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