Showing papers on "Circulant matrix published in 1992"
TL;DR: Beylkin and Rokhlin this paper presented exact and explicit representations of the differential operators in orthonormal bases of compactly supported wavelets as well as the representations of Hilbert transform and fractional derivatives.
Abstract: This paper describes exact and explicit representations of the differential operators, ${{d^n } / {dx^n }}$, $n = 1,2, \cdots $, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring $O(N\log N)$ operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length $N = 2^n $ is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from $O(N)$ to $O(\log ^2 N)$ significant entries.
614 citations
TL;DR: It is proved that both inherit nonsingularity and positive-definiteness from A, and fast algorithms for finding superoptimal preconditioners are suggested.
Abstract: While applying iterative methods to solve a linear algebraic system with matrix A, one often uses some preconditioner C. Two kinds of preconditioners are investigated: the “optimal” one, which minimizes $\| C - A \|_F$, and the “superoptimal” one, which minimizes $\| I - C^{ - 1} A \|_F $. It is proved that both inherit nonsingularity and positive-definiteness from A. Fast algorithms for finding superoptimal preconditioners are suggested that take $O(n^2 \log _2 n)$ operations in case of arbitrary A of order n, and only $O(n\log _2 n)$ operations in case of Toeplitz or doubly Toeplitz A.
201 citations
TL;DR: In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems by the conjugate gradient method for different families of circulant preconditioner systems.
Abstract: In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems $Tx = b$ by the conjugate gradient method for different families of circulant preconditioner...
104 citations
Book•
01 May 1992
TL;DR: In this article, a general H-function of two variables and the solution of its convergence problem are discussed. But they do not consider the convergence problem of general integral convolutions.
Abstract: General H-function of two variables and the solution of its convergence problem main properties, series presentations and characteristic of the H-function H-function with the third characteristic and its particular cases G-function of two variables table of special cases of the G-function one-dimensional H-transform in spaces -1(L) and -1c,gamma(L) and its composition structure classical Laplace convolution and its new properties general integral convolution for H-function transform existence and factorization property of the convolution new examples of convolution for classical integral transforms generalized integral convolution general Leibniz rules and their integral analogs.
92 citations
TL;DR: In this paper, the authors proposed new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of the matrices.
Abstract: In this paper are suggested new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom and Heinig-Rost formulas and are useful for complexity analysis.
89 citations
TL;DR: It is proven that if the convolution product converges to f uniformly, then the circulant preconditioned Toeplitz systems will have a clustered spectrum.
Abstract: Circulant preconditioners for Hermitian Toeplitz systems are considered from the viewpoint of function theory. It is shown that some well-known circulant preconditioners can be derived from convoluting the generating function f of the Toeplitz matrix with famous kernels like the Dirichlet and the Fejer kernels. Several circulant preconditioners are then constructed using this approach. Finally, it is proven that if the convolution product converges to f uniformly, then the circulant preconditioned Toeplitz systems will have a clustered spectrum.
49 citations
TL;DR: If instead the generating function is only piecewise continuous, then for all [epsilon] sufficiently small, there are O(log n) eigenvalues of C[sup [minus]1][ sub n]T[sub n] that lie outside the interval (1 - [ep silon], 1 + [epSilon]).
Abstract: The authors consider the solution of n-by-n Toeplitz systems T[sub n]x = b by preconditioned conjugate gradient methods. The preconditioner C[sub n] is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes [parallel]B[sub n] - T[sub n][parallel][sub F] over all circulant matrices B[sub n]. For Toeplitz matrices generated by positive 2[pi]-periodic continuous functions, they have shown earlier that the spectrum of the preconditioned system C[sup [minus]1][sub n]T[sub n] is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear. However, in this paper, they show that if instead the generating function is only piecewise continuous, then for all [epsilon] sufficiently small, there are O(log n) eigenvalues of C[sup [minus]1][sub n]T[sub n] that lie outside the interval (1 - [epsilon], 1 + [epsilon]). In particular, the spectrum of C[sup [minus]1][sub n]T[sub n] cannot be clustered around 1. Numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general. 20 refs.
40 citations
TL;DR: Ada´m's conjecture about isomorphisms of circulant graphs with degree 4 is proven, it is shown that, for a simple family of exceptions, the graph is sufficient to determine up to isomorphism the group and the set giving the edges.
Abstract: The correspondence between finite abelian groups and their Cayley graphs is studied in the case of degree 4. We show that, but for a simple family of exceptions, the graph is sufficient to determine up to isomorphism the group and the set giving the edges. Ada´m's conjecture about isomorphisms of circulant graphs with degree 4 is thus proven.
28 citations
TL;DR: In this research, the spectral clustering property of the preconditioned matrix R^{ - 1} T with T generated by two-dimensional rational functions $T(z_x ,z_y )$ of order $(p_x,q-x,p_y ,q-y)$ is examined and it is shown that the eigenvalues of $R^{ -1} T$ are clustered around unity.
Abstract: Research on preconditioning Toeplitz matrices with circulant matrices has been active recently. The preconditioning technique can be easily generalized to block Toeplitz matrices. That is, for a block Toeplitz matrix T consisting of $N \times N$ blocks with $M \times M$ elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spectral clustering property of the preconditioned matrix $R^{ - 1} T$ with T generated by two-dimensional rational functions $T(z_x ,z_y )$ of order $(p_x ,q_x ,p_y ,q_y )$ is examined. It is shown that the eigenvalues of $R^{ - 1} T$ are clustered around unity except at most $O(M\gamma _y + N\gamma _x )$ outliers, where $\gamma _x = \max (p_x ,q_x )$ and $\gamma _y = \max (p_y ,q_y )$. Furthermore, if T is separable, the outliers are clustered together such that $R^{ - 1} T$ has at most $(2\gamma _x + 1)(2\gamma _y + 1)$ asymptotic distinct eigenvalues. The superior convergence behavior of the preconditioned c...
TL;DR: In this article, the authors analytically study a certain class of non-uniform multiconductor transmission lines (NMTL), described by circulant impedance and admittance matrices.
Abstract: The authors analytically study a certain class of nonuniform multiconductor transmission lines (NMTL). This class is described by circulant impedance and admittance matrices. Due to the properties of circulant matrices (which form a special set of normal matrices) it is shown that the NMTL equations can be simultaneously diagonalized with the aid of the position-independent so-called Fourier matrix. Thereby, the NMTL equations decouple completely and the resulting equations for the modal components of the voltages and currents may be solved with techniques and methods already known from the analyses of single nonuniform transmission lines. Application examples covering the high-frequency propagation on NMTL and an exact solution (in the frequency domain) a coupled transmission line model for a unit cell of a periodic array of wave launchers are presented. The methods are also applied to a diverging pair of conductors above a perfectly conducting plane. Special interest, however, is focused on C/sub N/ and C/sub Nc/ symmetry of the NMTL configurations. >
TL;DR: The authors propose using the correlated cosine structure (CCS) for the computation of the discrete cosine transform (DCT) and show that there exists a close relationship between the CCS and the DCT.
Abstract: The authors propose using the correlated cosine structure (CCS) for the computation of the discrete cosine transform (DCT) This structure has circulant property and is most suitable for the hardware realization They show that there exists a close relationship between the CCS and the DCT In such a case, a 2/sup m/ length DCT can be decomposed recursively into shorter length CCS and DCT This new approach results in very simple and straightforward structure and gives the minimum number of multiplications for its realization >
TL;DR: A general algebraic method for finding the eigenvectors and the Eigenvalues of multilevel circulants is given and several classes of graphs can be analyzed using this scheme.
Abstract: A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.
TL;DR: An efficient solution of the max-cut problem in circulant graphs Cn,r is obtained so that it is shown that there is always a maximum cut of a particular shape, called a t-regular cut.
TL;DR: In this article, the block structure of k-circulant matrices A of order n (k≥2, k¦n) is investigated and statements enabling to reduce a series of problems with the matrices B+AT to analogous problems with matrices of lower order, namely the blocks of B and AT.
Abstract: The block structure of k-circulant matrices A of order n (k≥2, k¦n) is investigated and statements, enabling to reduce a series of problems with the matrices A+AT to analogous problems with matrices of lower order, namely the blocks of the matrices A and AT, are proved. The spectrum and the number of spanning trees of an undirected de Bruijn graph are obtained.
TL;DR: A special class of matrices W that are a generalization of reflexive and antireflexive matrices are introduced, their fundamental properties are developed, and a decomposition method associated with W is presented.
Abstract: In this paper a special class of matrices W in $\mathcal{C}^{n \times n} $ that are a generalization of reflexive and antireflexive matrices are introduced, their fundamental properties are developed, and a decomposition method associated with W is presented. The matrices W have the relation $W = e^{i\theta } P^ * WP,\,i = \sqrt { - 1} ,\theta \in \mathcal{R}$, where e is the exponential function and P an $n \times n$ unitary matrix with the property $P^k = I,\,k \geq 1$. The superscript $ * $ denotes the conjugate transpose and I is the identity matrix. It is assumed that k is finite and is the smallest positive integer for which the relation holds. The matrices W are referred to as circulative matrices of degree $\theta $ with respect to P. Embedded in this class of matrices are two special types of matrices U and V , $U = P^ * UP$ and $V = - P^ * VP$, which bear a great resemblance to reflexive matrices and antireflexive matrices, respectively. The matrices U and V are simply called circulative matrice...
TL;DR: Circulant codes related to QR codes are studied and an expression for the weight of a codeword in the code is given in terms of a Legendre sum.
TL;DR: The numerical stability of this algorithm is studied and it is shown that the algorithm is weakly stable; i.e., if the circulant matrix is well conditioned, then the computed solution is close to the exact solution.
Abstract: The solution of a set of linear equations involving a circulant matrix is easily accomplished with an algorithm based on fast Fourier transforms. The numerical stability of this algorithm is studied. It is shown that the algorithm is weakly stable; i.e., if the circulant matrix is well conditioned, then the computed solution is close to the exact solution. On the other hand, it is shown that the algorithm is not strongly stable—the computed solution is not necessarily the solution of a nearby circular deconvolution problem.
TL;DR: This paper constructs circulantGH(p2;Zp) for all primesp, and extends the earlier result that there exist circulants GH(p;Z p) forall odd primes p.
Abstract: The only known circulant ordinary Hadamard matrix is developed from the initial row-1, 1, 1, 1. Letp be a prime, and letZ p denote the cyclic group of orderp. In this paper, we construct circulantGH(p 2;Z p ) for all primesp. Whenp is odd, this result also extends the earlier result that there exist circulantGH(p;Z p ) for all odd primesp. Other families ofGH-matrices which are developed modulo a group are discussed.
TL;DR: For second-order hyperbolic equations with initial and Dirichlet boundary conditions, it is proved that the condition number of the preconditioned system is ofO(α) orO(m), where α is the quotient between the time and space steps andm is the number of interior gridpoints in each direction.
Abstract: Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO(α) orO(m), where α is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.
TL;DR: In this paper, the convergence rate of the preconditioned conjugate gradient method for Hermitian Toeplitz systems has been shown to converge superlinearly.
TL;DR: The aim of this paper is to consider skewcirculant preconditioners, and to show that in this case the condition number ofO(n) can easily be shown without using the somewhat unsatisfactory shift ρ/p/n2.
Abstract: In a recent paper Chan and Chan study the use of circulant preconditioners for the solution of elliptic problems. They prove that circulant preconditioners can be chosen so that the condition number of the preconditioned system can be reduced fromO(n 2 ) toO(n). In addition, using the Fast Fourier Transform, the computation of the preconditioner is highly parallelizable. To obtain their result, Chan and Chan introduce a shift ?/p/n 2 for some ?>0. The aim of this paper is to consider skewcirculant preconditioners, and to show that in this case the condition number ofO(n) can easily be shown without using the somewhat unsatisfactory shift ?/p/n 2. Furthermore, our estimates are more precise.
TL;DR: 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, and how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots.
Abstract: We show 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, we prove 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. We then extend the Tardos scheme to our case, obtaining a running time which is independent of the objective and right-hand side data. As a consequence of these results, we are able to show that LPs with real circulant coefficient matrices can be solved in strongly polynomial time. Finally, we show how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots.
30 Nov 1992
TL;DR: It is shown that regularization can easily be incorporated into the preconditioned conjugate gradient algorithm using square nonsingular circulant-block and related preconditionsers, constructed from the blocks of the rectangular matrix T.
Abstract: Discretized 2-D deconvolution problems arising, e.g., in image restoration and seismic tomography, can be formulated as 1eas squares compuaions, mm lib— Tx112 where T is often a large-scale rectangular Toeplitz-block matrix. We consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant-block and related preconditioners, constructed from the blocks of the rectangular matrix T. Preconditioning with such matrices allows efficient implementation using the 1-D or 2-D Fast Fourier Transform (FFT). It is well known that the resolution of ill-posed deconvolution problems can be substantially improved by regularization to compensate for their ill-posed nature. We show that regularization can easily be incorporated into our preconditioners, and we report on numerical experiments on a Cray Y-MP. The experiments illustrate good convergence properties of these FET—based preconditioned iterations.
TL;DR: A (70,35) circulant code was previously characterized in terms of the incidence matrix of a Hadamard difference set based on the twin prime product 35 but the weight distribution of this code is computed and extended to a (72,36) d=12 self-dual code which is not equivalent to the extended quadratic residue code.
01 Jan 1992
TL;DR: By solving a system of linear algebraic equations combined with the asymptotics of wavelet coefficients, this paper arrives at an effective method for computing the nonstandard form of convolution operators.
Abstract: This paper describes exact and explicit representations of the differential operators, dn/dXn, n = 1,2,.--, in orthonormal bases of compactly supported wavelets as well as the rep- resentations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators. Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring O(N log N) operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length N = 2n is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see (G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141-183)) may be reduced from O(N) to O(log2 N) significant entries. The computation of the nonstandard forms of many other operators reduces to solving a simple system of linear algebraic equations. Among such operators are fractional derivatives, Hilbert and Riesz transforms, and other operators for which analytic expressions are available. For convolution operators, there are significant simplifications in computing the nonstandard form since the vanishing moments of the autocorrelation function of the scaling function simplify the quadrature formu- las. Moreover, by solving a system of linear algebraic equations combined with the asymptotics of wavelet coefficients, we arrive at an effective method for computing the nonstandard form of convolution operators. As examples, we compute the nonstan- dard forms of the Hilbert transform and fractional derivatives. The generalization of this method for multidimensional convolution operators is straightforward.
TL;DR: This decomposition of the circulant representation of the resulting matrix offers insight into the process of 2-D convolution and has applications in image processing.