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Chapter 1 Numerical Algorithms This opening chapter introduces the basic concepts of numerical algorithms and scientific comput- ing. We begin with a general, brief introduction to the field in Section 1.1. This is followed by the more substantial Sections 1.2 and 1.3. Section 1.2 discusses the basic errors that may be encountered when applying numerical algorithms. Section 1.3 is concerned with essential properties of such algorithms and the appraisal of the results they produce. We get to the “meat” of the material in later chapters. 1.1 Scientific computing Scientific computing is a discipline concerned with the development and study of numerical al- gorithms for solving mathematical problems that arise in various disciplines in science and engin- eering. Typically, the starting point is a given mathematical model which has been formulated in an attempt to explain and understand an observed phenomenon in biology, chemistry, physics, eco- nomics, or any other scientific or engineering discipline. We will concentrate on those mathematical models which are continuous (or piecewise continuous ) and are difficult or impossible to solve ana- lytically; this is usually the case in practice. Relevant application areas within computer science and related engineering fields include graphics, vision and motion analysis, image and signal processing, search engines and data mining, machine learning, and hybrid and embedded systems. In order to solve such a model approximately on a computer, the continuous or piecewise continuous problem is approximated by a discrete one. Functions are approximated by finite arrays of values. Algorithms are then sought which approximately solve the mathematical problem effi- ciently, accurately, and reliably. This is the heart of scientific computing. Numerical analysis may be viewed as the theory behind such algorithms. The next step after devising suitable algorithms is their implementation. This leads to ques- tions involving programming languages, data structures, computing architectures, etc. The big pic- ture is depicted in Figure 1.1. The set of requirements that good scientific computing algorithms must satisfy, which seems elementary and obvious, may actually pose rather difficult and complex practical challenges. The main purpose of this book is to equip you with basic methods and analysis tools for handling such challenges as they arise in future endeavors. 1 Downloaded 08/20/18 to 132.174.255.3. Redistribution subject to SIAM license or copyright; see
2 Chapter 1. Numerical Algorithms Observed phenomenon Mathematical model Discretization Solution algorithm Efficiency Accuracy Robustness Implementation Programming environment Data structures Computing architecture Figure 1.1. Scientific computing.

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