Showing papers on "Circulant matrix published in 1991"

A variant of the Gohberg-Semencul formula involving circulant matrices

Abstract: The Gohberg–Semencul formula expresses the inverse of a Toeplitz matrix as the difference of products of lower triangular and upper triangular Toeplitz matrices. In this paper the idea of cyclic displacement structure is used to show that the upper triangular matrices in this formula can be replaced by circulant matrices. The resulting computational savings afforded by this modified formula is discussed.

The spectra of super-optimal circulant preconditioned Toeplitz systems

Abstract: The solutions of Hermitian positive-definite Toeplitz systems $A_n x = b$ by the preconditioned conjugate gradient method are studied. The preconditioner, called the “super-optimal” preconditioner, is the circulant matrix $T_n $ that minimizes $\| {I - C_n^{ - 1} A_n } \|_F $ over all circulant matrices $C_n $. The convergence rate is known to be governed by the distribution of the eigenvalues of $T_n^{-1}A_n $. For n-by-n Toeplitz matrix $A_n$ with entries being Fourier coefficients of a positive function in the Wiener class, the asymptotic behaviour of the eigenvalues of the preconditioned matrix $T_n^{-1}A_n $ is found as n increases, and it is proved that they are clustered around one.

Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems

Abstract: We study the solutions of Toeplitz systemsA n x=b by the preconditioned conjugate gradient method. Then ×n matrixA n is of the forma 0 I+H n wherea 0 is a real number,I is the identity matrix andH n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC n and the skew-circulant matrixS n whereA n =1/2(C n +S n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC −1 n An andS −1 n A n . For Toeplitz matricesA n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC n andS n and prove that the singular values ofC −1 n A n andS −1 n A n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.

The prime factors of Wendt’s binomial circulant determinant

Abstract: Wendt’s binomial circulant determinant, W„, is the determinant of an m by m circulant matrix of integers, with (i, j)th entry (i, ™.i) whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m < 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = mp + l are odd primes, 3 does not divide m, and m < 200, then the first case of Fermat’s Last Theorem is true for exponent p. © 1991 American Mathematical Society.

Parallel Complexity of Linear System Solution

Abstract: This book presents the most important parallel algorithms for the solution of linear systems. Despite the evolution and significance of the field of parallel solution of linear systems, no book is completely dedicated to the subject. People interested in the themes covered by this book belong to two different groups: numerical linear algebra and theoretical computer science, and this is the first effort to produce a useful tool for both. The book is organized as follows: after introducing the general features of parallel algorithms and the most important models of parallel computation, the authors analyze the complexity of solving linear systems in the circuit, PRAM, distributed, and VLSI models. The approach covers both the general case (i.e. dense linear systems without structure) and many important special cases (i.e. banded, sparse, Toeplitz, circulant linear systems).

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.

Connectivities and diameters of circulant graphs

Abstract: Let S = {al, az, . . . , ak} be a set of integers such that 0 < a1 < . . < ak < (n + 1)/2 and let the vertices of an n-vertex graph be labelled 0,1,2,. . . , n 1. Then the circulant graph C(n, S) has i f a1 , i f az, . . . , i f ak (mod n) adjacent to each vertex i. In the design of networks, connectivity of the underlying graphs is of interest as a measure of network reliability and vulnerability, while diameter is a measure of transmission delay. We examine the connectivity properties of circulant graphs, and present some results on the diameter of these graphs.

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.

Some fast Toeplitz least-squares algorithms

Abstract: We study fast preconditioned conjugate gradient (PCG) methods for solving least squares problems mm Ifb—Tx112, where T is an m x n Toeplitz matrix of rank n. Two circulant preconditioners are suggested: one, denoted by P, isbased on a block partitioning of T and the other, denoted by N, is based on the displacement representation of T'T. Each is obtained without forming T'T. We prove formally that for a wide class of of problems the PCG method with P converges in a small number of iterations independent of m and n, so that the computational cost of solving such Toeplitz least squares problems is O(m log n). Numerical experiments in using both P and N are reported, indicating similar good convergence properties for each preconditioner. 1 Introduction We consider the problem of solving least squares problems where the data matrix is Toeplitz, i.e. has constant diagonals. Toeplitz least squares problems occur in a variety of applications, especially in signal and image processing

Efficient hybrid finite element-boundary element method for 3-dimensional open-boundary field problems

Abstract: Highly storage-efficient choices of boundary meshes are devised for hybrid finite element-boundary element analysis in three dimensions, and a computationally efficient algorithm to solve the resulting matrix equations is developed. The boundary mesh makes use of cylindrical symmetry to produce block circulant boundary element matrices with proper choices of basis functions, and results in up to 64-fold reduction in matrix storage. The block circulant nature of the boundary element matrices provides a numerically efficient algorithm for solution of the coupled matrix equations. To demonstrate its efficiency, this hybrid scheme is applied to a three-dimensional magnetostatic field problem, proving storage and computational efficiency. This hybrid scheme is more advantageous than the pure finite-element method with approximate boundary conditions. and allows large three-dimensional open boundary field problems to be solved in small workstations. >

Multiresolution transform matrices

Abstract: Matrix descriptions of subband coders are introduced and used to obtain transform matrices of pyramidal and related algorithms. These matrices are generalizations of the Haar matrix, and they are all shown to have a stratified structure related to circulant matrix structure. An implementation of a class of biorthogonal digital filter banks (DFBs) with an efficient predictive structure is introduced. DFBs and pyramids optimal for quantization are also introduced. >

Template decomposition and inversion over hexagonally sampled images

Abstract: The family of real-valued circulant templates on nXm rectangular images is isomorphic to a quotient ring of the ring of real polynomials in two variables. Template decomposition is equivalent to factoring the corresponding polynomial. Template invertibility corresponds to polynomial invertibility in the quotient ring. Factoring and inverting are more difficult for polynomials in two variables than for those in one. Hexagonally sampled images have properties which simplify these operations. Hexagons organize themselves naturally into a hierarchy of snowflake-shaped regions. These tile the plane and consequently yield a simple definition of circulancy. Unlike the circulancy of rectangles in the plane, which yields a toroidal topology, the hexagonal analogue yields the topology of a circle. As a result, circulant templates are mapped isomorphically into a quotient of the ring of polynomials in one variable. These polynomials are products of linear factors over the complex numbers. A polynomial will be invertible in the quotient ring whenever each of its linear factors is invertible. This results in a simple criterion for template invertibility.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Optimal power distribution in robust, passive, fiber optic local communication networks with circulant topologies

Abstract: The power distribution of circulant topologies for passive fiber optic local communication networks (FOLCN) in the shot-noise limited case is analyzed. This class of topologies is chosen for its symmetries in both logical structure and power distribution, as well as for its mathematical tractability. Analysis according to a mathematical model for power distribution allows comparison of various circulant topologies, where circulance refers to a particular topological symmetry, and demonstrates that a high degree of connectivity aids even distribution of power. Appropriate topological design yields a network that can support thousands of users within a 50-dB loss budget, when low loss components are used. >

Polynomial algorithms for LP over a subring of the algebraic integers with applications to LP with circulant matrices

Abstract: It is shown that a modified variant of the interior point method can solve linear programs (LPs) whose coefficients are real numbers from a subring of the algebraic integers. By defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring, it is proved that the modified algorithm runs in time polynomial in the encoding size of the input coefficients, the dimension of the problem, and the order of the subring. The Tardos scheme is then extended to this case, yielding a running time that is independent of the objective and right-hand side data. As a consequence of these results, it is shown that LPs with real circulant coefficient matrices can be solved in strongly polynomial time. It is also shown how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots. >

Minimum-phase LU factorization preconditioner for Toeplitz matrices

Abstract: A new preconditioner is proposed for the solution of an N X N Toeplitz system TNx equals b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN is obtained based on factorizing the generating function T(z) into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, (parallel)FN-1(parallel) is bounded. For rational Toeplitz TN with generating function T(z) equals A(z-1)/B(z-1) + C(z)/D(z), where A(z), B(z), C(z), and D(z) are polynomials of orders p1, q1, p2, and q2, we show that the eigenvalues of FN-1TN are repeated exactly at 1 except at most (alpha) F outliers, where (alpha) F depends on p1, q1, p2, q2, and the number (omega) of the roots of T(z) equals A(z-1)D(z) + B(z-1)C(z) outside the unit circle. A preconditioner KN in circulant form generalized from the symmetric case is also presented for comparison.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A fast algorithm for half-cyclic convolution

Abstract: The definition of half-cyclic convolution is introduced. It is shown that the computation for cyclic convolution can be carried out based on the half-cyclic convolution, which is more general. The algorithm for half-cyclic convolution then can be used to build an algorithm for cyclic convolution and the fast Fourier transform (FFT), so that some problems in the algorithms for cyclic convolution and the FFT can be solved. An efficient and well-structured algorithm for half-cyclic convolution has been designed, called the Winograd-like algorithm. >

The discovery and enumeration of representative symbols for circulant tournaments

Abstract: This paper is concerned with finding and enumerating symbols to represent all circulant tournaments with n vertices up to isomorphism. A formula giving the exact number of such symbols is known for n prime and the derivation of this formula is reviewed. That result is then developed to produce a formula for the case where n is the product of two distinct primes and for giving an upper bound when n is the product of two identical primes. Some general techniques for testing isomorphism are also described. The paper describes a simple application of Burnside's lemma suitable for inclusion in an undergraduate course.

The spectra of super-optimal circulant preconditioned

Abstract: The solutions of Hermitian positive-definite Toeplitz systems A,x = b by the preconditioned conjugate gradient method are studied. The preconditioner, called the "super-optimal" preconditioner, is

Center for Digital Signal Processing Research Bureau of Engineering Research

Abstract: Matrix descriptions of subband coders are introduced and used to obtain transform matrices of pyramidal and related algorithms. These matrices are generalizations of the Haar matrix, and they are all shown to have a stratified structure related to circulant matrix structure. An implementation of a class of biorthogonal digital filter banks (DFB's) with an efficient predictive structure is introduced. DFB's and pyramids optimal for quantization are also introduced.

Use of smoothing in class 1 frequency domain estimation matrix filters

Abstract: It is observed that the performance of the frequency-domain Wiener estimation filters relies heavily on the type of smoothing algorithm and the characteristics of the signal. Thus, the designer or engineer must carefully select the optimum system regarding the implementation domain, smoothing algorithm, and computation complexity. Simulations showed that a frequency-domain Wiener estimation filter using diagonal smoothing perfectly estimates a narrowband signal. On the other hand, circulant smoothing is proved to be superior when estimating wideband signals. Both Toeplitz and circulant smoothing perform well for time-limited signals, thus necessitating a tradeoff between vector sparseness and MSE. It was found that diagonal smoothing yields the most computationally efficient structure. Therefore, advantage must be taken of this fact; for example, if the Fourier transformation is used, filters performing well under frequency-domain circulant smoothing can be replaced with time-domain filters using diagonal smoothing techniques. >

Computing eigenvalues of banded symmetric

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 matrixmwhich is trivial since the eigenvectors of circulants are known a priori and the eigenvalues can be obtained by Fourier transformmthe 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 + r2)), where n is the order and r the bandwidth of T.