Fast Transform Based Preconditioners for Toeplitz Equations

Boman, Eugene; Koltracht, I.

doi:10.1137/s0895479893254269

Cited by 32 publications

(14 citation statements)

References 10 publications

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“…Thus the matrix-vector product T n v can be computed in O(n log n) real operations. The details can be found in [3,4]. Hence the cost of each Lanczos iteration is of O(n log n) operations.…”

Section: The Convergence Of Preconditioned Lanczos Methodsmentioning

confidence: 99%

“…In this paper, we employ the sine transform-based preconditioner to precondition the Toeplitz matrix in the Lanczos algorithm. The idea of using preconditioned conjugate gradient methods with sine transform-based preconditioners for solving Toeplitz systems has been studied recently by various researchers, see [2,3,5,11]. In these papers, the symmetric Toeplitz matrix T n is assumed to be generated by a 2π-periodic even function.…”

Section: End Formentioning

confidence: 99%

“…The authors of [2,3,5,11] proposed the use of matrices that can be diagonalized by the sine transform matrix Ψ n to precondition Toeplitz matrices in conjugate gradient iterations. The motivation is to exploit the fast inversion of these sine transform-based matrices.…”

Section: Sine Transform Approximationsmentioning

confidence: 99%

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Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

2000

SIAM J. Sci. Comput.

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In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive 2π-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.

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Section: The Convergence Of Preconditioned Lanczos Methodsmentioning

confidence: 99%

Section: End Formentioning

confidence: 99%

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Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

2000

SIAM J. Sci. Comput.

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“…This observation allows us to suggest a variety of new O(m log m) real-arithmetic algorithms for the multiplication of a Toeplitz matrix by a vector, using any of eight versions of discrete cosine or sine transforms. For the DST-I case such an algorithm was suggested earlier in [BK95].…”

Section: Fast Real-arithmetic Multiplication Of a Toeplitz Matrix By mentioning

confidence: 99%

Displacement Structure Approach to Discrete-Trigonometric-Transform Based Preconditioners of G.Strang Type and of T.Chan Type

Kailath

Olshevsky

2005

SIAM J. Matrix Anal. & Appl.

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In this paper we use a displacement structure approach to design a class of new preconditioners for the conjugate gradient method applied to the solution of large Toeplitz linear equations. Explicit formulas are suggested for the Strang-type and for the T.Chan-type preconditioners belonging to any of eight classes of matrices diagonalized by the corresponding discrete cosine or sine transforms. Under the standard Wiener class assumption the clustering property is established for all of these preconditioners, guaranteeing rapid convergence of the preconditioned conjugate gradient method. All the computations related to the new preconditioners can be done in real arithmetic and to fully exploit this advantageous property one has to suggest a fast real-arithmetic algorithm for multiplication of a Toeplitz matrix by a vector. It turns out that the obtained formulas for the Strang-type preconditioners allow a number of representations for Toeplitz matrices leading to a wide variety of real-arithmetic multiplication algorithms based on any of eight discrete cosine or sine transforms. Recently, transformations of Toeplitz matrices to Vandermonde-like or Cauchy-like matrices have been found to be useful in developing accurate direct methods for Toeplitz linear equations. In this paper we suggest to further extend the range of the transformation approach by exploring it for iterative methods; this technique allowed us to reduce the complexity of each iteration of the preconditioned conjugate gradient method. We conclude the paper with a suggestion on how to exploit the displacement structure to efficiently organize numerical experiments in a numerically reliable way.

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“…Circulant preconditioners have been intensively studied by a number of researchers, see for instance Reference [1] and the references therein. Besides circulant preconditioners, there are fast trigonometric transform based preconditioners (see References [4][5][6] for instance) and Hartley transform based preconditioners (see Reference [7] for instance). By using proper choice of the preconditioning matrix P n , the preconditioned matrix P −1 n T n has spectrum clustered around 1, it follows that the PCG method will converge very fast.…”

Section: Introductionmentioning

confidence: 99%

Inverse Toeplitz preconditioners for Hermitian Toeplitz systems

Lin

Ching

2004

Numerical Linear Algebra App

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SUMMARYIn this paper we consider solving Hermitian Toeplitz systems Tnx = b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2 -periodic function f, i.e. Tn = T n[f]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if f is positive then the spectrum of T n[1=f]T n[f] is clustered around 1. We prove that the trigonometric polynomial q N , where N ¡n. We also extend our method to construct e cient preconditioners for Tn when f has ÿnite zeros of even orders. We prove that with our preconditioners, the preconditioned matrix has spectrum clustered around 1. It follows that the PCG methods converge very fast when applied to solve the preconditioned systems. Numerical results are given to demonstrate the e ciency of our preconditioners.

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Fast Transform Based Preconditioners for Toeplitz Equations

Cited by 32 publications

References 10 publications

Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

Displacement Structure Approach to Discrete-Trigonometric-Transform Based Preconditioners of G.Strang Type and of T.Chan Type

Inverse Toeplitz preconditioners for Hermitian Toeplitz systems

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