This is a mathematics textbook, an introduction to numerical analysis, suitable for upper level students in a university course. This text is well suited for students of Applied Mathematics.
This is an introduction to the mathematical procedures needed to solve scientific problems using computers. All of the computing can be done with a desktop IBM compatible computer.
This text has been written for students who have completed two years of University level mathematics including calculus and linear algebra or the equivalent. Such a student has been exposed to enough mathematics to be acquainted with traditional approaches for solving many mathematical problems. This is an introduction to the theory and methods needed to use computers to solve mathematical problens with an emphasis on problems in the sciences.
Excerpt
You the student have studied the problem of finding roots of an equation f(x) = 0 or the problem of solving a linear system of equations. You have been exposed to many key mathematical concepts and theorems in calculus, linear algebra, and possibly differential equat ions. All of this study is necessary preparation for doing the numerical mathematics discussed in this text. The methods for solving problems numerically which are discussed in this book have been developed and analyzed by use of the mathematical theories developed in these preceding courses of study.
Hence, the first essential point to observe is that numerical methods are not a substitute for theoretical mathematics. They are tools to be used along with all of the other mathematics you know. So, as each problem type is introduced we will begin our study with some review of the theoretical ideas connected with such problems. Numerical methods will be developed from this theoretical base and contrasted with purely theoretical or so-called analytical solutions, when such solutions exist. In some cases you will find that numerical algorithms have been developed for certain problems because no analytical solution is known. In other cases you will find that numerical methods are preferred to solve a problem because the analytical solution is not suitable for machine computation of numerical answers to questions which arise, or is less convenient to use than some simple numerical algorithm. In all cases theoretical results are useful in the interpretation or analysis of machine output. Without any theoretical framework to fall back upon some machine outputs would be incomprehensible.
In fact, in some of the recent work in applied mathematics researchers are forced to rely upon numerical results from machines in the study of problems which do not have an adequate theory developed yet. This is an undesirable situation, but unavoidable due to the complexity of the problems. In these cases it is usually hoped that machine results will aid in the development of some theoretical results.
This is an introduction to the theory and methods needed to use computers to solve mathematical problens with an emphasis on problems in the sciences.
