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Overview
Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. NUMERICAL MATHEMATICS AND COMPUTING, 7th Edition also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors.
- UPDATED! The Solving Systems of Linear Equations chapter has been moved earlier in the text to provide more clarity throughout the text.
- NEW! Exercises, computer exercises, and application exercises have been added to the text.
- NEW! A section of Fourier Series and Fast Fourier Transforms has been added.
- The first two chapters in the previous edition on Mathematical Preliminaries, Taylor Series, Oating-Point Representation, and Errors have been combined into a single introductory chapter to allow instructors and students to move quickly.
- Some sections and material have been re-moved from the new edition such as the introductory section on numerical integration. Some material and many bibliographical items have been moved from the textbook to the website.
- The two chapters, in the previous edition, on Ordinary Differential Equations have been combined into one chapter.
- Comprehensive, Current and Cutting Edge: Completely updated, the new edition includes new sections and material on such topics as the modified false position method, the conjugate gradient method, Simpsons method, and more.
- Hands-On Applications: Giving students myriad opportunities to put chapter concepts into real practice, additional exercises involving applications are presented throughout.
- References: Citation to recent references reflects the latest developments from the field.
- Appendices: Reorganized and revamped, new appendices offer a wealth of supplemental material, including advice on good programming practices, coverage of numbers in different bases, details on IEEE floating-point arithmetic, and discussions of linear algebra concepts and notation.
1. MATHEMATICAL PRELIMINARIES AND FLOATING-POINT REPRESENTATION.
Introduction, Mathematical Preliminaries. Floating-Point Representation. Loss of Significance.
2. LINEAR SYSTEMS.
Naive Gaussian Elimination. Gaussian Elimination with Scaled Partial Pivoting. Tridiagonal and Banded Systems.
3. NONLINEAR EQUATIONS.
Bisection Method. Newton’s Method, Secant Method.
4. INTERPOLATION AND NUMBERICAL DIFFERENTIATION.
Polynomial Interpolation. Errors in Polynomial Interpolation. Estimating Derivatives and Richardson Extrapolation.
5. NUMERICAL INTEGRATION.
Trapezoid Method. Romberg Algorithm. Simpson’s Rules and Newton-Cotes Rules. Gaussian Quadrature Formulas.
6. SPLINE FUNCTIONS.
First-Degree and Second-Degree Splines. Natural Cubic Splines. B Splines: Interpolation and Approximation.
7. INITIAL VALUES PROBLEMS.
Taylor Series Methods. Runge-Kutta Methods. Adaptive Runge-Kutta and Multistep Methods. Methods for First and Higher-Order Systems. Adams-Bashforth-Moulton Methods.
8. MORE ON LINEAR SYSTEMS.
Matrix Factorizations. Eigenvalues and Eigenvectors. Power Method. Iterative Solutions of Linear Systems.
9. LEAST SQUARES METHODS AND FOURIER SERIES.
Method of Least Squares. Orthogonal Systems and Chebyshev Polynomials. Examples of the Least-Squares Principle. Fourier Series.
10. MONTE CARLO METHODS AND SIMULATION.
Random Numbers. Estimation of Areas and Volumes by Monte Carlo Techniques. Simulation.
11. BOUNDARY-VALUE PROBLEMS.
Shooting Method. A Discretization Method.
12. PARTIAL DIFFERENTIAL EQUATIONS.
Parabolic Problems. Hyperbolic Problems. Elliptic Problems.
13. MINIMIZATION OF FUNTIONS.
One-Variable Case. Multivariable Case.
14. LINEAR PROGRAMMING PROBLEMS.
Standard Forms and Duality. Simplex Method, Inconsistent Linear Systems.
APPENDIX A. ADVICE ON GOOD PROGRAMMING PRACTICES.
Programming Suggestions.
APPENDIX B. REPRESENTATION OF NUMBERS IN DIFFERENT BASES.
Representation of Numbers in Different Bases.
APPENDIX C. ADDITIONAL DETAILS ON IEEE FLOATING-POINT ARITHMETIC.
More on IEEE Standard Floating-Point Arithmetic.
APPENDIX D. LINEAR ALGEBRA CONCEPTS AND NOTATION.
Elementary Concepts.
ANSWERS FOR SELECTED EXERCISES.
BIBLIOGRAPHY.
INDEX.
Introduction, Mathematical Preliminaries. Floating-Point Representation. Loss of Significance.
2. LINEAR SYSTEMS.
Naive Gaussian Elimination. Gaussian Elimination with Scaled Partial Pivoting. Tridiagonal and Banded Systems.
3. NONLINEAR EQUATIONS.
Bisection Method. Newton’s Method, Secant Method.
4. INTERPOLATION AND NUMBERICAL DIFFERENTIATION.
Polynomial Interpolation. Errors in Polynomial Interpolation. Estimating Derivatives and Richardson Extrapolation.
5. NUMERICAL INTEGRATION.
Trapezoid Method. Romberg Algorithm. Simpson’s Rules and Newton-Cotes Rules. Gaussian Quadrature Formulas.
6. SPLINE FUNCTIONS.
First-Degree and Second-Degree Splines. Natural Cubic Splines. B Splines: Interpolation and Approximation.
7. INITIAL VALUES PROBLEMS.
Taylor Series Methods. Runge-Kutta Methods. Adaptive Runge-Kutta and Multistep Methods. Methods for First and Higher-Order Systems. Adams-Bashforth-Moulton Methods.
8. MORE ON LINEAR SYSTEMS.
Matrix Factorizations. Eigenvalues and Eigenvectors. Power Method. Iterative Solutions of Linear Systems.
9. LEAST SQUARES METHODS AND FOURIER SERIES.
Method of Least Squares. Orthogonal Systems and Chebyshev Polynomials. Examples of the Least-Squares Principle. Fourier Series.
10. MONTE CARLO METHODS AND SIMULATION.
Random Numbers. Estimation of Areas and Volumes by Monte Carlo Techniques. Simulation.
11. BOUNDARY-VALUE PROBLEMS.
Shooting Method. A Discretization Method.
12. PARTIAL DIFFERENTIAL EQUATIONS.
Parabolic Problems. Hyperbolic Problems. Elliptic Problems.
13. MINIMIZATION OF FUNTIONS.
One-Variable Case. Multivariable Case.
14. LINEAR PROGRAMMING PROBLEMS.
Standard Forms and Duality. Simplex Method, Inconsistent Linear Systems.
APPENDIX A. ADVICE ON GOOD PROGRAMMING PRACTICES.
Programming Suggestions.
APPENDIX B. REPRESENTATION OF NUMBERS IN DIFFERENT BASES.
Representation of Numbers in Different Bases.
APPENDIX C. ADDITIONAL DETAILS ON IEEE FLOATING-POINT ARITHMETIC.
More on IEEE Standard Floating-Point Arithmetic.
APPENDIX D. LINEAR ALGEBRA CONCEPTS AND NOTATION.
Elementary Concepts.
ANSWERS FOR SELECTED EXERCISES.
BIBLIOGRAPHY.
INDEX.
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