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Index der MATLAB Programme

interpol

dividif hermpol par -spline bernstein bezier midpnte trapeze simps one newteot trapmode romberg simpadpt redmidpt redtrap midptr2d traptr2d eoeflege eoeflagu eoefherm zplege zplagu zpherm dft

Lagrangesehes Polynom unter Verwendung der Newtonsehen Darstellungsformel 9 Newtonsehe dividierte Differenzen 10 Oskulierendes Polynom 17 Parametrisehe Splines 34 Bernstein-Polynome . . 36 Bezier-Kurven. . . . . . 36 Zusammengesetzte Mittelpunktsformel 51 Zusammengesetzte Trapezregel . . . . 53 Zusammengesetzte Cavalieri-Simpson-Formel 54 Gesehlossene Newton-Cotes-Formeln . . . 59 Zusammengesetzte korrigierte Trapezregel 64 Romberg-Integration. . . . . . . . . 67 Adaptive Cavalieri-Simpson-Formel . 74 Mittelpunktreduktionsformel .... 82 Trapezreduktionsformel . . . . . . . 82 Mittelpunktregel auf einem Dreieek . 84 Trapezregel auf einem Dreieek. . . . 84 Koeffizienten der Legendre-Polynome . 108 Koeffizienten der Laguerre-Polynome . 108 Koeffizienten der Hermite-Polynome 109 Koeffizienten der GauB-Legendre-Formel 109 Koeffizienten der GauB-Laguerre-Formel 109 Koeffizienten der GauB-Hermite-Formel 109 Diskrete Fouriertransformation . . . . . 118

324 Index of MATLAB Programs

idft fftree eompdiff multistep predeor ellfem femmatr Hlerror artvise sgvise bern thetameth pardglegl

ipeidgO ipeidgl

Inverse diskrete Fouriertransformation 118 FFT-Algorithmus in der rekursiven Version 120 Kompakte Differenzen Sehemen . 125 Lineare Mehrsehrittverfahren . . 173 Pradiktor-Korrektor-Verfahren 191 Lineare FE fur Zweipunkt-RWP 244 Konstruktion der Steifigkeitsmatrix . 244 Bereehnung der HI-Norm des Fehlers 245 Kunstliehe Viskositat. . . . . . . . . 258 Optimale kunstliehe Viskositat . . . . 258 Auswertung der Bernoulli-Funktion. . 259 B-Methode fur die Warmeleitungsgleiehung 280 dG(I)eG(I)-Methode fur die Warmeleitungs-gleichung ............. 284 dG(O) implizites Euler-Verfahren 311 dG(I) implizites Euler-Verfahren 311

Index

A-Stabilitat 164 Abhiingigkeitsgebiet 289 - numerisches 295 A bschatzung - a-posteriori 58, 69, 72 - a-priori 58,69,72 Adaptivitat - Runge-Kutta-Verfahren Approximation - beste polynomiale 4

B-Splines 29 - parametrische 37 Bahn 207 Bandbreite 132 Bernoulli - Funktion 254 - Zahlen 66 Bezier-Kurve 35 Bezier-Polygon 35

CFL - Bedingung 295 - Zahl 295 charakteristische - Kurven 286

196

- Polygon 35 - Variablen 288

Dahlquist - erste Schranke 183 - zweite Schranke 183 diskrete - Fourier-Transformation 117 - Laplace-Transformation 138 - Legendre-Transformation 106 - Tschebyscheff-Transformation diskretes - Maximumprinzip 300 - Skalarprodukt 103 Dispersion 127,302 Dissipation 300 Distribution 234 - Ableitung einer 234 dividierte Differenzen 8

Eigenfunktionen 277 Exaktheitsgrad 48,56,57,83,98 exponentiell gefittet 254

Fehler - globaler Abbruchfehler 156 - Interpolationsfehler 3

104

326 Index

- lokaler Abbruchfehler 156,293 - Quadraturfehler 48 finite Differenzen 219 - kompakte 124 - Riickwfu"tsdifferenz 123 - Vorwartsdifferenz 122 - zentrale 122, 123 finite Elemente 22 - unstetige 282, 308 finite upwind Differenz 252 Fourierkoeffizient - diskreter 116 Fourierkoeffizienten 115 Freiheitsgrad 239 Funktion - Gamma 213 - Gewichts- 93 - Greensche 218 - Haar 140 - Stabilitatsfunktion 201

Galerkin - finite Elemente Methode 41,238 - - stabilisierte 255 - Methode 231 - pseudospektrale Approximation

279 - verallgemeinerte Methode 247 Gibbsches Phanomen 118 Gleichung - Differenzengleichung 165, 183 - Warmeleitung 269, 280 Gronwall-Lemma 154, 159

Hauptwurzel der Eins

Integration - adaptive 68 - automatische 68 - mehrdimensionale - nicht adaptive 68 Interpolation - Hermite 15 - oskulierende 16 - stiickweise 12 - Taylorsche 45 - zweidimensionale Interpolationsknoten - stiickweise 20

116

80

18 2

kleinste Quadrate 95 - diskrete 110 Knoten - GauE 104 - GauE-Lobatto Knotenpolynome Konditionszahl

102, 105 3

- Interpolation 6 Konsistenz 156, 177, 194

Lagrange - Darstellung 3 - Interpolation 2 Laplace-Operator 260 Lebesguekonstante 5,6 line are Regression 111

Massekonzentration 276 Matrix - Interpolation 4 - Massematrix 275 - Steifigkeitsmatrix 235 - Vandermondesche 44 Matrixmuster 263 Maximumprinzip 218, 220 - diskretes 254 Methode - {I-Methode 272 - Crank-Nicolson 281 - Euler-riickwarts/zentral 293 - Euler-vorwarts/nichtzentral 292 - Euler-vorwarts/zentral 291 - finite Elemente 261 - Lax-Friedrichs 291,297 - Lax-Wendroff 292, 297 - Leap-Frog 300 - Monte-Carlo 84 - Newmark 300 - Reduktionsformel 80 - Richardson-Extrapolation 64 - Romberg-Integration 66,87 - upwind 292, 296 Minimaxeigenschaft 96

Norm - Maximumnorm 4 - streng konvexe 111 numerischer Fluss 291

Operator - elliptischer 290 - hyperbolischer 290 - parabolischer 290

Pecietzahl 248 - lokale 250 Pade Approximation 46 Phasenraum 208 Phasenwinkel 301 Poisson-Gleichung 260 Polynom - Basis 3 - Bernstein 35 - bester Approximation 113 - Fourier 114 - Hermite 107 - interpolierendes 2 - Laguerre 107 - orthogonales 93 - stiickweise Lagrange 21 Problem - Cauchy 151 - steifes 204 - verallgemeinertes Eigenwertproblem

277 pseudo-spektrale - Ableitung 128 - Differentiationsmatrix 129

Quadraturformel 47 - auf Dreiecken 82 - Cavalieri-Simpson 53, 77, 78, 87 - - zusammengesetzte 53 - Cavalieri-Simpson-Regel 62 - GauB 99 - - auf Dreiecken 84 - GauB-Kronrod 70 - GauB-Lobatto 100, 103 - GauB-Radau - - auf Dreiecken 84 - Hermite 48, 62 - Lagrange 48 - Mittelpunktsregel 49, 62 - - auf Dreiecken 83 - - zusammengesetzte 50 - Newton-Cotes 54 - - zusammengesetzte 60 - pseudo-zufiillige 86

Index 327

- Trapezregel 51,62,118 - - auf Dreiecken 83 - - korrigierte 63 - - zusammengesetzte 52

Randbedingung - Dirichlet 227 - Neumann 227,270 - Robin 267 Raum - Sobolev 230 Reduktionsformel - Mittelpunkts- 81 - Trapez- 81 Referenzdreieck 19 Reihe - Fourier 94, 270 - Legendre 97 - Tschebyscheff 96 Runge's Gegenbeispiel 5, 18, 28

Schranke - von de la Vallee-Poussin 113 schwache - Formulierung 232 - Lasung 232,287 Semidiskretisierung SinguIare Integrale Spline

272,274 75

- eindimensionaler 23 - Kardinal 26 - kubischer 24 - natiirlicher 24 - nicht-ein-Knoten 25 - parametrischer 34 - periodischer 23 Stabilitat - absolute 162, 183, 185 -- Bereich der 162 - asymptotische 153 - der Interpolation 6 - Ljapunov 153 - Nullstabilitat 179, 185 - relative 185 starke Formulierung 234 statistischer Mittelwert 85 Stern 125 Stetigkeitsmodul 62 System

328 Index

- hyperbolisches 288 - - streng 288

Theorem - Tschebyscheff 113 Transformation - Fourier 130 - - schnelle 104 - Laplace 135 - Zeta 137 Triangulation 19,261

Ungleichung - Cauchy-Schwarz 14, 256 - Dreiecks- 257

- Poincare - Young

223,256 230

Verfahren - Crank-Nicolson - Euler-Riickwarts - Euler-Vorwarts - Heun 155 - Leap-Frog 293 - Merson 215

155 155

155

- modifiziertes Euler 214 - Newmark 293 - Nystron 214 Verstarkungsfaktor 298 Verstarkungsfehler 301

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