Trigonometric Preconditioners for Block Toeplitz Systems

Article (PDF Available) · April 1998with 48 Reads
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
DOI: 10.1007/978-3-0348-8871-4_18
Abstract
This paper is concerned with the solution of a system of linear equations T M, N X=b, where T M, N denotes a positive definite doubly symmetric block-Toeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strang-type trigonometric preconditioners P M, N of T M, N from the Fejér and Fourier sum of f, respectively. Using relations between trigonometric transforms and Toeplitz matrices, we prove that for all ε > 0 and sufficiently large M, N, at most O(M) + O(N) eigenvalues of lie outside the interval (1 — ε, l + ε) such that the preconditioned conjugate gradient method converges in at most O(M) + O(N) steps.
• ... Then, the matrices A N (f) have real entries such that we restrict our attention to trigonometric preconditioners. Table 5 Both matrices are ill{conditioned and the CG{method without preconditioning, with Strang{type{preconditioning or with optimal trigonometric preconditioning converges very slow (see 22,25]). Our preconditioning determined by (4.7) leads to the number of iterations in Table 5.5. ...
Article
Full-text available
This paper is concerned with the solution of systems of linear equations A Nx = b, where$$\{ A_N \} _{N \in \mathbb{N}}$$ denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f ∈ C 2π with zeros. We construct positive definite Hermitian preconditioners M N such that the eigenvalues of M N−1A N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.
• ... ρn is a block-Toeplitz-Toeplitz-block matrix. Note that we can avoid the complex arithmetic introduced by the FFT by using fast Toeplitz matrix times vector multiplications based on trigonometric transforms (see Algorithm 3 of [25]). ...
Article
Full-text available
We develop a new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires arithmetic operations. In case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity if either the points y j or the points x k are “reasonably uniformly distributed”. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions.
• ... As mentioned in the previous section, the optimal preconditioner in A F N coincides with our preconditioner (2.10) defined with respect to the Fejér kernel B 1,N and with w N = 0 in (2.7). It is easy to check (see [33]) that the optimal preconditioner in A O N , where O N ∈ {C IV N , S IV N } is equal to our preconditioner M N (f N , O N ) in (5.1) defined with respect to O N and with respect to the Fejér kernel. Unfortunately, the Fejér kernel preconditioners do not lead to a fast convergence of the PCG-method if the generating function f of A N has a zero of order 2s ≥ 2. In contrast to these results, the optimal preconditioners in A O N with O N defined by the DCT I -III or by the DST I -III do not coincide with the corresponding Fejér kernel preconditioner M N (f N , O N ) in (5.1). ...
Article
Full-text available
. In this paper, we are interested in the iterative solution of ill{conditioned Toeplitz systems generated by continuous non{negative real{valued functions f with a nite number of zeros. We construct new w{circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f KN with a suitable positive reproducing kernel KN . By the restriction to positive kernels we obtain positive denite preconditioners. Moreover, if f has only zeros of even order 2s, then we can prove that the property R t 2k KN (t) dt CN 2k (k = 0; : : : ; s) of the kernel is necessary and sucient to ensure the convergence of the PCG{method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were conrmed by numerical tests. 1991 Mathematics Subject Classication. 65F10, 65F15, 65T10. Key words and phrases. Ill{conditioned Toeplitz matrices , CG{method, preconditioners, reproducing kernels. 1 Intr...
• Chapter
In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.
• Article
Full-text available
We develop a new algorithm for the fast computation of discrete sums f(yj) :=∑k=1N αkK(yj - xk) (j = 1,...,M) based on the recently developed fast Fourier transform (FFT) at nonequispaced knots. Our algorithm, in particular our regularization procedure, is simply structured and can be easily adapted to different kernels K. Our method utilizes the widely known FFT and can consequently incorporate advanced FFT implementations. In summary, it requires O(N log N + M) arithmetic operations. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions.
• Article
The discrete Fourier-cosine transform (cos-DFT), the discrete Fourier-sine transform (sin-DFT) and the discrete cosine transform (DCT) are closely related to the discrete Fourier transform (DFT) of real-valued sequences. This paper describes a general method for constructing fast algorithms for the cos-DFT, the sin-DFT and the DCT, which is based on polynomial arithmetic with Chebyshev polynomials and on the Chinese Remainder Theorem.
• Article
The optimal circulant preconditioner for a given matrix A is defined to be the minimizer of ∥C − A∥F over the set of all circulant matrices C. Here ∥·∥F is the Frobenius norm. Optimal circulant preconditioners have been proved to be good preconditioners in solving Toeplitz systems with the preconditioned conjugate gradient method. In this paper, we construct an optimal sine transform based preconditioner which is defined to be the minimizer of ∥B − A∥F over the set of matrices B that can be diagonalized by sine transforms. We will prove that for general n-by-n matrices A, these optimal preconditioners can be constructed in O(n2) real operations and in O(n) real operations if A is Toeplitz. We will also show that the convergence properties of these optimal sine transform preconditioners are the same as that of the optimal circulant ones when they are employed to solve Toeplitz systems. Numerical examples are given to support our convergence analysis.
• Article
A classical scheme for multiplying polynomials is given by the Cauchy product formula. Faster methods for computing this product have been developed using circular convolution and fast Fourier transform algorithms. From the numerical point of view the Chebyshev expansion of polynomials is preferred to the monomial form. We develop a direct scheme for multiplication of polynomials in Chebyshev form as well as a fast algorithm using discrete cosine transforms. This approach leads to a new convolution operation and a new type of circulant matrices, both related to the discrete cosine transform. Extensions to bivariate polynomial products are also discussed.
• Article
Research on preconditioning Toeplitz matrices with circulant matrices has been active recently. The preconditioning technique can be easily generalized to block Toeplitz matrices. That is, for a block Toeplitz matrix T consisting of $N \times N$ blocks with $M \times M$ elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spectral clustering property of the preconditioned matrix $R^{ - 1} T$ with T generated by two-dimensional rational functions $T(z_x ,z_y )$ of order $(p_x ,q_x ,p_y ,q_y )$ is examined. It is shown that the eigenvalues of $R^{ - 1} T$ are clustered around unity except at most $O(M\gamma _y + N\gamma _x )$ outliers, where $\gamma _x = \max (p_x ,q_x )$ and $\gamma _y = \max (p_y ,q_y )$. Furthermore, if T is separable, the outliers are clustered together such that $R^{ - 1} T$ has at most $(2\gamma _x + 1)(2\gamma _y + 1)$ asymptotic distinct eigenvalues. The superior convergence behavior of the preconditioned c...
• Article
We present a new preconditioner for $n\times n$ symmetric, positive definite Toeplitz systems. This preconditioner is an element of the $n$--dimensional vector space of matrices that are diagonalized by the discrete sine transform. Conditions are given for which the preconditioner is positive definite and for which the preconditioned system has asymptotically clustered eigenvalues. The diagonal form of the preconditioner can be calculated in $O(n\log(n))$ operations if $n=2^k-1.$ Thus only $n$ additional parameters need be stored. Moreover, complex arithmetic is not needed. To use the preconditioner effectively, we develop a new technique for computing a fast convolution using the discrete sine transform (also requiring only real arithmetic). The results of numerical experimentation with this preconditioner are presented. Our preconditioner is comparable, and in some cases superior, to the standard circulant preconditioner of Tony Chan. Possible generalizations for other fast transforms are also indicated.
• Article
Given a Toeplitz matrix A, the authors derive an optimal circulant preconditioner C in the sense of minimizing norm of C - A/sub F/. It is in general different from the one proposed earlier except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of Câ»Â¹ A and comparably in terms of clustering the spectrum around unity.
• Article
The iterative solution of a block Toeplitz linear system by the conjugate gradient method is analyzed, the preconditioning step being solved by means of a discrete sine transform. Convergence properties are established and compared to the behavior of the block circulant preconditioner recently proposed in the literature. As in the scalar case, the new approach takes advantage if the system is ill conditioned.
• Article
In [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 948–966], Ku and Kuo proposed and analysed a block circulant preconditioner $R_{mn}$ for solving a family of block Toeplitz systems $T_{mn} v = b$. For a special class of block matrices called the quadrantally symmetric Toeplitz matrices, they proved that the eigenvalues of $R_{mn}^{ - 1} T_{mn}$ are clustered around one except at most $O(m + n)$ outliers with $T_{mn}$ generated by a two-dimensional rational function. The superior convergence rate of the preconditioned conjugate gradient (PCG) method is explained by the clustering property of the spectrum of $R_{mn}^{ - 1} T_{mn}$. However, in their analysis, there is no discussion on the positive definiteness of the matrix $T_{mn}$, and the preconditioner $R_{mn}$ is assumed to be invertible. In this paper, we give some results on these two aspects. Under the assumptions in the above-referenced paper, we prove that if the generating function $f(x,y)$ of $T_{mn}$ is positive, then $T_{mn}$ is posit...
• Article
In contrast to the usual (and successful) direct methods for Toeplitz systems Ax = b, we propose an algorithm based on the conjugate gradient method. The preconditioner is a circulant, so that all matrices have constant diagonals and all matrix-vector multiplications use the Fast Fourier Transform. We also suggest a technique for the eigenvalue problem, where current methods are less satisfactory. If the first indications are supported by further experiment, this new approach may have useful applications—including nearly Toeplitz systems, and parallel computations.