1,721,061 research outputs found
Extrapolation methods: theory and practice
No full textThis volume is a self-contained, exhaustive exposition of the extrapolation methods theory, and of the various algorithms and procedures for accelerating the convergence of scalar and vector sequences. Many subroutines (written in FORTRAN 77) with instructions for their use are provided on a floppy disk in order to demonstrate to those working with sequences the advantages of the use of extrapolation methods. Many numerical examples showing the effectiveness of the procedures and a consequent chapter on applications are also provided - including some never before published results and applica
Transpose-free Lanczos-type algorithms for nonsymmetric linear systems
No full textThe method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB
On the zeros of various kinds of orthogonal polynomials
No full textRecently, several generalizations of the notion of orthogonal polynomials appeared in the literature. The aim of this paper is to study their zeros
A look-ahead strategy for the implementation of some old and new extrapolation methods
No full textSequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of the G-transformation and the Shanks transformation (that is the -algorithm of Wynn), there is a connection with formal orthogonal polynomials. In this paper, this connection is exploited in order to propose a look-ahead strategy for the implementation of these two transformations. This strategy, which is quite similar to the strategy used for treating the same type of problems in Lanczos-based methods for solving systems of linear equations, consists in jumping over the polynomials which do not exist, thus avoiding a division by zero (breakdown) in the algorithms, and over those which could be badly computed (near-breakdown) thus leading to a better numerical stability. Numerical examples illustrate the procedure
Extrapolation and prediction of sequences in a vector space
No full textThe aim of this paper is to present a general theoretical framework for the extrapolation and the prediction of sequences of elements in a vector space. Then, particular cases are studied and recursive algorithms for implementing some of the procedures obtained are discussed. Possible extensions of this work are evoked
These strange fractions which never end or the history of continued fractions
No full textIn this paper, we will present a short account on the history of continued fractions, these fractions which never end. First, we will tell some stories for introducing Fibonacci numbers and see how continued fractions could be obtained from them. Then, mathematical properties and applications of continued fractions will be given. Finally, their development through the centuries will be described
Le classement des pages du web par les moteurs de recherche
No full textNous allons exposer les mathématiques qui, dans les moteurs de recherche, se cachent derrière le classement des pages du web selon leur ordre de pertinence décroissante. Puis nous verrons quelles sont les méthodes d’analyse numérique qui sont utilisées pour effectuer ce classement, comment en accélérer la convergence et comment des procédures d’extrapolation permettent de les améliorer
Orthogonal polynomials of dimension -1 in the non definite case
No full textOrthogonal polynomials of dimension are particular case of vector orthogonal polynomials which are, themselves, a particular case of biorthogonal polynomials. In this paper, we give the three-term recurrence relationship satisfied by these polynomials in the non-definite case, that is when some of them do not exist. Orthogonal polynomials of dimension generalize orthogonal polynomials on the unit circle
Convergence acceleration of Kaczmarz's method
No full textThe method of alternating projections (MAP) is an iterative procedure for finding the projection of a point on the intersection of closed subspaces of a Hilbert space. The convergence of this method is usually slow, and several methods for its acceleration have already been proposed. In this work, we consider a special MAP, namely Kaczmarz’s method for
solving consistent systems of linear equations. The convergence of this method is discussed. After giving its matrix formulation and its projection properties, we consider several procedures for accelerating its convergence. They are based on sequence transformations whose kernels contain sequences of the same form as the sequence of vectors generated by Kaczmarz’s method. Acceleration can be achieved either directly, that is without modifying the sequence obtained by the method, or by restarting it from the vector obtained by acceleration. Numerical examples show the effectiveness of both procedures
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