Ana C. Matos

A derivation of extrapolation algorithms based on error estimates

In this paper, we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolation processes. It is shown that this approach leads to a unified derivation of many extrapolation algorithms and related devices, to general results about their kernels and that it opens the way to many new algorithms. Their convergence and acceleration properties could also be studied within this framework.

Recursive computation of Padé–Legendre approximants and some acceleration properties

Summary. In this paper, after recalling the two definitions of the generalizations of the Padé approximants to orthogonal series, we will define the Padé–Legendre approximants of a Legendre series. We will propose two algorithms for the recursive computation of some sequences of these approximants. We will also estimate the speed of convergence of the columns of the Padé–Legendre table from the asymptotic behaviour of the coefficients of the Legendre series. Finally we will illustrate these results with some numerical examples.

Acceleration property for the columns of the E-algorithm

A convergence acceleration result for the E-algorithm is proved for sequences such that the error has an asymptotic expansion on a scale of comparison for which a determinantal relation holds. This result is also generalized to the vector case.

Least-squares orthogonal polynomials

Abstract In this paper the new notion of least-squares orthogonal polynomials is introduced. Their existence and uniqueness is studied and some methods for their recursive computation are given. The location of their zeros in the case of a positive definite functional is studied. The paper ends with applications to Pade-type approximation and Gaussian quadrature methods.

Acceleration methods based on convergence tests

SummaryIn this paper we propose an acceleration method based on a general convergence test and depending on an auxiliary sequence (xn). For different choices of (xn) we obtain some known and some new transformations and for each one sufficient conditions for acceleration are given.

Polishing Domestic Wastewater on a Subsurface Flow Constructed Wetland: Organic Matter Removal and Microbial Monitoring

Microbial monitoring of constructed wetlands (CWs) treating domestic wastewater is generally scarce, despite the need of more knowledge about its biocenosis. The sanitation quality of a wastewater treated in a CW is a crucial aspect, mainly when the receiving water body is used as a swimming and/or recreation area. The present study was carried out in a horizontal subsurface flow CW planted with Phragmites australis receiving pre-treated domestic wastewater (mean flow 50 m3 day−1), from a population of about 300 inhabitants. The monitoring programme undertaken during the first year operation, revealed removal efficiencies of 61% BOD5, 44% COD, and 65% TSS for inlet water with ca. 90 mg L−1 BOD5, 157 mg L−1 COD, and 17 mg L−1 TSS. Total Coliform (TC) and Faecal Coliform (FC) bacteria were removed from wastewater (mean inlet values of 5 × 106 CFU 100 mL−1 TC and of 9 × 105CFU 100 mL−1 FC), with efficiencies of 92 and 97%, respectively. The dynamics of microbial communities established in the system assessed by polymerase chain reaction-denaturing gradient gel electrophoresis (PCR-DGGE), had revealed a high bacterial diversity within the system, with no relevant differences in composition at the CW inlet and outlet but exhibiting temporal differences in bacterial communities.

Some convergence results for the generalized Padé-type approximants

The aim of this paper is to give some convergence results for some sequences of generalized Padé-type approximants. We will consider two types of interpolatory functionals: one corresponding to Langrange and Hermite interpolation and the other corresponding to orthogonal expansions. For these two cases we will give sufficient conditions on the generating functionG(x, t) and on the linear functionalc in order to obtain the convergence of the corresponding sequence of generalized Padé-type approximants. Some examples are given.

Acceleration results for the vector E-algorithm

AbstractIn this paper we are going to study the convergence and acceleration properties of the vector E-algorithm when applied to some families of vector sequences of the form

Extrapolation algorithms based on the asymptotic expansion of the inverse of the error; application to continued fractions

S_n - S = \sum\limits_{i = 0}^k {or} {\text{ }}S_n - S \sim \sum\limits_{i = 0}^\infty {a_i g_i (n)}