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From the Preface:Numerical analysis has advanced greatly since it began as a way of creating methods to approximate answers to mathematical questions. This book aims to bring students closer to the frontier regarding the numerical methods that are used. But this book is not only about newer, as well as “classical”, numerical methods. Rather the aim is to also explain how and why they work, or fail to work. This means that there is a significant amount of theory to be understood. Simple analyses can result in methods that usually work, but can then fail in certain circumstances, sometimes catastrophically. The causes of success of a numerical algorithm and its failure, are both important. Without understanding the underlying theory, the reasons for a method's success and failure remain mysterious, and we do not have a means to determine how to fix the problem(s). ......
The aim is to present numerical methods and their analysis in the context of modern applications and models. For example, the standard asymptotic error analysis of differential equations gives no advantage to implicit methods, which have a much larger computational cost. But for “stiff” problems there is a clear, and often decisive, advantage to implicit methods. While “stiffness” can be hard to quantify, it is also common in applications. We also wish to emphasize multivariate problems alongside single-variable problems: multivariate problems are crucial for partial differential equations, optimization, and integration over high-dimensional spaces. We deal with issues regarding randomness, including pseudo-random number generators, stochastic differential equations, and randomized algorithms. Stochastic differential equations meet a need for incorporating randomness into differential equations. High-dimensional integration is needed for studying questions and models in data science and simulation.
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