《Scientific Computing Vol. III》- 数值逼近与积分无水印PDF

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"Scientific Computing Vol. III – Approximation and Integration 是一本专注于数值计算中的逼近与积分的英文原版书籍，由John A. Trangenstein教授编写。这本书是Texts in Computational Science and Engineering系列的一部分，由多位知名学者共同编辑。书中涵盖了关于数值积分的多个方面，包括Riemann积分和梯形法则等常见积分方法。" 在科学计算领域，逼近与积分是不可或缺的基础部分。本书可能深入探讨了以下关键知识点： 1. **数值积分**: 数值积分是解决无法解析求解的积分问题的数学工具。书中提到的Riemann积分是一种基本的数值积分方法，通过将积分区间划分为小的子区间，然后对每个子区间的平均值乘以宽度求和来近似原积分。 2. **Riemann积分**: Riemann积分是通过对区间进行分段，然后用每个小区间上的函数值乘以小区间长度来估计积分值。在书中，`quadrature_rule=F77NAME(riemann)`表明可能讨论了在Fortran语言环境中实现Riemann积分的算法，`order=1`表示这是第一阶方法，意味着最基础的划分方式。 3. **梯形法则**: 梯形法则是一种更精确的数值积分方法，它假设每个小区间上函数曲线可以用一个直角梯形近似，然后计算这些梯形的面积总和。书中提及`case TRAPEZOIDAL:`可能详细介绍了这一方法的理论基础和实际应用。 4. **数值计算的误差分析**: 书中可能会讨论如何评估和控制数值积分的误差，包括绝对误差和相对误差，并分析不同积分方法的收敛性。 5. **高级积分技术**: 可能还包括高斯积分、辛积分方法或其他高级积分技巧，这些方法可以提高积分精度，特别是在处理复杂函数时。 6. **软件实现与编程**: 书中可能还涉及如何在实际编程环境中（如提到的FoxitReader、PDF-XChangeViewer、SumatraPDF和Firefox）使用这些算法，以及可能提供的额外材料，可能包含实际代码示例。 7. **应用领域**: 科学计算中的积分方法广泛应用于物理、工程、经济等领域，书中可能举例说明这些方法的实际应用及其重要性。这本资源对于理解数值积分的理论和实践，以及如何在实际问题中应用这些知识，都是非常有价值的。读者可以通过美国亚马逊官网获取更多关于此书的详细信息，或者访问Springer提供的额外在线资源。

Contents for Volume 1 xvii

3.4.4 Triangular..................................................... 155

3.4.5 Trapezoidal................................................... 159

3.4.6 Quasi-Triangular ............................................. 163

3.4.7 Permutations.................................................. 165

3.4.8 Orthogonal.................................................... 167

3.5 Norms .................................................................. 171

3.5.1 Vector Norms................................................. 171

3.5.2 Subordinate Matrix Norms .................................. 175

3.5.3 Consistent Matrix Norms .................................... 180

3.5.4 Hadamard Inequality ......................................... 184

3.6 PerturbationAnalysis.................................................. 184

3.6.1 Absolute Error................................................ 184

3.6.2 Condition Number ........................................... 188

3.6.3 RelativeError................................................. 189

3.6.4 A Posteriori Error Estimate.................................. 193

3.7 GaussianFactorization ................................................ 197

3.7.1 No Pivoting................................................... 197

3.7.2 Full Pivoting.................................................. 201

3.7.3 Partial Pivoting ............................................... 208

3.8 Rounding Errors ....................................................... 217

3.8.1 ErrorAccumulation.......................................... 218

3.8.2 Inner Products ................................................ 225

3.8.3 Matrix Products .............................................. 227

3.8.4 ForwardErrorEstimates..................................... 230

3.8.5 Factorization Errors.......................................... 231

3.8.6 Triangular System Errors .................................... 239

3.8.7 Condition Number Estimates ................................ 244

3.8.8 Case Study.................................................... 250

3.9 Improvements.......................................................... 253

3.9.1 Scaling........................................................ 253

3.9.2 IterativeImprovement........................................ 257

3.9.3 Case Study.................................................... 260

3.10 Inverses................................................................. 261

3.11 Determinants........................................................... 263

3.12 Object-OrientedProgramming........................................ 264

3.12.1 Class Design.................................................. 264

3.12.2 Vector Class .................................................. 266

3.12.3 Matrix Class.................................................. 266

3.12.4 GaussianFactorizationClass................................. 267

3.13 Special Matrices ....................................................... 267

3.13.1 Square ........................................................ 268

3.13.2 Symmetric.................................................... 269

3.13.3 SymmetricPositive........................................... 274

xviii Contents for Volume 1

3.13.4 Diagonally Dominant ........................................ 283

3.13.5 Tridiagonal ................................................... 286

3.13.6 Band .......................................................... 288

4 Scientiﬁc Visualization ...................................................... 291

4.1 Overview............................................................... 291

4.2 Pixelsand Color ....................................................... 292

4.3 CoordinateSystems ................................................... 296

4.3.1 2D............................................................. 296

4.3.2 3D............................................................. 296

4.3.3 HomogeneousCoordinates .................................. 297

4.3.4 WindowTransformation..................................... 298

4.3.5 ModelViewTransformation................................. 299

4.3.6 Projection Transformations.................................. 300

4.3.7 TransformingNormals....................................... 301

4.4 InteractiveRotations................................................... 302

4.4.1 Quaternions................................................... 303

4.4.2 Trackballs..................................................... 305

4.5 LightingModels ....................................................... 307

4.6 ViewingFunctions..................................................... 307

4.6.1 Functionsof One Variable................................... 307

4.6.2 Functionsof Two Variables.................................. 309

4.6.3 Functionsof ThreeVariables ................................ 314

4.7 GraphicsSoftware ..................................................... 317

4.7.1 2D Graphics .................................................. 318

4.7.2 3D Graphics .................................................. 320

4.8 GraphicsClasses....................................................... 321

4.8.1 2D Graphics .................................................. 322

4.8.2 3D Graphics .................................................. 323

4.9 Graphical User Interfaces ............................................. 324

5 Nonlinear Equations ......................................................... 329

5.1 Overview............................................................... 329

5.2 Well-Posedness ........................................................ 331

5.2.1 Existence ..................................................... 331

5.2.2 Uniqueness ................................................... 333

5.2.3 PerturbationTheory.......................................... 333

5.3 Bisection ............................................................... 336

5.3.1 Development ................................................. 336

5.3.2 Analysis ...................................................... 337

5.3.3 TerminationCriteria.......................................... 337

5.4 Newton’sMethod...................................................... 340

5.4.1 Algorithm Development ..................................... 341

5.4.2 ConvergenceBehavior....................................... 342

5.4.3 ConvergenceRates........................................... 344

5.4.4 ConvexFunctions ............................................ 346

Contents for Volume 1 xix

5.4.5 KantorovichTheorem........................................ 347

5.4.6 ContractiveMappings........................................ 352

5.4.7 Rounding Errors.............................................. 353

5.4.8 TerminationCriteria.......................................... 356

5.4.9 ApproximateDerivatives .................................... 361

5.5 Secant Method ......................................................... 370

5.5.1 Algorithm Development ..................................... 370

5.5.2 ErrorAnalysis................................................ 371

5.5.3 OrderofConvergence........................................ 374

5.5.4 RelativeEfﬁciency ........................................... 375

5.5.5 Rounding Errors.............................................. 379

5.6 GlobalConvergence................................................... 383

5.6.1 Bracketing .................................................... 383

5.6.2 Signiﬁcant Change........................................... 384

5.6.3 Regula Falsi .................................................. 386

5.6.4 IllinoisAlgorithm ............................................ 387

5.6.5 Muller-TraubMethod........................................ 388

5.6.6 RationalInterpolation........................................ 391

5.6.7 InverseQuadraticInterpolation.............................. 393

5.6.8 Summary ..................................................... 395

5.7 Minimization........................................................... 400

5.7.1 Necessary andSufﬁcientConditions........................ 401

5.7.2 PerturbationAnalysis ........................................ 402

5.7.3 Newton’sMethod ............................................ 404

5.7.4 Line Searches................................................. 404

5.7.5 HermiteCubicMinimization................................ 412

5.7.6 Derivative-Free Methods..................................... 415

5.7.7 Stochastic Optimization ..................................... 425

6 Least Squares Problems ..................................................... 429

6.1 Overview............................................................... 429

6.2 ExampleProblems..................................................... 431

6.3 Existenceand Uniqueness............................................. 433

6.4 Pseudo-Inverses........................................................ 437

6.5 PerturbationAnalysis.................................................. 441

6.6 A Posteriori ErrorEstimates .......................................... 453

6.7 HouseholderQRFactorization........................................ 456

6.7.1 Successive Reﬂection ........................................ 456

6.7.2 Pivoting....................................................... 467

6.7.3 Condition Number Estimates ................................ 468

6.7.4 Least SquaresProblems...................................... 470

6.7.5 Rounding Errors.............................................. 480

6.7.6 IterativeImprovement........................................ 490

Contents for Volume 2

1 Eigenvalues and Eigenvectors .............................................. 1

1.1 Overview............................................................... 2

1.2 Fundamental Concepts ................................................ 3

1.2.1 Examples ..................................................... 3

1.2.2 Characteristic Polynomial ................................... 5

1.2.3 Inverses and Hermitians ..................................... 9

1.2.4 Fundamental Theorem of Algebra .......................... 10

1.2.5 Gerschgorin Circle Theorem ................................ 12

1.2.6 Traceand Determinant....................................... 14

1.2.7 InvariantSubspace ........................................... 17

1.2.8 ChangeofBasis .............................................. 18

1.2.9 Similarity Transformations .................................. 20

1.2.10 SpectralRadius............................................... 21

1.2.11 Distance Between Subspaces ................................ 22

1.3 Hermitian Matrices .................................................... 38

1.3.1 Theory ........................................................ 38

1.3.2 PerturbationAnalysis ........................................ 44

1.3.3 Symmetric Tridiagonal Matrices ............................ 53

1.3.4 Sturm Sequences............................................. 55

1.3.5 Bisection...................................................... 65

1.3.6 Power Method................................................ 67

1.3.7 QL andQR Algorithms...................................... 73

1.3.8 Divide and Conquer .......................................... 80

1.3.9 dqds ........................................................... 84

1.3.10 SolverComparison........................................... 89

1.3.11 Householder Tridiagonalization ............................. 90

1.3.12 LanczosProcess.............................................. 92

1.3.13 SymmetricEigenvalueProblems............................ 97

1.4 GeneralMatrices....................................................... 98

1.4.1 Theory ........................................................ 98

1.4.2 PerturbationAnalysis ........................................ 123

xxi

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