Central orderings for the Newton interpolation formula

The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017. https://doi.org/10.1016/j.jat.2017.07.005; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included

Optimal stability of the Lagrange formula and conditioning of the Newton formula

A pointwise condition number associated to a representation of an interpolation operator is introduced. It is proved that the Lagrange formula is optimal with respect to this conditioning. For other representations of the interpolation operator, an upper bound for the conditioning is derived. A quantitative measure in terms of the Skeel condition number is used to compare the conditioning with the Lagrange representation. The conditioning of the Newton representation is considered for increasing nodes and for nodes in Leja order. For the polynomial Newton formula with n+1 equidistant nodes in increasing order, it is proved that 3n is the best uniform bound of its conditioning and it is attained at the last node. Numerical experiments are included

Optimal interval length for the collocation of the Newton interpolation basis

It is known that the Lagrange interpolation problem at equidistant nodes is ill-conditioned. We explore the influence of the interval length in the computation of divided differences of the Newton interpolation formula. Condition numbers are computed for lower triangular matrices associated to the Newton interpolation formula at equidistant nodes. We consider the collocation matrices L and PL of the monic Newton basis and a normalized Newton basis, so that PL is the lower triangular Pascal matrix. In contrast to L, PL does not depend on the interval length, and we show that the Skeel condition number of the (n + 1) × (n + 1) lower triangular Pascal matrix is 3n. The 8-norm condition number of the collocation matrix L of the monic Newton basis is computed in terms of the interval length. The minimum asymptotic growth rate is achieved for intervals of length 3

Inverse central ordering for the Newton interpolation formula

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples

CARACTERISATION D’HUILES D’OLIVE PRODUITES DANS DES COOPERATIVE PILOTES (LAKRARMA ET KENINE) AU NIVEAU DU MAROC ORIENTAL

In Eastern Morocco, olive tree growing has agreat economic and social importance. Itoccupies about 10% of the national olivegrowingsurface. Olive oil is the principalproduct; more than 70% of the olive productionis intended to trituration. In this region most ofthe olive orchards were established andcultivated by traditional methods, and olive oilextraction is generally carried out by artisanalmethods which can compromise its quality.Classical methods of analysis were used for apreliminary characterisation of olive oilproduced by regional cooperatives (Lakrarma,Kenine localised in Taourirt-Tafoughalt corridor). Analyses relating to free acidity, peroxide index, chlorophyll content and standardvalues of absorption, were realized. In comparison with International Olive Oil Council our results showed that analyzed olive oils have physico-chemical characteristics of a virgin orextra virgin olive oil.Au niveau du Maroc oriental l’oléiculturerevêt une grande importance économique et sociale, elle représente un peu moins de 10% de la superficie oléicole nationale. L’huile d’olive est le principal produit, du fait que plus de 70% de la production oléicole est destinée à la trituration. Dans cette région, la filière reste encore artisanale et mal organisée. L’extraction de l’huile d’olive s’effectue généralement par des méthodes traditionnelles qui peuvent compromettre sa qualité. Pour la caractérisation d’huile d’olive, provenant des coopératives régionales (Lakrarma, Kenine) du couloir Taourirt Tafoughalt, des analyses physicochimiques concernant l’acidité libre, l’indice de peroxyde, la teneur en chlorophylle et les valeurs standards d'absorption en UV ont été réalisées. Les résultats obtenus et selon la norme commerciale du Conseil Oléicole International, montrent que les huiles analysées possèdent des caractéristiques physicochimiques d’huile d’olive vierge à extra vierge

Isly Huile d’Olive Vierge Analyse des Triglycérides et Composition en Acides Gras

Physicochemical characteristics of several olive oils produced in oriental region of Morocco, was determined over three consecutives years harvest periods. Isly olive oil, distinguished by a particular appreciated taste, participated and won the 2010 1st national HOVE*-prize. To complete chemical description of Isly olive oil, fatty acid and triacylglycerol compositions, have been performed and may constitute a genuine fingerprint of Isly olive oil, which aims to be labeled as registered designed origin olive oil (Isly-ROD olive oil) from the eastern region of Morocco.La caractérisation physico-chimique de plusieurs huiles d'olive produites dans la région orientale du Maroc a été réalisée sur trois compagnes oléicoles consécutives. Isly est  une huile d'olive qui s’est distinguée par un goût particulièrement apprécié. Elle a participé et remporté le premier prix national HOVE* en 2010. Pour compléter la caractérisation chimique de cette huile d'olive Isly, les compositions en acides gras et triglycérides ont été réalisées. Ces derniers peuvent constituer une véritable empreinte digitale caractéristique de cette huile Isly qui cherche à se faire labelliser IGP « indication géographique protégée ».* HOVE: Huile d’Olive Vierge Extr