TY - BOOK AU - Butt,Rizwan TI - Introduction to numerical analysis using MATLAB / T2 - Mathematics series SN - 9781934015230 (hardcover with CDROM : alk. paper) U1 - 518 22 PY - 2010/// CY - Sudbury, Mass. PB - Jones and Bartlett Publishers KW - Numerical analysis KW - Data processing N1 - Includes bibliographical references (p. 801-808) and index; 1. Contents 2. 1 Number Systems and Errors 1 3. 1.1 4. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5. 1.2 6. Number Representation and Base of Numbers . . . . . . . . . . . . . 2 7. 1.2.1 8. Normalized Floating-point Representation . . . . . . . . . . . 4 9. 1.2.2Roundingand 10. Chopping. . . . . . . . . . . . . . . . . . . . . 6 11. 1.3 12. Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 13. 1.4Sources 14. Of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 15. 1.4.1Human 16. Error. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 17. 1.4.2 18. Truncation Error . . . . . . . . . . . . . . . . . . . . . . . . . 9 19. 1.4.3Round-o- 20. Error . . . . . . . . . . . . . . . . . . . . . . . . .10 21. 1.5 22. E-ect of Round-o- Errors in Arithmetic Operation . . . . . . . . . . 10 23. 1.5.1 24. Rounding o- Errors in Addition and Subtraction . . . . . . . 11 25. 1.5.2 26. Rounding o- Errors in Multiplication . . . . . . . . . . . . . 12 27. 1.5.3 28. Roundingo- Errors in Division . . . . . . . . . . . . . . . . . 14 29. 1.5.4 30. Rounding o- Errors in Powers and roots . . . . . . . . . . . . 16 31. 1.6Summary 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 33. 1.7Exercises 34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 35. 2 Solution of Nonlinear Equations 21 36. 2.1 37. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 38. 2.2Methodof 39. Bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 40. 2.3 41. False Position Method . . . . . . . . . . . . . . . . . . . . . . . . . . 29 42. 2.4Fixed-Point 43. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 44. 2.5Newton's 45. Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 46. 2.6Secant 47. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 48. 2.7Multiplicityofa 49. Root . . . . . . . . . . . . . . . . . . . . . . . . . . 55 50. 2.8 51. Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . . 63 52. 2.9 53. Acceleration of Convergence . . . . . . . . . . . . . . . . . . . . . . . 76 54. 2.10 55. Systems of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . 80 56. 2.10.1 57. Newton's Method. . . . . . . . . . . . . . . . . . . . . . . . . 81 58. Iii 59. Iv Contents 60. 2.11 61. Rootsofpolynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . 87 62. 2.11.1 63. Horner'sMethod . . . . . . . . . . . . . . . . . . . . . . . . . 88 64. 2.11.2 65. Muller'sMethod . . . . . . . . . . . . . . . . . . . . . . . . . 94 66. 2.11.3 67. Bairstow'sMethod. . . . . . . . . . . . . . . . . . . . . . . . 98 68. 2.12 69. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 70. 2.13 71. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 72. 3 Systems of Linear Equations 113 73. 3.1 74. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 75. 3.1.1 76. Linearsysteminmatrixnotation . . . . . . . . . . . . . . .117 77. 3.2 78. Propertiesofmatricesanddeterminant . . . . . . . . . . . . . . . .119 79. 3.2.1 80. Introductionofmatrices. . . . . . . . . . . . . . . . . . . . .120 81. 3.2.2 82. Somespecialmatrixforms . . . . . . . . . . . . . . . . . . .124 83. 3.2.3 84. Thedeterminantofmatrix . . . . . . . . . . . . . . . . . . .133 85. 3.3 86. Numericalmethodsforlinearsystems. . . . . . . . . . . . . . . . .140 87. 3.4 88. Directmethodsforlinearsystems . . . . . . . . . . . . . . . . . . .140 89. 3.4.1 90. Cramer'sRule . . . . . . . . . . . . . . . . . . . . . . . . . .140 91. 3.4.2 92. Gaussianeliminationmethod. . . . . . . . . . . . . . . . . . 143 93. 3.4.3 94. Pivotingstrategies . . . . . . . . . . . . . . . . . . . . . . . .156 95. 3.4.4 96. Gauss-jordanmethod . . . . . . . . . . . . . . . . . . . . . .162 97. 3.4.5 98. Ludecompositionmethod . . . . . . . . . . . . . . . . . . . 166 99. 3.4.6 100. Tridiagonalsystemsoflinearequations . . . . . . . . . . . . 187 101. 3.5 102. Normsofvectorsandmatrices . . . . . . . . . . . . . . . . . . . . .191 103. 3.5.1 104. Vectornorms. . . . . . . . . . . . . . . . . . . . . . . . . . .191 105. 3.5.2 106. Matrixnorms. . . . . . . . . . . . . . . . . . . . . . . . . . .193 107. 3.6 108. Iterativemethodsforsolvinglinearsystems . . . . . . . . . . . . . 196 109. 3.6.1 110. Jacobiiterativemethod . . . . . . . . . . . . . . . . . . . . .197 111. 3.6.2 112. Gauss-seideliterativemethod . . . . . . . . . . . . . . . . . 202 113. 3.6.3 114. Convergencecriteria. . . . . . . . . . . . . . . . . . . . . . .207 115. 3.7 116. Eigenvaluesandeigenvectors . . . . . . . . . . . . . . . . . . . . . .212 117. 3.7.1 118. Successiveover-relaxationmethod . . . . . . . . . . . . . . 220 119. 3.7.2 120. Conjugategradientmethod. . . . . . . . . . . . . . . . . . . 226 121. 3.8 122. Conditioningoflinearsystems . . . . . . . . . . . . . . . . . . . . .231 123. 3.8.1 124. Errorsinsolvinglinearsystems . . . . . . . . . . . . . . . . 231 125. 3.8.2 126. Iterativere-nement . . . . . . . . . . . . . . . . . . . . . . .242 127. 3.9 128. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244 129. 3.10 130. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245 131. 4 Approximating Functions 259 132. 4.1 133. Introduction................................ 259 134. 4.2 135. Polynomialinterpolationforunevenintervals............. 262 136. 4.2.1 137. Lagrangeinterpolatingpolynomials .............. 263 138. 4.2.2 139. Newton'sGeneralInterpolatingFormula............ 274 140. 4.2.3 141. Aitken'sMethod ......................... 291 142. 4.3 143. Polynomialinterpolationforevenintervals .............. 294 144. 4.3.1 145. Forward-di-erences........................ 295 146. 4.3.2 147. Backward-di-erences....................... 300 148. 4.3.3 149. Central-di-erences ........................ 303 150. 4.4 151. Interpolationwithsplinefunctions................... 308 152. 4.4.1 153. Naturalcubicspline....................... 316 154. 4.4.2 155. Clampedspline.......................... 317 156. 4.5 157. Leastsquaresapproximation...................... 320 158. 4.5.1 159. Linearleastsquares....................... 321 160. 4.5.2 161. Polynomialleastsquares.................... 326 162. 4.5.3 163. Nonlinearleastsquares..................... 330 164. 4.5.4 165. Leastsquaresplane ....................... 338 166. 4.5.5 167. Overdeterminedlinearsystems ................ 340 168. 4.5.6 169. Leastsquareswithqrdecomposition............. 343 170. 4.5.7 171. Leastsquareswithsingularvaluedecomposition ...... 348 172. 4.6 173. Summary ................................. 353 174. 4.7 175. Exercises ................................. 354 176. 5 Di-erentiation and Integration 367 177. 5.1 178. Introduction................................ 367 179. 5.2 180. Numericaldi-erentiation ........................ 367 181. 5.3 182. Numericaldi-erentiationformulas................... 369 183. 5.3.1 184. Firstderivativesformulas.................... 369 185. 5.3.2 186. Secondderivativesformulas .................. 386 187. 5.4 188. Formulasforcomputingderivatives .................. 393 189. 5.4.1 190. Centraldi-erenceformulas................... 393 191. 5.4.2 192. Forwardandbackwarddi-erenceformulas.......... 394 193. 5.5 194. Numericalintegration .......................... 395 195. 5.6 196. Newton-cotesformulas ......................... 397 197. 5.6.1 198. Closednewton-cotesformulas................. 397 199. 5.6.2 200. Opennewton-cotesformulas.................. 424 201. 5.7 202. Repeateduseofthetrapezoidalrule ................. 428 203. 5.8 204. Rombergintegration........................... 430 205. 5.9 206. Gaussianquadratures .......................... 433 207. 5.10Summary 208. ................................. 438 209. 5.11Exercises 210. ................................. 439 211. 6 Ordinary Di-erential Equations 447 212. 6.1 213. Introduction................................ 447 214. 6.1.1 215. Classi-cationofdi-erentialequations............. 449 216. 6.2 217. Numericalmethodsforsolvingivp .................. 451 218. 6.3 219. Single-stepmethodsforivp ...................... 452 220. 6.3.1 221. Euler'sMethod.......................... 452 222. 6.3.2 223. Analysisoftheeuler'smethod................. 455 224. 6.3.3 225. Higher-ordertaylormethods.................. 457 226. 6.3.4 227. Runge-kuttamethods...................... 460 228. 6.3.5 229. Third-orderrunge-kuttamethod............... 468 230. 6.3.6 231. Fourth-orderrunge-kuttamethod .............. 469 232. 6.3.7 233. Fifth-orderrunge-kuttamethod ............... 472 234. 6.3.8 235. Runge-Kutta-mersonmethod.................. 473 236. 6.3.9 237. Runge-Kutta-Lawson'sFifth-ordermethod.......... 475 238. 6.3.10Runge-Kutta-butchersixth-ordermethod 239. .......... 475 240. 6.3.11Runge-Kutta-fehlbergmethod 241. ................. 476 242. 6.4 243. Multi-stepsmethodsforivp ...................... 478 244. 6.5 245. Predictor-correctormethods ...................... 485 246. 6.5.1 247. Milne-simpsonmethod ..................... 486 248. 6.5.2 249. Adams-Bashforth-moultonmethod............... 489 250. 6.6 251. Systemsofsimultaneousode ..................... 494 252. 6.7 253. Higher-orderdi-erentialequations .................. 498 254. 6.8 255. Boundary-valueproblems........................ 500 256. 6.8.1 257. Theshootingmethod ...................... 501 258. 6.8.2 259. Thenonlinearshootingmethod ................ 505 260. 6.8.3 261. Thefinitedi-erencemethod.................. 507 262. 6.9 263. Summary ................................. 511 264. 6.10Exercises 265. ................................. 512 266. 7 Eigenvalues and Eigenvectors 517 267. 7.1 268. Introduction................................ 517 269. 7.2 270. Linearalgebraandeigenvaluesproblems ............... 527 271. 7.3 272. Diagonalizationofmatrices ....................... 530 273. 7.4 274. Basicpropertiesofeigenvalueproblems................ 543 275. 7.5 276. Someimportantresultsofeigenvalueproblems ........... 560 277. 7.6 278. Numericalmethodsforeigenvalueproblems ............. 563 279. 7.7 280. Vectoriterativemethodsforeigenvalues................ 564 281. Contents v 282. Vi Contents 283. 7.7.1 284. Powermethod .......................... 564 285. Contents vii 286. 7.7.2 287. Inversepowermethod...................... 569 288. 7.7.3 289. Shiftedinversepowermethod.................. 572 290. 7.8 291. Locationofeigenvalues ......................... 576 292. 7.8.1 293. Gerschgorincirclestheorem .................. 576 294. 7.8.2 295. Rayleighquotient ........................ 578 296. 7.9 297. Intermediateeigenvalues......................... 580 298. 7.10eigenvaluesofsymmetricmatrices................... 583 299. 7.10.1jacobimethod 300. .......................... 584 301. 7.10.2sturmsequenceiteration 302. .................... 590 303. 7.10.3Given'sMethod.......................... 594 304. 7.10.4Householder'sMethod...................... 598 305. 7.11matrixdecompositionmethods..................... 602 306. 7.11.1qrmethod............................ 602 307. 7.11.2lrmethod............................ 606 308. 7.11.3upperhessenbergform..................... 609 309. 7.11.4singularvaluedecomposition.................. 616 310. 7.12Summary 311. ................................. 623 312. 7.13Exercises 313. ................................. 624 314. Appendices 634 315. A Some Mathematical Preliminaries 635 316. B Introduction of MATLAB 659 317. C Answers to Selected Exercises 697 318. Bibliography 726 319. Index 727 ER -