부산대학교 도서관

In the January/February 2002 issue of SIAM News, Professor Lloyd N. Trefethen, head of the Numerical Analysis Group at Oxford University, proposed a contest which consisted of ten challenging problems in numerical computing. Each problem was stated in at most three simple sentences and had a single real number as a solution. The objective was to compute each number to as many digits of precision as possible. Scoring for the contest would be simple: each correct digit of the answer, up to ten per problem, would earn a single point. Professor Trefethen warned that the problems were hard and indicated that he would be impressed if anyone managed to score even 50 points. With the gauntlet thrown down, the scientific community rose to the challenge. By the submission deadline in mid-May of 2002, entries had been received from 94 teams (roughly 180 contestants) from 25 countries. Twenty teams scored a perfect 100 points, and another five teams scored 99 points. \par Written by several contestants who, either individually or as a member of a team, achieved a perfect score, the book under review takes the reader on a fascinating journey through the world of numerical mathematics inspired by Trefethen's 100-digit challenge. In the opening chapter, the names of all the first prize winners (those who achieved a perfect score) and all the second prize winners (those who achieved a score of 99 points) are listed, and statistics on the number of correct digits obtained for each of the ten contest problems are presented. There is also a brief and enlightening interview with Professor Trefethen, in which he discusses various aspects of the contest, and an interesting collection of comments solicited from the prize winning contestants. \par Each of the next ten chapters is devoted to a single contest problem. The authors' first objective is to present several techniques for calculating each answer to at least ten digits of precision. For example, one of the contest problems requires the calculation of an improper integral with a highly oscillatory integrand. The presented solution techniques include: (i) subdivide the integration interval at either the zeros or the extrema of the integrand, use Romberg integration to evaluate the integrals over each of the resulting subintervals and iterate Aitken's $\Delta^2$-method to accelerate convergence; (ii) replace the real integral by a complex contour integral with an appropriately chosen integration contour and then employ a canned integration routine; (iii) transform the integral of an unbounded function over a finite interval into an integral of a bounded function over a semi-infinite interval, perform integration by parts several times and evaluate the resulting integral using Romberg integration; (iv) represent the integral as a divergent infinite series, replace the series with an appropriate complex contour integral and evaluate using the trapezoidal rule; and (v) transform the original integral to a Fourier integral and evaluate the Fourier integral using the Ooura-Mori method. The authors' second objective in each chapter is to validate the correctness of each calculated digit. This objective is achieved through various combinations of carefully designed computer experiments, a posteriori error estimates and computer-assisted proofs based on interval arithmetic. \par The book closes with four appendices. The first discusses the various convergence acceleration schemes (e.g., Richardson extrapolation, Aitken's method, the rho algorithm and the theta algorithm) that play a central role in the main chapters. In the second appendix, the authors summarize their efforts to calculate the answers to ten-thousand digits of precision. This extreme precision is achieved for all but one of the contest problems. For the remaining problem, which requires calculating the norm of an infinite matrix, only 273 digits have been validated. The third appendix contains code for solving all ten contest problems in a variety of computing environments, including C, PARI/GP, MATLAB, INTLAB and Mathematica. More extensive code, as well as additional material related to the book, can be found on the book's website. The final appendix lists twenty-two problems similar to those from the contest. These problems are presented ``to help readers experience first-hand the excitement, frustration, and joy of working on a challenging numerical problem''.