The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different problems from different applicative areas. In this paper we survey the main features of cyclic reduction, relate it to properties of analytic functions, recall its extension to solving more general finite and infinite linear systems, and different kinds of nonlinear matrix equations, including algebraic Riccati equations, with applications to Markov chains, queueing models and transport theory. Some new results concerning the convergence properties of cyclic reduction and its applicability are proved under very weak assumptions. New formulae for overcoming breakdown are provided.     

Computing curve intersection by means of simultaneous iterations

A new algorithm is proposed for computing the intersection of two plane curves given in rational parametric form. It relies on the Ehrlich–Aberth iteration complemented with some computational tools like the properties of Sylvester and Bézout matrices, a stopping criterion based on the concept of pseudo-zero, an inclusion result and the choice of initial approximations based on the Newton polygon. The algorithm is implemented as a Fortran 95 module. From the numerical experiments performed with a wide set of test problems it shows a better robustness and stability with respect to the Manocha–Demmel approach based on eigenvalue computation. In fact, the algorithm provides better approximations in terms of the relative error and performs successfully in many critical cases where the eigenvalue computation fails.     

A formula for the Euler characteristic of $\overline{{\cal M}}_{2,n}$

In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves. The proof relies on quite elementary methods, such as the enumeration of the graphs involved in a suitable stratification of . Received September 8, 1998; in final form July 1, 1999 / Published online December 8, 2000     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

Influence of intrinsic nucleon motion on energy spectra and angular distributions for O-induced reactions at 20 MeV/A

Energy spectra and angular distributions for peripheral reactions induced by ¹⁶O on ²⁰⁸Pb at 20 MeV/A can be explained in terms of the intrinsic nucleon motion of the excited projectile, in a manner similar to the interpretation of projectile fragmentation reactions at 2.1 GeV/A.     

The unitary completion and QR iterations for a class of structured matrices

We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower triangular part of a special form. For such a specified part the unitary completion is a structured matrix, and we derive in this paper the formulas for its structure. Next we apply the unitary completion method to the solution of the eigenvalue problem for a class of structured matrices via structured QR iterations.

     

Structural and spectroscopic properties of pure and doped Ba6Ti2Nb8O30 tungsten bronze

Pure and doped Ba(6)Ti(2)Nb(8)O(30) (BTN), obtained by substituting M = Cr, Mn, or Fe on the Ti site (Ba(6)Ti(2-x) M(x)Nb(8)O(30), x = 0.06 and 0.18) and Y and Fe on the Ba and Ti sites, respectively (Ba(6-x)Y(x)Ti(2-x)Fe(x)Nb(8)O(30), x= 0.18), are synthesized. The influence of cation doping on the local structure, the cation oxidation state, and the possible defect formation able to maintain the charge neutrality are investigated by spectroscopic (electron paramagnetic resonance (EPR) and micro-Raman), structural (X-ray powder diffraction) and transport (impedance spectroscopy, thermoelectric power) measurements, in the temperature range of 300-1200 K in air and N(2) flow. Starting from the valence state of the doping ions (Fe(3+), Cr(3+), and Mn(2+)), determined by EPR, and from thermoelectric power measurements, evidencing a negative charge transport, different charge-compensating defect equilibria, based on the creation of positive electron holes or oxygen vacancies and electrons, are discussed to interpret the conductivity results.     

Physical frames along circular orbits in stationary axisymmetric spacetimes

Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi–Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes. Dedicated to Bahram Mashhoon for his 60th birthday.