Showing papers on "Circulant matrix published in 1981"

On Certain Parallel Toeplitz Linear System Solvers

Abstract: In this paper we describe three algorithms for solving positive-definite banded Toeplitz systems of linear equations on parallel computers. Assuming we have $4mn$ processors, where n is the order of the system and m is the number of super- or subdiagonals, each system may be solved in $O(m\log n)$ time steps. Numerical experiments that compare the behavior of these algorithms in solving pentadiagonal Toeplitz systems are presented.

Circulant orbitals for atoms and molecules

Abstract: Circulant orbitals ϕn for a closed-shell system are the orbitals obtained when the N canonical orthonormal Hartree-Fock orbitals λ[unk] are subjected to a unitary transformation which is the discrete Fourier transformation: ϕn = 1/√N Σ[unk]λ[unk]ω(n-1)([unk]-1), where ω = exp(2πi/N). Electron densities associated with the orbitals ϕn are each close to the average total electron density. The Fock matrix, diagonal for canonical orbitals, for circulant orbitals is a Hermitian circulant matrix, em, m+q = 1/N Σ[unk]e[unk]ωq([unk]-1), where the e[unk] are the canonical orbital energies. The states ^Fϕn are uniformly distributed on the surface of a sphere in Hilbert space.

The $v \times v $$(0,1,-1)$-Circulant Equation $AA^T = v I - J$

Abstract: An electromechanical pulse generator has been proposed (J. P. Craig and R. Saeks, An electromechanical pulse generator, Proc. 1st IEEE International Pulsed Power Conference, Institute of Electrical and Electronics Engineers, 1976, pp. IIB 7-1–IIB 7-4) which is equivalent to finding a $v \times v $ circulant matrix A with entries from $\{ 0,1, - 1 \}$ such that $AA^T = v I - J$. In this earlier work it is reported that if v is an odd prime and the entries in the first row of A are the Legendre symbols $(j/v )$, $0\leqq j\leqq v - 1$, then A is a solution. It was conjectured that A exists only if $v $ is an odd prime and that the solution is unique up to cyclic permutation of the columns and multiplication of A by $ - 1$. In this paper, using convolution products, Fourier transforms and number theory, we settle these two conjectures affirmatively.

m-adic (polyadic) invariance and its applications

Abstract: The concept of m-adic invariance allows approximation of a linear time-invariant (LTI) system by a linear m-adic invariant (LMI) system or, equivalently, approximation of a circulant matrix by a super-circulant matrix. The approximation reduces the number of multiplies required for computing N-point cyclic convolution to 2(\log_{m}N - 1)N , where N = mn. The error introduced by the approximation can be removed, if desired, by subsequent processing. In one concrete case, determination of a small number of noncontiguous frequencies, this approach-approximation and subsequent correction-can effect substantial savings in a number of multiplies compared to both fast Fourier transform (FFT) algorithm and direct discrete Fourier transform (DDFT). These applications are preceded by a tutorial presentation of concepts which are basic to m-adic invariant systems.

Limit-cycle predictions in sampled data systems

Abstract: For sampled data systems, it is possible to express discrete time convolution in terms of appropriate matrix multiplication. A limiting process then yields the steady-state response to a periodic input. The matrix involved in this operation is a circulant matrix based on the impulse response of the system. Circulant matrices are known to have useful structural properties and permit the association of the frequency domain with the time domain. The use of circulants in the analysis of nonlinear systems provides the means for converting the unwieldy nonlinear equations of continuous systems to simple matrix multiplications. It is then possible to apply numerical range techniques and rederive the circle criterion. The direct application of this approach yields a criterion which is less conservative than that obtained by the simple application of the small-gains theorem. Use of the approach in conjunction with a loop transformation, on the other hand, provides an alternative derivation of the circle criterion for discrete systems. The method can be extended to multivariable systems, and, because of its association with the time domain, it permits the assessment of system stability in the face of imperfect system descriptions, namely truncated impulse responses.

On real factorizations of symmetric circulant sparse matrices

Abstract: For the matricesA mentioned in the headline we determine the limit points up to which there is possible a real factorization of the formA=QQ T . HereQ=(q ij ) is a circulant matrix, where from the elementsq ij andq ji withi≠j always one element is vanishing.