This textbook is written for a course in numerical methods for engineering undergraduates.
In 2002, National Science Foundation funded a prototype proposal on Holistic Numerical Methods to develop various resources for typical numerical methods topics of interpolation and nonlinear equations. With the success of this proposal, NSF continued to fund the proposal for other topics of numerical methods via three more multi-university grants in 2004-07, 2008-12, 2009-11. This funding has so far resulted in complete resources for a comprehensive course in Numerical Methods. Revision of the resources continued with funding from more NSF grants in 2013-16, 2016-20, 2020-23. These resources include textbook chapters, PowerPoint presentations, worksheets in MATLAB, MATHEMATICA, Maple and MathCAD, multiple-choice tests, experiments, digital audiovisual lectures, and a blog.
Go to the main website http://nm.mathforcollege.com and click on the numerical method of your choice. You will have access to PowerPoint presentations, worksheets, simulations, additional examples, and multiple-choice tests. In addition, we have uploaded broadcast quality (https://nm.mathforcollege.com/audiovisual-digital-lectures/) instructional audiovisual content for most of the course content. A blog on numerical methods and MATLAB is also available at http://blog.autarkaw.com.
The textbook consists of eleven topics:
1. Introduction to Scientific Computing
3. Nonlinear Equations
4. Simultaneous Linear Equations
8. Ordinary Differential Equations
- Partial Differential Equations
11. Fast Fourier Transforms
Each subtopic in these topics is covered in several separate chapters because we decided to keep the chapters concise and independent. This modular nature of chapters allows you to customize the book based on your needs. Supplemental material is always available from the website.
The chapters in the book are numbered as Chapter XX.YY. The XX stands for the main topic, while YY is the chapter number within that topic.
Chapter 01.YY introduces scientific computing by taking a real-life example to show that solving an engineering problem requires one to develop a mathematical model, solve the model, and then implement the corresponding solution. This content is followed by a discussion of sources of numerical error and their measurement, binary, and floating-point representation of numbers, propagation of errors, and Taylor series.
Chapters 02.YY starts with physical applications of numerical differentiation followed by a just-in-time primer on differential calculus, numerical differentiation of continuous functions and functions that are only given at discrete points.
Chapter 03.YY starts with physical applications of solution of nonlinear equations, a primer on quadratic and cubic equations, and numerical methods of solving nonlinear equations, including bisection, Newton Raphson, secant methods. Simultaneous nonlinear equations are covered as well.
Chapter 04.YY starts with physical applications of solution of simultaneous linear equations, a just-in-time primer for matrix algebra, and numerical methods such as Gaussian elimination, LU decomposition and Gauss-Seidel methods.
Chapter 05.YY starts with physical applications of interpolation. It starts with the background of interpolation, followed by numerical methods of the direct method, Newton divided difference interpolation, Lagrange interpolation, and spline interpolation. Also included are chapters on extrapolation being a bad idea and how to numerically calculate the length of a curve.
Chapter 06.YY starts with physical application of regression. Background content for regression, include simple statistics, minimum of functions, and partial derivatives. The content includes linear regression, nonlinear regression and adequacy of linear regression models.
Chapter 07.YY starts with physical applications of numerical integration. It starts with a primer of integral calculus followed by numerical methods of trapezoidal rule, Simpson’s 1/3rd rule, Simpson’s 3/8 rule, Gauss quadrature rule, and integrating functions that are given at discrete points.
Chapter 08.YY begins with physical applications of numerical solution of ordinary differential equation. It begins with a primer for ordinary differential equations followed by numerical methods of Euler’s, Runge-Kutta 2nd order, Runge-Kutta 4th order, shooting method, and finite difference methods.
Chapter 9 on optimization will be added soon.
Chapter 10 on the numerical solutions of partial differential equations will be added soon.
Chapter 11 on the fast Fourier transforms will be added soon.
Numerical methods used to solve the mathematical procedure are shown with complete treatment and examples. Most chapters are followed by a multiple-choice test and a problem set. Comprehensive solutions to individual multiple-choice tests are available via a link at the end of each multiple-choice question set. The final answers to the problem set are given at the end of each problem.
We would like to thank - Sri Harsha Garapati, Luke Snyder, Eric Marvella, Sue Britten, and Matthew Emmons for reformatting and typing the textbook. Sean Rodby’s meticulous proofreading has been critical in maintaining the accuracy of the contents of the book. We would like to thank Cuong Nguyen, Praveen Chalasani, Michael Keteltas, and Luke Snyder for contributions to the textbook. The conversion of the original files from Word to markdown was a considerable task, and Kaw would like to thank Jonas Fernandes, Bharath Pulaparthi, Jayendra Patel, and Gregory Sims for the detailed work involved. Kaw would like to thank his spouse, Sherrie and children Candace and Angelie for their encouragement in writing this textbook.
We would like to thank Professors Melvin Corley of Louisiana Technical University, Tianxia Zhao of Indiana University-Purdue University, Fort Wayne and Xudong Jia of California State Polytechnic University for reviewing the contents of the textbook content of the first edition of the textbook.
We would appreciate feedback, questions, or comments that you may have on the book or the numerical methods project. You can contact the first author, Autar Kaw, via
Mailing Address: Department of Mechanical Engineering Department, University of South Florida, 4202 East Fowler Avenue ENG030, Tampa, FL 33620-5350.