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Showing papers on "Circulant matrix published in 1990"


Journal ArticleDOI
TL;DR: A new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented, which is quite competitive with Gaussian elimination and with the modified double sweep method.
Abstract: In this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented. This special kind of system appears in many applications. After an appropriate partition of the system and the elimination of the last unknown, we apply the Woodbury formula to arrive at a very efficient and stable method, which is quite competitive with Gaussian elimination and with the modified double sweep method.

43 citations



Journal ArticleDOI
TL;DR: It is shown how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular Toeplitz matrices, and as sumsof products of matrices derived from finite groups (group matrices) and matrices which are “close” to group matrices.

20 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the index of convergence of C equals the exponent of a related primitive circulant Boolean matrix, and that the maximum density of C is the same as that of the corresponding non-circulant matrix.

17 citations


Journal ArticleDOI
TL;DR: It is shown that arbitrarily accurate approximate solutions can be computed in $O(n\log n)$ arithmetic operations for large n, provided that $z(t)$ is sufficiently smooth.
Abstract: Let $\{ z_j \} _{j = 0}^{n - 1} $ be a set of distinct points in the complex plane $\mathbb{C}$, and introduce the $n \times n$ matrix $A_n = [a_{jk} ]_{j,k = 0}^{n - 1} $, $a_{jk} = (z_j - z_k )^{ - 1} ,\, j e k$ and $a_{jj} = 0$. Recently Golub and Trummer raised the question of whether or not, for an arbitrary vector $x \in \mathbb{C}^n $, $A{\bf x}$ can be computed in fewer than $O(n^2 )$ arithmetic operations by using the structure of $A_n $. In this paper it is assumed that there is a smooth $2\pi $-periodic bijective function $z(t)$ such that $z_j = z(2\pi (j - 1)/n),\, j = 1(1)n$, and shown that when n increases, there is a sequence of matrices of low rank $\tilde A_n ,\, n = 1,2,3, \cdots $ such that $\tilde A_n \to A_n $ as $n \to \infty $ and $\tilde A_n {\bf x}$ can be computed in $O(n\log n)$ arithmetic operations. The method to construct the matrices $\tilde A_n $ is then used in a fast solution scheme for Fredholm integral equations of the second kind with smooth periodic kernels. The integral equations are discretized by the trapezoidal rule using the nodes $z_j = z(2\pi j/n)$, $0 \leqq j < n$, and it is shown that arbitrarily accurate approximate solutions can be computed in $O(n\log n)$ arithmetic operations for large n, provided that $z(t)$ is sufficiently smooth. When the asymptotic analysis is not applicable, fast iterative $O(n^2 )$ solution methods are obtained. The scheme is applied to the solution of a Fredholm integral equation of the second kind of plane potential theory and Cauchy singular integral equations.

15 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that if m ≠ 2, then the generalised prism K2 mxCn is a circulant iff n is odd.
Abstract: Graph products of circulants are studied. It is shown that if G and H are circulants and gcd(v(G), v(H)) = 1, then every B-product of G and H is again a circulant. We prove that if m ≠ 2, then the generalised prism K2 mxCn is a circulant iff n is odd. A similar result is deduced for the conjunction. We also prove that Cp x Cq is a circulant iff p and q are relatively prime. We close by showing that the composition of two circulants is again a circulant and explicitly describe the resultant circulant's jump sequence in terms of the constituent circulants' jump sequences.

15 citations


Journal ArticleDOI
TL;DR: A set of alternative smoothing operators to the usual relaxation methods for multigrid algorithms is presented and analyzed in terms of frequency domain behavior and results prove effective in solving Frobenius matrix norm minimization problem.
Abstract: A set of alternative smoothing operators to the usual relaxation methods for multigrid algorithms is presented and analyzed in terms of frequency domain behavior. The operations presented are inherently parallel and fit well onto hypercube multiprocessors: they can be readily calculated and applied in a parallel manner. We start by interpreting multigrid smoothers as approximate inverses. In particular, a least squares approximate inverse obtained by solving a Frobenius matrix norm minimization problem proves effective. This approximate inverse also has a least squares interpretation in the frequency domain for the special case of circulant operators, or in the case of local mode Fourier analysis for the discrete operator in the central part of the domain over which the discretization is performed. Experimental results are presented for one and two dimensional problems. Convergence rates as determined by direct iteration are compared with local mode Fourier analysis results.

13 citations


Journal ArticleDOI
TL;DR: In this article, the convergence of C n as n tends to infinity in terms of the set U = { j : c j 0} for Markov chains having C as transition matrix.

3 citations



Book ChapterDOI
03 Jan 1990
TL;DR: Initial numerical results are depicted on the feasibility of circulant preconditioned iterative methods for the indefinite symmetric case of Toeplitz systems of linear equations.
Abstract: Stable fast direct methods for solving symmetric positive-definite Toeplitz systems of linear equations have been known for a number of years. Recently, a conjugate-gradient method with circulant preconditioning has been proposed as an effective means for solving these equations. For the (non-singular) indefinite case, the only stable algorithms that appear to be known are the general O( n 3 ) direct methods, such as LU decomposition, which do not exploit the Toeplitz structure. We depict here some initial numerical results on the feasibility of circulant preconditioned iterative methods for the indefinite symmetric case.

3 citations


Proceedings ArticleDOI
01 Jul 1990
TL;DR: In this paper, a method to analyze the linear imaging characteristics of rotationally invariant, radially invariant tomographic imaging systems using singular value decomposition (SVD) is presented.
Abstract: We describe a method to analyze the linear imaging characteristics of rotationally invariant, radially variant tomographic imaging systems using singular value decomposition (SVD). When the projection measurements from such a system are assumed to be samples from independent and identically distributed multi-normal random variables, the best estimate of the emission intensity is given by the unweighted least squares estimator. The noise amplification of this estimator is inversely proportional to the singular values of the normal matrix used to model projection and backprojection. After choosing an acceptable noise amplification, the new method can determine the number of parameters arid hence the number of pixels that should be estimated from data acquired from an existing system with a fixed number of angles and projection bins. Conversely, for the design of a new system, the number of angles and projection bins necessary for a given number of pixels and noise amplification can be determined. In general, computing the SVD of the projection normal matrix has cubic computational complexity. However, the projection normal matrix for this class of rotationally invariant, radially variant systems has a block circulant form. A fast parallel algorithm to compute the SVD of this block circulant matrix makes the singular value analysis practical by asymptotically reducing the computation complexity of the method by a multiplicative factor equal to the number of angles squared.

Journal ArticleDOI
TL;DR: The conditions under which a symmetric circulant matrix C, with entries from a finite field F, can be factored over F as C=AA′, where A is a circular matrix and A′ denotes the transpose of A.