Showing papers on "Circulant matrix published in 2007"
Book•
22 Nov 2007
TL;DR: The present work presents a unified treatment from kernels system for block Toeplitz systems and some examples of such systems have been described in detail in the literature.
Abstract: Preface 1. Introduction 2. Circulant preconditioners 3. Unified treatment from kernels 4. Ill-conditioned Toeplitz systems 5. Block Toeplitz systems A. M-files used in the book Bibliography.
310 citations
TL;DR: In this paper, closed-form formulae for Kirchhoff index and resistance distances of circulant graphs are derived in terms of Laplacian spectrum and eigenvectors.
Abstract: The resistance distance rij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf (G) is the sum of resistance distances between all pairs of vertices. In this work, closed-form formulae for Kirchhoff index and resistance distances of circulant graphs are derived in terms of Laplacian spectrum and eigenvectors. Special formulae are also given for four classes of circulant graphscomplete graphs, complete graphs minus a perfect matching, cycles, Mobius ladders Mp � . In particular, the
115 citations
TL;DR: In this article, it was shown that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral).
Abstract: The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.
88 citations
TL;DR: It is proved that the convergence rate is independent of the system dimension even in presence of asymptotical ill-conditioning and the proposed V-cycle Multigrid method is optimal.
Abstract: We analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant, tau, Hartley, or are of Toeplitz type. In the case of matrix algebra linear systems, we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill-conditioning (this happens iff f takes the zero value). More precisely, if the d-level coefficient matrix has partial dimension nr at level r, with $${r=1, \ldots, d}$$, then the size of the system is $${{N(\varvec{n})=\prod_{r=1}^d n_r}}$$, $${\varvec{n}=(n_1, \ldots, n_d)}$$, and O(N(n)) operations are required by the considered V-cycle Multigrid in order to compute the solution within a fixed accuracy. Since the total arithmetic cost is asymptotically equivalent to the one of a matrix-vector product, the proposed method is optimal. Some numerical experiments concerning linear systems arising in 2D and 3D applications are considered and discussed.
76 citations
TL;DR: An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented and a new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented.
Abstract: An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant graphs, is one of the main contributions of this work. A detailed analysis of some degree-four circulant graphs has allowed us to detail the weight distribution for these signal spaces. A new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented. Finally, we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six.
59 citations
TL;DR: In this article, it was shown that for the case where the first row is a palindrome, the limiting spectral measure converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian.
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.
54 citations
TL;DR: In this paper, a large class of quantum dxd states which are positive under partial transposition (so called PPT states) were constructed based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure.
Abstract: We construct a large class of quantum dxd states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure - that is why we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well-known examples of PPT states from the literature and gives rise to a huge family of completely new states.
50 citations
01 Jan 2007
TL;DR: A List of Symbols Used x as discussed by the authors is a list of symbols used in the past and present of symbols in the English language, including symbols used for symbol-based communication.
Abstract: ix List of Symbols Used x
44 citations
TL;DR: For 3-regular and 4-regular circulant graphs, the necessary and sufficient conditions for the existence of efficient dominating sets are given and their exact structure is described according to the relationship between chords.
42 citations
TL;DR: This work generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant code, and Gaborit Quadratic Double Circular codes into self-dual codes from the adjacency matrices of these schemes.
Abstract: Two class association schemes consist of either strongly regular
graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary
codes attached to rank three groups.
39 citations
TL;DR: An efficient algorithm for the approximation of functions and their derivatives using radial basis functions (RBFs) is proposed, capable of solving large-scale problems with more than 100@?000 interpolation points in two dimensions.
TL;DR: It is shown that the linear convolution required in block filtering can be decomposed into a sum of skew-cyclic convolutions, which results in computational savings over the traditional overlap-add and overlap-save algorithms.
Abstract: It is shown that the linear convolution required in block filtering can be decomposed into a sum of skew-cyclic convolutions. Such convolutions can be realized efficiently with half-length complex transforms when the signals are real. This method results in computational savings over the traditional overlap-add and overlap-save algorithms. It is also more economical than fast parallel finite impulse response (FIR) filter structures for longer filter lengths
TL;DR: It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations.
TL;DR: This work proposes an efficient matrix decomposition algorithm for the Method of Fundamental Solutions when applied to three-dimensional boundary value problems governed by elliptic systems of partial differential equations and makes use of fast Fourier transforms.
TL;DR: In this article, the authors give a self-contained modern linear stability analysis of a system of n equal-mass bodies in circular orbit about a single more massive body, and they show that these systems are always unstable for 2 ≤ n ≤ 6, and for n > 6 they are stable provided that the central mass is massive enough.
Abstract: We give a self-contained modern linear stability analysis of a system of n equal-mass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear approximation, compute all of the eigenvalues of the linear stability matrix, and finally derive inequalities that guarantee that none of these eigenvalues have a positive real part. In the end we rederive the result that Maxwell found for large n in his seminal paper on the nature and stability of Saturn's rings, which was published 150 years ago. In addition, we identify the exact matrix that defines the linearized system even when n is not large. This matrix is then investigated numerically (by computer) to find stability inequalities. Furthermore, using properties of circulant matrices, the eigenvalues of the large 4n × 4n matrix are computed by solving n quartic equations, which further facilitates the investigation of stability. Finally, we implement an n-body simulator, and we verify that the threshold mass ratios that we derive mathematically or numerically do indeed identify the threshold between stability and instability. Throughout the paper we consider only the planar n-body problem so that the analysis can be carried out purely in complex notation, which makes the equations and derivations more compact, more elegant, and therefore, we hope, more transparent. The result is a fresh analysis that shows that these systems are always unstable for 2 ≤ n ≤ 6, and for n > 6 they are stable provided that the central mass is massive enough. We give an explicit formula for this mass-ratio threshold.
TL;DR: This work is mailed to journal of the American Medical Association, Obstetricians and Gynaecologists, 2nd Ed.
TL;DR: This work investigates the recursive inversion of matrices with circulant blocks using the discrete Fourier transform and an implementation of the algorithm in MATLAB and numerical results are presented to demonstrate the efficiency in terms of CPU time of this approach.
TL;DR: A double circulant self-dual [32, 16] code over $\mathbb F_5$ which has a higher minimum weight than the previously best known linear code with these parameters is presented.
Abstract: In this paper, we consider double circulant and quasi-twisted selfdual codes over $\mathbb F_5$ and $\mathbb F_7$. We determine the highest minimum weights for such codes of lengths up to 34 for $\mathbb F_5$ and up to 28 for $\mathbb F_7$, and classify the codes with these minimum weights. In particular, we give a double circulant self-dual [32, 16] code over $\mathbb F_5$ which has a higher minimum weight than the previously best known linear code with these parameters. In addition, a self-dual code over $\mathbb F_7$ is presented which has a higher minimum weight than the previously best known self-dual code for length 28.
TL;DR: This paper shows that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if d=pn (p a prime number and n any integer) there exists d+1 mutually unbiased bases in Cd.
Abstract: In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results.
TL;DR: In this article, the authors studied the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks, and showed that the spectral measure for Wigners with k-weakly dependent entries need not be the semicircle law in the limit.
Abstract: We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with k-weakly dependent entries need not to be the semicircle law in the limit.
24 Jun 2007
TL;DR: An explicit construction of type-II codes which guarantees girth at least 6 is presented, and a necessary and sufficient condition for a QC matrix with one or more rows of circulants, to be fullrank is derived.
Abstract: Type-II quasi-cyclic (QC) LDPC codes are constructed from combinations of weight-0, weight-1 and weight-2 circulant matrices. The structure of cycles of length 2n are investigated, and necessary and sufficient conditions for a type-II QC LDPC parity check matrix H to have girth at least 2(n + 1) are given. An explicit construction of type-II codes which guarantees girth at least 6 is presented. A necessary and sufficient condition for a QC matrix with one or more rows of circulants, to be fullrank is derived.
TL;DR: A minimax mean-squared error (MSE) approach in which the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region is sought and it is shown that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently.
Abstract: We consider the problem of estimating a vector ${\bf x}$ in the linear model ${\bf A}{\bf x} \approx {\bf y}$, where ${\bf A}$ is a block circulant (BC) matrix with $N$ blocks and ${\bf x}$ is assumed to have a weighted norm bound. In the case where both ${\bf A}$ and ${\bf y}$ are subjected to noise, we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region. For an arbitrary choice of weighting, we show that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently. For a Euclidean norm bound on ${\bf x}$, the SDP is reduced to a simple convex program with $N+1$ unknowns. Finally, we demonstrate through an image deblurring example the potential of the minimax MSE approach in comparison with other conventional methods.
TL;DR: In this paper, a Kronecker product-based technique was applied to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for corresponding linear systems.
TL;DR: The circulant G = C(n, S), where S subset of Z(n) {0, is the graph with vertex set Z n and edge set E n = {x, x + s} vertical bar x is an element of Z n, s is an elements of s} as mentioned in this paper.
Abstract: The circulant G = C(n, S), where S subset of Z(n) {0}, is the graph with vertex set Z(n) and edge set E(G) = {{x, x + s} vertical bar x is an element of Z(n), s is an element of s}. It is shown that for n odd, every 6-regular connected circulant C(n, S) is decomposable into Hamilton cycles. (c) 2006 Wiley Periodicals, Inc.
Journal Article•
TL;DR: In this paper, the Modified Pell, Jacobsthal and Jacob Sthal-Lucas Numbers with the modified Pell, Negacyclic and Semicirculant Matrices with the Modified Pells and Jacobsthals were used.
Abstract: Circulant, Negacyclic and Semicirculant Matrices with the Modified Pell, Jacobsthal and Jacobsthal-Lucas Numbers
Posted Content•
TL;DR: In this paper, the authors studied the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks, and showed that the spectral measure for a Hermitians Circulant blockmatrix with weakly dependent entries does not need to be the semicircle law in the limit.
Abstract: We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k-$weakly dependent entries need not to be the semicircle law in the limit.
Posted Content•
TL;DR: In this paper, the authors consider n-by-n circulant matrices having entries 0 and 1 and prove that permutation matrices P,Q such that A =PBQ can be identified with sets of residues mod n, corresponding to columns in which the top row contains an entry.
Abstract: We consider n-by-n circulant matrices having entries 0 and 1 Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1 Let A and B be two such matrices, and suppose that the corresponding residue sets S_A and S_B have size at most 3 We prove that the following are equivalent: (1) there are integers u,v mod n, with u a unit, such that S_A = uS_B + v; (2) there are permutation matrices P,Q such that A=PBQ Our proof relies on some new results about vanishing sums of roots of unity We give examples showing this result is not always true for denser circulants, as well as results showing it continues to hold in some situations We also explain how our problem relates to the Adam problem on isomorphisms of circulant directed graphs
01 Oct 2007
TL;DR: A decoding method for cyclic codes up to the Hartmann-Tzeng bound using the discrete Fourier transform and how to make a submatrix of the circulant matrix from an error vector is considered.
Abstract: We consider a decoding method for cyclic codes up to the Hartmann-Tzeng bound using the discrete Fourier transform. Indeed we propose how to make a submatrix of the circulant matrix from an error vector for this decoding method. Moreover an example of a binary cyclic code, which could not be corrected by BCH decoding but be correctable by proposed decoding, is given. It is expected that this decoding method induces universal understanding for decoding method up to the Roos bound and the shift bound.
TL;DR: The present correspondence derives new theoretical results regarding the convergence of a single entry of a continuous function of a Toeplitz matrix.
Abstract: Toeplitz matrices and functions of Toeplitz matrices (such as the inverse of a Toeplitz matrix, powers of a Toeplitz matrix or the exponential of a Toeplitz matrix) arise in many different theoretical and applied fields. They can be found in the mathematical modeling of problems where some kind of shift invariance occurs in terms of space or time. R. M. Gray's excellent tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szegouml distribution theory on the asymptotic behavior of continuous functions of Toeplitz matrices. His asymptotic results, widely used in engineering due to the simplicity of its mathematical proofs, do not concern individual entries of these matrices but rather, they describe an "average" behavior. However, there are important applications where the asymptotic expressions of interest are directly related to the convergence of a single entry of a continuous function of a Toeplitz matrix. Using similar mathematical tools and to gain insight into the solutions of this sort of problems, the present correspondence derives new theoretical results regarding the convergence of these entries
01 Oct 2007
TL;DR: In this paper, a cylindrical image representation is proposed to preserve both in-plane and axial symmetries between the tubes of response for a given camera, leading to a system matrix having a block-circulant structure.
Abstract: Iterative image reconstruction methods based on an accurate and fully three-dimensional (3D) system probability matrix are well-known to provide images of higher quality. However, the size of the system matrix and the computation burden often make such methods impractical. To address this problem, we proposed to use a cylindrical image representation that preserves both in-plane and axial symmetries between the tubes of response for a given camera, leading to a system matrix having a block-circulant structure. For 3D image reconstruction, such a system matrix can be structured into a block-circulant matrix where blocks are themselves block-circulant. By storing only non-redundant parts of the block-circulant matrix, memory requirements can be reduced by a factor equivalent to the total number of system symmetries. The block-circulant system matrix can be stored in the Fourier domain representation to accelerate the forward and back projection steps of the iterative image reconstruction methods. When represented in the Fourier domain, the system matrix sparsity is reduced compared to the spatial domain representation, but some null values are still preserved.