부산대학교 도서관

In the January/February 2002 issue of SIAM News, Professor LloydN. Trefethen, head of the Numerical Analysis Group at OxfordUniversity, proposed a contest which consisted of ten challengingproblems in numerical computing. Each problem was stated in at mostthree simple sentences and had a single real number as a solution. Theobjective was to compute each number to as many digits of precision aspossible. Scoring for the contest would be simple: each correct digitof the answer, up to ten per problem, would earn a singlepoint. Professor Trefethen warned that the problems were hard andindicated that he would be impressed if anyone managed to score even50 points. With the gauntlet thrown down, the scientific communityrose to the challenge. By the submission deadline in mid-May of 2002,entries had been received from 94 teams (roughly 180 contestants) from25 countries. Twenty teams scored a perfect 100 points, and anotherfive teams scored 99 points.\parWritten by several contestants who, either individually or as a memberof a team, achieved a perfect score, the book under review takes thereader on a fascinating journey through the world of numericalmathematics inspired by Trefethen's 100-digit challenge. In theopening chapter, the names of all the first prize winners (those whoachieved a perfect score) and all the second prize winners (those whoachieved a score of 99 points) are listed, and statistics on thenumber of correct digits obtained for each of the ten contest problemsare presented. There is also a brief and enlightening interview withProfessor Trefethen, in which he discusses various aspects of thecontest, and an interesting collection of comments solicited from theprize winning contestants.\parEach of the next ten chapters is devoted to a single contestproblem. The authors' first objective is to present several techniquesfor calculating each answer to at least ten digits of precision. Forexample, one of the contest problems requires the calculation of animproper integral with a highly oscillatory integrand. The presentedsolution techniques include: (i) subdivide the integration interval ateither the zeros or the extrema of the integrand, use Rombergintegration to evaluate the integrals over each of the resultingsubintervals and iterate Aitken's $\Delta^2$-method to accelerateconvergence; (ii) replace the real integral by a complex contourintegral with an appropriately chosen integration contour and thenemploy a canned integration routine; (iii) transform the integral ofan unbounded function over a finite interval into an integral of abounded function over a semi-infinite interval, perform integration byparts several times and evaluate the resulting integral using Rombergintegration; (iv) represent the integral as a divergent infiniteseries, replace the series with an appropriate complex contourintegral and evaluate using the trapezoidal rule; and (v) transformthe original integral to a Fourier integral and evaluate the Fourierintegral using the Ooura-Mori method. The authors' second objective ineach chapter is to validate the correctness of each calculateddigit. This objective is achieved through various combinations ofcarefully designed computer experiments, a posteriori error estimatesand computer-assisted proofs based on interval arithmetic.\parThe book closes with four appendices. The first discusses the variousconvergence acceleration schemes (e.g., Richardson extrapolation,Aitken's method, the rho algorithm and the theta algorithm) that playa central role in the main chapters. In the second appendix, theauthors summarize their efforts to calculate the answers toten-thousand digits of precision. This extreme precision is achievedfor all but one of the contest problems. For the remaining problem,which requires calculating the norm of an infinite matrix, only 273digits have been validated. The third appendix contains code forsolving all ten contest problems in a variety of computingenvironments, including C, PARI/GP, MATLAB, INTLAB andMathematica. More extensive code, as well as additional materialrelated to the book, can be found on the book's website. The finalappendix lists twenty-two problems similar to those from thecontest. These problems are presented ``to help readers experiencefirst-hand the excitement, frustration, and joy of working on achallenging numerical problem''.