This book has evolved over many years from lecture notes that accompany cer- certain upper-division courses in mathematics and computer sciences at our university. These courses introduce students to the algorithms and methods that are commonly needed in scientific computing. The mathematical underpinnings of these methods are emphasized as much as their algorithmic aspects. The students have been diverse: mathematics, engineering, science, and computer science undergraduates, as well as graduate students from various disciplines. Portions of the book also have been used to lay the groundwork in several graduate courses devoted to special topics in numerical analysis, such as the numerical solution of differential equations, numerical linear algebra, and approximation theory. Our approach has always been to treat the subject from a mathematical point of view, with attention given to its rich offering of theorems, proofs, and interesting ideas. From these arise many computational procedures and intriguing questions of computer science. Of course, our motivation comes from the practical world of scientific computing, which dictates the choice of topics and the manner of treating each. For example, with some topics it is more instructive to discuss the theoretical foundations of the subject and not attempt to analyze algorithms in detail. In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them-although we offer a blanket admonishment to use well-tested software, such as from program libraries, on problems arising from applications.

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