《Scientific Computing Vol. I》线性与非线性方程无水印PDF

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"Scientific Computing Vol. I – Linear and Nonlinear Equations 是一本关于科学计算的书籍，专注于线性和非线性方程的求解。这本书由John A. Trangenstein撰写，他是杜克大学数学系的荣誉教授。书中可能涵盖了数值方法、迭代法以及在科学和工程计算中的应用等内容。此版本为英文无水印原版pdf，可在多个PDF阅读器上正常打开。若存在侵权问题，可以联系提供者或CDN进行删除。更多关于本书的详细信息可以在美国亚马逊官网查询。此外，读者还可以在Springer出版社的系列网站上找到额外的相关材料。" 此资源涉及到的知识点包括： 1. **科学计算**：科学计算是应用数学的一个分支，它使用计算机来解决科学研究和工程问题中的复杂计算。这通常涉及数值分析、优化算法、数据处理等技术。 2. **线性方程**：线性方程组是形如ax + b = 0的一类方程，其中a和b是常数，x是变量。它们在数学和工程中有广泛应用，例如电路分析、物理模型等。求解线性方程的常见方法有高斯消元法、克拉默法则、矩阵求逆等。 3. **非线性方程**：非线性方程是指那些不是线性关系的方程，即方程的解不能表示为常数与变量的线性组合。求解非线性方程通常需要迭代方法，如牛顿法、拟牛顿法、二分法等。 4. **牛顿法（Newton's Method）**：牛顿法是一种迭代优化方法，用于求解非线性方程。它基于函数的泰勒展开，通过不断线性化目标函数并求其根来逼近原方程的实根。文中提到的"nsteps_newton"和"elapsed_newton"可能与牛顿法的迭代次数和计算时间有关。 5. **数值方法**：数值方法是处理数学问题的一种近似方法，特别是在无法获得解析解的情况下。这些方法包括但不限于插值、积分、微分方程的求解等。 6. **迭代法**：在解决非线性问题时，迭代法是一种常用策略。通过反复应用一个迭代公式，逐步接近解。文中提到的"dxold"和"nsteps_newton"可能与迭代过程中的步长和迭代次数有关。 7. **计算机编程在科学计算中的应用**：科学计算书籍通常会包含如何使用编程语言实现这些算法，如C、C++、Python等。书中的"OPERATOR_NEW_BRACKET"可能是指内存分配操作，用于存储迭代过程中的数据。 8. **软件工具**：书中提到的FoxitReader、PDF-XChangeViewer、SumatraPDF和Firefox都是能够打开PDF文档的软件，这些工具在学术界和工程领域中非常常见，用于阅读和管理电子资料。 9. **出版信息**：此书是Springer出版社“Texts in Computational Science and Engineering”系列的一部分，具有ISBN编号，并且可以在Springer的网站上找到额外的教材资源。 10. **学科交叉**：科学计算涵盖了许多科学和工程领域，如物理学、化学、生物科学、地球科学等，因此，理解和应用这些计算方法对于跨学科研究至关重要。

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

xvii

xviii Contents for Volume 2

1.4.3 Condition Numbers .......................................... 130

1.4.4 A PosterioriEstimates ....................................... 132

1.4.5 Rayleigh Quotients........................................... 132

1.4.6 Power Method................................................ 133

1.4.7 Orthogonal Iteration ......................................... 139

1.4.8 UpperHessenbergMatrices ................................. 144

1.4.9 General Eigenproblems...................................... 159

1.5 Singular Value Decomposition........................................ 160

1.5.1 Existence ..................................................... 161

1.5.2 NormalMatrices ............................................. 163

1.5.3 Pseudo-Inverse ............................................... 165

1.5.4 Range and Nullspace......................................... 168

1.5.5 Norms......................................................... 169

1.5.6 MinimaxTheorem ........................................... 170

1.5.7 PerturbationTheory.......................................... 172

1.5.8 Householder Bidiagonalization.............................. 178

1.5.9 Implicit QR Algorithm....................................... 180

1.5.10 General Singular Values ..................................... 182

1.5.11 Least SquaresProblems...................................... 184

1.6 LinearRecurrences .................................................... 185

1.6.1 Solutionof Linear Recurrences ............................. 185

1.6.2 Powers of Matrices........................................... 186

1.6.3 Examples ..................................................... 188

1.6.4 Stability ....................................................... 191

1.7 Functionsof Matrices ................................................. 193

1.7.1 ConvergentMatrix Series.................................... 193

1.7.2 Matrix Exponentials ......................................... 194

1.7.3 Matrix SquareRoots ......................................... 198

2 Iterative Linear Algebra .................................................... 203

2.1 Overview............................................................... 203

2.2 Sparse Matrices........................................................ 204

2.3 NeumannSeries........................................................ 205

2.4 IterativeImprovement................................................. 208

2.4.1 Theory ........................................................ 208

2.4.2 Richardson’sIteration........................................ 210

2.4.3 JacobiIteration ............................................... 216

2.4.4 Gauss-SeidelIteration ....................................... 220

2.4.5 Successive Over-Relaxation................................. 224

2.4.6 Termination Criteria for Iterative Methods ................. 228

2.5 Gradient Methods...................................................... 232

2.5.1 Matrix Inner Product ......................................... 234

2.5.2 Steepest Descent ............................................. 235

2.5.3 Conjugate Gradients ......................................... 238

Contents for Volume 2 xix

2.5.4 PreconditionedConjugateGradients........................ 249

2.5.5 BiconjugateGradients ....................................... 253

2.6 Minimum Residual Methods .......................................... 272

2.6.1 Orthomin ..................................................... 273

2.6.2 GMRES....................................................... 280

2.6.3 QMR.......................................................... 290

2.7 Multigrid ............................................................... 293

2.7.1 V-Cycle ....................................................... 294

2.7.2 W-Cycle ...................................................... 295

2.7.3 Work Estimate................................................ 296

2.7.4 Convergence.................................................. 298

2.7.5 Prolongation .................................................. 301

2.7.6 Multigrid Debugging Techniques ........................... 302

3 Nonlinear Systems ........................................................... 307

3.1 Overview............................................................... 308

3.2 Calculus................................................................ 309

3.2.1 Functionals ................................................... 309

3.2.2 Vector-ValuedFunctions..................................... 311

3.2.3 Second-Order Derivatives ................................... 316

3.2.4 LocalMinima ................................................ 319

3.2.5 Existence andUniqueness ................................... 322

3.2.6 PerturbationAnalysis ........................................ 332

3.3 ConvergenceRates .................................................... 337

3.4 Newton’sMethod...................................................... 338

3.4.1 LocalModel.................................................. 339

3.4.2 ConvergenceBehavior....................................... 339

3.4.3 Finite DifferenceJacobian................................... 341

3.4.4 Rounding Errors.............................................. 345

3.4.5 MaximumAttainable Accuracy ............................. 349

3.4.6 IterativeLinear Solvers ...................................... 353

3.4.7 Software ...................................................... 354

3.5 UnconstrainedMinimization.......................................... 356

3.5.1 DescentDirections ........................................... 356

3.5.2 Newton’sMethod ............................................ 357

3.5.3 Step Selection ................................................ 365

3.6 GlobalConvergence................................................... 381

3.6.1 TrustRegions................................................. 381

3.6.2 Nonlinear Systems ........................................... 396

3.7 Quasi-Newton Methods ............................................... 400

3.7.1 Broyden’sMethod............................................ 400

3.7.2 LocalConvergence........................................... 402

3.7.3 NumericalImplementation .................................. 412

3.7.4 UnconstrainedMinimization ................................ 416

xx Contents for Volume 2

3.8 Krylov Subspace Iterative Methods................................... 418

3.8.1 Conjugate Gradients ......................................... 419

3.8.2 Fletcher-Reeves .............................................. 420

3.8.3 GMRES....................................................... 420

3.9 Direct Search Methods ................................................ 422

3.9.1 AlternatingVariableMethod ................................ 423

3.9.2 SimplexSearch............................................... 423

3.10 StochasticMinimization............................................... 425

3.10.1 Random Search............................................... 425

3.10.2 Simulated Annealing......................................... 427

3.11 Test Problems.......................................................... 428

3.12 Case Study: Lennard Jones Potential ................................. 429

3.12.1 Two Atoms ................................................... 430

3.12.2 ThreeAtoms.................................................. 431

3.12.3 FourAtoms................................................... 431

3.12.4 FiveAtoms ................................................... 431

4 Constrained Optimization .................................................. 433

4.1 Overview............................................................... 433

4.2 LinearProgramming................................................... 435

4.2.1 ConvertingtoStandard Form................................ 436

4.2.2 Feasibility of Standard Form ................................ 439

4.2.3 Fundamental Theorem ....................................... 440

4.2.4 SimplexMethod.............................................. 442

4.2.5 Initial Basic Feasible Guess ................................. 446

4.2.6 SimplexTableau.............................................. 447

4.2.7 InverseUpdates .............................................. 450

4.2.8 Factorization Updates........................................ 451

4.2.9 Duality ........................................................ 453

4.2.10 ComplementarySlackness................................... 464

4.2.11 PerturbationAnalysis ........................................ 469

4.2.12 DualSimplexMethod........................................ 482

4.2.13 Karmarkar’sAlgorithm ...................................... 490

4.3 Kuhn-Tucker Theory .................................................. 499

4.3.1 Hypersurfaces and Tangent Planes .......................... 500

4.3.2 Equality Constraints ......................................... 503

4.3.3 Kuhn-Tucker Theorems...................................... 507

4.3.4 Lagrangian.................................................... 514

4.3.5 Duality ........................................................ 516

4.3.6 PerturbationAnalysis ........................................ 518

4.4 Quadratic Programming............................................... 520

4.4.1 Theory ........................................................ 521

4.4.2 ConstrainedObjective ....................................... 522

4.4.3 PositiveConstrained Hessian................................ 523

4.4.4 Nonnegative Constrained Hessian........................... 523

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