Showing papers on "Circulant matrix published in 2005"

Eigenvectors of block circulant and alternating circulant matrices

Abstract: The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant matrices, which reduce to block circulant matrices with square submatrices of order 2.

V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems

Abstract: In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or $\tau$ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r=1,...,d, then the size of the algebraic system is $N(n)=\prod_{r=1}^d n_r$, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.

Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants

Abstract: A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970's. It is shown that a connected arc transitive circulant ? of order n is one of the following: a complete graph Kn, a lexicographic product $\Sigma [{\bar K}_b]$ , a deleted lexicographic product $\Sigma [{\bar K}_b] - b\Sigma$ , where ? is a smaller arc transitive circulant, or ? is a normal circulant, that is, Auta? has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.

The Field Descent Method

Abstract: We obtain a broadly applicable decomposition of group ring elements into a "subfield part" and a "kernel part". Applications include the verification of Lander's conjecture for all difference sets whose order is a power of a prime >3 and for all McFarland, Spence and Chen/Davis/Jedwab difference sets. We obtain a new general exponent bound for difference sets. We show that there is no circulant Hadamard matrix of order v with 4

Multigrid Methods for Multilevel Circulant Matrices

Abstract: We introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form $f(x_j^{[n]})$, where $f$ is continuous and independent of $n=(n_1,\ldots,n_d)$, and $x_j^{[n]} \equiv 2\pi j/n = (2\pi j_1/n_1, \ldots, 2\pi j_d/n_d)$, $0 \le j_r \le n_r - 1$. The interest of the proposed technique pertains to the multilevel banded case, where the total cost is optimal, i.e., $O(N)$ arithmetic operations (ops), $N=\prod_{r=1}^d n_r$, instead of $O(N\log N)$ ops arising from the use of FFTs. In fact, multilevel banded circulants are used as preconditioners for elliptic and parabolic PDEs (with Dirichlet or periodic boundary conditions) and for some two-dimensional image restoration problems where the point spread function (PSF) is numerically banded, so that the overall cost is reduced from $O(k(\varepsilon,n)N \log N)$ to $O(k(\varepsilon,n)N)$, where $k(\varepsilon,n)$ is the number of PCG iterations to reach the solution within an accuracy of $\varepsilon$. Several numerical experiments concerning one-rank regularized circulant discretization of elliptic $2q$-differential operators over one-dimensional and two-dimensional square domains with mixed boundary conditions are performed and discussed.

Longest Induced Cycles in Circulant Graphs

Abstract: In this paper we study the length of the longest induced cycle in the unit circulant graph $X_n = Cay({\Bbb Z}_n; {\Bbb Z}_n^*)$, where ${\Bbb Z}_n^*$ is the group of units in ${\Bbb Z}_n$. Using residues modulo the primes dividing $n$, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing $n$, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in $X_n$. We also see that if $n$ has $r$ distinct prime divisors, $X_n$ always contains an induced cycle of length $2^r+2$, improving the $r \ln r$ lower bound of Berrezbeitia and Giudici. Moreover, we extend our results for $X_n$ to conjunctions of complete $k_i$-partite graphs, where $k_i$ need not be finite, and also to unit circulant graphs on any quotient of a Dedekind domain.

A fast wavelet algorithm for image deblurring

Abstract: We present a nonlinear fully adaptive wavelet algorithm which can recover a blurred image (n?n) observed in white noise with O(n 2 (logn) 2 ) steps. Our method exploits both the natural representation of the convolution operator in the Fourier domain and the typical characterisation of Besov classes in the wavelet domain. A particular feature of our method includes "cycle-spinning" band-limited wavelet approximations over all circulant shifts. The speed and the accuracy of the algorithm is illustrated with numerical examples of image deblurring. All figures presented in this paper are reproducible using the WaveD software package.

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Abstract: Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy's interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices.

Use of discrete Fourier transforms in the 1D‐Var retrieval problem

Abstract: For some purposes, observation error covariance matrices having a symmetric Toeplitz form can be well approximated by circulant matrices. This amounts to modifying correlations so that they are periodic on the scale of the observing domain. The inverse of a circulant matrix can be evaluated efficiently with a discrete Fourier transform, as can circulant matrix–vector and matrix–matrix products. This could be useful for 1D-Var retrievals of measurements made with high-spectral-resolution infrared sounders, when the observation vector is large and the errors are correlated. Two 1D-Var simulation studies indicate that symmetric Toeplitz observation error covariance matrices can in this context be accurately approximated with circulant matrices, although some care is required when the correlations have a Gaussian fall-off. The simulation studies also show that assuming the observation errors are uncorrelated, when they are in fact correlated, can give misleading ‘information content’ estimates. This may be important for channel selection calculations when the errors are assumed to be uncorrelated. © Crown copyright, 2005. Royal Meteorological Society

On the Regularizing Power of Multigrid-type Algorithms

Abstract: We consider the deblurring problem of noisy and blurred images in the case of known space invariant point spread functions with four choices of boundary conditions. We combine an algebraic multigrid previously defined ad hoc for structured matrices related to space invariant operators (Toeplitz, circulants, trigonometric matrix algebras, etc.) and the classical geometric multigrid studied in the partial differential equations context. The resulting technique is parameterized in order to have more degrees of freedom: a simple choice of the parameters allows us to devise a quite powerful regularizing method. It defines an iterative regularizing method where the smoother itself has to be an iterative regularizing method (e.g., conjugate gradient, Landweber, conjugate gradient for normal equations, etc.). More precisely, with respect to the smoother, the regularization properties are improved and the total complexity is lower. Furthermore, in several cases, when it is directly applied to the system $A{\bf f}={\bf g}$, the quality of the restored image is comparable with that of all the best known techniques for the normal equations $A^TA{\bf f}=A^T{\bf g}$, but the related convergence is substantially faster. Finally, the associated curves of the relative errors versus the iteration numbers are "flatter" with respect to the smoother (the estimation of the stop iteration is less crucial). Therefore, we can choose multigrid procedures which are much more efficient than classical techniques without losing accuracy in the restored image (as often occurs when using preconditioning). Several numerical experiments show the effectiveness of our proposals.

On the weight enumerators of duadic and quadratic residue codes

Abstract: In this correspondence, we compute the weight enumerators of various quadratic residue codes over F/sub 2/ and F/sub 3/, together with certain codes of related families like the duadic and the quadratic double circulant codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduce by usual classical methods for self-dual and formally self-dual codes over F/sub 2/ and F/sub 3/ their associated, previously unknown, weight enumerators. We compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes.

A Global Solution for the Structured Total Least Squares Problem with Block Circulant Matrices

Abstract: We study the structured total least squares (STLS) problem of system of linear equations Ax = b, where A has a block circulant structure with N blocks. We show that by applying the discrete Fourier transform (DFT), the STLS problem decomposes into N unstructured total least squares (TLS) problems. The N solutions of these problems are then assembled to generate the optimal global solution of the STLS problem. Similar results are obtained for elementary block circulant matrices. Here the optimal solution is obtained by assembling two solutions: one of an unstructured TLS problem and the second of a multidimensional TLS problem.

A multigrid for image deblurring with Tikhonov regularization

Abstract: In the resolution of certain image deblurring problems with given boundary conditions we obtain two-level structured linear systems. In the case of shift-invariant point spread function with Dirichlet (zero) boundary conditions, the blurring matrices are block Toeplitz matrices with Toeplitz blocks. If the periodic boundary conditions are used, then the involved structures become block circulant with circulant blocks. Furthermore, Gaussian-like point spread functions usually lead to numerically banded matrices which are ill-conditioned since they are associated to generating functions that vanish in a neighbourhood of (π,π). We solve such systems by applying a multigrid method. The proposed technique shows an optimality property, i.e. its cost is of O(N) arithmetic operations (like matrix–vector product), where N is the size of the linear system. In the case of images affected by noise we use two Tikhonov regularization techniques to reduce the noise effects. Copyright © 2005 John Wiley & Sons, Ltd.

Well-Covered Vector Spaces of Graphs

Abstract: For any field ${\bf F}$, the set of all functions $f : V(G) \rightarrow {\bf F}$ whose sum on each maximal independent set is constant forms a vector space over ${\bf F}$ In this paper, we show that the dimension can vary depending on the characteristic of the field We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, such as nonempty circulant graphs of prime order, have bounded dimension

On Invariance of Cyclic Group Symmetries in Multiagent Formations

Abstract: This paper explores how the interconnection topology among individuals of a multiagent system influences symmetry in its trajectories. It is shown how circulant connectivity preserves cyclic group symmetries in a formation of simple planar integrators. Moreover, it is revealed to what extent circulant connectivity is necessary in order that symmetric formations remain invariant under the system's dynamics.

Applications of the Generalized Fourier Transform in Numerical Linear Algebra

Abstract: Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential. The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices. Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.

On the perfect t-dominating set problem in circulant graphs and codes over gaussian integers

Abstract: The basis for designing error-correcting codes for two dimensional signal sets is considered in this paper. Both, algebraic and graph-theoretical approaches are employed in this research for establishing the fundamentals of these codes. We give a solution to the t-dominating set problem in a subfamily of degree four circulant graphs which directly provides perfect codes over the Gaussian integers. In order to show the applicability of our results, simple examples for designing different coding schemes are also presented

A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

Abstract: We consider the approximate solution of axisymmetric biharmonic problems using a boundary-type meshless method, the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation. For such problems, the coefficient matrix of the linear system defining the approximate solution has a block circulant structure. This structure is exploited to formulate a matrix decomposition method employing fast Fourier transforms for the efficient solution of the system. The results of several numerical examples are presented.

Factorized Banded Inverse Preconditioners for Matrices with Toeplitz Structure

Abstract: In this paper, we study factorized banded inverse preconditioners for matrices with Toeplitz structure. We show that if a Toeplitz matrix T has certain off-diagonal decay property, then the factorized banded inverse preconditioner approximates T-1 accurately, and the spectra of these preconditioned matrices are clustered around 1. In nonlinear image restoration applications, Toeplitz-related systems of the form I + T* D T are required to solve, where D is a positive nonconstant diagonal matrix. We construct inverse preconditioners for such matrices. Numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners. A two-dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner.

On the Complexity of the Preconditioned Conjugate Gradient Algorithm for Solving Toeplitz Systems with a Fisher-Hartwig Singularity

Abstract: The Toeplitz matrix $T_n$ with generating function $f ( \omega ) = |1 - e ^{-i \omega}|^{-2d} h( \omega )$, where $d \in (-\frac{1}{2}, \frac{1}{2})\setminus \{0\}$ and $h(\omega)$ is positive, continuous on $[-\pi,\pi]$, and differentiable on $[-\pi,\pi]\setminus\{0\}$, has a Fisher--Hartwig singularity [M. E. Fisher and R. E. Hartwig (1968), Adv. Chem. Phys., 32, pp. 190--225]. The complexity of the preconditioned conjugate gradient (PCG) algorithm is known [R. H. Chan and M. Ng (1996), SIAM Rev., 38, pp. 427--482] to be $O(n\log n)$ for Toeplitz systems when $d = 0$. However, the effect on the PCG algorithm of the Fisher--Hartwig singularity in $T_n$ has not been explored in the literature. We show that the complexity of the conjugate gradient (CG) algorithm for solving $T_n x=b$ without any preconditioning grows asymptotically as $n^{1+|d|}\log (n)$. With T. Chan's optimal circulant preconditioner $C_n$ [T. Chan (1988), SIAM J. Sci. Statist. Comput., 9, pp. 766--771], the complexity of the PCG algorithm is $O(n\log^3(n))$.

Permutation Equivalence Classes of Kronecker Products of Unitary Fourier Matrices

Abstract: Kronecker products of unitary Fourier matrices play important role in solving multilevel circulant systems by a multidimensional Fast Fourier Transform. They are also special cases of complex Hadamard (Zeilinger) matrices arising in many problems of mathematics and theoretical physics. The main result of the paper is splitting the set of all kronecker products of unitary Fourier matrices into permutation equivalence classes. The choice of permutation equivalence to relate the products is motivated by the quantum information theory problem of constructing maximally entangled bases of finite dimensional quantum systems. Permutation inequivalent products can be used to construct inequivalent, in a certain sense, maximally entangled bases.

Splitting iterations for circulant-plus-diagonal systems

Abstract: We consider the system of linear equations (C + iD)x=b, where C is a circulant matrix and D is a real diagonal matrix We study the technique for constructing the normal/skew-Hermitian splitting for such coefficient matrices Theoretical results show that if the eigenvalues of C have positive real part, the splitting method converges to the exact solution of the system of linear equations When the eigenvalues of C have non-negative real part, the convergence conditions are also given We present a successive overrelaxation acceleration scheme for the proposed splitting iteration Numerical examples are given to illustrate the effectiveness of the method Copyright © 2005 John Wiley & Sons, Ltd