5 Numerical and Symbolic Computation

The chapter on Airy functions in the prototype Web site includes four subsections on the general subject of Computations; see Table 2. The first, Methods of Computation, gives a list of general approaches that have been used to construct algorithms, with references to the literature. The approaches identified in this subsection are distinguished by their generality, i.e. in principle they can be combined to achieve any degree of precision because they start from analytical definitions of the functions: series expansions, differential equations, integral representations, and representations in terms of other functions. Numerical considerations such as convergence, accuracy and stability are mentioned briefly. The second subsection, Tables, gives references to published numerical tabulations. Such tables are of occasional use in validating mathematical software but almost always their suitability for this purpose is severely limited by their inadequate precision and range in comparison to the capabilities of current software. Because of the existence of numerical software, tables are rarely used today for their original purpose of providing function values for pencil-and-paper calculations by interpolation between the tabulated entries. Therefore, the DLMF will not include voluminous static tables, which occupied over one-half the pages of AMS 55.

The third Computations subsection is Approximations. Here, references are given to papers that provide fixed finite-precision approximations, usually for restricted ranges of the independent variables. These are valuable when, as is often the case, their execution speed is fast in comparison to other methods. They are often found at the heart of numerical subroutines in software libraries. The fourth subsection, Software, is the one likely to be the most often consulted by DLMF users. Here distinctions are made among programs that have been constructed and published by an original author; libraries that have been produced by gathering programs and imposing uniform conventions with respect to documentation, style and handling of errors; and systems that provide an interactive command-line interface. The subsection provides a classification and listing of published and commercially available software, complete with the pertinent restrictions on the ranges of the independent variables and the precision of the computed results. It also provides immediate access to documentation, and even to source code, via links to GAMS (http://gams.nist.gov/) (the Guide to Available Mathematical Software) and Netlib (http://www.netlib.org/); see also [3].

Eventually the DLMF will include a more interactive facility for computing numerical values of special functions. This will allow a user to specify precision and the ranges of independent variables quite arbitrarily. The actual computation may take place on dedicated computers at NIST or, alternatively, in JAVA code downloaded to the user's local environment. A number of research problems remain to be solved before such a facility can be put fully into place, even for a small selected subset of mathematical functions. The most important of these problems is the requisite error analysis, which is very demanding analytically, and which must be put into a computable form. This is essential to be able to assure that the computed results are accurate to the precision specified. Nevertheless, a prototype facility for selected functions is under construction that will apply within certain limits of precision. It will have a strong likelihood (but not a guarantee) that the precision criterion has been met.

One useful purpose for such a facility is a ``software test service for special functions;'' see [6]. Another is to support user-driven visualizations of mathematical functions in which ranges of independent variables are specified by the DLMF user.

The role of symbolic computation in the DLMF is still being discussed. One possible role is to provide a way to determine mathematical equivalence of expressions, for example when a user is searching for a mathematical formula which he or she expresses in a form that is different from but equivalent to a formula that is encoded in the DLMF database.