Showing papers on "Circulant matrix published in 1989"

Toeplitz equations by conjugate gradients with circulant preconditioner

Abstract: This paper studies the solution of symmetric positive definite Toeplitz systems $Ax = b$ by the preconditioned conjugate gradient method. The preconditioner is a circulant matrix C that copies the middle diagonals of A, and each iteration uses the Fast Fourier Transform. Convergence is governed by the eigenvalues of $C^{ - 1} A$–a Toeplitz-circulant eigenvalue problem—and it is fast if those eigenvalues are clustered. The limiting behavior of the eigenvalues is found as the dimension increases, and it is proved that they cluster around $\lambda = 1$. For a wide class of problems the error after q conjugate gradient steps decreases as $r^{q^2 } $.

Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation

Abstract: The class of nonnegative definite Toeplitz matrices that can be embedded in nonnegative definite circulant matrices of a larger size is characterized. An equivalent characterization in terms of the spectrum of the underlying process is also presented, together with the corresponding extremal processes. It is shown that a given finite-duration sequence rho can be extended to be the covariance of a periodic stationary processes whenever the Toeplitz matrix R generated by this sequence is strictly positive definite. The sequence rho =1, cos alpha , cos 2 alpha with ( alpha / pi ) irrational, which has a unique nonperiodic extension as a covariance sequence, demonstrates that the strictness is needed. A simple constructive proof supplies a bound on the abovementioned period in terms of the minimal eigenvalue of R. It also yields, under the same conditions, an extension of rho to covariances that eventually decay to zero. For the maximum-likelihood estimate of the covariance of a stationary Gaussian process, the extension length required for using the estimate-maximize iterative algorithm is determined. >

The spectrum of a family of circulant preconditioned Toeplitz systems

Abstract: The solutions of symmetric positive definite Toeplitz systems $Ax = b$ are studied by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm $\| {C - A} \|_F $ [T. Chan, “An Optimal Circulant Preconditioner for Toeplitz Systems,” UCLA Department of Mathematics, CAM Report 87-06, June 1987]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of $C^{ - 1} A$. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, this paper establishes the invertibility of C, finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix $C^{ - 1} A$ as the dimension increases and proves that they are clustered around 1.

An automorphic approach to the design of fault-tolerant multiprocessors

Abstract: A general and systematic approach to the design of fault-tolerant multiprocessors modeled by graphs is developed. The approach is based on graph automorphisms and is applicable to any graph structure and any degree of fault tolerance. In addition, it incorporates other useful design criteria such as incremental design, low redundancy, and efficient reconfigurability. The authors apply their approach directly to a class of regular multiprocessor graphs termed 'circulant'. For noncirculant graphs, they give an algorithm to construct their circulant edge supergraphs efficiently. They show that the automorphic design method is amenable to efficient implementation using switched redundant links. An application of the foregoing theory to the design of a fault-tolerant hypercube multiprocessor is described. >

Deblurring random time-varying blur.

Abstract: The problem of restoring a constant image distorted by a system of random time-varying point-spread functions is studied. The restoration is based on a finite number of images that are observed in a finite period of time. Two features distinguish this problem. The first is that of the signal–noise dependency, and the second is the availability of large amounts of data. The Wiener criterion approach is used to solve the signal–noise-dependency problem. The problem of data size is also alleviated. For the case of time–space separability, a Karhunen–Loeve transformation is used to reduce the computations to the size of a single-frame problem. For the case in which the noise is stationary in time and in space, a solution based on the direct form of the Wiener filter is presented. The amount of computations here is reduced considerably by the use of fast Fourier transforms and circulant matrix approximations whenever they are valid.

The radon transform on Z n

Abstract: The Radon transform on $\mathbb{Z}_n $ that arises in the analysis of directional data and circular time series replaces each value $f(k)$ of a function f by the average value of f over the translate of a set S by k. For general S the discrete Fourier transform is used to characterize the null space and range of the transform and to calculate a (generalized) inverse transform. Explicit forms of the coefficients in the inversion formula are obtained in the two cases $S = \{ - r, + r\} $ and the symmetric moving average $S = \{ - r, \ldots , + r\} $. We show that the proportion of all choices S of size t giving invertible transforms is nearly unity when min $(t,n - t)$ is large.

A new interpretation of the compass gradient edge operators

Abstract: Two-dimensional 3×3 compass gradient operators are commonly used in the edge detection and usually detect eight compass directional components. In this paper, we present a new interpretation of the relationship between the resulting eight gradient components and the eight intensity values of neighboring pixels which are covered by the 2-dimensional 3×3 mask. We show that 8-directional edge values can be expressed by using a circulant matrix. And a circulant matrix can be diagonalized by using the Fourier transform matrix. By using the above two relations, we present the new interpretation of compass gradient operators such as Sobel, Prewitt, and Kirsch operators.

Macrotasking the singluar value decomposition of block circulant matrices on the Cray-2

Abstract: A parallel algorithm to compute the singular value decomposition (SVD) of block circulant matrices on the Cray-2 is described. For a block circulant form described by M blocks with m x n elements in each block, the computation time using an SVD algorithm for general matrices has a lower bound O(M3min(m, n)mn). Using a combination of fast Fourier transform (FFT) and SVD steps, the computation time for block circulant singular value decomposition (BCSVD) has a lower bound O(Mmin(m, n)mn); a relative savings of ~ M2. Memory usage bounds are reduced from T(M2mn) to T(Mmn); a relative savings of ~ M. For M = m = n = 64, this decreases the computation time from approximately 12 hours to 30 seconds and memory usage is reduced from 768 megabytes to 12 megabytes. The BCSVD algorithm partitions well into n macrotasks with a granularity of T(mM log M) for the FFT portion of the algorithm. The SVD portion of the algorithm partitions into M macrotasks with a granularity of O(min(m, n)mn). Again, for the case where M = m = n = 64, the FFT granularity is 29ms and the SVD granularity is 428ms. A speedup of 3.06 was achieved by using a prescheduled partitioning of tasks. The process creation overhead was 2.63ms. Using a more elaborate self-scheduling method with four synchronizing server processes, a speedup of 3.25 was observed with four processors available. The server synchronization overhead was 0.32ms. Relative memory overhead in both cases was about 4% for data space and 40% for code space.

Fast algorithms to QR factor circulant matrices

Abstract: The authors introduce a fast algorithm for computing the QR factors of a complex vertically circulant matrix C. The complexity of the algorithm is 5mn+2n/sup 2/+O(m) complex-fixed point operations or 3mn+n/sup 2/+O(m) complex floating-point operations, where m is the number of rows in C and n is the number of columns (it is supposed that m >

Computing eigenvalues of banded symmetric Toeplitz matrices

Abstract: A recently proposed algorithm for the computation of the eigenvalues of symmetric banded Toeplitz matrices is investigated.The basic idea of the algorithm is to embed the Toeplitz matrix in a symmetric circulant matrix of higher order. After having computed the spectral decomposition of the circulant matrix—which is trivial since the eigenvectors of circulants are known a priori and the eigenvalues can be obtained by Fourier transform—the Toeplitz eigenvalue problem is treated as a restricted eigenvalue problem. In doing this, use can be made of the theory for intermediate problems of Weinstein and Aronszajn and its recent refinements.Since the main part of the proposed algorithm consists of independent searches for zeros in disjoint real intervals, the algorithm is well suited for parallel computers. Implementations of the algorithm are discussed and some numerical results are given. The (sequential) complexity of the computation of s eigenvalues of the Toeplitz matrix T is $O(sr(n + r^2 ))$, where n is the order and rthe bandwidth of T.

Absolute bounds on optimal cost for a class of set covering problems

Abstract: We study the class of (m constraint,n variable) set covering problems which have no more thank variables represented in each constraint. Letd denote the maximum column sum in the constraint matrix, letr=[m/d]−1, and letZ g denote the cost of a greedy heuristic solution. Then we prove $$\begin{gathered} Zg \leqslant 1 + r + m - d - \left[ {mk \cdot MAX\left\{ {\frac{{2r}}{{2n - r - 1}},\ln \frac{n}{{n - r}}} \right\}} \right. \hfill \\ \left. { - kd \cdot MIN\left\{ {\frac{{r(r + 1)}}{{2(n - r)}},n \cdot \ln \frac{{n - 1}}{{n - r - 1}} - 1} \right\}} \right]. \hfill \\ \end{gathered} $$ This provides the firsta priori nontrivial upper bound discovered on heuristic solution cost (and thus on optimal solution cost) for the set covering problem. An example demonstrates that this bound is attainable, both for a greedy heuristic solution and for the optimal solution. Numerical examples show that this bound is substantially better than existing bounds for many problem instances. An important subclass of these problems occurs when the constraint matrix is a circulant, in which casem=n andk=d=[αη] for some 0<α<1. For this subclass we prove $$\mathop {\lim }\limits_{n \to \infty } Zg/n \leqslant \frac{{\alpha ^2 }}{2}[1/\alpha ][1/\alpha ].$$

Hartley transform as a similarity transform-2D case

Abstract: Consideration is given to one-dimensional and two-dimensional discrete Hartley transforms as similarity transforms. The structure of the matrix is presented which could be diagonalized by the above transform. In particular it is demonstrated that the Hartley transform can diagonalize structured matrices richer than circulant matrices. >

Cyclic inequalities and discrete fourier analysis

Abstract: Several cyclic inequalities on differences of finite sequences will be proved in the context of the circulant matrix theory and the operator norms. Beyond the classical lp -norms (1p∞), their mean oscillations are discussed.

Agarwal-Cooley Convolution Algorithm

Abstract: The cyclic convolution algorithms of chapter 6 are efficient for special small block lengths, but as the size of the block length increases, other methods are required First as discussed in chapter 6, these algorithms keep the number of required multiplications small, but can require many additions Also, each size requires a different algorithm There is no uniform structure that can be repeatedly called upon In this chapter, a technique similar to the Good-Thomas PFA will be developed to decompose a large size cyclic convolution into several small size cyclic convolutions which in turn can be evaluated using the Winograd cyclic convolution algorithm These ideas were introduced by Agarwal and Cooley [1] in 1977 As in the Good-Thomas PFA, the CRT is used to define an indexing of data This indexing changes a one-dimensional cyclic convolution into a two-dimensional cyclic convolution We will see how to compute a two-dimensional cyclic convolution by ‘nesting’ a fast algorithm for one-dimensional cyclic convolution inside another fast algorithm for one-dimensional cyclic convolution There are several two-dimensional cyclic convolution algorithms which although important will not be discussed These can be found in [2]

The discrete fourier transform and cyclic convolution on integral lattices

Abstract: A discrete Fourier transform and a cyclic convolution are constructed on an arbitrary integral lattice. The construction includes as a special case the usual discrete Fourier transform and the usual cyclic convolution. Applications to questions of interpolation of functions and digital signal processing are considered. Methods in the spectral theory of automorphic functions are used to investigate questions in approximation of arbitrary lattices by integral lattices.Bibliography: 14 titles.

Linear and Cyclic Convolution

Abstract: Linear convolution is one of the most frequent computations carried out in digital signal processing (DSP). The standard method for computing a linear convolution is to use the convolution theorem which replaces the computation by FFT of correspondingsize. In the last ten years, theoretically better convolution algorithms have been developed. The Winograd Small Convolution algorithm [1] is the most efficient as measured by the number of multiplications.