Society for Industrial and Applied Mathematics, 2004, -319 pp.This book will take you on a thrilling tour of some of the most important and powerful areas of contemporary numerical mathematics. A first unusual feature is that the tour is organized by problems, not methods: it is extremely valuable to realize that numerical problems often yield to a wide variety of methods. For example, we solve a random-walk problem (Chapter 6) by several different techniques, such as large-scale linear algebra, a three-term recursion obtained by symbolic computations, elliptic integrals and the arithmetic-geometric mean, and Fourier analysis. We do so in IEEE arithmetic to full accuracy and, at the extreme, in high-precision arithmetic to get 10,000 digits. A second unusual feature is that we very carefully try to justify the validity of every single digit of a numerical answer, using methods ranging from carefully designed computer experiments and a posteriori error estimates to computer-assisted proofs based on interval arithmetic. In the real world, the first two methods are usually adequate and give the desired confidence in the answer. Interval methods, while nicely rigorous, would most often not provide any additional benefit. Yet it sometimes happens that one of the best approaches to a problem is one that provides proof along the way (this occurs in Chapter 4), a point that has considerable mathematical interest. A main theme of the book is that there are usually two options for solving a numerical problem: either use a brute force method running overnight and unsupervised on a high- performance workstation with lots of memory, or spend your days thinking harder, with the help of mathematical theory and a good library, in the hope of coming up with a clever method that will solve the problem in less than a second on common hardware. Of course, in practice these two options of attacking a problem will scale differently with problem size and difficulty, and your choice will depend on such resources as your time, interest, and knowledge and the computer power at your disposal. One noteworthy case, where a detour guided by theory leads to an approach that is ultimately much more efficient than the direct path, is illustrated on the cover of this book. That diagram (taken from Chapter 1) illustrates that many problems about real numbers can be made much, much easier by stepping outside the real axis and taking a route through the complex plane.The Story A Twisted Tail Reliability amid Chaos How Far Away Is Infinity? Think Globally, Act Locally A Complex Optimization Biasing for a Fair Return Too Large to Be Easy, Too Small to Be Hard In the Moment of Heat Gradus ad Parnassum Hitting the Ends A Convergence Acceleration B Extreme Digit-Hunting C Code D More Problems
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