By Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jörg Waldvogel
This e-book takes readers on an exciting journey of a few of crucial and robust parts of latest numerical arithmetic. The journey is geared up alongside the ten difficulties of the SIAM 100-Digit problem, a competition posed by means of Nick Trefethen of Oxford college within the January/February 2002 factor of SIAM information. the full tale of the competition in addition to a full of life interview with Nick Trefethen also are included.
The authors, participants of groups that solved all 10 difficulties, convey intimately a number of methods for fixing each one challenge, starting from user-friendly to stylish, from brute-force to schemes that may be scaled to supply hundreds of thousands of digits of accuracy and which can resolve even better similar difficulties. The authors contact on almost each significant means of glossy numerical research: matrix computation, iterative linear equipment, restrict extrapolation and convergence acceleration, numerical quadrature, contour integration, discretization of PDEs, worldwide optimization, Monte Carlo and evolutionary algorithms, blunders keep watch over, period and high-precision mathematics, and lots of more.
The SIAM 100-Digit problem: A examine in High-Accuracy Numerical Computing provides concrete examples of the way to justify the validity of each unmarried digit of a numerical solution. tools variety from rigorously designed laptop experiments to a posteriori errors estimates and computer-assisted proofs in accordance with period arithmetic.
This e-book will reduction readers in constructing problem-solving abilities for making sensible approach choices. The chapters should be learn independently. Appendices A and B comprise uncomplicated tools of convergence acceleration and information of computing the strategies to very excessive accuracy. complete code for all of the equipment, examples, tables, and figures is given (partly in Appendix C, thoroughly at the accompanying internet page).The code is written in numerous languages, together with Mathematica, Matlab, Maple, C, Octave, and PARI/GP. Appendix D deals a pattern of extra tough difficulties when you grasp a few of the concepts mentioned the following.
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Additional info for The SIAM 100-digit challenge. Study in high-accuracy numerical computing
1. 99526291944335416087109 19 Working Backwards Because the motion of the photon in the direction of negative time follows exactly the same rules as in the positive direction, it is natural to investigate backwards trajectories to learn something about the number of correct digits in a numerical solution of the problem. 1, that we are working only with machine precision. Then a natural idea is to take the computed point after 10 seconds (call it Q) and also the direction of the ﬁnal segment and run the algorithm backwards to see how close the result (call it P∗ ) is to the starting postion P .
Such an approach allows one to get 100 digits in a fraction of a second. 44 Chapter 2. 5. The true trajectory of the photon up to time 2000, obtained by interval arithmetic with initial intervals having diameter of 10−5460 . Norm Last@ReliableTrajectory StartIntervalPrecision 1 1 , , 1, 0 , 10, AccuracyGoal 2 10 127 100, //IntervalForm Initial condition interval radius is 10−128 . 9952629194433541608903118094267216210294669227341543498032088580 7298617962283063209917498189761887606190 599926 Indeed, this method works quite well even for 10000 digits (see Appendix B).
46 Chapter 2. Reliability amid Chaos √ √ And this explains the gap between 14/3 and 5 − 2/3. Further observations are possible if one uses the facts about uniform distribution from billiard theory alluded to above. 448. This would mean that 935 is the expected number of such segments among the 2086 segments in the time-2000 trajectory; in fact, there are 925 such segments, nicely consistent with the prediction. We leave more investigations along these lines—perhaps including variation in the mirror radius—to the reader.
The SIAM 100-digit challenge. Study in high-accuracy numerical computing by Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jörg Waldvogel