A general extrapolation procedure revisited

TheE-algorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into theE-algorithm, and allows one to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed.     

Avoiding breakdown and near-breakdown in Lanczos type algorithms

Lanczos type algorithms form a wide and interesting class of iterative methods for solving systems of linear equations. One of their main interest is that they provide the exact answer in at mostn steps wheren is the dimension of the system. However a breakdown can occur in these algorithms due to a division by a zero scalar product. After recalling the so-called method of recursive zoom (MRZ) which allows to jump over such breakdown we propose two new variants. Then the method and its variants are extended to treat the case of a near-breakdown due to a division by a scalar product whose absolute value is small which is the reason for an important propagation of rounding errors in the method. Programming the various algorithms is then analyzed and explained. Numerical results illustrating the processes are discussed. The subroutines corresponding to the algorithms described can be obtained vianetlib.     

Hybrid procedures for solving linear systems

Summary. In this paper, we introduce the notion of hybrid procedures for solving a system of linear equations. A hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one. Thus the combination only depends on one parameter whose value is chosen in order to minimize the Euclidean norm of the residual vector obtained by the hybrid procedure. Properties of such procedures are studied in detail. The two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods. Several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed. Numerical experiments illustrate the various procedures. Received October 21, 1992/Revised version received May 28, 1993     

Forme confluente de l'ε-algorithme topologique

This paper deals with a confluent form of the topological ε-algorithm which is a method to accelerate the convergence of a sequence of elements of a topological vector space. After giving the rules of the algorithm it is related to some generalizations of the functional Hankel determinants. Some properties and some results about it are proved. An interpretation of the algorithm is given. The last paragraph is devoted with convergence results about the confluent form of the topological ε-algorithm. A parameter is introduced in the algorithm to accelerate the convergence. The optimal value of this parameter is caracterized. By estimating this optimal value, the confluent form of the ?-algorithm is obtained. The paper ends with a remark about the confluent form of the topological ?-algorithm.     

Addendum to “Avoiding breakdown and near-breakdown in Lanczos type algorithms”

Some new numerical results for the near-breakdown free version of Lanczos method are reported.     

Block Descent Methods and Hybrid Procedures for Linear Systems

In this paper, we define and study several types of block descent methods for the simultaneous solution of a system of linear equations with several right hand sides. Then, improved block EN methods will be proposed. Finally, block hybrid and minimal residual smoothing procedures will be considered.     

Use of half-cell barriers to eliminate junction clogging and thermal hysteresis in silver/silver chloride reference electrodes

The junctions of Ag/AgCl electrodes rapidly become clogged with precipitated silver chloride during use. This causes loss of flow and slow response with thick junctions and may cause offset and stirring error with thin junctions. This problem can be eliminated by using a pure potassium chloride junction electrolyte, together with suitable barriers to impede migration of silver chloride from the half-cell. Diffusional transport through a mechanical barrier is inversely proportional to its electrical resistance when filled with electrolyte. Based on this principle, very effective barriers to half-cell diffusion and flow were fabricated from microporous glass. Alternatively, a cation-selective membrane was used to block soluble silver chloride complexes without a significant increase in electrode resistance. Thermal drift and hysteresis are not inherent features of the Ag/Cl couple and are eliminated by proper design of the half-cell.     

Convergence acceleration of some sequences by the ɛ-algorithm

Summary It is proved that the sequences formed by the columns and the diagonals of the array of the -algorithm converge faster than the initial sequence when it is either a non logarithmic totally monotonic sequence or a converging totally oscillating sequence.     

Extrapolation methods for PageRank computations

The mathematical problem behind Web search is the computation of the nonnegative left eigenvector of a stochastic matrix P corresponding to the dominant eigenvalue 1. This vector is called the PageRank vector. Since the matrix P is ill-conditioned, the computation of PageRank is difficult and the matrix P is replaced by $P(c)=cP+(1?c)E$, where E is a rank one matrix and c a parameter. The dominant left eigenvector of $P(c)$ is denoted by PageRank$(c)$. This vector can be computed for several values of c and then extrapolated at the point $c=1$. In this Note, we construct special extrapolation methods for this problem. They are based on the mathematical analysis of the vector PageRank$(c)$. To cite this article: C. Brezinski et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).