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The purpose of this book is to develop the understanding of basic numerical methods and their implementations as software that are necessary for solving fundamental mathematical problems by numerical means. It is designed for the person who wants to do numerical computing. Through the examples and exercises, the reader studies the behavior of solutions of the mathematical problem along with an algorithm for solving the problem. Experience and understanding of the algorithm are gained through hand computation and practice solving problems with a computer implementation. It is essential that the reader understand how the codes provided work, precisely what they do, and what their limitations are. The codes provided are powerful, yet simple enough for pedagogical use. The reader is exposed to the art of numerical computing as well as the science.
The book is intended for a one-semester course, requiring only calculus and a modest acquaintance with FORTRAN, C, C  , or Matlab. These constraints of background and time have important implications: the book focuses on the problems that are most common in practice and accessible with the background assumed. By concentrating on one effective algorithm for each basic task, it is possible to develop the fundamental theory in a brief, elementary way. There are ample exercises, and codes are provided to reduce the time otherwise required for programming and debugging. The intended audience includes engineers, scientists, and anyone else interested in scientific programming. The level is upper-division undergraduate to beginning graduate and there is adequate material for a one semester to two quarter course. Numerical analysis blends mathematics, programming, and a considerable amount of art. We provide programs with the book that illustrate this. They are more than mere implementations in a particular language of the algorithms presented, but they are not production-grade software. To appreciate the subject fully, it will be necessary to study the codes provided and gain experience solving problems first with these programs and then with production-grade software.
Many exercises are provided in varying degrees of difficulty. Some are designed to get the reader to think about the text material and to test understanding, while others are purely computational in nature. Problem sets may involve hand calculation, algebraic derivations, straightforward computer solution, or more sophisticated computing exercises.
The algorithms that we study and implement in the book are designed to avoid severe roundoff errors (arising from the finite number of digits available on computers and calculators), estimate truncation errors (arising from mathematical approximations), and give some indication of the sensitivity of the problem to errors in the data. In Chapter 1 we give some basic definitions of errors arising in computations and study roundoff errors through some simple but illuminating computations. Chapter 2 deals with one of the most frequently occurring problems in scientific computation, the solution of linear systems of equations. In Chapter 3 we deal with the problem of interpolation, one of the most fundamental and widely used tools in numerical computation. In Chapter 4 we study methods for finding solutions to nonlinear equations. Numerical integration is taken up in Chapter 5 and the numerical solution of ordinary differential equations is examined in Chapter 6_. Each chapter contains a case study that illustrates how to combine analysis with computation for the topic of that chapter. Before taking up the various mathematical problems and procedures for solving them numerically, we need to discuss briefly programming languages and acquisition of software.
Errors and Floating Point Arithmetic
Systems of Linear Equations
Interpolation
Roots of Nonlinear Equations
Numerical Integration
Ordinary Differential Equations
Notation and Some Theorems from the Calculus
Answers to Selected Exercises