COMMUNICATION New Directions in Numerical Computation Tobin A. Driscoll, Endre Süli, and Alex Townsend, Editors Nick Trefethen In August 2015 a distinguished collec- tion of numerical analysts gathered at Oxford to celebrate the sixtieth birth- day of Nick Trefethen FRS and consider the future of numerical analysis. Some of the plenary speakers provided short essays for Notices. The full collection is online. 1 In a 1992 essay, “The Definition of Numerical Analysis”, 2 Trefethen writes of the field, “[O]ur central mission is to compute quantities that are typically uncomputable, from an analytical point of view, and to do it at lightning speed.” These essays explore a few of the particulars of that mission. Jean-Paul Berrut A Puzzling Question about Numerical Analysis Why are so many academic mathematicians con- tent with piecewise smooth approximations to solutions to functional equations when those so- lutions are known a priori to be smooth? Chebfun impressively and beautifully demonstrates the effectiveness of smooth one-dimensional approx- imants. These may oscillate more than splines of comparable accuracy, but their convergence is faster and automatically adjusts to the smooth- ness of the underlying function. In contrast, the Tobin A. Driscoll is professor of mathematical sciences at the University of Delaware. His email address is driscoll@udel.edu. 1 tobydriscoll.net/newdirections2015. 2 LN Trefethen, “The Definition of Numerical Analysis” SIAM News, November 1992. For permission to reprint this article, please contact: reprint-permission@ams.org. DOI: http://dx.doi.org/10.1090/noti1363 Photo courtesy of Burchard Kaup. Jean-Paul Berrut use of smooth func- tions in two and higher dimensions was, until recently, limited to rela- tively simple domains. Fundamental research on infinitely smooth two-dimensional inter- polants may lead to interesting new ap- proaches, yet the number of scientists working on them ap- pears to be surprisingly small. Bengt Fornberg The Method—Not the Machine It used to be said that improvements in sci- entific computing capabilities originate about equally from advances in algorithms and in hard- ware. During the last decade or two, the focus has shifted to building heroic-scale supercom- puting facilities. Maybe this is partly because processing characteristics are quantifiable and easy to show in lists, with national and world records falling incessantly. However, the largest systems require inordinate amounts of power and infrastructure, and they also become obsolete very quickly. Yet algorithmic opportunities are as expansive as ever. Endre Süli is professor of numerical analysis at the Uni- versity of Oxford. His email address is suli@maths.ox. ac.uk. Alex Townsend is assistant professor of mathematics at Cornell University. His email address is ajt@mit.edu. Jean-Paul Berrut is professor of mathematics at the Université de Fribourg. His email address is jean-paul. berrut@unifr.ch. Bengt Fornberg is professor of applied mathematics at the University of Colorado. His email address is fornberg@colorado.edu. 398 Notices of the AMS Volume 63, Number 4