Showing papers on "Circulant matrix published in 1985"
TL;DR: In this paper, the minimum diameter, maximum connectivity circulant problem is considered and several results are given for the general case and a simple solution is derived for the connectivity four case.
Abstract: It is well known that maximum connectivity graphs play an important role in the design of reliable networks. The class of symmetric graphs called circulants is known to contain such maximum connectivity graphs. Although not all circulants have this maximum connectivity property, those that do have a great variation in their diameters. Since diameter is a measure of transmission delay, the minimum diameter, maximum connectivity circulant problem is considered here. Several results are given for the general case and a simple solution is derived for the connectivity four case.
116 citations
TL;DR: The algorithm is in essence a fast implementation of the Trench algorithm in reverse and involves imbedding of the given matrix in a cyclic matrix and a fast HD (half-divisor) algorithm to compute the first row of the inverse matrix.
Abstract: A fast algorithm for the solution of a Toeplitz system of equations is presented. The algorithm requires order N(\log N)^{2} computations where N is the number of equations. For banded Toeplitz matrices the order of computations is reduced to only N \log N + m(\log m)^{2} where 2m is the maximum number of nonzero principal subdiagonals of the Toeplitz matrix. The algorithm is in essence a fast implementation of the Trench algorithm in reverse. Thus, the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast HD (half-divisor) algorithm to compute the first row of the inverse matrix. The desired solution is then obtained directly from the first row by applying fast Fourier transform techniques in order N \log N computations. Finally, the extension of the algorithm to block Toeplitz matrices is also presented.
73 citations
TL;DR: On etudie les solutions periodiques non triviales de familles parametrisees d'equations differentielles a retard as mentioned in this paper, les solutions are non triviale de families.
36 citations
24 citations
TL;DR: A complete classification of self-dual doublycirculant codes of any length over GF q is presented, generalizing the results on orthogonal circulant matrices obtained by MacWilliams [5].
24 citations
TL;DR: The circulant approximation of vector and quadratic forms involving the inverse of a Toeplitz covariance matrix R is addressed and a result is presented which increases the rate of convergence of the average matrix error under certain conditions on \undertilde{r} , the vector which defines R.
Abstract: The circulant approximation of vector and quadratic forms involving the inverse of a Toeplitz covariance matrix R is addressed. First, a result is presented which increases the rate of convergence of the average matrix error under certain conditions on \undertilde{r} , the vector which defines R. Concerning vector and quadratic operations using R-1, it is noted that if \undertilde{x} is AR(p), then the p-banded, near-Toeplitz structure of R-1results in an O(1/N)-type mean convergence of associated errors.
24 citations
TL;DR: Circulant matrices of order t with elements circulants of order s are used for the construction of D-optimal saturated designs of order N = 2st and the optimal design is constructed for the first time.
19 citations
9 citations
TL;DR: In this paper, it is shown that if the product of every two circulant matrices is again one, then they are very nearly group matrices, and it is also possible to determine the styles of circulants preserved under multiplication by group matrix.
Abstract: Consider an arbitrary style of "circulant" matrix, with rows derived by fixed permutations from the first row. If the product of every two such matrices is again one, then they are very nearly group matrices. Specifically, there is a group G and a subordinate index set S such that the indices correspond to G×S and the matrix entry in row (g s) and column (ǵś) is the entry that occurs in the first row in column (g-1ǵ,ś). It is also possible to determine the styles of circulant preserved under multiplication by group matrices.
7 citations
TL;DR: A constructive characterization of all self dual 2n circulant codes over F2r is given, and the number of these codes, as well as an outline of an algorithm to construct them, are given.
Abstract: We give a constructive characterization of all self dual 2n circulant codes over F2r. This construction generalizes those of F. J. MacWilliams [3], and G.F.M. Beenker [1]. Our method is original. We also give the number of these codes, as well as an outline of an algorithm to construct them. At last, we give several codes we have obtained by software at AAECC Lab.
TL;DR: In this article, the analysis of band circulant matrices which occur in the periodic spline interpolation theory with equispaced knots with explicit bounds on the inverse matrices are given.
TL;DR: In this article, the authors enumerate the nonisomorphic classes of 2-circulant graphs X(S, q, F) such that |S| = m and |F| = k.
Abstract: Alspach and Sutcliffe call a graph X(S, q, F) 2-circulant if it consists of two isomorphic copies of circulant graphs X(p, S) and X(p, qS) on p vertices with “cross-edges” joining one another in a prescribed manner. In this paper, we enumerate the nonisomorphic classes of 2-circulant graphs X(S, q, F) such that |S| = m and |F| = k. We also determine a necessary and sufficient condition for a 2-circulant graph to be a GRR. The nonisomorphic classes of GRR on 2p vertices are also enumerated.
TL;DR: In this article, a constant-denominator perturbation formalism was developed employing a basis of circulant orbitals and a projected Moller-Plesset partitioning of the Hamiltonian operator.
Abstract: A constant-denominator perturbation formalism is developed employing a basis of circulant orbitals and a projected Moller–Plesset partitioning of the Hamiltonian operator. A formal justification for the classical Unsold approximation is thereby provided. A calculation of correlation energy in the beryllium atom is carried out, and the results are compared with results obtained by the full configuration interaction method and conventional Moller–Plesset perturbation theory.
TL;DR: In this paper, a factorisation procedure for banded symmetric matrices which occur repeatedly in the solution of ordinary/partial differential equations under periodic boundary conditions is described, and the numerical solution to the derived special linear systems can then be obtained efficiently by a sequence of simple forward and back substitution processes.
Abstract: A factorisation procedure is described for certain banded symmetric matrices which occur repeatedly in the solution of ordinary/partial differential equations under periodic boundary conditions. The numerical solution to the derived special linear systems can then be obtained efficiently by a sequence of simple forward and back substitution processes.
TL;DR: In this article, the problem of finding sufficient conditions on an integral p×p circulant C which ensure that C can be factored in the form C1C'1 (where'denotes transpose) with C 1 an integral circulants is referred to, and it is conjectured that such a factorization is possible if C is unimodular and positive definite symmetric and 12(p-1) is also a prime.
01 Aug 1985
TL;DR: The solution of certain Toeplitz linear systems is considered and the methods presented here are more efficient than the Cholesky decomposition method and are based on the circulant factorization of the bandedcirculant matrix, the use of the Woodbury formula and algebraic perturbation method.
Abstract: : The solution of certain Toeplitz linear systems is considered in this paper. This kind of system is encountered when we solve certain partial differential equations by finite difference techniques and approximate functions using higher order splines. The methods presented here are more efficient than the Cholesky decomposition method and are based on the circulant factorization of the banded circulant matrix, the use of the Woodbury formula and algebraic perturbation method. Additional keywords: Boundary value problems; FORTRAN. (Author)
TL;DR: The class of kernels of convolution integral operators is defined in this article, and a criterion for a measurable function to belong to this class is given, and the question of the behavior of the Fourier transform of a kernel (the symbol) in is considered.
Abstract: The class of kernels of convolution integral operators is defined, and a criterion for a measurable function to belong to is given. The question of the behavior of the Fourier transform of a kernel (the symbol) in is considered, and it is shown that in the sense of order the symbol can vanish at infinity in an arbitrarily slow manner, and more slowly than any power in the mean.Bibliography: 6 titles.
01 Apr 1985
TL;DR: A new 2-D array architecture which fully exploits features of large block circulant sparse matrices, which have all non-zero elements within an elliptical or general closed strip, is introduced for the iterative solution of linear systems involving such matrices.
Abstract: Large block circulant sparse matrices whose submatrices have all non-zero elements within an elliptical or general closed strip, appear in Tomographic image reconstruction from projections A new 2-D array architecture which fully exploits these features is introduced for the iterative solution of linear systems involving such matrices Each sub row-vector product is computed in parallel by one row of simple processing units and pipelined to give a high utilisation ratio A two level controller is used having one Main Control Unit (MCU) and identical simple Sub-Control Units (SCU) - one for each row of the array This allows appropriate flow of the data and partial products, which in effect split and straighten the elliptical strip to produce two strip diagonal matrices The VLSI designs for the MCU and SCU have been completed