Prologue ...................................................... xix
PI Overview ................................................. xix
P2 Numerical Analysis Software .............................. xxi
P3 Textbooks and Monographs ................................. xxi
P3.1 Selected Textbooks on Numerical Analysis ........... xxi
P3.2 Monographs and Books on Specialized Topics ....... xxiii
P4 Journals ................................................ xxvi
1 Machine Arithmetic and Related Matters ....................... 1
1.1 Real Numbers, Machine Numbers, and Rounding ............. 2
1.1.1 Real Numbers ..................................... 2
1.1.2 Machine Numbers .................................. 3
1.1.3 Rounding ......................................... 5
1.2 Machine Arithmetic ...................................... 7
1.2.1 A Model of Machine Arithmetic .................... 7
1.2.2 Error Propagation in Arithmetic Operations:
Cancellation Error ............................... 8
1.3 The Condition of a Problem ............................. 11
1.3.1 Condition Numbers ............................... 13
1.3.2 Examples ........................................ 16
1.4 The Condition of an Algorithm .......................... 24
1.5 Computer Solution of a Problem; Overall Error .......... 27
1.6 Notes to Chapter 1 ..................................... 28
Exercises and Machine Assignments to Chapter 1 .............. 31
Exercises ................................................... 31
Machine Assignments ......................................... 39
Selected Solutions to Exercises ............................. 44
Selected Solutions to Machine Assignments ................... 48
2 Approximation and Interpolation ............................. 55
2.1 Least Squares Approximation ............................ 59
2.1.1 Inner Products .................................. 59
2.1.2 The Normal Equations ............................ 61
2.1.1 Least Squares Error; Convergence ................ 64
2.1.4 Examples of Orthogonal Systems .................. 67
2.2 Polynomial Interpolation ............................... 73
2.2.1 Lagrange Interpolation Formula: Interpolation
Operator ........................................ 74
2.2.2 Interpolation Error ............................. 77
2.2.3 Convergence ..................................... 81
2.2.4 Chebyshev Polynomials and Nodes ................. 86
2.2.5 Barycentric Formula ............................. 91
2.2.6 Newton's Formula ................................ 93
2.2.7 Hermite Interpolation ........................... 97
2.2.8 Inverse Interpolation .......................... 100
2.3 Approximation and Interpolation by Spline Functions ... 101
2.3.1 Interpolation by Piecewise Linear Functions .... 102
2.3.2 A Basis for S10(Δ) ............................. 104
2.3.3 Least Squares Approximation .................... 106
2.3.4 Interpolation by Cubic Splines ................. 107
2.3.5 Minimality Properties of Cubic Spline
Interpolants ................................... 110
2.4 Notes to Chapter 2 .................................... 112
Exercises and Machine Assignments to Chapter 2 ............. 118
Exercises .................................................. 118
Machine Assignments ........................................ 134
Selected Solutions to Exercises ............................ 138
Selected Solutions to Machine Assignments .................. 150
3 Numerical Differentiation and Integration .................. 159
3.1 Numerical Differentiation ............................. 159
3.1.1 A General Differentiation Formula for
Unequally Spaced Points ........................ 159
3.1.2 Examples ....................................... 161
3.1.3 Numerical Differentiation with Perturbed Data .. 163
3.2 Numerical Integration ................................. 165
3.2.1 The Composite Trapezoidal and Simpson's Rules .. 165
3.2.2 (Weighted) Newton-Cotes and Gauss Formulae ..... 169
3.2.3 Properties of Gaussian Quadrature Rules ........ 175
3.2.4 Some Applications of the Gauss Quadrature
Rule ........................................... 178
3.2.5 Approximation of Linear Functionals: Method
of Interpolation vs. Method of Undetermined
Coefficients ................................... 182
3.2.6 Peano Representation of Linear Functionals ..... 187
3.2.7 Extrapolation Methods .......................... 190
3.3 Notes to Chapter 3 .................................... 195
Exercises and Machine Assignments to Chapter 3 ............. 200
Exercises .................................................. 200
Machine Assignments ........................................ 214
Selected Solutions to Exercises ............................ 219
Selected Solutions to Machine Assignments .................. 232
4 Nonlinear Equations ........................................ 253
4.l Examples .............................................. 254
4.1.1 A Transcendental Equation ...................... 254
4.1.2 A Two-Point Boundary Value Problem ............. 254
4.1.3 A Nonlinear Integral Equation .................. 256
4.1.4 s-Orthogonal Polynomials ....................... 257
4.2 Iteration, Convergence, and Efficiency ................ 258
4.3 The Methods of Bisection and Sturm Sequences .......... 261
4.3.1 Bisection Method ............................... 261
4.3.2 Method of Sturm Sequences ...................... 264
4.4 Method of False Position .............................. 266
4.5 Secant Method ......................................... 269
4.6 Newton's Method ....................................... 274
4.7 Fixed Point Iteration ................................. 278
4.8 Algebraic Equations ................................... 280
4.8.1 Newton's Method Applied to an Algebraic
Equation ....................................... 280
4.8.2 An Accelerated Newton Method for Equations
with Real Roots ................................ 282
4.9 Systems of Nonlinear Equations ........................ 284
4.9.1 Contraction Mapping Principle .................. 284
4.9.2 Newton's Method for Systems of Equations ....... 285
4.10 Notes to Chapter 4 .................................... 287
Exercises and Machine Assignments to Chapter 4 ............. 292
Exercises .................................................. 292
Machine Assignments ........................................ 302
Selected Solutions to Exercises ............................ 306
Selected Solutions to Machine Assignments .................. 318
5 Initial Value Problems for ODEs: One-Step Methods .......... 325
5.1 Examples .............................................. 326
5.2 Types of Differential Equations ....................... 328
5.3 Existence and Uniqueness .............................. 331
5.4 Numerical Methods ..................................... 332
5.5 Local Description of One-Step Methods ................. 333
5.6 Examples of One-Step Methods .......................... 335
5.6.1 Euler's Method ................................. 335
5.6.2 Method of Taylor Expansion ..................... 336
5.6.3 Improved Euler Methods ......................... 337
5.6.4 Second-Order Two-Stage Methods ................. 339
5.6.5 Runge-Kutta Methods ............................ 341
5.7 Global Description of One-Step Methods ................ 343
5.7.1 Stability ...................................... 344
5.7.2 Convergence .................................... 347
5.7.3 Asymptotics of Global Error .................... 348
5.8 Error Monitoring and Step Control ..................... 352
5.8.1 Estimation of Global Error ..................... 352
5.8.2 Truncation Error Estimates ..................... 354
5.8.3 Step Control ................................... 357
5.9 Stiff Problems ........................................ 360
5.9.1 A-Stability .................................... 361
5.9.2 Pade Approximation ............................. 362
5.9.3 Examples of A-Stable One-Step Methods .......... 367
5.9.4 Regions of Absolute Stability .................. 370
5.10 Notes to Chapter 5 .................................... 371
Exercises and Machine Assignments to Chapter 5 ............. 378
Exercises .................................................. 378
Machine Assignments ........................................ 383
Selected Solutions to Exercises ............................ 387
Selected Solutions to Machine Assignments .................. 392
6 Initial Value Problems for ODEs: Multistep Methods ......... 399
6.1 Local Description of Multistep Methods ................ 399
6.1.1 Explicit and Implicit Methods .................. 399
6.1.2 Local Accuracy ................................. 401
6.1.3 Polynomial Degree vs. Order .................... 405
6.2 Examples of Multistep Methods ......................... 408
6.2.1 Adams-Bashforth Method ......................... 409
6.2.2 Adams-Moulton Method ........................... 412
6.2.3 Predictor-Corrector Methods .................... 413
6.3 Global Description of Multistep Methods ............... 416
6.3.1 Linear Difference Equations .................... 416
6.3.2 Stability and Root Condition ................... 420
6.3.3 Convergence .................................... 424
6.3.4 Asymptotics of Global Error .................... 426
6.3.5 Estimation of Global Error ..................... 430
6.4 Analytic Theory of Order and Stability ................ 433
6.4.1 Analytic Characterization of Order ............. 433
6.4.2 Stable Methods of Maximum Order ................ 441
6.4.3 Applications ................................... 446
6.5 Stiff Problems ........................................ 450
6.5.1 A-Stability .................................... 450
6.5.2 A(α)-Stability ................................. 452
6.6 Notes to Chapter 6 .................................... 453
Exercises and Machine Assignments to Chapter 6 ............. 456
Exercises .................................................. 456
Machine Assignments ........................................ 459
Selected Solutions to Exercises ............................ 461
7 Two-Point Boundary Value Problems for ODEs ................. 471
7.1 Existence and Uniqueness .............................. 474
7.1.1 Examples ....................................... 474
7.1.2 A Scalar Boundary Value Problem ................ 476
7.1.3 General Linear and Nonlinear Systems ........... 481
7.2 Initial Value Techniques .............................. 482
7.2.1 Shooting Method for a Scalar Boundary Value
Problem ........................................ 483
7.2.2 Linear and Nonlinear Systems ................... 485
7.2.3 Parallel Shooting .............................. 490
7.3 Finite Difference Methods ............................. 494
7.3.1 Linear Second-Order Equations .................. 494
7.3.2 Nonlinear Second-Order Equations ............... 500
7.4 Variational Methods ................................... 503
7.4.1 Variational Formulation ........................ 503
7.4.2 The Extremal Problem ........................... 506
7.4.3 Approximate Solution of the Extremal Problem ... 507
7.5 Notes to Chapter 7 .................................... 509
Exercises and Machine Assignments to Chapter 7 ............. 512
Exercises .................................................. 512
Machine Assignments ........................................ 518
Selected Solutions to Exercises ............................ 521
Selected Solutions to Machine Assignments .................. 532
References .................................................... 543
Index ......................................................... 571
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