Showing papers on "Circulant matrix published in 2004"
Book•
16 Dec 2004TL;DR: This chapter discusses Iterative methods, a method for solving the differential equations of toeplitz systems, and its applications to ordinary and partial differential equations.
Abstract: 1 Notations and definitions 2 Iterative methods THEORY 3 Toeplitz systems 4 Circulant preconditioners 5 Non-circulant type preconditioners 6 Ill-conditioned Toeplitz systems 7 Structured systems APPLICATIONS 8 Applications to ordinary and partial differential equations 9 Applications to queuing networks 10 Applications to signal processing 11 Applications to image processing 12 Applications to integral equations
236 citations
TL;DR: This letter introduces a combinatorial construction of girth-eight high-rate low-density parity-check codes based on integer lattices and the resulting matrix of parity checks is an array of circulant matrices.
Abstract: This letter introduces a combinatorial construction of girth-eight high-rate low-density parity-check codes based on integer lattices. The parity-check matrix of a code is defined as a point-line incidence matrix of a 1-configuration based on a rectangular integer lattice, and the girth-eight property is achieved by a judicious selection of sets of parallel lines included in a configuration. A class of codes with a wide range of lengths and column weights is obtained. The resulting matrix of parity checks is an array of circulant matrices.
194 citations
Dissertation•
01 Jan 2004
TL;DR: In this paper, Behbahani et al. showed that the OD(l2; 1,1, 1, 9) is the only orthogonal design constructible from 16 circulant matrices, whenever n > 1 is an odd integer, and showed that for each integer n for which 4n is the order of a Hadamard matrix and 8n 2 + 1 is a prime, there is a productive regular matrix of order 16n 2 (8n 2 ) 2.
Abstract: On Orthogonal Matrices Majid Behbahani Depar tment of Mathemat i c s and Computer Science Universi ty of Lethbridge M. Sc. Thesis , 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these ma trices, we show that the OD(l2; 1,1,1, 9) is the only orthogonal design constructible from 16 circulant matrices of type OD(4n; 1,1, l , 4n — 3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design con structible from 16 circulant matrices of order 12 on 4 variables is the OD(12; 1 ,1,1, 9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To com plement our studies we reproduce an important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matrices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n 2 — 1 is a prime, there is a productive regular Hadamard matrix of order 16n 2 (8n 2 — l ) 2 . As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n 2 — 1) — 1, 4n(8n 2 — 1) + 1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work.
138 citations
TL;DR: The isomorphism problem for Cayley graphs over the cyclic group has been solved in this article, and the main result of this paper is that the Cayley graph is a Cayley cyclic graph.
Abstract: The isomorphism problem for circulant graphs (Cayley graphs over the cyclic group) which has been open since 1967 is completely solved in this paper. The main result of the paper gives an efficient isomorphism criterion for circulant graphs of arbitrary order. This result also solves an isomorphism problem for colored circulant graphs and some classes of cyclic codes.
114 citations
27 Jun 2004
TL;DR: A method is presented for constructing LDPC codes with excellent performance, simple hardware implementation, low encoder complexity, and which can be concisely documented.
Abstract: A method is presented for constructing LDPC codes with excellent performance, simple hardware implementation, low encoder complexity, and which can be concisely documented. The simple code structure is achieved by using a base graph, expanded with circulants. The base graph is chosen by computer search using simulated annealing, driven by density evolution's decoding threshold as determined by the reciprocal channel approximation. To build a full parity check matrix, each edge of the base graph is replaced by a circulant permutation, chosen to maximize loop length by using a Viterbi-like algorithm.
82 citations
TL;DR: A novel family of structured low-density parity-check codes with block-circulant parity- check matrices that consist of permutation blocks is proposed, which tend to outperform many other known structured LDPCs of comparable rate and length.
Abstract: A novel family of structured low-density parity-check (LDPC) codes with block-circulant parity-check matrices that consist of permutation blocks is proposed. The codes from this family are based on new combinatorial objects termed cycle-invariant difference sets, and they have low storage requirements, fast encoding algorithms, and girth of at least six. Most importantly, they tend to outperform many other known structured LDPCs of comparable rate and length.
81 citations
TL;DR: A circulant is a Cayley digraph over a finite cyclic group as discussed by the authors, and the classification of arc-transitive circulants is shown in this paper.
Abstract: A circulant is a Cayley digraph over a finite cyclic group. The classification of arc-transitive circulants is shown. The result follows from earlier descriptions of Schur rings over cyclic groups.
64 citations
TL;DR: A readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum is derived and an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices is proposed.
54 citations
TL;DR: It is proved that the recursive circulant graphs G(2m, 4), proposed by Park and Chwa, are super-connected if and only if m ≠ 2.
42 citations
TL;DR: In this article, an algorithm was constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time, and the consistency of the algorithm was based on a new result on the structure of Schur rings over a finite cyclic group.
Abstract: An algorithm is constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time. This algorithm also yields a cycle base of an arbitrary solvable permutation group. The consistency of the algorithm is based on a new result on the structure of Schur rings over a finite cyclic group.
42 citations
TL;DR: In this paper, the pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered and it is shown that when a certain winding number or twist condition is satisfied, related to Hormander's commutator condition for partial differential equations, e-pseudoeigenvectors of such matrices for exponentially small values of e exist in the form of localized wave packets.
Abstract: The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hormander’s commutator condition for partial differential equations, e-pseudoeigenvectors of such matrices for exponentially small values of e exist in the form of localized wave packets. The symbol need not be smooth with respect to x, just differentiable at a point (or less). c � 2004 Wiley Periodicals, Inc.
TL;DR: Familiar concepts from control engineering such as bandwidth and stability margin are extended into the two-dimensional frequency domain, providing intuitive measures of closed-loop performance and robustness.
TL;DR: An infinite sequence of rules Q(4, δ) is constructed that has a limit rho index of 27/34 ≈ 0.79, an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.
Abstract: We introduce the class of skew-circulant lattice rules. These are s-dimensional lattice rules that may be generated by the rows of an s × s skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for δ = d + 1 ≤ 47, we have constructed an infinite sequence of rules Q(4, δ) that has a limit rho index of 27/34 ≈ 0.79. This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.
TL;DR: Some formulas, asymptotic behaviors and linear recurrence relations for the number of its spanning trees are obtained and some known results on the ones in even valent circulant graphs C"n(a"1, a"2,...,a"k) are improved.
TL;DR: In this article, the determinant of the p×p circulant matrix whose first row is (1, −x,0, 0, −y,1, 0) is the sum of the absolute values of the monomials in the polynomial Φ.
Abstract: We study the polynomial\(\Phi \left( {x,y} \right) = \prod {_{j = 0}^{p - 1} \left( {1 - xw^j - yw^{qj} } \right)} \), where ω is a primitivepth root of unity. This polynomial arises in CR geometry [1]. We show that it is the determinant of thep×p circulant matrix whose first row is (1, −x,0,…,0,−y,0,…,0), the −y being in positionq+1. Therefore, the coefficients of this polynomial Φ are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomialxrys in Φ have the same sign, and we determine that sign. We prove that a monomialxrys appears in Φ if and only ifp dividesr+sq. Finally, we show that the size of the largest coefficient of the monomials in Φ grows exponentially withp, by proving that the permanent of the circulant whose first row is (1, 1, 0, …, 0, 1, 0, …, 0) is the sum of the absolute values of the monomials in the polynomial Φ.
TL;DR: Lower bounds of order n log n are proved for both the problem of multiplying polynomials of degree n, and of dividing polynomers with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers, to establish a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.
Abstract: We prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305--306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.
Journal Article•
TL;DR: This paper derives a straightforward combinatorial proof of the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n, and generalizes this technique to show that the numbers of Hamiltonian Cycles, EulerianCycles and Euleria Orientations in circular graphs also satisfy recurrence relations.
Abstract: It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs' adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circulant graph into smaller ones, our technique is to instead decompose a large circulant graph into different step graph cases and then construct a recurrence relation on the step graphs. We then generalize this technique to show that the numbers of Hamiltonian Cycles, Eulerian Cycles and Eulerian Orientations in circulant graphs also satisfy recurrence relations.
TL;DR: The classification of binary optimal linear [ n , n /2] codes up to length 28 is complete and it is shown that the smallest length for which an optimal rate has a trivial automorphism group is 14.
Patent•
08 Nov 2004TL;DR: In this article, an LDPC encoding method in a digital communication system is provided, in which a parity-check matrix H having a plurality of circulant matrices as elements is first generated.
Abstract: An LDPC encoding method in a digital communication system is provided, in which a parity-check matrix H having a plurality of circulant matrices as elements is first generated. A generation matrix G is generated using the parity-check matrix. Information bits are then encoded using the generation matrix G.
Patent•
08 Jan 2004TL;DR: In this paper, the received signals are sampled and processed using the combined code block diagonal matrix and the channel response block diagonal matrices with a Cholesky algorithm to produce spread symbols.
Abstract: Symbols are to be recovered from signals received in a shared spectrum. Codes of the signals received in the shared spectrum are processed using a block Fourier transform (FT 34), producing a code block diagonal matrix. A channel response of the received signals is estimated. The channel response is extended and modified (36) to produce a block circulant matrix and a block FT (38) is taken, producing a channel response block diagonal matrix. The code block diagonal matrix is combined (40, 44, 46) with the channel response block diagonal matrix. The received signals are sampled and processed using the combined code block diagonal matrix and the channel response block diagonal matrix with a Cholesky algorithm. A block inverse FT (60) is performed on a result of the Cholesky algorithm to produce spread symbols. The spread symbols are despread to recover symbols of the received signals.
TL;DR: If an A k 1, k 2 -stable BVM is used, it is shown that the authors' preconditioner is invertible and the spectrum of the preconditionsed matrix is clustered, so it follows that when the GMRES method is applied to solving the precONDitioned systems, the method would converge fast.
Abstract: We consider the solution of differential equations with multidelays by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. If an A k 1 , k 2 -stable BVM is used, we show that our preconditioner is invertible and the spectrum of the preconditioned matrix is clustered. It follows that when the GMRES method is applied to solving the preconditioned systems, the method would converge fast. Numerical results are given to show the effectiveness of our methods.
20 Dec 2004
TL;DR: It is shown how that the number of spanning trees for all classes of n node circulant graphs satisfies a recurrence relation in n even when the jumps are non-constant (but linear) in the graph size.
Abstract: Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recurrence relation in n For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few special cases such as the Mobius ladder had been studied but no general results were known.
In this note we show how that the number of spanning trees for all classes of n node circulant graphs satisfies a recurrence relation in n even when the jumps are non-constant (but linear) in the graph size The technique developed is very general and can be used to show that many other structures of these circulant graphs, e.g., number of Hamiltonian Cycles, Eulerian Cycles, Eulerian Orientations, etc., also satisfy recurrence relations.
The technique presented for deriving the recurrence relations is very mechanical and, for circulant graphs with small jump parameters, can easily be quickly implemented on a computer We illustrate this by deriving recurrence relations counting all of the structures listed above for various circulant graphs.
TL;DR: A lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p/spl equiv//spl plusmn/3(mod8) is given, which improves on the square root bound obtained by Calderbank and Beenker, using a completely different technique.
Abstract: We give a lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p/spl equiv//spl plusmn/3(mod8). This bound improves on the square root bound obtained by Calderbank and Beenker, using a completely different technique. The key to our estimates is to apply a result by Helleseth, to which we give a new and shorter proof. Combining this result with the Weil bound leads to the improvement of the Calderbank and Beenker bound. For large primes p, their bound is of order /spl radic/(2p) while our new improved bound is of order 2/spl radic/p. The results can be extended to any prime power q and the modifications of the proofs are briefly indicated.
27 Sep 2004
TL;DR: In this paper, a new interpolatory subdivision for quadrilateral meshes is presented, which employs a /spl radic/2 split operator to refine a given control mesh such that the face number of the refined mesh is doubled after each refinement.
Abstract: This paper presents a new interpolatory subdivision for quadrilateral meshes. The proposed scheme employs a /spl radic/2 split operator to refine a given control mesh such that the face number of the refined mesh is doubled after each refinement. For regular meshes, the smallest mask is chosen to calculate newly inserted vertices and special rules are developed to compute the F-vertices for irregular faces based on the Fourier analysis of block circulant matrices. Numerical analysis manifests that the scheme yields globally C1 continuous limit surfaces. Finally, an extension to arbitrary polygonal meshes is considered.
Journal Article•
TL;DR: A fast Fourier transform algorithm for the production of the permutation factor circulant matrices of order n based on the fast Fouriers transform (FFT) was presented, and arithmetric complexity is O(nlog_2n).
Abstract: A fast Fourier transform algorithm for the production of the permutation factor circulant matrices of order n based on the fast Fourier transform(FFT) was presented, and arithmetric complexity is O(nlog_2n).
TL;DR: It is proved that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-governing circulants.
TL;DR: A block-circulant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks and alleviates the critical slowing down that is especially severe close to the critical point.
Abstract: Critical slowing down associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block-circulant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present algorithm per iteration is O(rslog s) + delops, where the stiffness matrix A is partitioned into r × r blocks such that each block is an s × s matrix, and delops represents the operational count associated with solving a block-diagonal matrix with r × r dense matrix blocks. This algorithm using the block-circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient algorithm, and alleviates the critical slowing down that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.
TL;DR: In this paper, it was shown that when n is a prime power, d(n) e p(n), i.e., the number of terms in the determinant matrix of A, is polynomial.
Abstract: Let A e (aij) be the generic n × n circulant matrix given by aij e xi + j, with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n) e p(n).
Patent•
28 May 2004
TL;DR: In this paper, a circulant matrix is calculated that represents a transformation of the signal on the base grid to a complex signal having real and imaginary parts representing the signal having transformed amplitude-frequency characteristics.
Abstract: A method of performing a direct discrete transformation of a signal includes reading amplitude-frequency characteristics of a signal from a sensor. The signal is associated with a base grid to form a digital representation of a function that has a discrete Fourier representation. A circulant matrix is calculated that represents a transformation of the signal on the base grid to a complex signal having real and imaginary parts representing the signal having transformed amplitude-frequency characteristics. The signal on the base grid is then transformed with the circulant matrix to obtain real and imaginary parts of the signal having transformed amplitude-frequency characteristics. The real and imaginary parts of the signal are then written to a display or a storage device.
15 Apr 2004
TL;DR: This paper proposes an easy and efficient way to generate the matrix M directly by Monte Carlo simulation, using the natural pixel matrix M, which has several advantages over using the conventional system matrix A.
Abstract: In discrete detector PET, natural pixels are image basis functions calculated from responses of detector pairs. By using reconstruction with natural pixels the discretization of the object into a predefined grid can be avoided. Instead of trying to find the object x from system matrix A and measured data p by solving Ax = p, one solves Mq = AA/sup T/q = p. The backprojection of the final q results in the estimated image of the object x. In previous work natural pixels were calculated by intersections of strip functions or combinations of strip functions. However, in PET systems the detector response is not correctly described by a strip function. In PET the detector response is better described by a Gaussian tube. It is quite difficult to calculate the intersections of all possible lines of response (LOR) in the scanner. This paper proposes an easy and efficient way to generate the matrix M directly by Monte Carlo simulation. Using the natural pixel matrix M has several advantages over using the conventional system matrix A. The discretisation of the object into pixels, which requires an angular dependent projector and backprojector, is avoided. Due to rotational symmetry in the PET scanner the matrix M is block circulant and only the first block row needs to be stored. The block circulant property of the matrix allows use of fast direct methods to calculate the solution. Iterative methods to solve the system are preferred because of numerical stability.