From Numerical Analysis to Computational Science
The modern development of numerical computing is driven by the rapid increase in computer performance. The present exponential growth approximately follows Moore’s law, doubling in capacity every eighteen months. Numerical computing has, of course, been p
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1. Introduction The modern development of numerical computing is driven by the rapid increase in computer performance. The present exponential growth approximate! y follows Moore's law, doubling in capacity every eighteen months. Numerical computing has, of course, been part of mathematics for a very long time. Algorithms by the names of Euclid, Newton and Gauss, originally designed for computation "by hand", are still used today in computer simulations. The electronic computer originated from the intense research and development done during the second world war. In the early applications of these computers the computational techniques that were designed for calculation by pencil and paper or tables and mechanical machines were directly implemented on the new devices. Together with a deeper understanding of the computational processes new algorithms soon emerged. The foundation of modem numerical analysis was built in the period from the late forties to the late fifties. It became justifiable to view numerical analysis as an emerging separate discipline of mathematics. Even the name, numerical analysis, originated during this period and was coined by the National Bureau of Standards in the name of its laboratory at UCLA, the Institute for Numerical Analysis. Basic concepts in numerical analysis became well defined and started to be understood during this time: • • • • • •
numerical algorithm iteration and recursion stability local polynomial approximation convergence computational complexity
The emerging capability of iteratively repeating a set of computational operations thousands or millions of times required carefully chosen algorithms. A theory of stability became necessary. All of the basic concepts were important but the development of the new mathematical theory of stability had the most immediate impact. The pioneers in the development of stability theory for finite difference approximations of partial differential equations were von Neumann, Lax and Kreiss, see e.g. the text by Richtmyer and Morton [16]. In numerical linear algebra the early analysis by Wilkinson was fundamental, [19] and the paper B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
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c mputational cience for the lote\1 .~hift in paradigms, that is 110\1' occuring at the tum of the century. We 11se the label
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[7] by Dahlquist gave the foundation for stability and convergence theory of numerical methods for ordinary differential equations. The development of numerical computing has been gradual and based on the improvements of both algorithms and computers. Therefore, labeling different periods becomes somewhat artificial. However, it is still useful to talk about a new area, often called scientific computing, which developed a couple of decades after the foundation of numerical analysis. The SIAM Journal of Scientific and Statistical Computing was started in 1980. Numerical analysis has always been strongly linked to mathematics, applications and th
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