Nash - Scientific Computing with PCs (523165), страница 41
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It seems likely thatCALGO, as the Collection is referred to, will become a purely electronic product in the future.Most scientific subroutine libraries are aimed at the large machine market. International Mathematical andStatistical Libraries Inc. (IMSL) and the Numerical Algorithms Group Inc. (NAG) have been primarilyFORTRAN subroutine libraries targeted to users in centers with at least one scientific programmingadvisor. Subsets are marketed, however (Nash J C 1986c, 1986d).
One collection of quality mathematicalsoftware for PCs is the Scientific Desk from C. Abaci Inc., P.O.Box 5715, Raleigh NC 27650, (919) 832-4847.The product is aimed at FORTRAN users.Statistical packages are far too numerous to mentions in detail. Over the years there have been severalcatalogs and attempts at reviews of packages. In 1987/88, when the Boston Computer Society attemptedto review as many statistical packages for MS-DOS machines as possible in a study led by Dr. RichardGoldstein.
There were over 200 products considered. A review of selected statistical and forecastingsoftware by PC Magazine in their 1989 issue considered several dozen products, of which we had a handin reviewing three (Nash J C and Walker-Smith, 1989a). The American Statistician regularly publishesreviews of statistical software (e.g., J C Nash, 1992). The Current Index to Statistics may be useful infinding such reviews.Our own use of statistical packages has not always been driven by scientific considerations. We useMINITAB for teaching purposes because the University of Ottawa has a site license for this product anda student version has been available at a modest price.
Moreover, its time series tools — rare in studentpackages — help us teach forecasting.For research purposes, we like Stata for easy-to-control but powerful graphical and data manipulationfunctions, along with an interpreted command language that extends functionality. Other users may prefera fully menu-driven package. For forecasting applications, an example is MESOSAUR (thanks to Dr.MaryAnn Hill, SYSTAT Technical Director, for review copies of MESOSAUR and SYSTAT).As a general numerical package, we use MATLAB. Originally designed for teaching, it has become aprofessional product, then returned to its roots with a new student version. In the sphere of symbolicmathematics as well as computation and graphics, there seems to be healthy competition from Maple andMathematica, with DERIVE occupying an interesting niche (Nash, 1995).Quality software of a scientific nature for specific problems is much more difficult to recommend.
There13: PROBLEM FORMULATION117are many products — far too many to analyze in any detail — and these often have a very small base ofuser experience on which to judge user satisfaction. Nevertheless, it is possible to guide users to sourcesof information about software products in their own subject area.First, as already mentioned, other users with similar needs can point out their success or failure in findingor using software, thereby saving us the time and trouble to duplicate their effort. This is very much the"networking" approach to gathering information. It works. Some researchers have started distributingregular newsletters by electronic mail.
One of the more successful has been the NANET Newsletter (forNumerical Analysis NETwork). There are similar electronic exchanges (or list servers) for other subjects.Second, computer magazines or professional trade publications may have occasional reviews of products.We have already mentioned the reviews of statistical software (Nash J C and Walker-Smith, 1989a) andLotus 1-2-3 add-ins (Nash J C, 1991b) in PC Magazine.Third, journals on the subject of the application of interest may publish announcements and reviews ofproducts. In the area of statistics, the American Statistician and AMSTAT News have been useful to theauthors for keeping up with the marketplace.Fourth, scientific conferences often have product demonstration exhibits.
These can be a very useful sourceof information, since the developers are usually on hand and can respond to detailed questions.Fifth, public database services, such as the Knowledge Index (from DIALOG Information Services Inc.)and BRS After Dark, allow the user to search bibliographic abstracts for reviews of software.
The art ofefficiently selecting wanted references and avoiding others takes some experience. Some libraries andinformation companies such as our own will carry out searches for a fee, which may be less costly forinexperienced searchers. Some database services also support computer conferences between like-mindedusers. Without a moderator we believe these to be of little value. Similarly, we do not often use bulletinboards.A different form of database is beginning to appear using the academic and commercial electronic mailnetworks. There are several mechanisms for access and file transfer. One of the most useful has beenNETLIB (Dongarra and Grosse, 1987) which contains codes and data relating to numerical mathematics.More recently similar services in other subjects have begun operations.
The academic networks alsosupport several file transfer mechanisms that are beginning to have a role in the distribution of reportsand software. Related developments include list servers and various network search aids such as GOPHERand ARCHIE (Krol, 1992).Finally, books are published that contain programs for diverse applications, both general and particular.Some, including two of our own works (Nash J C and Walker-Smith, 1987; Nash J C 1990d) and one bya relative (Kahaner, Moler and Nash S G 1989) include diskettes. We feel that published softwaredirectories are a waste of good paper. They are out of date too quickly and at best provide only a partiallist of potentially interesting software.
The effort to reduce this list of possibilities to a workable small setis not, we think, worth expending.We find computer stores and agencies poorly informed about scientific software. With the trend to sealedpackages, they will rarely offer demonstrations. Mail-order sources limit the time to ask for refunds. Wemay lose shipping costs, customs duties, or "restocking fees".
Worse, we spend time trying to getrecalcitrant programs to work as claimed.Exhausting all of the above routes to "known" methods does not mean that they do not exist. We mustdecide how much time and effort to spend in finding existing methods before we begin to develop ourown. Or we may make do with a "poor" method until something better is available, e.g., using a functionminimizer like Hooke and Jeeves for least squares problems as in Section 13.1.Even if we find a source of some software, we need to know that it will work with our PC configurationand our problem, and we still have to get it.
There may be delays or no response to our order. Our usual118Copyright © 1984, 1994 J C & M M NashNash Information Services Inc., 1975 Bel Air Drive, Ottawa, ON K2C 0X1 CanadaSCIENTIFIC COMPUTING WITH PCsCopy for:Dr. Dobb’s Journalpractice is to make enquiries for information first to gauge the speed with which an order is likely to befilled. Unless we are convinced a supplier is reliable, we will not order software.13.6 Verifying ResultsGood formulation for a problem lets us more easily verify that the results we obtain are correct.
For"standard" problems, many tests are available, so the sub-problem solution method(s) can be checkedbefore we go to a full solution. Since many sub-problems are well known, the dangers and characteristicsare discussed in documentation and other literature, so there is less danger of building a program ofrotten blocks.Also, we may be able to obtain subroutines already written for the sub-problems, or use a package likeMATLAB where the blocks are available as simple commands.
Even if we cannot simply get machinereadable code ready-made for our PC, the break-up of the problem into well-defined parts will usuallybe a help to getting correct answers. Results obtained by different approaches can be compared. Byformulating a problem two different ways, we have a natural check on results. Even if a method is tooslow for production use, it may still be a valuable tool for comparison purposes.It may seem obvious that answers should, where possible, be tested, but users may not know what testsmay be appropriate. We need to ask how well the proposed solution satisfies the mathematical problemrepresenting the real problem.
Are there subsidiary conditions or identities that must be satisfied by theanswers? What results are expected from experience? By users? By alternative solution methods? Wherepossible, it is a good idea to draw pictures, that is, to graph the results. We must also be aware of thepossibility of multiple solutions.The rest of this section includes some suggestions of tests that may be applied in specific areas.Approximation of functionsCody and Waite (1980) give several identity and comparison tests for the common special functions.
Froma less computational point of view, Abramowitz and Stegun (1965) provide a wealth of tables andformulas.Linear equationsAx=bFor these equations, we can compute the residuals r = b - A x. The condition number of A can beestimated to provide a measure of the likely precision of the computed solution (Forsythe, Malcolm andMoler, 1977, p. 42ff., Kahaner, Moler and Nash S G, 1989, p.
68).Linear least squares approximationAx≈bResiduals r = b - A x should always be calculated. Methods which compute solutions via the singularvalue decomposition (Nash, 1990d) or eigenvalue methods provide a measure of collinearity in thecolumns of Belsley, Kuh and Welsch (1980) have aptly titled their book Regression Diagnostics.Algebraic eigenvalue problemAx=exThis problem yields so many identities that there is no excuse for not testing output from an algebraiceigenvalue calculation (Wilkinson, 1965).Differential equationsWe can check the adherence of a solution to the original problem by approximating derivatives.
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