By Demailly Jean-Pierre

Demailly J.-P. examine numerique et equations differentielles (EDP Sciences, 2006)(ISBN 286883891X)(fr)(345s)_MN_

Best mathematics books

Mathematics and general relativity: proceedings of the AMS-IMS-SIAM joint summer research conference held June 22-28, 1986 with support from the National Science Foundation

This quantity comprises the court cases of the AMS-IMS-SIAM Joint summer time study convention normally Relativity, held in June 1986 on the college of California, Santa Cruz. common relativity is without doubt one of the such a lot winning alliances of arithmetic and physics. It presents us with a concept of gravity which consents with all experimentation and remark to this point.

Mathematics: A Simple Tool for Geologists, Second Edition

This ebook is for college kids who didn't stick to arithmetic via to the top in their tuition careers, and graduates and pros who're trying to find a refresher path. This new version comprises many new difficulties and in addition has linked spreadsheets designed to enhance scholars' realizing. those spreadsheets is usually used to resolve a few of the difficulties scholars are inclined to come upon in the course of the rest of their geological careers.

Additional resources for Analyse numérique et équations différentielles

Example text

N + 1)| cos θ − cos θi | tn+1 (xi ) = (n + 1) Minorons la quantit´e cos θ − cos θi = −2 sin θ + θi θ − θi sin . 2 2 49 II – Approximation polynomiale des fonctions num´ eriques y y= 2 π t 1 y = sin t 0 π/2 Pour t ∈ 0, π2 on a sin t ≥ 2 π ∈ θ+θi 2 sin θi θi +π 2, 2 θ + θi ≥ min 2 Comme sin θi = 2 sin θi 2 cos t t, or θ − θi π π ∈ − , , 2 2 2 Par ailleurs π avec sin θi 2 sin donc θi 2 ≤ π 2 et θi +π 2 θi θi + π , sin 2 2 ≤ 2 min sin |li (cos θ)| ≤ π 2 |θ − θi | θ − θi ≥ . 2 π 2 ≥ π 2, donc = min sin θi θi 2 , cos 2 θi θi , cos .

On peut d’abord choisir α > 0 tel que f − f χα 2 < 2ε ; α ´etant ainsi ﬁx´e, on peut choisir n0 tel que n > n0 entraˆıne f χα − rα,n 2 < 2ε et donc f − rn 2 < ε. Mise en œuvre num´ erique – Si les polynˆomes pn sont connus, le calcul des rn est possible d`es lors qu’on sait ´evaluer les int´egrales f, pk : les m´ethodes d’int´egration num´erique feront pr´ecis´ement l’objet du prochain chapitre. Si les polynˆ omes pn ne sont pas connus, on peut les calculer num´eriquement par la formule de r´ecurrence du th´eor`eme 2.

N=k=0 Comme tn a pour coeﬃcient directeur 2n−1 si n ≥ 1, on en d´eduit p0 (x) = t0 (x) = 1 pn (x) = 21−n tn (x) si n ≥ 1. On sait que tn a n z´eros distincts dans ] − 1, 1[. On va voir que c’est une propri´et´e g´en´erale des polynˆ omes orthogonaux. Th´ eor` eme 3 – Pour tout poids w sur ]a, b[, le polynˆome pn poss`ede n z´eros distincts dans l’intervalle ]a, b[. D´ emonstration. Soient x1 , . . , xk les z´eros distincts de pn contenus dans ]a, b[ et m1 , . . , mk leurs multiplicit´es respectives.