The SIAM 100-digit challenge of Bronemann et al: A review

Mogadalai Gururajan — Fri, 12 Jan 2007 22:11:38 -0500

Suppose if somebody asked you the following question, and more importantly, wanted the answer to an accuracy of 100-digits:

Problem A: A particle at the center of a 10 x 1 rectangle undergoes Brownian motion (i.e., two-dimensional random walk with infinitesimal step lengths) until it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?

Or, this question (again, demanding the answer to an accuracy of 100-digits):

Problem B: A square plate [-1,1]x[-1x1] is at a temperature u = 0. At time t=0 the temperature is increased to u=5 along one of the four sides while being held at u=0 along the other three sides, and heat then flows into the plate according to u_t = \nabla u. When does the temperature reach u=1 at the center of the plate?

You might be tempted to answer back "What's the point?", and, probably, you are justified. However, if you did that, you would have missed some wonderful oppurtunity to learn (and/or teach), among other things, the following lessons:

For Problem A, for example:

a solution can not be obtained using Monte Carlo to the required accuracy;
it is possible, however, to obtain the solution to 10 digits accurcay using MATLAB in less than a second on a 2GHz PC;
or, even upto 10,000 significant digits in about 2 seconds on a 2GHz PC.

Similarly, for Problem B, for example,

a finite element method with second-order Lagrange elements in more accurate than one with fourth-order elements (which, apparently, is due to the jumps of the Dirichlet boundary conditions);
standard finite difference scheme can give a more accurate solution than some of the commercial FEM packages; and,
a solution which is accurate to 10,000 digits can be obtained (using Mathematica) on a 2GHz PC.

Intrigued? Then, you should pick (or, if possible, buy yourself a personal copy) of The SIAM 100-digit challenge: A study in high-accuracy numerical computing by Folkmar Bornemann, Dirk Laurie, Stan Wagon and Joerg Waldvogel, SIAM (2004). There are not two but ten problems in the book; and, the book could also be a nice place to look for projects for a numerical methods course, to introduce software like MATHEMATICA/MATLAB for solving problems of considerable complexity, and/or (as the foreward of the book suggests) to give a flavour of modern Numerical Analysis.

Do not forget to visit the SIAM page for the book, which, among other things leads you to reproducible scientific computations, and the remarkable story behind the book (the many versions of it). And, finally, here is the page of Prof. Nick Trefethen, who started it all; and, on a personal note, here is the story of how I came to know of this amazing book!

Have fun, and a happy computational weekend!

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The SIAM 100-digit challenge of Bronemann et al: A review