Showing papers on "Circulant matrix published in 1997"
TL;DR: This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over $m+1$ equispaced points on a line can be produced at the cost of an initial FFT of length $2m$ with each new realization requiring an additionalFFT of the same length.
Abstract: Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid $\Omega$. This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over $m+1$ equispaced points on a line can be produced at the cost of an initial FFT of length $2m$ with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an $(m+1)\times(m+1) $ Toeplitz correlation matrix $R$ can be embedded in a nonnegative definite $2M\times2M$ circulant matrix $S$, exact realizations of the normal multivariate $y \sim {\cal N}(0,R)$ can be generated via FFTs of length $2M$. Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which $M = m$ is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated.
573 citations
TL;DR: It is proved that for even r > 2 the diameter of the MC(r, k) network is kr 2 − ⌊ K 2 ⌋ which is smaller than the diameterof the corresponding torus.
151 citations
TL;DR: The general case of a lossless FDN feedback matrix is shown to be any matrix having unit-modulus eigenvalues and linearly independent eigenvectors, and a special class of FDNs using circulant matrices is proposed.
Abstract: The feedback delay network (FDN) has been proposed for digital reverberation, The digital waveguide network (DWN) is also proposed with similar advantages. This paper notes that the commonly used FDN with an N/spl times/N orthogonal feedback matrix is isomorphic to a normalized digital waveguide network consisting of one scattering junction joining N reflectively terminated branches. Generalizations of FDNs and DWNs are discussed. The general case of a lossless FDN feedback matrix is shown to be any matrix having unit-modulus eigenvalues and linearly independent eigenvectors. A special class of FDNs using circulant matrices is proposed. These structures can be efficiently implemented and allow control of the time and frequency behavior. Applications of circulant feedback delay networks in audio signal processing are discussed.
99 citations
TL;DR: It is proved that Adam's conjecture remains also true if the number of vertices of a graph is twice square-free.
79 citations
TL;DR: This paper considers as preconditioners band-Toeplitz matrices generated by trigonometric polynomials g of fixed degree l to devise a polynomial g which has some analytical properties of f, is easily computable and is such that the corresponding preconditionsed system has a condition number bounded by a constant independent of n.
Abstract: In this paper we are concerned with the solution of n x n Hermitian Toeplitz systems with nonnegative generating functions f. The preconditioned conjugate gradient (PCG) method with the well-known circulant preconditioners fails in the case where f has zeros. In this paper we consider as preconditioners band-Toeplitz matrices generated by trigonometric polynomials g of fixed degree l. We use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f, is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n. For each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy. We obtain different estimates of l for which the total cost of the proposed PCG methods is optimal and the related rates of convergence are superlinear. Finally, for the most economical strategy, we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems.
67 citations
TL;DR: A direct scheme for multiplication of polynomials in Chebyshev form as well as a fast algorithm using discrete cosine transforms are developed, which leads to a new convolution operation and a new type of circulant matrices, both related to the discrete Cosine transform.
56 citations
TL;DR: In this paper, closed-form expressions of the inverse of a block-circulant matrix (BCM) are derived following two alternative procedures, which are based on the discrete Fourier transform (DFT).
Abstract: Closed-form expressions of the inverse of a block-circulant matrix (BCM) are well known. Such expressions are derived here following two alternative procedures, which are based on the discrete Fourier transform (DFT) and appear to be simple and brief when compared to previously published procedures. In the case of circulant matrices of scalars, the formulas of inversion are derived using the theory of circular arrays and this approach suggested an extension of the DFT to the BCM case.
53 citations
01 Jan 1997
TL;DR: In this article, the authors proved the existence of continuous families of complex Hadamard matrices of certain prime dimensions, n = 7, 13, 19, 31, 79 and 79.
Abstract: One proves the existence of continuous families of complex Hadamard matrices of certain prime dimensions, n = 7, 13, 19, 31, 79 This result implies the existence in the corresponding matrix algebras $M\sb{n}$(C) of uncountably many non isomorphic orthogonal pairs of maximal abelian $\sp\*$-subalgebras (MASA's) in the sense of SPopa A main point of interest of our result is that it might produce examples of one parameter families of nonisomorphic subfactors of same index n and same graph $A\sb{\infty}$ The numerical and symbolic computations that motivated this class of examples, as well as their formal construction are presented We are also exploring the possibility of extending this construction to all the primes of the form $n = 3\sp{2k} - 2, (k \ge 1)$
For any prime dimension n one proves that the standard complex Hadamard matrix (which corresponds to the matrix of the Fourier transform of vectors in C$\sp{n}$) is an isolated point among the normalized complex Hadamard matrices of the same dimension We also draw connections to the recent finiteness results of UHaagerup regarding circulant complex Hadamard matrices
43 citations
TL;DR: It is shown that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n, $\delta _k(n) = k$ and the lower bound is obtained by use of circulant graphs.
Abstract: Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.
32 citations
TL;DR: A class of double circulant codes over Z4 leads to an extremal even unimodular 40–dimensional lattice, and it is conjectured that there should be “Nine more constructions of the Leech lattice”.
Abstract: With the help of some new results about weight enumerators of self-dual codes over Z\!Z_4 we investigate a class of double circulant codes over Z\!Z_4, one of which leads to an extremal even unimodular 40-dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech lattice”
32 citations
TL;DR: This work presents particular choices of the feedback coefficients, namely Galois sequences arranged in a circulant matrix, to produce a maximum echo density in the time response, which gives implementations having a low number of multipliers and the resulting circuit can be efficiently pipelined.
Abstract: Feedback delay networks are widely used for simulating the diffuse part of reverberation in a room. We present particular choices of the feedback coefficients, namely Galois sequences arranged in a circulant matrix, to produce a maximum echo density in the time response. These specific sets of coefficients give implementations having a low number of multipliers, and the resulting circuit can be efficiently pipelined. The resulting networks are compared with other efficient implementations.
TL;DR: It is shown that the circulant graph G(2m, 4) is Hamiltonian decomposable, and a recursive construction method is proposed, which is a partial answer to a problem posed by B. Alspach.
TL;DR: The iterative solution of a block Toeplitz linear system by the conjugate gradient method is analyzed, the preconditioning step being solved by means of a discrete sine transform to take advantage if the system is ill conditioned.
Abstract: The iterative solution of a block Toeplitz linear system by the conjugate gradient method is analyzed, the preconditioning step being solved by means of a discrete sine transform. Convergence properties are established and compared to the behavior of the block circulant preconditioner recently proposed in the literature. As in the scalar case, the new approach takes advantage if the system is ill conditioned.
TL;DR: It is shown that all Hamiltonian cycles of a circulant 2-digraph are periodic and two simple algorithms are derived for solving the sum and bottleneck versions of CTSP forcirculant distance matrices with two nonzero stripes.
TL;DR: All extremal double circulant formally self-dual even codes which are not self- dual are classified and the existence of near-extremal formallySelf-duAL even codes are investigated.
Abstract: Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.
TL;DR: It is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circular code and its shadow through a relationship between bordered and pure doublecirculant codes.
Abstract: In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.
TL;DR: The preconditioned conjugate gradient method is applied to solving linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators with Dirichlet boundary conditions and the numerical results show that the method converges faster than the MILU and MINV methods.
Abstract: We consider applying the preconditioned conjugate gradient (PCG) method to solving linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators with Dirichlet boundary conditions Let (L + Σ)Σ−1(Lt + Σ) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix Σ We propose a preconditioner M = (Lˆ + Σˆ)Σˆ−1(Lˆt + Σˆ) with block diagonal matrix Σˆ and lower block triangular matrix Lˆ The diagonal blocks of Σˆ and the subdiagonal blocks of Lˆ are respectively the optimal sine transform approximations to the diagonal blocks of Σ and the subdiagonal blocks of L We show that for two-dimensional domains, the construction cost of M and the cost for each iteration of the PCG algorithm are of order O(n2 log n) Furthermore, for rectangular regions, we show that the condition number of the preconditioned system M−1A is of order O(1) In contrast, the system preconditioned by the MILU and MINV methods are of order O(n) We will also show that M can be obtained from A by taking the optimal sine transform approximations of each sub-block of A Thus, the construction of M is similar to that of Level-1 circulant preconditioners Our numerical results on two-dimensional square and L-shaped domains show that our method converges faster than the MILU and MINV methods Extension to higher-dimensional domains will also be discussed © 1997 John Wiley & Sons, Ltd
TL;DR: In this paper, the authors characterised those hyponormal Toeplitz operators on the Hardy space of the unit circle that have polynomial symbols with circulant-type sets of coefficients.
Abstract: This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.
TL;DR: The 2-peripatetic Salesman Problem (2-PSP for short) is the problem where 2 edge-disjoint Hamiltonian cycles of minimal to tallength are required as discussed by the authors.
Abstract: The 2-peripatetic Salesman Problem (2-PSP for short) is the problem. where 2 edge-disjoint Hamiltonian cycles of minimal to tallength are required. We have investigated whether the 2-PSP, either the sum version or the bottleneck version can also be wellsolved in known well-solved cases of the TSP. We show how to polynomially solve the 2-PSP in the following non-trivial cases: — Problems on Small matrices with distinct values for the sum as wellx as for the bottleneck criterion; — Problems on asymmetric Circulant matrices with a prime number of cities again for both criteria; — Problems with the bottleneck criterion on max-distribution matrices with the number of negative a-values equal to the number of nonnegative b-values.
TL;DR: The Wiener–Hopf equations with high-order quadrature rules by preconditioned conjugate gradient (PCG) methods are considered and it is shown that with the proper choice of kernel functions for the precONDitioners, the resulting preconditionsed equations will have clustered spectra and therefore can be solved by the PCG method with superlinear convergence rate.
Abstract: We consider solving the Wiener--Hopf equations with high-order quadrature rules by preconditioned conjugate gradient (PCG) methods. We propose using convolution operators as preconditioners for these equations. We will show that with the proper choice of kernel functions for the preconditioners, the resulting preconditioned equations will have clustered spectra and therefore can be solved by the PCG method with superlinear convergence rate. Moreover, the discretization of these equations by high-order quadrature rules leads to matrix systems that involve only Toeplitz or diagonal matrix--vector multiplications and hence can be computed efficiently by FFTs. Numerical results are given to illustrate the fast convergence of the method and the improvement on accuracy by using higher-order quadrature rule. We also compare the performance of our preconditioners with the circulant integral operators.
TL;DR: It is proved that the preconditioned linear system has singular values clustered around one when the number of inventory levels tends to infinity, which means conjugate-gradient methods will converge very fast when applied to solving the preconding linear system.
TL;DR: This paper provides a classification of length 24 double circulant Type I codes over ℤ4 with minimum Euclidean weight 12 that determine the odd Leech lattice, which is a unique 24-dimensional odd unimodular lattice with minimum norm 3.
Abstract: Recently, alternative constructions of the Leech lattice and the shorter Leech lattice have been discovered using self-dual codes over ℤ4. In this paper, we provide a classification of length 24 double circulant Type I codes over ℤ4 with minimum Euclidean weight 12. These codes determine (via Construction A4) the odd Leech lattice, which is a unique 24-dimensional odd unimodular lattice with minimum norm 3.
26 Oct 1997
TL;DR: New preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems are described, which lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration.
Abstract: Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shift-invariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shift-invariant (i.e. approximately block-Toeplitz or block-circulant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging), the Hessian of the (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems. Compared to diagonal or Fourier preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. Applications to position emission tomography (PET) illustrate the method.
TL;DR: Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.
Abstract: The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ<√2e(n+1)+2, whereN=n
2 is the size of the discrete problem, ande stands for the ratio of the anisotropy. Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.
TL;DR: In this article, the condition number of a block-Toeplitz operator was shown to be the lowest upper bound for the condition of corresponding block circulant matrices of all possible sizes.
Abstract: The recursive algorithm of a (fast) discrete wavelet transform, as well as its generalizations, can be described as repeated applications of block-Toeplitz operators or, in the case of periodized wavelets, multiplications by block circulant matrices. Singular values of a block circulant matrix are the singular values of some matrix trigonometric series evaluated at certain points. The norm of a block-Toeplitz operator is then the essential supremum of the largest singular value curve of this series. For all reasonable wavelets, the condition number of a block-Toeplitz operator thus is the lowest upper bound for the condition of corresponding block circulant matrices of all possible sizes. In the last section, these results are used to study conditioning of biorthogonal wavelets based on B-splines.
TL;DR: In this article, the maximum number of spanning trees in 2k-regular circulant graphs (k > 1) on n vertices is estimated as follows: fk(n) = ((2k)/(rk))n(1+o(1)), where
Abstract: The following asymptotic estimation of the maximum number of spanning trees fk(n) in 2k-regular circulant graphs (k > 1) on n vertices is the main result of this paper: fk(n) = ((2k)/(rk))n(1+o(1)), where
© 1997 John Wiley & Sons, Inc. Networks 30:47–56, 1997
TL;DR: An infinite family of vertex critical 4-dichromatic circulant tournaments is presented, answering the problem posed by Neumann-Lara and Urrutia (1984).
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TL;DR: It is shown that infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit are constructed.
Abstract: We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself.
18 Dec 1997
TL;DR: The parallel decentralized node-invariant broadcast and p-gossiping algorithms are proposed which provide a minimum execution time and a minimum of message loading of a network during message-passing.
Abstract: In this paper we consider the broadcast and gossiping problems in circulant networks. The circulant graphs are studied extensively as reliable interconnection networks for the multiprocessor systems. We consider gossiping in the store-and-forward, full-duplex and shouting model for the case when communicating nodes can exchange up to a fixed number p of packets at each round of gossiping (p-gossiping). A general method for evaluation of the lower bounds for p-gossiping in circulant graphs is established. The parallel decentralized node-invariant broadcast and p-gossiping algorithms are proposed which provide a minimum execution time and a minimum of message loading of a network during message-passing.