Rent 9780534955328 | Numerical Analysis (6th Ed)

Summary Table of Contents

This well-respected text gives an introduction to the modern approximation techniques and explains how, why, and when the techniques can be expected to work. The authors focus on building students' intuition to help them understand why the techniques presented work in general, and why, in some situations, they fail. With a wealth of examples and exercises, the text demonstrates the relevance of numerical analysis to a variety of disciplines and provides ample practice for students. The applications chosen demonstrate concisely how numerical methods can be, and often must be, applied in real-life situations. In this edition, the presentation has been fine-tuned to make the book even more useful to the instructor and more interesting to the reader. Overall, students gain a theoretical understanding of, and a firm basis for future study of, numerical analysis and scientific computing. A more applied text with a different menu of topics is the authors' highly regarded NUMERICAL METHODS, Third Edition.

1 Mathematical Preliminaries

(46)

1.1 Review of Calculus

(14)

... MORE

1.2 Roundoff Errors and Computer Arithmetic

(14)

1.3 Algorithms and Convergence

(10)

1.4 Numerical Software

(7)

2 Solutions of Equations in One Variable

(57)

2.1 The Bisection Method

(6)

2.2 Fixed-Point Iteration

(11)

2.3 The Newton-Raphson Method

(13)

2.4 Error Analysis for Iterative Methods

(9)

2.5 Accelerating Convergence

(4)

2.6 Zeros of Polynomials and Muller's Method

(11)

2.7 Survey of Methods and Software

102

(2)

3 Interpolation and Polynomial Approximation

104

(63)

3.1 Interpolation and the Lagrange Polynomial

107

(16)

3.2 Divided Differences

123

(12)

3.3 Hermite Interpolation

135

(8)

3.4 Cubic Spline Interpolation

143

(15)

3.5 Parametric Curves

158

(7)

3.6 Survey of Methods and Software

165

(2)

4 Numerical Differentiation and Integration

167

(85)

4.1 Numerical Differentiation

168

(12)

4.2 Richardson's Extrapolation

180

(8)

4.3 Elements of Numerical Integration

188

(11)

4.4 Composite Numerical Integration

199

(10)

4.5 Romberg Integration

209

(6)

4.6 Adaptive Quadrature Methods

215

(7)

4.7 Gaussian Quadrature

222

(7)

4.8 Multiple Integrals

229

(15)

4.9 Improper Integrals

244

(6)

4.10 Survey of Methods and Software

250

(2)

5 Initial-Value Problems for Ordinary Differential Equations

252

(98)

5.1 The Elementary Theory of Initial-Value Problems

254

(5)

5.2 Euler's Method

259

(10)

5.3 Higher-Order Taylor Methods

269

(7)

5.4 Runge-Kutta Methods

276

(10)

5.5 Error Control and the Runge-Kutta-Fehlberg Method

286

(8)

5.6 Multistep Methods

294

(12)

5.7 Variable Step-Size Multistep Methods

306

(5)

5.8 Extrapolation Methods

311

(7)

5.9 Higher-Order Equations and Systems of Differential Equations

318

(12)

5.10 Stability

330

(10)

5.11 Stiff Differential Equations

340

(8)

5.12 Survey of Methods and Software

348

(2)

6 Direct Methods for Solving Linear Systems

350

(73)

6.1 Linear Systems of Equations

351

(15)

6.2 Pivoting Strategies

366

(10)

6.3 Linear Algebra and Matrix Inversion

376

(13)

6.4 The Determinant of a Matrix

389

(5)

6.5 Matrix Factorization

394

(10)

6.6 Special Types of Matrices

404

(15)

6.7 Survey of Methods and Software

419

(4)

7 Iterative Techniques in Matrix Algebra

423

(50)

7.1 Norms of Vectors and Matrices

424

(12)

7.2 Eigenvalues and Eigenvectors

436

(7)

7.3 Iterative Techniques for Solving Linear Systems

443

(18)

7.4 Error Estimates and Iterative Refinement

461

(10)

7.5 Survey of Methods and Software

471

(2)

8 Approximation Theory

473

(66)

8.1 Discrete Least Squares Approximation

474

(13)

8.2 Orthogonal Polynomials and Least Squares Approximation

487

(9)

8.3 Chebyshev Polynomials and Economization of Power Series

496

(11)

8.4 Rational Function Approximation

507

(11)

8.5 Trigonometric Polynomial Approximation

518

(8)

8.6 Fast Fourier Transforms

526

(11)

8.7 Survey of Methods and Software

537

(2)

9 Approximating Eigenvalues

539

(49)

9.1 Linear Algebra and Eigenvalues

540

(8)

9.2 The Power Method

548

(17)

9.3 Householder's Method

565

(8)

9.4 The QR Algorithm

573

(13)

9.5 Survey of Methods and Software

586

(2)

10 Numerical Solutions of Nonlinear Systems of Equations

588

(36)

10.1 Fixed Points for Functions of Several Variables

589

(10)

10.2 Newton's Method

599

(8)

10.3 Quasi-Newton Methods

607

(7)

10.4 Steepest Descent Techniques

614

(7)

10.5 Survey of Methods and Software

621

(3)

11 Boundary-Value Problems for Ordinary Differential Equations

624

(46)

11.1 The Linear Shooting Method

625

(7)

11.2 The Shooting Method for Nonlinear Problems

632

(7)

11.3 Finite-Difference Methods for Linear Problems

639

(7)

11.4 Finite-Difference Methods for Nonlinear Problems

646

(6)

11.5 The Rayleigh-Ritz Method

652

(16)

11.6 Survey of Methods and Software

668

(2)

12 Numerical Solutions to Partial-Differential Equations

670

(55)

12.1 Elliptic Partial-Differential Equations

673

(9)

12.2 Parabolic Partial-Differential Equations

682

(16)

12.3 Hyperbolic Partial-Differential Equations

698

(9)

12.4 An Introduction to the Finite-Element Method

707

(15)

12.5 Survey of Methods and Software

722

(3)

Bibliography

725

(10)

Answers to Selected Exercises

735

(68)

Index

803

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Numerical Analysis (6th Ed)

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