Showing papers on "Circulant matrix published in 1996"

Integral formulation of the measured equation of invariance: a novel sparse matrix boundary element method

Abstract: A novel integral formulation of the measured equation of invariance method is derived from the reciprocity theorem and implemented for perfectly conducting (PEC) 2-D scattering problems. This formulation uses the electric and magnetic Green's functions of the environment to obtain a matrix equation for the induced surface current with the same number of unknowns as the conventional boundary element-method of moments (BE-MoM) approach. However, the matrix that must be inverted in the new formulation is sparse and circulant, with only three non-zero elements per row. Sample results for two-dimensional TM and TE problems with perfectly conducting scatterers show enormous CPU time and memory savings over the conventional BEM-MoM approach. The new formulation has important advantages over the original finite difference formulation of MEI, but also shares some of its limitations.

Which circulant preconditioner is better

Abstract: The eigenvalue clustering of matrices S n −1 A n and C n −1 A n is experimentally studied, where A n , S n and C n respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function f(x). Some illustrations are given to show how the clustering depends on the smoothness of f(x) and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product CAC, where A is a Toeplitz matrix and C is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of CAC is not greater than 4, provided that C and A are symmetric.

Toeplitz-circulant Preconditioners for Toeplitz Systems and Their Applications to Queueing Networks with Batch Arrivals

Abstract: The preconditioned conjugate gradient method is employed to solve Toeplitz systems $T_n {\bf x} ={\bf b}$ where the generating functions of the $n$-by-$n$ Toeplitz matrices $T_n$ are functions with zeros. In this case, circulant preconditioners are known to give poor convergence, whereas band-Toeplitz preconditioners offer only linear convergence and can handle only real-valued functions with zeros of even orders. We propose here preconditioners which are products of band-Toeplitz matrices and circulant matrices. The band-Toeplitz matrices are used to cope with the zeros of the given generating function and the circulant matrices are used to speed up the convergence rate of the algorithm. Our preconditioner can handle complex-valued functions with zeros of arbitrary orders. We prove that the preconditioned Toeplitz matrices have singular values clustered around 1 for large $n$. We apply our preconditioners to solve the stationary probability distribution vectors of Markovian queueing models with batch arrivals. We show that if the number of servers is fixed independent of the queue size $n$, then the preconditioners are invertible and the preconditioned matrices have singular values clustered around 1 for large $n$. Numerical results are given to illustrate the fast convergence of our methods.

A novel method for calculating the convolution sum of two finite length sequences

Abstract: Some students find convolution difficult to understand and compute when first learning. This paper presents a direct method of computing the discrete linear convolution of two finite length sequences. The approach is easy to learn because of the similarities to computing the multiplication of two numbers by a pencil and paper calculation. This method allows students to quickly verify their answers obtained by graphical convolution or from a computer program and easily extends to deconvolution and circular convolution as shown in this paper. When this convolution method was taught in a discrete signals and systems course, the students' understanding of convolution significantly improved.

A fast and exact method for multidimensional Gaussian stochastic simulations: Extension to realizations conditioned on direct and indirect measurements

Abstract: Recently Dietnch and Newsam [1993] derived a fast and exact method for generating unconditional realizations of a stationary, multidimensional Gaussian random field on a rectangular sampling grid. The method is based on embedding the random field covariance matrix in a larger positive definite matrix with circulant/block circulant structure. The circulant structure of the embedding matrix means that a square root of this matrix can be efficiently computed by the fast Fourier transform; realizations are then generated by multiplying vectors of white noise by this square root. This paper extends the method to generating realizations conditioned on direct and/or indirect measurements of the field collected at an arbitrary set of scattered data points.

Normal Toeplitz Matrices

Abstract: It is well known from the work of Brown and Halmos [J. Reine Angew. Math., 213 (1963/1964), pp. 89--102] that an infinite Toeplitz matrix is normal if and only if it is a rotation and translation of a Hermitian Toeplitz matrix. In the present article we prove that all finite normal Toeplitz matrices are either generalised circulants or are obtained from Hermitian Toeplitz matrices by rotation and translation.

The existence of selfcomplementary circulant graphs

Abstract: All values of n for which there exist a selfcomplementary circulant graph of order n are determined.

Critical sets in back circulant latin squares

Abstract: To date very few families of critical sets for latin squares are known. In this paper a new family of critical sets for back circulant latin squares is identified. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.

Convolution using the undecimated discrete wavelet transform

Abstract: Convolution is one of the most widely used digital signal processing operations. It can be implemented using the fast Fourier transform (FFT) with a computational complexity of O (N log N). The undecimated discrete wavelet transform (UDWT) is linear and shift invariant, so it can also be used to implement convolution. We propose a scheme to implement the convolution using the UDWT, and study its advantages and limitations.

On the isomorphisms and automorphism groups of circulants

Abstract: Denote by C n (S) the circulant graph (or digraph). Let M be a minimal generating element subset of Z n , the cyclic group of integers modulo n, and $$\tilde M = \left\{ {\left. {m, - m} \right|m \in M} \right\}$$ In this paper, we discuss the problems about the automorphism group and isomorphisms of C n (S). When $$M \subseteq S \subseteq \tilde M$$ , we determine the automorphism group of C n (S) and prove that for any T ⊆ Z n , C n (S) ? C n (T) if and only if T = ?S, where ? is an integer relatively prime to n. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.

Fast recursive least squares adaptive filtering by fast Fourier transform-based conjugate gradient iterations

Abstract: Recursive least squares (RLS) estimations are used extensively in many signal processing and control applications. In this paper we consider RLS with sliding data windows involving multiple (rank k) updating and downdating computations. The least squares estimator can be found by solving a near-Toeplitz matrix system at each step. Our approach is to employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Here we iterate in the time domain (using Toeplitz matrix-vector multiplications) and precondition in the Fourier domain, so that the fast Fourier transform (FFT) is used throughout the computations. The circulant preconditioners are derived from the spectral properties of the given input stochastic process. When the input stochastic process is stationary, we prove that with probability 1, the spectrum of the preconditioned system is clustered around 1 and the method converges superlinearly provided that a sufficient number of data samples are taken, i.e.,...

Circulant weighing designs

Abstract: Algebraic techniques are employed to obtain necessary conditions for the existence of certain families of circulant weighing designs. As an application we rule out the existence of many circulant weighing designs. In particular, we show that there does not exist a circulant weighing matrix of order 43 for any weight. We also prove two conjectures of Yosef Strassler. © 1996 John Wiley & Sons, Inc. Disciplines Physical Sciences and Mathematics Publication Details Arasu K T and Seberry J, Circulant weighing designs, Journal of Combinatorial Designs, 4 (1996), 439-447. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1122 Circulant Weighing Designs K. T. Arasu* Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435 Jennifer Seberryt Department of Computer Science, University of Wollongong, NSW 2522, Australia ABSTRACT Algebraic techniques are employed to obtain necessary conditions for the existence of certain families of circulant weighing designs. As an application we rule out the existence of many circulant weighing designs. In particular, we show that there does not exist a circulant weighing matrix of order 43 for any weight. We also prove two conjectures of Yosef Strassler. © 1996 John Wiley & Sons, Inc.Algebraic techniques are employed to obtain necessary conditions for the existence of certain families of circulant weighing designs. As an application we rule out the existence of many circulant weighing designs. In particular, we show that there does not exist a circulant weighing matrix of order 43 for any weight. We also prove two conjectures of Yosef Strassler. © 1996 John Wiley & Sons, Inc.

Circulant matrices and the stability of a class of CNNs

Abstract: In this paper we show that feedback matrices of ring CNNs are block circulants; as special cases, for example, feedback matrices of one-dimensional ring CNNs are circulant matrices. Circulants and their close relations the block circulants possess many pleasant properties which allow one to describe their spectrum completely. After deriving the spectrum of the feedback operator, we discuss conditions for a CNN to be contractive, ensuring global asymptotic stability.

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

Abstract: We study methods for solving the constrained and weighted least squares problem minx\(\min _x \tfrac{1}{2}\left( {b - Ax} \right)^T W\left( {b - Ax} \right)\) by the preconditioned conjugate gradient (PCG) method. HereW = diag (ω1, ⋯, ωm) with ω1 ≥ ⋯ ≥ ωm ≥ 0, andAT = [T1T, ⋯,TkT] with Toeplitz blocksTl eRn × n,l = 1, ⋯,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemMλ +Ax =b, AT λ = 0, whereM =W−1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.

Necessary and sufficient conditions for some two variable orthogonal designs in order 36

Abstract: We give new sets of sequences with entries from {0, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d with zero autocorrelation function. We show that the necessary conditions are sufficient for the existence two variable orthogonal designs constructed from circulant matrices in order 36. Further we show that the necessary conditions for the existence of an OD(36;S1,S2) are sufficient except possibly for the following five cases: (3,29) (11,20) (11,21) (13,19) (15,17). Disciplines Physical Sciences and Mathematics Publication Details Koukouvinos C and Seberry J, Necessary and sufficient conditions for some two variable orthogonal designs in order 36, Congressus Numerantium, 114 (1996), 129-140. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1127 Necessary and sufficient conditions for some two variable orthogonal designs in order 36 Christos Koukouvinos; Nikos Platis; Jennifer Seberry~ Dedicated to Ralph Gordon Stanton on his 70th birthday Abstract We give new sets of sequences with entries from {O, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d with zero autocorrelation function. We show that the necessary conditions are sufficient for the existence two variable orthogonal designs constructed from circulant matrices in order 36. Further we show that the necessary conditions for the existence of an 0 D( 36; SI, S2) are sufficient except possibly for the following five cases: (3,29) (11,20) (11,21) (13,19) (15,17).We give new sets of sequences with entries from {O, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d with zero autocorrelation function. We show that the necessary conditions are sufficient for the existence two variable orthogonal designs constructed from circulant matrices in order 36. Further we show that the necessary conditions for the existence of an 0 D( 36; SI, S2) are sufficient except possibly for the following five cases: (3,29) (11,20) (11,21) (13,19) (15,17).

New algorithm for constructing fault-tolerant solutions of the circulant graph configuration

Abstract: The circulant graph configuration has been used to model several important parallel architectures (such as rings and meshes; for examples). Recently, a new method was developed to construct a k-fault-tolerant solution for any given circulant graph, where k is the number of faulty nodes to be tolerated. A generalization of this method is presented in this paper. We propose a new algorithm which (unlike the earlier method) constructs a large family of k-fault-tolerant solutions, for any given circulant graph (and any value of k). These solutions will then be compared to select the one with the least node-degree. Our algorithm is very efficient to implement, since it requires only a polynomial time to generate and search the solutions. Moreover, our method has useful applications to other parallel architectures, as demonstrated in the paper. We shall examine the application of the method to the problem of designing k-fault-tolerant extensions of (2- and 3-dimensional) meshes; and show that the solutions obtained are very efficient.

On Edge Transitive Circulant Graphs

Abstract: A classification is given for edge transitive circulant graphs whose complements are also edge transitive. In particular, it is shown that the only self-complementary symmetric circulant graphs are certain Paley graphs with a prime number of vertices. Similar results on digraphs are also obtained.

A practical algorithm to design fast and optimal band-Toeplitz preconditioners for hermitian toeplitz systems

Abstract: In this paper we are concerned with the iterative solution ofn×n Hermitian Toeplitz systems by means of preconditioned conjugate gradient (PCG) methods. In many applications [9] such as signal processing [24], differential equations [39], linear prediction of stationary processes [18], the related Toeplitz systems have the formAn(f)x=b where the symbolf, the generating function, is anL1 function and the entries ofAn(f) along thek-th diagnonal coincide with thek-th Fourier coefficient off. When the essential range of the generating function has a convex hull containing zero, the matricesAn(f) are asymptotically ill-conditioned [21, 33, 28] and circulant or Hartley preconditioners do not work [15]. For this difficult case the only optimal preconditioners in the sense of [3, 29] are found in the τ algebra [15, 35] and especially in the band Toeplitz matrix class [7, 16]. In particular the band Toeplitz preconditioning strategy has been shown to be the most flexible one since it allows one to treat the nonnegative case [7, 16, 11, 31], the nondefinite one [27, 30, 34, 26]. On the other hand, the main criticism to this approach is surely the assumption that we must know the position and the order of the zeros off: in some applicative fields this is a feasible assumption, in other applications it is merely a theoretical possibility. Therefore, we discuss an economical technique in order to discover the sign off, the position of the possible zeros of the generating function and to evaluate approximately the order of these zeros. Finally, we exhibit some numerical experiments which confirm the effectiveness of the proposed idea.

Critical sets in direct products of back circulant Latin squares

Abstract: To date very Few families of critical sets for latin squares are known. The only previously known method for constructing critical sets involves taking a critical set which is known to satisfy certain strong initial conditions and using a doubling construction. This construction can be applied to the known critical sets in back circulant latin squares of even order. However, the doubling construction cannot be applied to critical sets in back circulant latin squares of odd order. In this paper a family of critical sets is identified for latin squares which are the product of the latin square of order 2 with a back circulant latin square of odd order. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.