Etudes sur les ε- et ρ-algorithmes

Some methods of accelerating the convergence of sequences are studied herein. It is shown that, in some special cases, the convergence of the -algorithm of Wynn is assured. A modification of this algorithm is then proposed which may still accelerate the convergence toward the limit. Finally, a new -algorithm is given in which the dependence with respect to a parameter is explicitely introduced.     

On interpolatory multivariate Padé-type approximants

It is proved that higher order interpolatory Padé-type approximants in two variables do not exist.     

Look-ahead in Bi-CGSTAB and other product methods for linear systems

The Lanczos method for solvingAx = b consists in constructing the sequence of vectorsx _k such thatr _k =b – Ax _k =P _k (A)r ₀ whereP _k is the orthogonal polynomial of degree at mostk with respect to the linear functionalc whose moments arec( _i ) =c _i = (y, A ⁱ r ₀).In this paper we discuss how to avoid breakdown and near-breakdown in a whole class of methods defined byr _k =Q _k (A)P _k (A)r ₀,Q _k being a given polynomial. In particular, the case of the Bi-CGSTAB algorithm is treated in detail. Some other choices of the polynomialsQ _k are also studied.     

Fission fragment angular distributions in 16O+181Ta

Time of flight and energy of fission fragments were measured using pulsed beam. Fission fragment mass and energy integrated angular distributions were extracted. Fission fragment anisotropy was explained in the framework of saddle point model.     

Rational extrapolation for the PageRank vector

An important problem in web search is to determine the importance of each page. From the mathematical point of view, this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1. Since this matrix is neither stochastic nor irreducible, the power method has convergence problems. So, the matrix is replaced by a convex combination, depending on a parameter , with a rank one matrix. Its left principal eigenvector now depends on , and it is the PageRank vector we are looking for. However, when is close to 1, the problem is ill-conditioned, and the power method converges slowly. So, the idea developed in this paper consists in computing the PageRank vector for several values of , and then to extrapolate them, by a conveniently chosen rational function, at a point near 1. The choice of this extrapolating function is based on the mathematical expression of the PageRank vector as a function of . Numerical experiments end the paper.

     

Extrapolation Algorithms for Filtering Series of Functions,and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the -algorithm) or by approximating the series by a rational function (in particular, a Padé approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.     

Treatment of near-breakdown in the CGS algorithm

Lanczos' method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x _k ) such thatr _k =b–Ax _k =P _k (A)r ₀ wherer ₀=b–Ax ₀.P _k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr _k =P _k ² (A)r₀. In the recurrence relation forP _k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.     

Some results in the theory of the vector ε-algorithm

The solution of systems of linear equations by mean of the vector ε-algorithm is investigated. The cases with a singular matrix are studied which show that, under certain assumptions on the minimal polynomial of the matrix, a solution can be obtained. Some theorems concerning the application of the ε-algorithm to vectors satisfying a matrix difference equation are proved. These results generalize results on the scalar ε-algorithm and some recent theorems on the vector ε-algorithm.