Download Numerische Mathematik 1: Eine Einführung — unter by Josef Stoer PDF

By Josef Stoer

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"Diese Einf?hrung zeichnet sich durch eine klare, intestine lesbare Darstellung aus und ist eine gelungene Synthese zwischen theoretischer Begr?ndung und praktischer Anwendung der behandelten Methoden. Deshalb ist sie auch zu einem Standardlehrbuch der Numerischen Mathematik geworden."
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"Unter den Numerik-Lehrb?chern ... sei auf das vorliegende Buch besonders hingewiesen, da hier bei allen Anspr?chen an mathematische Strenge das Schwergewicht auf die Bereitstellung von praktikablen Algorithmen nach neuesten Erkenntnissen mit vielen numerischen Beispielen und kritischen Beurteilungen liegt.
...F?r Veranstaltungen der Numerik und ihren Anwendungen in der Informatik findet der Lehrende viele Anregungen und gute Informationsm?glichkeiten."
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Additional resources for Numerische Mathematik 1: Eine Einführung — unter Berücksichtigung von Vorlesungen von F.L. Bauer

Sample text

Wahrend die Polynominterpolation friiher haufig zur Interpolation von Funktionswerten aus Tafelwerken benutzt wurde, ist diese Anwendung seit dem Autkommen der modemen elektronischen Digitalrechner selten geworden. Sie ist nach wie vor wichtig, urn Formeln fUr die numerische Integration von Funktionen abzuleiten. Neuerdings benutzt man sie und die rationale Interpolation (s. ) hiiufig im Rahmen von Algorithmen, die die Konvergenz J. Stoer, Numerische Mathematik 1 © Springer-Verlag Berlin Heidelberg 1999 42 2 Interpolation gewisser Folgen beschleunigen (Richardson-Verfahren, Extrapolationsalgorithmen).

Wegen b n := Sn-I. Wir wollen zwei verschiedene niitzliche Abschatzungen flir das Residuum r := c - al bl - ... - anbn geben. 6) folgenden Gleichungen So - c = O. tj) a' -s· __ J_ -a·b·ll· -JI+aj JJt"'J. anbn - Sn-I J' -12 1 - ...... n - . = 8Sn_l. (lsjl + lajbjl)], j=1 8' := (O falls an = 1. 1 sonst. 7). tj) D n-I] 1 (1 + ak) 8 ~. 9) e = I:>jbj(1 + J-tj) n j-I {1 + CXk)-1 + a nb n {1 + 8)-1 k=1 j=1 n n-I {1 + CXk)-I. IO) + j. 8) folgt insbesondere die Gutartigkeit unseres Algorithmus zur Berechnung von f3n.

3) f(n)(g) . • x n] = - - fur em n! • x n]. 1 Interpolation durch Polynome Beispiel: f(x) Xi = 55 = sinx : 7f 10 . i =0, 1,2,3,4,5, ·1, sinx - P(x) = (x - n = 5, -sin; xo)(x - x}) ... (x - x s )720' I I sinx - P(x)1 ::::: 720 I(x - xo)(x - x}) ... (x - xs)1 ; = Hx), = --:no . Iw(x)1 Aufierhalb des Intervalls I[xo, ... , xn] wiichst Iw(x)1 sehr schnell an. Eine Verwendung des Interpolationspolynoms P zur Approximation von I an einer Stelle x auBerhalb des Intervalls I[xo, ... , x n] - man spricht dann von Extrapolation - so lIte deshalb nach Moglichkeit vermieden werden.

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