COMMUNICATION New Directions in Numerical Computation

Tobin A. Driscoll, Endre Süli, and Alex Townsend, Editors In August 2015 a distinguished collec- use of smooth func- tion of numerical analysts gathered at tions in two and higher Oxford to celebrate the sixtieth birth- dimensions was, until day of Nick Trefethen FRS and consider recently, limited to rela- the future of numerical analysis. Some tively simple domains. of the plenary speakers provided short Fundamental research essays for Notices. The full collection is on infinitely smooth online.1 two-dimensional inter- In a 1992 essay, “The Definition of polants may lead to Numerical Analysis”,2 Trefethen writes interesting new ap- of the field, “[O]ur central mission is proaches, yet the number of scientists to compute quantities that are typically working on them ap-

uncomputable, from an analytical point Photo courtesy of Burchard Kaup. Nick Trefethen pears to be surprisingly of view, and to do it at lightning speed.” Jean-Paul Berrut small. These essays explore a few of the particulars of that mission. Bengt Fornberg

Jean-Paul Berrut The Method—Not the Machine It used to be said that improvements in sci- A Puzzling Question about Numerical entific computing capabilities originate about Analysis equally from advances in algorithms and in hard- Why are so many academic mathematicians con- ware. During the last decade or two, the focus tent with piecewise smooth approximations to has shifted to building heroic-scale supercom- solutions to functional equations when those so- puting facilities. Maybe this is partly because lutions are known a priori to be smooth? Chebfun processing characteristics are quantifiable and impressively and beautifully demonstrates the easy to show in lists, with national and world records falling incessantly. However, the largest effectiveness of smooth one-dimensional approx- systems require inordinate amounts of power and imants. These may oscillate more than splines infrastructure, and they also become obsolete of comparable accuracy, but their convergence is very quickly. Yet algorithmic opportunities are as faster and automatically adjusts to the smooth- expansive as ever. ness of the underlying function. In contrast, the Endre Süli is professor of numerical analysis at the Uni- Tobin A. Driscoll is professor of mathematical sciences versity of Oxford. His email address is [email protected]. at the University of Delaware. His email address is ac.uk. [email protected]. Alex Townsend is assistant professor of mathematics at Cornell University. His email address is [email protected]. 1tobydriscoll.net/newdirections2015. Jean-Paul Berrut is professor of mathematics at the 2LN Trefethen, “The Definition of Numerical Analysis” Université de Fribourg. His email address is jean-paul. SIAM News, November 1992. [email protected]. For permission to reprint this article, please contact: Bengt Fornberg is professor of applied mathematics [email protected]. at the University of Colorado. His email address is DOI: http://dx.doi.org/10.1090/noti1363 [email protected].

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Photo courtesy of Peter Hudston.