The seventh edition of this classic text has retained the features that make it popular, while updating its treatment and inclusion of Computer Algebra Systems and Programming Languages. Interesting and timely applications motivate and enhance readers'understanding of methods and analysis of results. This text incorporates a balance of theory with techniques and applications, including optional theory-based sections in each chapter. The exercise sets include additional challenging problems and projects which show practical applications of the material. Also, sections which discuss the use of computer algebra systems such as Maplereg;,Mathematicareg;, and MATLABreg;, facilitate the integration of technology in the course. Furthermore, the text incorporates programming material in both FORTRAN and C. The breadth of topics, such as partial differential equations, systems of nonlinear equations, and matrix algebra, provide comprehensive and flexible coverage of all aspects of numerical analysis. Preliminaries; Solving Nonlinear Equations; Solving Sets of Equations; Interpolation and Curve Fitting; Approximation of Functions; Numerical Differentiation and Integration; Numerical Solution of Ordinary Differential Equations; Optimization; Partial Differential Equations; Finite Element Analysis For all readers interested in applied numerical analysis.

Preface

ix

0 Preliminaries

1

(31)

Contents of This Chapter

1

(1)

0.1 Analysis Versus Numerical Analysis

2

(2)

0.2 Computers and Numerical Analysis

4

(2)

0.3 An Illustrative Example

6

(4)

0.4 Kinds of Errors in Numerical Procedures

10

(9)

0.5 Interval Arithmetic

19

(2)

0.6 Parallel and Distributed Computing

21

(5)

0.7 Measuring the Efficiency of Numerical Procedures

26

(2)

Exercises

28

(2)

Applied Problems and Projects

30

(2)

1 Solving Nonlinear Equations

32

(44)

Contents of This Chapter

33

(1)

1.1 Interval Halving (Bisection)

33

(5)

1.2 Linear Interpolation Methods

38

(4)

1.3 Newton's Method

42

(8)

1.4 Muller's Method

50

(4)

1.5 Fixed-Point Interation: x = g(x) Method

54

(6)

1.6 Multiple Roots

60

(3)

1.7 Nonlinear Systems

63

(4)

Exercises

67

(4)

Applied Problems and Projects

71

(5)

2 Solving Sets of Equations

Contents of This Chapter

76

(1)

2.1 Matrices and Vectors

77

(11)

2.2 Elimination Methods

88

(18)

2.3 The Inverse of a Matrix and Matrix Pathology

106

(4)

2.4 Ill-Conditioned Systems

110

(11)

2.5 Iterative Methods

121

(8)

2.6 Parallel Processing

129

(6)

Exercises

135

(6)

Applied Problems and Projects

141

(6)

3 Interpolation and Curve Fitting

147

(73)

Contents of This Chapter

148

(1)

3.1 Interpolating Polynomials

149

(8)

3.2 Divided Differences

157

(11)

3.3 Spline Curves

168

(11)

3.4 Bezier Curves and B-Splines Curves

179

(9)

3.5 Interpolating on a Surface

188

(11)

3.6 Least-Squares Approximations

199

(10)

Exercises

209

(6)

Applied Problems and Projects

215

(5)

4 Approximation of Functions

220

(36)

Contents of This Chapter

220

(1)

4.1 Chebyshev Polynomials and Chebyshev Series

221

(11)

4.2 Rational Function Approximations

232

(8)

4.3 Fourier Series

240

(12)

Exercises

252

(2)

Applied Problems and Projects

254

(2)

5 Numerical Differentiation and Integration

256

(73)

Contents of This Chapter

257

(1)

5.1 Differentiation with a Computer

258

(14)

5.2 Numerical Integration-The Trapezoidal Rule

272

(8)

5.3 Simpson's Rules

280

(5)

5.4 An Application of Numerical Integration-Fourier Series and Fourier Transforms

285

(12)

5.5 Adaptive Integration

297

(4)

5.6 Gaussian Quadrature

301

(6)

5.7 Multiple Integrals

307

(10)

5.8 Applications of Cubic Splines

317

(4)

Exercises

321

(5)

Applied Problems and Projects

326

(3)

6 Numerical Solution of Ordinary Differential Equations

329

(76)

Contents of This Chapter

330

(2)

6.1 The Taylor-Series Method

332

(3)

6.2 The Euter Method and Its Modifications

335

(5)

6.3 Runge-Kutta Methods

340

(7)

6.4 Multistep Methods

347

(12)

6.5 Higher-Order Equations and Systems

359

(5)

6.6 Stiff Equations

364

(2)

6.7 Boundary-Value Problems

366

(15)

6.8 Characteristic-Value Problems

381

(13)

Exercises

394

(5)

Applied Problems and Projects

399

(6)

7 Optimization

405

(56)

Contents of This Chapter

405

(1)

7.1 Finding the Minimum of y = f(x)

406

(11)

7.2 Minimizing a Function of Several Variables

417

(11)

7.3 Linear Programming

428

(14)

7.4 Nonlinear Programming

442

(7)

7.5 Other Optimizations

449

(4)

Exercises

453

(5)

Applied Problems and Projects

458

(3)

8 Partial-Differential Equations

461

(56)

Contents of This Chapter

463

(1)

8.1 Elliptic Equations

463

(18)

8.2 Parabolic Equations

481

(18)

8.3 Hyperbolic Equations

499

(10)

Exercises

509

(4)

Applied Problems and Projects

513

(4)

9 Finite-Element Analysis

517

(50)

Contents of This Chapter

518

(1)

9.1 Mathematical Background

518

(8)

9.2 Finite Elements for Ordinary-Differential Equations

526

(9)

9.3 Finite Elements for Partial-Differential Equations

535

(27)

Exercises

562

(2)

Applied Problems and Projects

564

(3)

Appendixes

A Some Basic Information from Calculus

567

(4)

B Software Resources

571

(4)

Answers to Selected Exercises

575

(24)

References

599

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Applied Numerical Analysis

9780321133045

0321133048

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