From rbk@usl.edu Wed May 7 16:34:25 1997 Received: from bp.ucs.usl.edu (root@bp.ucs.usl.edu [130.70.40.36]) by dkuug.dk (8.6.12/8.6.12) with SMTP id QAA06586 for ; Wed, 7 May 1997 16:34:21 +0200 Received: from rbk5287.usl.edu (liberty.usl.edu) (liberty.usl.edu [130.70.46.171]) by bp.ucs.usl.edu with SMTP id AA05724 (5.65c/IDA-1.4.4 for ); Wed, 7 May 1997 09:27:39 -0500 Message-Id: <2.2.32.19970507142700.0073e814@pop.usl.edu> X-Sender: rbk5287@pop.usl.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 07 May 1997 09:27:00 -0500 To: Michel.Olagnon@ifremer.fr, Michael.Metcalf@cern.ch, sc22wg5@dkuug.dk From: "R. Baker Kearfott" Subject: Re: (SC22WG5.1372) Floating-point arithmetic Cc: Bill.Walster@eng.sun.com At 11:23 AM 5/7/97 +0200, Michel.Olagnon@ifremer.fr wrote: >I strongly agree with Mike. I would appreciate if someone could point >out an application, other than benchmarking purposes of vendors marketing >teams, where standardization of interval arithmetic would be useful. >On the opposite, predictability is essential in many actual applications. > Actually, benchmarking purposes of vendor's marketing teams is a minor application within the full body of experience of interval arithmetic. Particularly successful applications have been in global optimization and constraint propagation. When interval arithmetic is viewed as a painless way of manipulating inequalities and constraints, it is very powerful. For example, in global optimization, bounds on the objective can be obtained that greatly facilitate elimination of parts of the search region. There is a group of people who have developed similar algorithms with non-interval methods, by doing ad-hoc classical (by-hand) mathematical analysis for each particular problem to obtain bounds on the variation. However, the non-interval techniques are more complicated, and they also usually are less efficient. Similarly, bounds on the values of expressions can be used to determine feasible regions for systems of constraints. Such systems of constraints arise in robot kinematics problems, among other things. Some strong actual applications have been in the chemical industry. For example, in the analysis of trayed towers, the models often have several solutions, and floating point methods only converge to one solution, not the GLOBAL optimum, and with no way to check whether it is the global optimum. However, the global optimum is the only one that is meaningful in a chemical engineering sense. In contrast, interval methods have successfully given the global optimum, with a mathematically airtight guarantee that it is, indeed, the global optimum. Many other industrial-strength problems are similar to the ones encountered in chemical engineering. I can list more successes and references upon demand. Another particularly attractive application is in adaptive quadrature. Instead of using a heuristic to estimate the error, the actual error can be bounded in a simple but airtight way. The result is an algorithm that is not only more efficient, but has GUARANTEED accuracy. A final application I would like to mention at this point is verification of the results of floating point computations. For example, a simple, quick interval computation can be applied to the result of a linear equation solution algorithm. Tiny intervals are constructed about the approximate solution to a linear algebraic system, within which interval arithmetic proves (with an airtight guarantee) that there is a unique solution to the linear system. For many classes of problems, the bounds so obtained are not only tighter than those implied by condition number estimation, but are also obtained in less computation time (than even the heuristic LINPACK condition number estimator). For a reference, see C. F. Korn and Ch. Ullrich, "Extending LINPACK by verification routines for linear systems," Mathematics and Computers in Simulation 39 (1-2), pp. 21-37, 1995. I'll stop here, but I can go into more detail, give references, and list more, if someone so desires. Best regards, Baker --------------------------------------------------------------- R. Baker Kearfott, rbk@usl.edu (318) 482-5346 (fax) (318) 482-5270 (work) (318) 981-9744 (home) URL: http://interval.usl.edu/kearfott.html Department of Mathematics, University of Southwestern Louisiana USL Box 4-1010, Lafayette, LA 70504-1010, USA ---------------------------------------------------------------