This book has evolved over many years from lecture notes that accompany 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.