block-Toeplitz-Toeplitz-block systems.…”

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confidence: 99%

Block Diagonal and Schur Complement Preconditioners for Block-Toeplitz Systems with Small Size Blocks

Ching¹,

Ng²,

Wen³

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and this fixed number depends only on the size of block. Hence conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners are considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method.

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“…, n, all the eigenvalues of A, counted with their multiplicities, numbered in non-decreasing way: in particular, λ 1 (A) = λ min (A) and λ n (A) = λ max (A). In [15] it was proved that, under the hypothesis that f (x, y) is continuous on the closed squareQ = [−π, π] 2 and not identically constant, for any n and m all the eigenvalues of A n,m lie in the open interval (min Q f, max Q f ), and it holds lim n,m→∞…”

Section: Introductionmentioning

confidence: 99%

“…the generating function depends on one variable only, and the associated matrices have scalar Toeplitz structure), while the study of the block case is a recent matter, motivated mainly by the need of finding good preconditioners for conjugate gradient and other iterative methods ( [15,3,4,5]). …”

Section: Introductionmentioning

confidence: 99%

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

Tilli¹

1997

Math. Comp.

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Abstract. We study the asymptotic behaviour of the eigenvalues of Hermitian n × n block Toeplitz matrices An,m, with m × m Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function f , and we study their eigenvalues for large n and m, relating their behaviour to some properties of f as a function; in particular we show that, for any fixed k, the first k eigenvalues of An,m tend to inf f , while the last k tend to sup f , so extending to the block case a well-known result due to Szegö. In the case the An,m's are positive-definite, we study the asymptotic spectrum of P −1 n,m An,m, where Pn,m is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system An,mx = b, obtaining strict estimates, when n and m are fixed, and exact limit values, when n and m tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

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Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems

Cited by 85 publications

References 15 publications

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Block Diagonal and Schur Complement Preconditioners for Block-Toeplitz Systems with Small Size Blocks

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

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