Nummath1 - Numerische Mathematik Vorlesung von Johann...

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Numerische Mathematik Vorlesung von Johann Linhart Wintersemester 2004/05
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Inhaltsverzeichnis 1 Einleitung 1 1 .1 F eh l e r an a ly s e ........................... 2 1 .2 K omp l ex i t s a s e ....................... 2 1 .3 L i t e r a tu r.............................. 3 2 Zahlendarstellungen 5 2.1 b -ad i s ch eEn tw i c k lun gr e e l l e rZ ah l en . ............. 5 2 .2 G l e i tk omm ad a r s t e l e e l l e l en. 6 2 .2 .1 Ab s chn e id ....................... 6 2 .2 Rund ......................... 6 2 .3 R e la t iv e rundab so lu t e rF l e r.............. 7 2 .4 Rundun g s f l e r...................... 7 2 .5 G l e i a a r i thm e t ik. ................. 8 3 Fehleranalyse 9 3 .1 V ek to r -undM a t r ixn o rm ................... 9 3 ond i t ione in e rAu fgab e..................... 1 2 3 .1 A l lg em e e s........................ 1 2 3 i t iond e rG rund r e chnun g sa r t ......... 1 4 3 .3 K i t e sl ea r enG l e i chun g s sy s t s ...... 1 7 3 i t e sA o r i thmu s................... 2 4 3.3.1 Unterschied zwischen der Kondition eines Algorithmus undd e rK i t e e ............ 2 4 3 .3 .2 V o rw ä r t san a s e...................... 2 6 iii
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iv INHALTSVERZEICHNIS 3 .3 .3 Rü c kw ä r t san a ly s e..................... 27 4 Lineare Gleichungssysteme 31 4 .1 D a sE l im in a t ion sv e r fah r en. ................... 31 4 .1 .1 P iv o t i s i e run g ....................... 32 4 .2 Z e i tk omp l ex i t...................... 33 4.1.3 Lineare Gleichungssysteme mit mehreren rechten Seiten 33 4 .2 L ea r eAu sg l e i ch s r e chnun g ................... 34 4 .2 r ob l em s t e l lun g...................... 4 .2 Äqu i l ib r i e g....................... 4 .3 N o rm a lg l e i chun g en . 35 4.2.4 QR -Z e r l egun 39 5 Interpolation 45 5 r l s t e l g.......................... 45 5 .2 Ex i s t en zundE ind eu t igk e i td e sIn t e rpo la t spo lyn om s.... 4 6 5 .3 F eh l e r ab s ä t zun g ........................ 47 5 .4 B e r e gd e t e t s............. 50 5.4.1 Die Lagrange’sche Form des Interpolationspolynoms . . 51 5 .4 .2 D a sN ev i l l e -S a.................... 53 5 .5 Ex t r apo t ion. .......................... 55 5.5.1 Extrapolation für x =0 ................. 5 .5 .2 Summ a t ione e rR e ih i t t e l sEx t r t ion . .... 5 7 6N um e r i s c h eD i f erenziation 59 6.1 Motivation. ............................ 5 9 6.2 Di f e r z ia t iond e t e t s........... 60 6 .1 Ab s ä t e sV e r r s f l e r s ........... 6 .2 F l e r an a s e ....................... 62 6 .3 Zw e i t eAb l e i tun g ..................... 64 6.3 Di f e r z t iondu r chEx t r t ..............
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INHALTSVERZEICHNIS v 7N um e r i s c h e I n t e g r a t i o n 6 7 7 .1 N ew ton -C o t e sF o rm e ln. ..................... 6 7 7.1.1 Geschlossene Newton-Cotes Formeln . . ........ 6 8 7.1.2 O f en eN o t e o e ............. 7 1 7.1.3 Rundungsfehler bei den Newton-Cotes Formeln . . . . 73 7 .2 Zu samm g e s e t z t eF o e ln . .................. 7 3 7 .3 R omb e r g - In t eg r a t ion . ...................... 7 6 8 Iterative Lösung von Gleichungen 81 8 .1 D a sK on t r ak t ion sp r in z ip. .................... 8 1 8 .1 .1 A l lg em e e s........................ 8 1 8.1.2 Anwendung des Kontraktionsprinzips im R s ...... 8 5 8 .3 K v e r g zo rdnun g.................... 8 9 8 .2 D a sN -V e r fah r en . 9 3 8 .3 Sp e z i e l l ee ind im s a l eI t e r a t sv e r r en. ......... 9 6 8 .3 .1 In t e rv a l lh a lb i e run g .................... 9 6 8 i eS ek an t enm e th od e................... 9 8 8.3.3 Das Newton-Verfahren bei mehrfachen Nullstellen . . . 98 8 .4 Nu l l s t e l l env onP o lyn om ................... 10 0 8 .4 l e e 0 8 a sH o rn e r -S ch a.................... 3 8 .3 D i eM e ev onMu l l e r................. 6 8 .4 D i e onB a i r s tow. ............... 8 8 .5 I t e r a t iv eL ö sun gv onl e a r enG l e i chun g s sy s t 11 2 8 .5 l e e 2 8.5.2 Das Jacobi-Verfahren (Gesamtschrittverfahren) . . . . 113 8.5.3 Das Gauß-Seidel-Verfahren (Einzelschrittverfahren) . . 115 8 .4 R e lax a t e r r 6
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vi INHALTSVERZEICHNIS
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Kapitel 1 Einleitung Am 2. 12. 2004 wurde ein zusätzlicher Abschnitt "Iterative Lösung von linearen Gleichungssystemen" angefügt!
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This note was uploaded on 10/20/2010 for the course AERONAUTIC A.E. taught by Professor Allwyn during the Spring '10 term at Anna University Chennai - Regional Office, Coimbatore.

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Nummath1 - Numerische Mathematik Vorlesung von Johann...

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