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NUMERICAL METHODS, Fourth Edition emphasizes the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Students learn why the numerical methods work, what kinds of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. The techniques are the same as those covered in the authors' top-selling Numerical Analysis text, but this text provides an overview for students who need to know the methods without having to perform the analysis. This concise approach still includes mathematical justifications, but only when they are necessary to understand the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the student that the method is reasonable both mathematically and computationally.
- New examples and exercises appear throughout the text, offering fresh options for assignments.
- Chapter 7, "Iterative Methods for Solving Linear Systems," includes a new section on Conjugate Gradient Methods.
- Chapter 10, "Solutions of Systems of Nonlinear Equations," includes a new section on Homotopy and Continuation Methods.
- Revised techniques for algorithms and programs are included in six languages: FORTRAN, Pascal, C, MAPLE, Mathematica, and MATLAB.
- All of the Maple material in the text is updated to conform with the newest release (Maple 7). All of the material on the CD that accompanies the book is updated to conform to the latest available versions of Maple, Mathematica, and MATLAB.
- This edition includes many more examples of Maple code.
- This text is designed for use in a one-semester course, but contains more material than needed. Instructors have flexibility in choosing topics and students gain a useful reference for future work.
- Worked examples using computer algebra systems help students understand why the software usually works, why it might fail, and what to do when a software program fails.
- The exercise sets include problems reflecting a wide range of difficulty as well as problems that offer good illustrations of the methods being discussed, while requiring little calculation.
- The book contains instructions for a wide range of popular computer algebra systems.
Introduction. Review of Calculus. Round-off Error and Computer Arithmetic. Errors in Scientific Computation. Computer Software.
2. SOLUTIONS OF EQUATIONS OF ONE VARIABLE.
Introduction. The Bisection Method. The Secant Method. Newton's Method. Error Analysis and Accelerating Convergence. Müller's Method. Survey of Methods and Software.
3. INTERPOLATION AND POLYNOMIAL APPROXIMATION.
Introduction. Lagrange Polynomials. Divided Differences. Hermite Interpolation. Spline Interpolation. Parametric Curves. Survey of Methods and Software.
4. NUMERICAL INTEGRATION AND DIFFERENTIATION.
Introduction. Basic Quadrature Rules. Composite Quadrature Rules. Romberg Integration. Gaussian Quadrature. Adaptive Quadrature. Multiple Integrals. Improper Integrals. Numerical Differentiation. Survey of Methods and Software.
5. NUMERICAL SOLUTION OF INITIAL-VALUE PROBLEMS.
Introduction. Taylor Methods. Runge-Kutta Methods. Predictor-Corrector Methods. Extrapolation Methods. Adaptive Techniques. Methods for Systems of Equations. Stiff Differentials Equations. Survey of Methods and Software.
6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS.
Introduction. Gaussian Elimination. Pivoting Strategies. Linear Algebra and Matrix Inversion. Matrix Factorization. Techniques for Special Matrices. Survey of Methods and Software.
7. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS.
Introduction. Convergence of Vectors. Eigenvalues and Eigenvectors. Conjugate Gradient Methods. The Jacobi and Gauss-Seidel Methods. The SOR Method. Error Bounds and Iterative Refinement. Survey of Methods and Software.
8. APPROXIMATION THEORY.
Introduction. Discrete Least Squares Approximation. Continuous Least Squares Approximation. Chebyshev Polynomials. Rational Function Approximation. Trigonometric Polynomial Approximation. Fast Fourier Transforms. Survey of Methods and Software.
9. APPROXIMATING EIGENVALUES.
Introduction. Isolating Eigenvalues. The Power Method. Householder's Method. The QR Method. Survey of Methods and Software.
10. SOLUTIONS OF SYSTEMS OF NONLINEAR EQUATIONS.
Introduction. Newton's Methods for Systems. Quasi-Newton Methods. The Steepest Descent Method. Survey of Methods and Software. Homotopy and Continuation Methods.
11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS.
Introduction. The Linear Shooting Method. Linear Finite Difference Methods. The Nonlinear Shooting Method. Nonlinear Finite-Difference Methods. Variational Techniques. Survey of Methods and Software.
12. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.
Introduction. Finite-Difference Methods for Elliptic Problems. Finite-Difference Methods for Parabolic Problems. Finite-Difference Methods for Hyperbolic Problems. Introduction to the Finite-Element Method. Survey of Methods and Software.
Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.
This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. www.cengage.com/solutionbuilder
Student Solutions Manual
Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Student Solutions Manual