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More About This Textbook
Overview
This comprehensive book includes over 800 problems including open ended, project type and design problems. Chapter topics include Introduction to Numerical Methods; Solution of Nonlinear Equations; Simultaneous Linear Algebraic Equations; Solution of Matrix Eigenvalue Problem; Curve Fitting and Interpolation; Statistical Methods; Numerical Differentiation; Numerical Integration; Numerical Solution of Ordinary Differential Equations: Initial Value Problems; Numerical Solution of Ordinary Differential Equations: Boundary Value Problems; Numerical Solution of Partial Differential Equations; Numerical Methods of Optimization ;Finite Element Method. This book is intended as a reference for numerical methods in engineering.
Editorial Reviews
From The Critics
Intended primarily for engineering students, this textbook concentrates of the application of numerical methods to engineering problems, using different approaches and a variety of software. It encourages the student to intelligently select among numerical approaches and software systems, and to experiment with the various parameters of the problem accordingly. Chapters concentrate on numerical methods, nonlinear equations, simultaneous linear algebraic equations, matrix eigenvalue problems, curve fitting and interpretation, statistical methods, numerical differentiation, numerical integration, initialvalue problems, boundaryvalue problems, partial differential equations, optimization, and the finiteelement method. Rao teaches at the University of Miami at Coral Gables. Annotation c. Book News, Inc., Portland, OR (booknews.com)Product Details
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Read an Excerpt
The use of numerical methods for the analysis, simulation, and design of engineering processes and systems has been increasing at a rapid rate in recent years. The availability of cheap highspeed computing power makes the numerical solution of even complex engineering problems economically feasible. In the face of ever increasing demands on engineering profession to perform better, the students who learn numerical methods in preparing to face the challenges of 21st century should learn not only the theory behind the methods, but also acquire skills to implement the methods for computer solution. In addition, the students should be aware of the many commercial software systems available and their use in the solution of engineering problems. Although a student may not learn all the numerical methods described in this book and use all the software systems available in any one course, he or she should be in a position to intelligently select and use suitable numerical methods and software systems as the need arises in practice.
The use of numerical methods in engineering can be considered partly science and partly art. Thus, a cookbooktype procedure will not be effective in learning the methods. A student should solve a problem using different approaches and a variety of software systems and experiment with the various parameters of the problem. The different results obtained through this process will form an experience base for selecting a suitable method and interpreting the results for a new problem. It is always desirable to compare and verify the results with other available solutions based on engineering judgment and intuition.
This book is intended for courses on numerical methods at the junior and senior level as well as at the beginning graduate level. The book also serves as a reference for numerical methods in engineering. Fortran and C programs, along with illustrative examples, are given in each chapter to implement many of the numerical methods discussed in that chapter. The use of commercial numerical softwaresâ€”MATLAB, MAPLE and MATHCADâ€”in the solution of practical problems is demonstrated in every chapter. Even when a program from a software package is used, we need to understand the basic principles, purpose, and limitations of the program. Often, in many engineering applications, an available standard program cannot be used directly; we need to adapt and modify it. This invariably requires a sound knowledge of the numerical method as well as some computational experience with the method. The book is aimed at presenting numerical methods along with their practical applications in a manner that helps students achieve the goals just outlined.
Organization
Applied Numerical Methods for Engineers is organized into 13 chapters and 6 appendices. Chapter 1 presents an overview of numerical methods, iterative processes, numerical errors, software available for numerical methods, programming languages, and the various aspects of computer program development. The methods of solving nonlinear equations are given in Chapter 2. The solution of sets of linear algebraic equations is presented in Chapter 3. Both direct and iterative methods are considered. The matrix eigenvalue problem is the topic of Chapter 4. Chapter 5 deals with the methods of curve fitting and interpolation. The probabilistic and statistical methods are considered in Chapter 6. The numerical differentiation and numerical integration are the topics of Chapters 7 and 8, respectively. The numerical solution of ordinary differential equations is considered in Chapters 9 and 10. While Chapter 9 presents the methods of solving initialvalue problems, Chapter 10 deals with the solution of boundaryvalue problems. The numerical solution of partial differential equations is considered in Chapter 11. The optimization and the finiteelement methods are presented in Chapters 12 and 13, respectively. Appendices A and B provide the basics of Fortran and C languages while Appendices C, D, and E summarize the basics of MAPLE, MATLAB, and MATHCAD, respectively. A review of matrix algebra is given in Appendix F. Finally, Appendix G presents tables of statistical distributions.
The material of the book provides flexible options for different types of numerical methods courses. A junior and senior level course may cover the basic techniques of Chapters 1, 2, 3, and 5 to 9. A firstlevel graduate course can cover Chapters 4, 10, 11, 12, and 13 as well. The prerequisites for using the text are elementary calculus, basic concepts of linear algebra, and an introduction to differential equations.
Each topic for Applied Numerical Methods for Engineers is selfcontained. In derivations and developments, steps needed for continuity of understanding have been included to aid the reader at the introductory level. Representative engineering applications are given at the beginning of each chapter so that the reader can appreciate the practical use and application of the numerical methods presented in that chapter. Many sample problems are solved by using several methods, and the results are compared, discussed, and general conclusions are drawn. Most of the algorithms described in the book are implemented in the form of Fortran and C codes and are made available at the Web site of the book. The use of different commercial software systems, as well as the programs available at the Web site of the book, is illustrated in each chapter.
Features
The specific features of the book include
The Fortran and C programs used in the book, answers to problems, solutions to review questions, and brief biographical information of scientists can be found at the web site of the book: http://www.prenhall.com/rao. Note that the programs and techniques presented in the book and at the web site are intended for use by students in learning the material. Although the material has been tested, no warranty is implied as to their accuracy. I would appreciate receiving any errors found in the book.
Table of Contents
1. Introduction to Numerical Methods.
Importance of Numerical Methods in Engineering. Computers. Computer Programming Languages. Data Representation. Programming Structure. Errors. Numerical Methods Considered. Software for Numerical Analysis. Use of Software Packages. Computer Programs.
2. Solution of Nonlinear Equations.
Introduction. Engineering Applications. Incremental Search Method. Bisection Method. NewtonRaphson Method. Secant Method. Regula Falsi Method. Fixed Point Iteration or Successive Substitution Method. Determination of Multiple Roots. Bairstow's Method. Muller's Method. NewtonRaphson Method for Simultaneous Nonlinear Equations. Unconstrained Minimization. Convergence of Methods. Choice of Method. Use of Software Packages. Computer Programs.
3. Solution of Simultaneous Linear Algebraic Equations.
Introduction. Engineering Applications. Vector and Matrix Norms. Basic Concepts of Solution. Linearly Independent Equations. IllConditioned Equations. Graphical Interpretation of the Solution. Solution Using Cramer's Rule. Gauss Elimination Method. GaussJordan Elimination Procedure. LU Decomposition Method. Jacobi Iteration Method. GaussSeidel Iteration Method. Relaxation Methods. Simultaneous Linear Equations with Complex Coefficients and Constants. Matrix Inversion. Equations with Special Form of Coefficient Matrix. Overdetermined, Underdetermined, and Homogeneous Equations. Comparative Efficiencies of Various Methods and Recommendations. Choice of the Method. Use of Software Packages. Computer Programs.
4. Solution of Matrix Eigenvalue Problem.
Introduction. Engineering Applications. Conversion of General Eigenvalue Problem to Standard Form. Methods of Solving Eigenvalue Problems. Solution of the Characteristic Polynomial Equations. Jacobi Method. Given's Method. Householder's Method. Eigenvalues of a Tridiagonal Matrix. Eigenvectors of a Tridiagonal Matrix. Power Method. Choice of Method. Use of Software Packages. Computer Programs.
5. Curve Fitting and Interpolation.
Introduction. Engineering Applications. CollocationPolynomial Fit. Interpolation. Lagrange Interpolation Formula. Newton's DividedDifference Interpolating Polynomials. Interpolation Using Chebysev Polynomials. Interpolation Using Splines. LeastSquares Regression. Curve Fitting with Multiple Variables. Choice of Method. Use of Software Packages. Computer Programs.
6. Statistical Methods.
Introduction. Engineering Applications. Basic Definitions. Histogram and Probability Density Function. Statistical Characteristics. Normal Distributions. Statistical Tests. ChiSquare Test for Distribution. Choice of Method. Use of Software Packages. Computer Programs.
7. Numerical Differentiation.
Introduction. Engineering Applications. Definition of the Derivative. Basic FiniteDifference Approximations. Using Taylor's Series Expansions. Using Difference Operators. Approximation of Derivatives Using Difference Operators. Using Differentiation of Interpolating Polynomials. FiniteDifference Approximations for Partial Derivatives. Choice of Method. Use of Software Packages. Computer Programs.
8. Numerical Integration.
Introduction. Engineering Applications. NewtonCotes Formulas. Simpson's Rule. General NewtonCotes Formulas. Richardson's Extrapolation. Romberg Integration. Gauss Quadrature. Integration with Unequal Segments. Numerical Integration of Improper Integrals. Numerical Integration in Two and ThreeDimensional Domains. Choice of Method. Use of Software Packages. Computer Programs.
9. Ordinary Differential Equations: InitialValue Problems.
Introduction. Engineering Applications. Simultaneous Differential Equations. Solution Concept. Euler's Method. Improvements and Modifications of Euler's Method. RungeKutta Methods. Multistep Methods. Adams Methods. PredictorCorrector Methods. Simultaneous Differential Equations. Stiff Equations. Choice of Method. Use of Software Packages. Computer Programs.
10. Ordinary Differential Equations: BoundaryValue Problems.
Introduction. Engineering Applications. Shooting Methods. Generalization to n Equations. FiniteDifference Methods. Solution of Nonlinear BoundaryValue Problems. Solution of Eigenvalue Problems. Choice of Method. Use of Software Packages. Computer Programs.
11. Partial Differential Equations.
Introduction. Engineering Applications. Initial and Boundary Conditions. Elliptic Partial Differential Equations. Parabolic Partial Differential Equations. CrankNicholson Method. Method of Lines. TwoDimensional Parabolic Problems. Hyperbolic Partial Differential Equations. Method of Characteristics. FiniteDifference Formulas in Polar Coordinate System. Choice of Method. Use of Software Packages. Computer Programs.
12. Optimization.
Introduction. Types of Optimization Problems. Engineering Applications. Optimization Methods from Differential Calculus. LinearProgramming Problem. Simplex Method. Search Methods for Nonlinear Optimization. Optimization of a Function of a Single Variable. Unconstrained Minimization of a Function of Several Variables. Constrained Minimization of a Function of Several Variables. Choice of Method. Use of Software Packages. Computer Programs.
13. FiniteElement Method.
Introduction. Engineering Applications. Discretization of the Domain. Interpolation Functions. Derivation of Element Characteristic Matrices and Vectors. Assemblage of Element Characteristics Matrices and Vectors. Solution of System Equations. Choice of Method. Use of Software Packages. Computer Programs.
Appendix A: Basics of Fortran 90.
Appendix B: Basics of C Language.
Appendix C: Basics of MAPLE.
Appendix D: Basics of MATLAB.
Appendix E: Basics of MathCAD.
Appendix F: Review of Matrix Algebra.
Appendix G: Statistical Tables.
Index.
Preface
The use of numerical methods for the analysis, simulation, and design of engineering processes and systems has been increasing at a rapid rate in recent years. The availability of cheap highspeed computing power makes the numerical solution of even complex engineering problems economically feasible. In the face of ever increasing demands on engineering profession to perform better, the students who learn numerical methods in preparing to face the challenges of 21st century should learn not only the theory behind the methods, but also acquire skills to implement the methods for computer solution. In addition, the students should be aware of the many commercial software systems available and their use in the solution of engineering problems. Although a student may not learn all the numerical methods described in this book and use all the software systems available in any one course, he or she should be in a position to intelligently select and use suitable numerical methods and software systems as the need arises in practice.
The use of numerical methods in engineering can be considered partly science and partly art. Thus, a cookbooktype procedure will not be effective in learning the methods. A student should solve a problem using different approaches and a variety of software systems and experiment with the various parameters of the problem. The different results obtained through this process will form an experience base for selecting a suitable method and interpreting the results for a new problem. It is always desirable to compare and verify the results with other available solutions based on engineering judgment and intuition.
This book is intended for courses on numerical methods at the junior and senior level as well as at the beginning graduate level. The book also serves as a reference for numerical methods in engineering. Fortran and C programs, along with illustrative examples, are given in each chapter to implement many of the numerical methods discussed in that chapter. The use of commercial numerical softwaresâ€”MATLAB, MAPLE and MATHCADâ€”in the solution of practical problems is demonstrated in every chapter. Even when a program from a software package is used, we need to understand the basic principles, purpose, and limitations of the program. Often, in many engineering applications, an available standard program cannot be used directly; we need to adapt and modify it. This invariably requires a sound knowledge of the numerical method as well as some computational experience with the method. The book is aimed at presenting numerical methods along with their practical applications in a manner that helps students achieve the goals just outlined.
Organization
Applied Numerical Methods for Engineers is organized into 13 chapters and 6 appendices. Chapter 1 presents an overview of numerical methods, iterative processes, numerical errors, software available for numerical methods, programming languages, and the various aspects of computer program development. The methods of solving nonlinear equations are given in Chapter 2. The solution of sets of linear algebraic equations is presented in Chapter 3. Both direct and iterative methods are considered. The matrix eigenvalue problem is the topic of Chapter 4. Chapter 5 deals with the methods of curve fitting and interpolation. The probabilistic and statistical methods are considered in Chapter 6. The numerical differentiation and numerical integration are the topics of Chapters 7 and 8, respectively. The numerical solution of ordinary differential equations is considered in Chapters 9 and 10. While Chapter 9 presents the methods of solving initialvalue problems, Chapter 10 deals with the solution of boundaryvalue problems. The numerical solution of partial differential equations is considered in Chapter 11. The optimization and the finiteelement methods are presented in Chapters 12 and 13, respectively. Appendices A and B provide the basics of Fortran and C languages while Appendices C, D, and E summarize the basics of MAPLE, MATLAB, and MATHCAD, respectively. A review of matrix algebra is given in Appendix F. Finally, Appendix G presents tables of statistical distributions.
The material of the book provides flexible options for different types of numerical methods courses. A junior and senior level course may cover the basic techniques of Chapters 1, 2, 3, and 5 to 9. A firstlevel graduate course can cover Chapters 4, 10, 11, 12, and 13 as well. The prerequisites for using the text are elementary calculus, basic concepts of linear algebra, and an introduction to differential equations.
Each topic for Applied Numerical Methods for Engineers is selfcontained. In derivations and developments, steps needed for continuity of understanding have been included to aid the reader at the introductory level. Representative engineering applications are given at the beginning of each chapter so that the reader can appreciate the practical use and application of the numerical methods presented in that chapter. Many sample problems are solved by using several methods, and the results are compared, discussed, and general conclusions are drawn. Most of the algorithms described in the book are implemented in the form of Fortran and C codes and are made available at the Web site of the book. The use of different commercial software systems, as well as the programs available at the Web site of the book, is illustrated in each chapter.
Features
The specific features of the book include
Web site of the book
The Fortran and C programs used in the book, answers to problems, solutions to review questions, and brief biographical information of scientists can be found at the web site of the book: http://www.prenhall.com/rao . Note that the programs and techniques presented in the book and at the web site are intended for use by students in learning the material. Although the material has been tested, no warranty is implied as to their accuracy. I would appreciate receiving any errors found in the book.