This book provides an introduction into some of the basic theories and techniques of Numerical Analysis. Its main purpose is to provide the basis for a first level college course in this field. However, it is written in a way that would help any reader outside of the classroom, with an appropriate background, to attain insight and a fundamental understanding into this field of mathematics.
The theory behind the various methods are explored and, where possible, derived in an intuitive manner. Traditionally, teaching this field relied on the student performing the repetitive steps of these procedures with pencil and paper or programming the algorithms on a computer using any one of a number of languages (e.g. FORTRAN, Pascal, or C). Here, these methods are demonstrated by implementing them in spreadsheets using Microsoft's Excel. The derivation of each of the spreadsheets is covered in depth so that the practical application of the theory is highlighted. With the use of examples, the student can see the numerical techniques actually converge to the problem's solution on their personal computer. The spreadsheets are generalized so that they can also be used by the student to solve other problems.
Spreadsheets, like Excel, lend themselves to performing repetitive steps without frustrating the student with the likelihood of making simple arithmetic errors which detract from seeing the beauty of these techniques in operation. In addition, basic spreadsheets are relatively user-friendly and easy to understand so the student does not need to learn or avail themselves of a traditional programming language in order to understand this topic.
The student only needs a basic understanding of spreadsheets in order to use this book. Whenever some of Excel's less common tools are used, an explanation is given to show how they are implemented. An example would be to show how a user-defined function is created.
There are four main sections to this book.
The first section covers finding roots of equations. The techniques are straight-forward and are generally described geometrically. Newton's Method does require an understanding of derivatives.
The second section deals with numerical methods for evaluating definite integrals. Clearly, an understanding of integral calculus is required to fully appreciate what is going on.
Techniques for performing some matrix operations are covered in the third Section. Mainly, it shows how the technique of transforming a matrix into reduced row echelon form can be employed to address many of these operations. Familiarity with inverses and determinants of matrices would be very useful. A lot of the operations addressed by the reduced echelon transformation (e.g. finding determinants) are actually built into Excel. The book demonstrates how these are used as well.
In the Fourth Section, iterative techniques for some simple examples of differential equations are covered. It deals with equations of the general form dy/dx=f(x, y) where there is one known value of x and y, it implements techniques that will allow one to find the value of y that is associated with some other value of x. So a basic understanding of differential equations is required to fully appreciate this Section.
Each of these Sections illustrates the techniques by walking through specific examples. By following the examples and illustrations, the reader will be able to reproduce and use the spreadsheets in other applications.
