Nash - Compact Numerical Methods for Computers (523163), страница 5
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In just over 300 lines of BASIC is the capability to solve linearequations, linear least squares, matrix inverse and generalised inverse, symmetric matrix eigenproblem and nonlinear least squares problems. The jokeback-fired in that the capability of this program, which ran on the Sinclair ZX81computer among other machines, is quite respectable.Kahaner, Moler and Nash (1989)—This numerical analysis textbook includes FORTRAN codes which illustrate thematerial presented. The authors have taken pains to choose this software forA starting point11quality. The user must, however, learn how to invoke the programs, as there isno user interface to assist in problem specification and input.Press et al (1986) Numerical Recipes—This is an ambitious collection of methods with wide capability.
Codes areoffered in FORTRAN, Pascal, and C. However, it appears to have been onlysuperficially tested and the examples presented are quite simple. It has beenheavily advertised.Many other products exist and more are appearing every month. Finding outabout them requires effort, the waste of which can sometimes be avoided by usingmodern online search tools. Sadly, more effort is required to determine the quality ofthe software, often after money has been spent.Finally on sources of software, readers should be aware of the Association forComputing Machinery (ACM) Transactions on Mathematical Software which publishes research papers and reports algorithms. The algorithms themselves are available after a delay of approximately 1 year on NETLIB and are published in full in theCollected Algorithms of the ACM. Unfortunately, many are now quite large programs, and the Transactions on Mathematical Software (TOMS) usually onlypublishes a summary of the codes, which is insufficient to produce a workingprogram.
Moreover, the programs are generally in FORTRAN.Other journals which publish algorithms in some form or other are AppliedStatistics (Journal of the Royal Statistical Society, Part C), the Society for Industrialand Applied Mathematics (SIAM) journals on Numerical Analysis and on Scientificand Statistical Computing, the Computer Journal (of the British Computer Society),as well as some of the specialist journals in computational statistics, physics,chemistry and engineering. Occasionally magazines, such as Byte or PC Magazine,include articles with interesting programs for scientific or mathematical problems.These may be of very variable quality depending on the authorship, but someexceptionally good material has appeared in magazines, which sometimes offer thecodes in machine-readable form, such as the Byte Information Exchange (BIX) anddisk ordering service.
The reader has, however, to be assiduous in verifying thequality of the programs.1.4. PROGRAMMING LANGUAGES USED AND STRUCTUREDPROGRAMMINGThe algorithms presented in this book are designed to be coded quickly and easily foroperation on a diverse collection of possible target machines in a variety ofprogramming languages. Originally, in preparing the first edition of the book, Iconsidered presenting programs in BASIC, but found at the time that the variousdialects of this language were not uniform in syntax. Since then, InternationalStandard Minimal BASIC (IS0 6373/ 1984) has been published, and most commonlyavailable BASICS will run Minimal BASIC without difficulty. The obstacle for the user isthat Minimal BASIC is too limited for most serious computing tasks, in that it lacksstring and file handling capabilities. Nevertheless, it is capable of demonstrating allthe algorithms in this book.12Compact numerical methods for computersAs this revision is being developed, efforts are ongoing to agree an internationalstandard for Full BASIC.
Sadly, in my opinion, these efforts do not reflect the majorityof existing commercial and scientific applications. which are coded in a dialect ofBASIC compatible with language processors from Microsoft Corporation or BorlandInternational (Turbo BASIC).Many programmers and users do not wish to use BASIC, however, for reasons quiteapart from capability.
They may prefer FORTRAN, APL, C, Pascal, or some otherprogramming language. On certain machinery, users may be forced to use theprogramming facilities provided. In the 1970s, most Hewlett-Packard desktopcomputers used exotic programming languages of their own, and this has continuedto the present on programmable calculators such as the HP 15C. Computers offeringparallel computation usually employ programming languages with special extensionsto allow the extra functionality to be exploited.As an author trying to serve this fragmented market, I have therefore wanted tokeep to my policy of presenting the algorithms in step-and-description form.However, implementations of the algorithms allow their testing, and direct publication of a working code limits the possibilities for typographical errors.
Therefore,in this edition, the step-and-description codes have been replaced by Turbo Pascalimplementations. A coupon for the diskette of the codes is included. Turbo Pascalhas a few disadvantages, notably some differences from International StandardPascal, but one of its merits (others are discussed in $1.6) is that it allows thealgorithms to be presented in a manner which is readable and structured.In recent years the concepts of structured and modular programming havebecome very popular, to the extent that one programming language (Modula-2) isfounded on such principles.
The interested reader is referred to Kernighan andPlauger (1974) or Yourdon (1975) for background, and to Riley (1988) for a moremodern exposition of these ideas. In my own work, I have found such conceptsextremely useful, and I recommend them to any practitioner who wishes to keep hisdebugging and reprogramming efforts to a minimum. Nevertheless, while modularity is relatively easy to impose at the level of individual tasks such as the decomposition of a matrix or the finding of the minimum of a function along a line, it is notalways reasonable to insist that the program avoid GOTO instructions. After all, inaimimg to keep memory requirements as low as possible, any program code whichcan do double duty is desirable. If possible, this should be collected into asubprogram.
In a number of cases this will not be feasible, since the code may have tobe entered at several points. Here the programmer has to make a judgement betweencompactness and readability of his program. I have opted for the former goal whensuch a decision has been necessary and have depended on comments and the essentialshortness of the code to prevent it from becoming incomprehensible.The coding of the algorithms in the book is not as compact as it might be in aspecific application.
In order to maintain a certain generality, I have chosen to allowvariables and parameters to be passed to procedures and functions from fairlygeneral driver programs. If algorithms are to be placed in-line in applications, it ispossible to remove some of this program ‘glue’. Furthermore, some features may notalways be necessary, for example, computation of eigenvectors in the Jacobi methodfor eigensolutions of a real symmetric matrix (algorithm 14).A starting point13It should also be noted that I have taken pains to make it easy to save a ‘copy’ ofthe screen output to a file by duplicating all the output statements, that is the ‘write’and ‘writeln’ commands, so that output is copied to a file which the user may name.(These statements are on the disk files, but deleted from the listings to reduce spaceand improve readability.) Input is allowed from an input file to allow examples to bepresented without the user needing to type the appropriate response other than thename of the relevant ‘example’ file.Furthermore, I have taken advantage of features within the MS-DOS operatingsystem, and supported by compiler directives in Turbo Pascal, which allow forpipelining of input and output.
This has allowed me to use batch files to automate therunning of tests.In the driver programs I have tried to include tests of the results of calculations, forexample, the residuals in eigenvalue computations. In practice, I believe it isworthwhile to perform these calculations. When memory is at a premium, they canbe performed ‘off-line’ in most cases. That is. the results can be saved to disk(backing storage) and the tests computed as a separate task, with data brought infrom the disk only as needed.These extra features use many extra bytes of code, but are, of course, easilydeleted. Indeed, for specific problems, 75% or more of the code can be removed.1.5.
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