Showing papers on "Circulant matrix published in 2000"
TL;DR: This paper develops a method for determining the exact number of spanning trees in (directed or undirected) circulant graphs that can exhibit a recurrence relation for the number of its spanning trees.
79 citations
TL;DR: An interconnection structure for multicomputer networks, called recursive circulant, is proposed, and a shortest-path routing algorithm in G(cdm,d) is developed, and various network metrics such as connectivity, diameter, mean internode distance, and visit ratio are analyzed.
68 citations
TL;DR: The P-circulants, a recently introduced circulant approximation for unsymmetric Toeplitz matrices, are shown to be more suitable sometimes than othercirculant matrices for the underlying block preconditioner.
Abstract: In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods, a class of implicit methods for the numerical integration of ODEs based on linear multistep formulas, are considered more in detail for initial value problems.
Theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems. Moreover, the P-circulants, a recently introduced circulant approximation for unsymmetric Toeplitz matrices, are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner.
53 citations
TL;DR: A computationally efficient algorithm for the eigenspace decomposition of correlated images that uses the theory of circulant matrices to automatically determine the dimension of the subspace required to represent an image with a guaranteed user-specified accuracy.
Abstract: We present a computationally efficient algorithm for the eigenspace decomposition of correlated images. Our approach is motivated by the fact that for a planar rotation of a two-dimensional (2-D) image, analytical expressions can be given for the eigendecomposition, based on the theory of circulant matrices. These analytical expressions turn out to be good first approximations of the eigendecomposition, even for three-dimensional (3-D) objects rotated about a single axis. In addition, the theory of circulant matrices yields good approximations to the eigendecomposition for images that result when objects are translated and scaled. We use these observations to automatically determine the dimension of the subspace required to represent an image with a guaranteed user-specified accuracy, as well as to quickly compute a basis for the subspace. Examples show that the algorithm performs very well on a number of test cases ranging from images of 3-D objects rotated about a single axis to arbitrary video sequences.
42 citations
40 citations
TL;DR: In this paper, the salient results in the algebra of circulant matrices which can be used in time-series analysis are presented and a statistical test for the stationarity or homogeneity of empirical processes is also presented.
Abstract: This paper sets forth some of the salient results in the algebra of circulant matrices which can be used in time-series analysis. It provides easy derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions. A statistical test for the stationarity or homogeneity of empirical processes is also presented.
38 citations
TL;DR: The purpose of this paper is to describe in detail the statistical properties of this multivariate model and the eigenstructure of the covariance matrix and the model is applied to some datasets to explore shape variability.
Abstract: SUMMARY Grenander & Miller (1994) describe a model for representing amorphous twodimensional objects with no obvious landmark. Each object is represented by n vertices around its perimeter, and is described by deforming an n-sided regular polygon using edge transformations. A multivariate normal distribution with a block circulant covariance matrix is used to model these edge transformations. The purpose of this paper is to describe in detail the statistical properties of this multivariate model and the eigenstructure of the covariance matrix. Various special cases of the model are considered, including articulated models and conditional Markov random field models. We consider maximum likelihood based inference and the model is applied to some datasets to explore shape variability.
34 citations
TL;DR: A new family of circulant preconditioners for ill-conditioned Hermitian Toeplitz systems A x= b is proposed, constructed by convolving the generating function f of A with the generalized Jackson kernels and it is shown that the convergence is superlinear.
Abstract: In this paper, we propose a new family of circulant preconditioners for ill-conditioned Hermitian Toeplitz systems A x= b. The preconditioners are constructed by convolving the generating function f of A with the generalized Jackson kernels. For an n-by-n Toeplitz matrix A, the construction of the preconditioners requires only the entries of A and does not require the explicit knowledge of f. When f is a nonnegative continuous function with a zero of order 2p, the condition number of A is known to grow as O(n2p). We show, however, that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p+1 outliers. Moreover, the smallest eigenvalue is uniformly bounded away from zero. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges linearly. The total complexity of solving the system is therefore of O(n log n) operations. In the case when f is positive, we show that the convergence is superlinear. Numerical results are included to illustrate the effectiveness of our new circulant preconditioners.
32 citations
TL;DR: It is proved that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast, and this is important for finding the steady state distribution of SANs.
Abstract: Stochastic Automata Networks (SANs) are widely used in modeling communication systems, manufacturing systems and computer systems. The SAN approach gives a more compact and efficient representation of the network when compared to the stochastic Petri nets approach. To find the steady state distribution of SANs, it requires solutions of linear systems involving the generator matrices of the SANs. Very often, direct methods such as the LU decomposition are inefficient because of the huge size of the generator matrices. An efficient algorithm should make use of the structure of the matrices. Iterative methods such as the conjugate gradient methods are possible choices. However, their convergence rates are slow in general and preconditioning is required. We note that the MILU and MINV based preconditioners are not appropriate because of their expensive construction cost. In this paper, we consider preconditioners obtained by circulant approximations of SANs. They have low construction cost and can be inverted efficiently. We prove that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast. Numerical results for three different SANs solved by CGS are given to illustrate the fast convergence of our method.
32 citations
TL;DR: An algorithm that can find all the eigenvalues of an n×n complex Hankel matrix in O (n 2 log n) operations is presented.
32 citations
TL;DR: In this paper, it was shown that for a field of characteristic ≠ 2, each square matrix with entries in K can be written as a product of circulant and diagonal matrices with entries.
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01 Feb 2000
TL;DR: It is shown that in this subclass, isomorphism is equivalent to Adam-isomorphism, and various results are obtained for the chromatic number, line-transitivity and the diameter.
Abstract: A subclass of the class of circulant graphs is considered. It is shown that in this subclass, isomorphism is equivalent to Adam-isomorphism. Various results are obtained for the chromatic number, line-transitivity and the diameter.
11 Jun 2000
TL;DR: A nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes is proposed and Numerical results are given to illustrate the effectiveness of this method.
Abstract: We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.
TL;DR: In this paper, the problem of finding a basis of orthogonal vectors of length p m in the Leech lattice for any even number m = 2p and p is a prime not equal to 11 was studied.
Abstract: We consider the problem of finding, for each even number m, a basis of orthogonal vectors of length p m in the Leech lattice. We give such a construction by means of double circulant codes whenever m = 2p and p is a prime not equal to 11. From this one can derive a construction for all even m not of the form 2 · 11 r .
TL;DR: In additive Gaussian noise, the Toeplitz covariance structure provides higher performance than the AR structure and has performance similar to that reported in the literature of a cepstral system on the same database.
TL;DR: It is shown that the circulant graph G(cdm,d) is Hamilton decomposable for all positive integers c,d, and m with c.
TL;DR: It is shown that a fully adaptive detector of coherent pulse trains embedded in compound-Gaussian clutter, whose covariance matrix is a circulant one, exhibits an acceptable loss with respect to previously proposed nonadaptive structures, even for small sizes of the estimation sample.
Abstract: We present a fully adaptive detector of coherent pulse trains embedded in compound-Gaussian clutter, whose covariance matrix is a circulant one. Remarkably, the proposed receiver ensures CFAR-ness with respect to the non-Gaussian noise distribution as well as to its temporal correlation. It also exhibits an acceptable loss with respect to previously proposed nonadaptive structures, even for small sizes of the estimation sample.
TL;DR: In this paper, a set of circulant graphs with p k vertices (k⩾1, p an odd prime) is decomposed into a collection of well-specified subsets and the counting of non-isomorphic circulants in each of the subsets is presented as an orbit enumeration problem of Polya type with respect to a certain Abelian group of multipliers.
01 Mar 2000
TL;DR: This paper provides the explicit formula for the Lovasz number of the union of two cycles, in two special cases, and a practically efficient algorithm, for the general case.
Abstract: The theta function of a graph, also known as the Lovasz number, has the remarkable property of being computable in polynomial time, despite being "sandwiched" between two hard to compute integers, i.e., clique and chromatic number. Very little is known about the explicit value of the theta function for special classes of graphs. In this paper we provide the explicit formula for the Lovasz number of the union of two cycles, in two special cases, and a practically efficient algorithm, for the general case.
TL;DR: The problem of computing the dichromatic number of a Zykov-sum of digraphs over a digraph D is reduced to that of computing a multicovering number of an hypergraph H1(D) associated to D in a natural way to construct an infinite family of pairwise non isomorphic vertex-critical k-dichromatic circulant tournaments.
Abstract: The dichromatic number dc(D) of a digraph D is the smallest number of colours needed to colour the vertices of D so that no monochromatic directed cycle is created. In this paper the problem of computing the dichromatic number of a Zykov-sum of digraphs over a digraph D is reduced to that of computing a multicovering number of an hypergraph H1(D) associated to D in a natural way. This result allows us to construct an infinite family of pairwise non isomorphic vertex-critical k-dichromatic circulant tournaments for every k ≥ 3, k 6= 7.
TL;DR: A method will be presented which will allow for problems of the above nature to be split into two separate systems which can be solved in parallel, and then combined and corrected to obtain a solution to the original system.
Abstract: In 1994, Yan and Chung produced a fast algorithm for solving a diagonally dominant symmetric Toeplitz tridiagonal system of linear equations Ax = b. In this work a method will be presented which will allow for problems of the above nature to be split into two separate systems which can be solved in parallel, and then combined and corrected to obtain a solution to the original system. An error analysis will be provided along with example cases and time comparison results.
TL;DR: A discrete image of several colors is viewed as a discrete random field obtained by clipping or quantizing a Gaussian random field at several levels by exploiting the block Toeplitz covariance matrix embedded into a block circulant matrix which is diagonalized by the fast Fourier transform.
Abstract: A discrete image of several colors is viewed as a discrete random field obtained by clipping or quantizing a Gaussian random field at several levels. Given a discrete image, parameters of the unobserved original Gaussian random field are estimated. Discrete images, statistically similar to the original image, are then obtained by generating different realizations of the Gaussian field and clipping them. To overcome the computational difficulties, the block Toeplitz covariance matrix of the Gaussian field is embedded into a block circulant matrix which is diagonalized by the fast Fourier transform. The Gibbs sampler is used to apply the stochastic EM algorithm for the estimation of the field's parameters.
Book•
01 Jan 2000
TL;DR: The Markov Cluster Process (MCL) as mentioned in this paper is a discrete stochastic uncoupling process for finite spaces, which is introduced in the context of Markov clusters.
Abstract:
ewcommand{\dpsd}{$\mathit{dpsd}$} A discrete stochastic uncoupling process for finite spaces is introduced, called the \emph{Markov Cluster Process} (MCL~process). The process takes a stochastic matrix as input, and then alternates flow expansion and flow inflation, each step defining a stochastic matrix in terms of the previous one. Flow expansion corresponds with taking the~$k^{th}$~power of a stochastic matrix, where~$k\in\N$. Flow inflation corresponds with a parametrized operator~$\Gamma_r$, $r\geq 0$, which maps the set of (column) stochastic matrices onto itself. The image~$\Gamma_r M$ is obtained by raising each entry in~$M$ to the~$r^{th}$~power and rescaling each column to have sum~$1$ again. In practice the process converges very fast towards a limit which is idempotent under both matrix multiplication and inflation, with quadratic convergence around the limit points. The limit is in general extremely sparse and the number of components of its associated graph may be larger than the number associated with the input matrix. This uncoupling is a desired effect as it reveals structure in the input matrix. The inflation operator~$\Gamma_r$ is shown to map the class of matrices which are diagonally similar to a symmetric matrix onto itself. The term \emph{diagonally positive semi-definite} (\dpsd) is used for matrices which are diagonally similar to a positive semi-definite matrix. It is shown that for $r\in\N$ and for~$M$ a stochastic \dpsd\ matrix, the image~$\Gamma_r M$ is again \dpsd. Determinantal inequalities satisfied by a \dpsd\ matrix~$M$ imply a natural ordering among the diagonal elements of~$M$, generalizing a mapping of nonnegative column allowable idempotent matrices onto overlapping clusterings. The spectrum of~$\Gamma_{\infty} M$, for \dpsd\ $M$, is of the form~$\{0^{n-k}, 1^k\}$, where~$k$ is the number of endclasses of the ordering associated with~$M$, and~$n$ is the dimension of~$M$. Reductions of \dpsd\ matrices are given, a connection with Hilbert''s distance and the contraction ratio defined for nonnegative matrices is discussed, and several conjectures are made.
05 Jun 2000
TL;DR: This paper introduces a generalized cyclic convolution which can be implemented via the conventional cyclic Convolution system by the discrete Fourier transform (DFT) with pre-multiplication for the input and post- multiplier for the output.
Abstract: This paper introduces a generalized cyclic convolution which can be implemented via the conventional cyclic convolution system by the discrete Fourier transform (DFT) with pre-multiplication for the input and post-multiplication for the output. The generalized cyclic convolution is applied for computing a negacyclic convolution. Comparison shows that the proposed implementation is more efficient and simpler in structure than other methods. The generalized cyclic convolution is also applied for the linear convolution by the modified Fermat number transform.
TL;DR: A class cation of 2-arc-transitive circulant digraphs is given and a description of the determinants of digraphy and their properties is given.
TL;DR: It is concluded that the preconditioned minimal residual method requires only ${cal O} (N \log ^2 N)$ arithmetical operations to achieve a solution of prescribed precision if the spectral condition numbers of the Toeplitz systems increase at most polynomial in N.
Abstract: This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes.
If the generating function is given, we prove that for any $\varepsilon >0$, only ${\cal O} (\log N)$ eigenvalues of our preconditioned Toeplitz systems of size N × N are not contained in $[-1-\varepsilon, -1+\varepsilon] \cup [1 -\varepsilon, 1+\varepsilon]$. The result can be modified for trigonometric preconditioners. We also suggest circulant preconditioners for the case that the generating function is not explicitly known and show that only ${\cal O} (\log N)$ absolute values of the eigenvalues of the preconditioned Toeplitz systems are not contained in a positive interval on the real axis.
Using the above results, we conclude that the preconditioned minimal residual method requires only ${\cal O} (N \log ^2 N)$ arithmetical operations to achieve a solution of prescribed precision if the spectral condition numbers of the Toeplitz systems increase at most polynomial in N. We present various numerical tests.
Journal Article•
TL;DR: This work gives new sets of sequences with entries from {0, ±a, ±b, ±c} on the commuting variables a, b, c and zero autocorrelation function to construct some new orthogonal de-signs.
Abstract: We give new sets of sequences with entries from {0, ±a, ±b, ±c} on the commuting variables a, b, c and zero autocorrelation function. Then we use these sequences to construct some new orthogonal de-signs. We show the necessary conditions for the existence of an OD(28; s1, s2, s3) plus the condition that (s1, s2, s3) ≠ (1,5,20) are sufficient conditions for the existence of an OD(28; s1, s2, s3). We also show the necessary conditions for the existence of an OD(28; s1, s2, s3) constructed using four circulant matrices are sufficient conditions for the existence of 4 — NPAF(s1, s2, s3) sequences of of length n for all lengths n ≥ 7. We establish asymptotic existence results for OD(4N; s1, s2) for 2 ≤ s1 + s2 ≤ 28. We show the necessary conditions for the existence of an OD(28; s1, s2) with 25 ≤ s1 + s2 ≤ 28, constructed using four circulant matrices, plus the condition that (s1, s2) ≠ (1,26), (2, 25), (7, 19), (8, 19) or (13, 14), are sufficient conditions for the existence of 4 — NPAF(s1, s2) sequences of of length n for all lengths n ≥ 7.
14 May 2000
TL;DR: Using the Hamiltonian circuit Latin square (HCLS), an O(m2) parallel routing algorithm is presented for constructing a set of m shortest and node-disjoint paths on recursive circulant network.
Abstract: In this paper, we investigate the routing of a message in recursive circulant, that is a key to the performance of this network. On recursive circulant network, we would like to transmit m packets from a source node to a destination node simultaneously along m paths, where the ith packet will traverse along the ith path (0/spl les/i/spl les/m-1). In order for all packets to arrive at the destination node quickly and securely, these m paths must be node-disjoint and the sum of lengths of paths be the smallest. Employing the Hamiltonian circuit Latin square (HCLS), we present O(m2) parallel routing algorithm for constructing a set of m shortest and node-disjoint paths.