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An Introduction to Numerical Analysis

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This introduction to numerical analysis was written for students in mathematics, the physical sciences, and engineering, at the upper undergraduate to beginning graduate level. Prerequisites for using the text are elementary calculus, linear algebra, and an introduction to differential equations. The student's level of mathematical maturity or experience with mathematics should be somewhat higher; I have found that most students do not attain the necessary level until their senior year. Finally, the student should have a knowledge of computer programming. The preferred language for most scientific programming is Fortran.

A truly effective use of numerical analysis in applications requires both a theoretical knowledge of the subject and computational experience with it. The theoretical knowledge should include an understanding of both the original problem being solved and of the numerical me.thods for its solution, including their derivation, error analysis, and an idea of when they will perform well or poorly. This kind of knowledge is necessary even if you are only considering using a package program from your computer center. You must still understand the program's purpose and limitations to know whether it applies to your particular situation or not. More importantly, a majority of problems cannot be solved by the simple application of a standard program. For such problems you must devise new numerical methods, and this is usually done by adapting standard numerical methods to the new situation. This requires a good theoretical foundation in numerical analysis, both to devise the new methods and to avoid certain numerical pitfalls that occur easily in a number of problem areas.

Computational experience is also very important. It ·gives a sense of reality to most theoretical discussions; and it brings out the important difference between the exact arithmetic implicit in most theoretical discussions and the finite-length arithmetic computation, whether on a computer or a hand calculator. The use of a computer also imposes constraints on the structure of numerical methods, constraints that are not evident and that seem unnecessary from· a strictly mathematical viewpoint. For example, iterative procedures are often preferred over direct procedures because of simpler programming requirements or computer memory size limitations, even though the direct procedure may seem simpler to explain and to use. Many numerical examples are ~ven in this text to illustrate these points, and there are a number of exercises that will give the student a variety of <...)mputational experience.

This Second Edition of a standard numerical analysis text retains organization of the original edition, but all sections have been revised, some extensively, and bibliographies have been updated. New topics covered include optimization, trigonometric interpolation and the fast Fourier transform, numerical differentiation, the method of lines, boundary value problems, the conjugate gradient method, and the least squares solutions of systems of linear equations. Contains many problems, some with solutions.

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