NISTIR 6297Mathematical andComputationalSciencesDivision

Information Technology Laboratory

Preprint

Design of a Digital Mathematical Library forScience, Technology and Education

Daniel W. Lozier, Bruce R. Miller and Bonita V.Saunders

February 1999

U. S. DEPARTMENT OF COMMERCE

Technology Administration

National Institute of Standards and Technology

Gaithersburg, MD 20899

PREPRINT

This paper has been accepted for presentation at theIEEE Advances in Digital Libraries Conference, Bal-timore, Maryland, May 19–21, 1999. It will appear infinal form in the conference proceedings to be publishedby the IEEE Computer Society.

Design of a Digital Mathematical Libraryfor Science, Technology and Education ∗

Daniel W. Lozier, Bruce R. Miller and Bonita V. Saunders

National Institute of Standards and Technology

100 Bureau Drive, Gaithersburg, MD 20899-8910

dlozier@nist.gov, bruce.miller@nist.gov, bonita.saunders@nist.gov

Abstract

The concept of a digital library is of proven worth be-cause of its ability to provide dramatic capabilities thatare impossible with traditional print media. We areinterested in providing such capabilities for scientific,technical and educational users of mathematical refer-ence data. Our attention is focused on the highly spe-cialized field of mathematics that is concerned with theproperties, application and computation of the elemen-tary and higher mathematical functions. Calling upondomain experts worldwide for assistance, the NationalInstitute of Standards and Technology is conducting anambitious project to construct, ab initio, a comprehen-sive and authoritative Web resource on this subject.The need to make effective use of the latest develop-ments in digital library research is a major focus, as isthe development of content. In this paper we discussour approach to such difficulties as the representation,display and manipulation of symbolic expressions, nu-merical data and graphical visualizations, and we de-scribe a prototype Web site that has been constructedto test, evaluate and advance the NIST Digital Libraryof Mathematical Functions project.

Keywords: Database, Digital Library, DocumentConversion, Mathematics, Special Functions, Visual-ization, World Wide Web.

1 Introduction

The body of knowledge in the sciences, engineering,and mathematics is vast and increasing at an exponen-tial rate. By reading specialized journals and attend-ing conferences regularly, an individual researcher can

∗This work is supported in part by the Defense AdvancedResearch Projects Agency under contract DAAH 04-95-1-0595.

keep abreast of developments within a relatively nar-row field. But when the need arises for results fromquite different fields, a researcher may have to investa tremendous amount of time immersed in the litera-ture; even then, he or she may not succeed in arriv-ing at, and assessing the reliability of, the pertinentinformation. Digital library technology, coupled withpainstaking development and validation of comprehen-sive reference data, has the potential to minimize thisgeneral problem.

Pure and applied mathematics are the most perva-sive disciplines in science and engineering. Mathemati-cal definition is the key to uniform and accurate utiliza-tion of technical knowledge. Up-to-date refinements offundamental mathematical techniques, such as approx-imation of functions, solution of ordinary and partialdifferential equations, and statistical analysis, providethe underpinning for all modern quantitative science.The increasing reliance of scientists and engineers onmathematical modeling and simulation, the growinguse of symbolic and numerical software, and the rapidlydeveloping capabilities of the Internet and World WideWeb, present challenges and opportunities for a com-prehensive standardization of mathematical knowledgewhich supports new levels of multidisciplinary commu-nication.

The authors are part of a NIST team effort tocollect, organize, validate, develop, and disseminatea comprehensive and evolving digital library pertain-ing to mathematical functions. This library is be-ing called the Digital Library of Mathematical Func-tions, or DLMF. The reason for beginning with thisparticular branch of applied mathematics, instead of,

Certain commercial software products are identified in thispaper. In no case does such identification imply recommenda-tion or endorsement by the National Institute of Standards andTechnology, nor does it imply that the products are among thebest available for the purposes they serve.

0

600

1200

1974

. .. .

1977

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. .. .. .

1980

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. .. .. .

1983

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. .. .. .

1986

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1989

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1992

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1995

. .

. .

Figure 1. Selected yearly citations to AMS 55

(black bars) compared with total citations scaled by

0.00147 (white bars). Data from Science Citation

Index.

say, numerical analysis, is NIST’s direct experiencewith the 1964 Handbook of Mathematical Functions [1],known also as AMS 55 (for Applied Mathematics Se-ries No. 55). This work has had unique influence amongindividuals who apply mathematics to the solution ofreal-world problems, e.g. engineers, physical scientists,and statisticians. Such users have come to regard AMS55 as the definitive source of reference information onthe “special functions” of mathematics.

AMS 55 is one of the most frequently cited worksin the scientific literature. Even though it is 40 yearsout of date (never having been revised), the numberof citations to it continues to rise annually, not onlyin absolute numbers but also as a fraction of the to-tal number of citations made in the sciences and engi-neering each year ; see Figure 1. Currently, about onceevery 1 1

2 hours of each working day some author, some-where, makes sufficient use of this handbook to list itas a reference. Moreover, the journals in which thesereferences appear range widely over the sciences andengineering; see Table 1.

The target date for completion of the public versionof the DLMF, which will be freely accessible from aWeb site at NIST, is late in 2002. A distinctive char-acteristic, in comparison with other initiatives in digi-tal library research and development, is the emphasison original development of detailed and authoritativecontent; see § 2 of this paper. An equally importantthrust is the dissemination of mathematical referencedata as a digital library on the Web with provisions forstate-of-the-art indexing, searching, navigation, cross-referencing, linking, downloading, and so on. A pro-

Table 1. Journals with highest number of citations

to AMS 55 in the 10-year period 1988–1997. Data

from Science Citation Index.

Cit. Journal

498 Phys. Rev. B: Condensed Matter Physics

462 Phys. Rev. A: Atomic, Molecular, Optical Phys.

381 Journal of Chemical Physics

262 J. Phys. A: Mathematical and General Physics

240 Phys. Rev. E: Statist. Phys., Plasmas, & Fluids

231 Journal of the Acoustical Society of America

205 Journal of Fluid Mechanics

183 Astrophysical Journal

182 Phys. Rev. D: Elementary Particles

153 J. Phys. B: Atomic, Molecular, & Optical Phys.

totype Web site with a newly written chapter on Airyfunctions [7] is described, and some of the issues in-volved in its construction are discussed in § 3. Ad-vanced interactive graphics in two and three dimen-sions are a valuable aid to qualitative understanding ofthe properties of mathematical functions; § 4 discussesissues associated with this topic. § 5 discusses the issueof numerical and symbolic computation, including thelocation and downloading of software; many users willwant easy-to-use support in these matters. Applicationand learning modules are the subject of § 6. These areauxiliary units tailored to the needs of fields outsidemathematics itself, with links to the DLMF Web site.This paper concludes with a few final remarks in § 7.

2 Content Development

The first step in content development is to establishthat a genuine need exists for a new initiative in specialfunctions.

The citation record shows (see Figure 1) that spe-cial functions are important in science and engineering,and that AMS 55 is meeting this need. However, itswriting was essentially finished by 1960, and in the in-tervening decades numerous advances in mathematicalknowledge have been made:

• New functions have entered the realm of practicalimportance. One example among many is Carl-son’s elliptic integrals which provide a unificationof a branch of special functions that had been un-changed since its inception in the first half of the19th century.

• New fields of application have emerged. For exam-ple, classical special functions are used in soliton

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theory and nonlinear dynamics.

• Analytical developments have occurred, e.g. inasymptotics and nonlinear sequence transforma-tions.

• New properties, including integral representations,integrals, addition formulas and generating func-tions, have been discovered.

• Numerical developments, e.g. interval analysis,Pade approximations, and boundary-value meth-ods, have come into being.

• Applications of computer algebra and symbolicshave come into wide use.

• An enormous increase in computing power has ren-dered obsolete many standard numerical processesof the 1950’s, such as table-making and interpola-tion, while at the same time increasing the valueof others, e.g. integration of defining differentialequations in the complex plane.

• Comprehensive software packages have been con-structed, both commercial and non-commercial,for generating special functions by sound numeri-cal procedures.

• The dissemination of information has been rev-olutionized consequent upon the introduction ofelectronic publishing and computer networks.

All of these developments are important, and all needto be accounted for in a modern compendium on specialfunctions.

Resources other than AMS 55 exist. How do theyrelate to the DLMF project? Two handbooks of greatimportance that are contemporaneous with AMS 55,and therefore just as out-of-date, are [4, 5]. More re-cent handbooks, such as [8, 9, 10], are much less com-prehensive and do not meet the overall need identifiedfor the DLMF, which is to provide a thorough and au-thoritative treatment of special functions as they applywithin and outside mathematics. In recent handbooks,a strong emphasis is often placed on computation, insome cases using numerical and programming meth-ods of lesser overall quality compared with standardprocedures in the better-known software libraries. An-other kind of resource is represented by current soft-ware packages, such as Macsyma, Maple, Mathe-matica and Matlab, that include substantial supportfor special functions as well as broad support of otheruseful mathematical fields. The fact that companiesexpend funds in this way for special functions is an-other indicator of their perceived importance. How-ever, it must be stressed that these systems do not

provide a substitute for an authoritative reference com-pendium. In fact, AMS 55 continues to serve as a sourceof reference information for users of these systems, aswell as for the system developers; the DLMF will con-tinue in serving this purpose.

The second step in content development is contentdefinition. Which topics should be included, and whichexcluded?

Our approach here is to call upon recognized do-main experts. An invitational workshop held at NISTin the summer of 1997 resulted in a tentative list of 34chapter headings. The majority treat individual func-tion classes, such as the elementary functions (expo-nential, hyperbolic and trigonometric functions, andtheir inverses) and higher functions (Airy functions,Bessel functions, Legendre functions, and so on); theremainder deal with closely associated topics such asalgebraic, analytical and numerical methods. Furtherrefinement of the exact content to be included will re-sult from regular meetings, augmented by email andtelephone communications, of an editorial board madeup of the four NIST editors and ten prestigious asso-ciate editors from the U.S. and abroad.

The third step in content development is the actualcontent generation, together with quality control. Forthe DLMF this will consist of contracting with highlyqualified authors who will be selected by the NIST ed-itors. The associate editors have all committed them-selves to provide substantial assistance to the NISTeditors in this and all other important editorial deci-sions in their respective subfields of expertise. Theywill also assist in reviewing the written material sub-mitted by the authors. After the content is put intothe DLMF Web site by NIST staff, the original au-thors will be asked to review it, and further review willbe arranged by contracts with qualified validators whoare independent of the original authors.

The entire project requires substantial funding, andits completion date depends on the funding rate. Ex-ternal funding is being sought to augment committedinternal NIST funding. Under the best possible fundingscenario, completion of the DLMF project will occur in2002. The DLMF will require a continuing low level offunding indefinitely. NIST is committed to continuingmaintenance under the auspices of its congressionallymandated Standard Reference Data Program.

3 Web Site

In this section, we address some of our designgoals for the DLMF Web site, and the implicationsthese goals have for the information architecture of theproject. We then give an overview of how we plan to

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implement that architecture, particularly, how we planto translate the material our authors will provide intoan appropriate internal representation which satisfiesthe needs of the site.

The bulk of this section describes the plans for theultimate implementation of the DLMF. However, wehave experimented with many of the features describedhere in a “mock up” of the eventual site (See http://dlmf.nist.gov/, particularly Chapter 11 on AiryFunctions[7].), which was prototyped using a highlymodified version of LaTeX2HTML.

3.1 Design Goals and their Implications

Let us begin by considering a sampling of the kindsof operations we wish to provide based on our experi-ences with AMS 55 and its users.

1. Browse through information on a particular spe-cial function.

2. Search for a particular kind of formula or infor-mation relating to a given property of a particularspecial function.

3. Find the solution of a particular differential equa-tion, or find the closed form of a particular seriesexpansion.

4. Obtain detailed information about the history,sources, derivation or bibliographic references fora particular formula or section.

5. Cut&paste a piece of mathematical, tabular orother text from the DLMF into an arbitrary userapplication in LATEX, Fortran, Mathematica,Postscript, or other format.

6. Print a quick reference guide of the main proper-ties of a special function, or of definitions of allspecial functions, or of any other specific propertyof a user-specified selection of special functions.

The first three of these user operations primarilyconcern how a user navigates the site, going from thetop of the site to some particular piece of informationwithin it. Such traditional tools as indices and a tableof contents are as important to this task as they everwere — although they are not as often found on theWeb as they should be. A clearly delineated hierarchywith non-overlapping topics at each level is essential forgenerating a table of contents that a user can quicklyand confidently navigate; this requirement is too of-ten ignored. In addition, extensive meta-informationindicating properties not only of sectional units, but

also of formulas and graphics, is needed to constructthe several multi-level indices that users might need.This latter information is also essential in constructingautomated search tools.

Meta-information attached to each unit or formulacan provide much more than indexing capabilities.It provides a natural mechanism to attach informa-tion about historical developments, derivations and soforth. This is information that would normally be hid-den from the user. But, when a user wishes to godeeper into a subject, that information can be collectedand presented, say, by following a hypertext link asso-ciated with each formula. Thus, augmenting the textwith abundant meta-information helps satisfy items 4and 5 on our list. See Figure 2 for an example ofthe meta-information associated with a formula in themockup site.

To be able to construct quick reference guides (item6) which collect a variety of different objects (e.g. sec-tions or formulas) satisfying given criteria suggests aslightly different requirement. Attributes of sectionsand formulas, defined in the meta-information or im-plied by the sectioning, must be usable in determiningthe role of the object in its context. In this way, wecan automatically find, for example, definitions of allfunctions in the DLMF; from these collected objectswe construct the desired virtual document. Broadlyspeaking, these attributes define different views of, orslices through, the document. Consequently, in addi-tion to assigning the appropriate attributes to all unitsand formulas, we must also take care that they are writ-ten in such a way as to be useful outside their originalcontext.

Taken together, these implications require us tocarefully lay out the overt, hierarchical organization,that is the default book-like arrangement, that DLMFauthors will need to adhere to. They also require aninternal representation based more on the semantics ofthe material than merely the presentation of it; thata unit is a subsection is less important than that theunit deals with a particular mathematical property ofa function. They also require us to develop severalappropriate vocabularies for classifying the entities asto their roles (e.g. definitions, addition theorems, se-ries expansions, . . . ), thier origination (e.g. references,derived from another formula, restrictions on applica-bility, . . . ) and so on.

3.2 Internal Document Format: XML

Given these considerations, we find that an in-ternal document format based on XML (eXtensibleMarkup Language: http://www.w3c.org/XML/), com-

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Figure 2. An example of a Meta-information page.

plemented by XSL (eXtensible Style Language: http://www.w3c.org/Style/XSL/), will be the most use-ful and flexible solution. The combination gives a di-rect method of presentation in browsers that supportit. It also supports translation to other needed for-mats, such as: HTMLwith MathML (Math MarkupLanguage: http://www.w3c.org/Math/), LATEX andPostscript.

Additionally, we will employ the XML applica-tion OpenMath (http://www.openmath.org/) (orthe content model of MathML) to represent the math-ematical formulas themselves. OpenMath allows us torepresent the semantics of the mathematical formula —what it means and not just how it looks. This aspectis essential to enable the use of these formulas in anyother way besides looking at them. Further, Open-Math is being developed as an interchange format formathematical information and reader modules are ex-pected for many different applications such as com-puter algebra, text formatting and graphics systems.

3.2.1 Document Structure We do not, in this re-port, describe a complete DTD (Document Type Def-inition. See the XML site for more information.). Wedo, however, give an overview of the main features ofthe anticipated document structure. In Table 2, we

give a schematic of the structure of a chapter on aparticular special function (there will be other typesof chapter in the DLMF, with different structural re-quirements). Sections at the lowest level shown willtypically include nested subsections and paragraphs,as well as formulas and graphical objects.

3.2.2 Meta-Information Attributes Everystructural unit described above, including formulasand graphical objects, can and should be annotatedwith a rich set of attributes. The following listdescribes some of the main classes of annotation thatwe are considering.

attribute: Names of properties satisfied by this unit.

citation: Citation of original information sources.

note: Additional commentary describing the unit.

The following would apply only to formulas.

derived from: Derivation of one formula from an-other.

applicability: Indicates that the formula is valid onlywhen arguments or variables are restricted to thespecified ranges.

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Table 2. Section headings for a special function chap-

ter.

Special FunctionIntroduction

OverviewNotationDefinitionGraphs and Visualizations

Mathematical PropertiesDifferential EquationsIntegral RepresentationsRelation to Other Functions

...Computations

Methods of ComputationTablesApproximationsSoftware

...References

3.2.3 Database of Sectional Units Finally, eachstructural unit (including graphics and formulas) willbe entered into a database, indexed (at least) by the an-notations described above. By this means, collectionsof such document fragments matching various searchcriteria can be found. Reassembling these pieces al-lows the construction of a wide variety of quick refer-ence guides or other distillations of material, whetherwe had planned them in advance or not.

3.3 Creation of XML Documents: Trans-lation from LATEX

There is one last complication to this grand plan,which is the question of how to create the XML docu-ments as described above. The chapters of the DLMFwill be written by various experts around the world.XML is rather new, and unlikely to be familiar tothese authors. In contrast, LATEX is the lingua francafor exchange of documents in the mathematical com-munity; most mathematical authors are familiar withit, and it generates beautifully typeset output. Whilethe markup of LATEX is more semantically orientedthan that of raw TEX, it is still primarily oriented to-wards presentation, and this introduces ambiguities;see §3.3.2, below.

It is conceivable that we might manually translate

the LATEX documents into the desired XML format.However, we may anticipate a period of editing andenhancing the information of each chapter. Since theauthor may not be prepared to deal with our XMLdocument, we would exchange corrected versions inLATEX format. Thus a document may have to be re-peatedly converted from LATEX to XML; indeed, manysubtle problems with a given document (missing meta-information, ambiguities, etc) may only be revealed bypost-processing of the XML! Consequently, we requirethis translation to be highly automated; if user inter-vention is required, we should at least be able to aug-ment the source with declarations which would auto-matically resolve those ambiguities on subsequent pro-cessing.

3.3.1 Style files For the most part, the issues in-volving document structure and meta-information willbe easily handled by the definition of a LATEX style filefor use by the authors. Macros defining our view of thedocument structure are easily defined, allowing authorsto process, print and proofread their drafts in the usualway. On the other hand, these macros will be recog-nized by our translation system and used to constructthe desired XML representation of the material.

Similarly, a set of macros such as

\attribute{addition_theorem}

are used to assign the various meta-information to thecontaining sectional unit or formula. When processedby LATEX, these generate indices or margin notes; theycan even be ignored if they are not being proofread.When processed by our translation system, however,they will be embedded in the XML output, and also ex-tracted to construct the required indices and databaseattributes.

3.3.2 Mathematical Ambiguities The ambigui-ties of LATEX are most apparent in the case of markupof mathematical formulas. As two simple examples,consider the expression f(x+y) (which would displayas f(x+y)), and \frac{df}{dx} (which would displayas df

dx ). In the first case, the juxtaposition of f and(x+y) might represent either function application ormultiplication. In the second case, this fraction con-struct is a common idiom for a derivative, but mightreally be intended to represent the (unsimplified!) frac-tion. Generally, human readers aware of the context ofthe formula will immediately recognize the intendedinterpretation. Computers often fare less well.

It is, of course, here that our translation softwarewill be most challenged. And given the high confidencelevel we hope to achieve with the DLMF, we are not

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inclined to rely on heuristic methods of analysis. Tothis end, we devise what may be more tedious, buthopefully more reliable, methods to disambiguate theformula.

To handle the first kind of problem, we require thatevery symbol used must be declared to clarify its typeas either a variable or function; a scalar, vector or ma-trix; real or complex; etc. Thus, if f is a function andx + y is not, we interpret f(x+y) as a function appli-cation; otherwise the expression represents a product.Given these declarations, a minimal type inference sys-tem can be expected to be sufficient for resolving thesecases.

The second kind of problem is handled simply by re-quiring the author to say what he means; macros suchas \deriv{f}{x} are introduced to express explicitlythat a derivative is intended. Nevertheless, we stillmust expend some effort to recognize expressions like\frac{df}{dx} in the document as potentially stand-ing for a derivative.

Ultimately, we must develop a rather complete setof such macros to disambiguate all the notations andusages that we expect to find in the author’s materials.These macros would be included in the provided stylefiles, and authors would be encouraged to use them.We do have the advantage, however, of not having todeal with all of mathematics, for which the richness ofnotations with multiple interpretations becomes quiteoverwhelming.

4 Graphics and Visualization

A Web-based digital library offers significant advan-tages over printed media for the presentation of infor-mative graphics. Whereas graphical representations in[1] were sparse and restricted to static 2D plots, in theDLMF, dynamic 3D visualizations of complex specialfunctions will complement 2D and 3D still images. Thejudicious use of graphics not only will help scientistsand other technical users gain a deeper understandingof special functions, it will also assist us in making someparts of the DLMF accessible to educators, studentsand others who want a short introduction to the field.

The development of effective graphical displays inthe DLMF presents several challenges. To insure uni-formity throughout the DLMF, we must come to aconsensus about the location, frequency, and type ofvisualizations to be placed in the system; but close col-laboration with individual authors will be required todetermine the specific needs for each chapter. A re-liable means of computing accurate data for the vi-sualizations must be found, and the author must de-cide which special features of a function, such as zeros,

poles, and other singularities, should be emphasized.At the same time, the plotting range and scaling of thefunction must be determined to illustrate best thosefeatures. Another issue is deciding the type of file for-mats to be used, while also considering the availabilityof any additional Web plug-ins that would be neededby the user. We address some of these challenges byexamining what we have done for the chapter on Airyfunctions in the prototype Web site.

4.1 Static and Dynamic Visualizations ofSpecial Functions

The author of the chapter on Airy functions[7] spec-ified which Airy functions should be displayed and sug-gested the ranges for the plots. To obtain reliable datawe used a double precision Fortran routine for the cal-culation of Airy functions written by D.E. Amos [2].In general, all of the authors of the DLMF shouldbe knowledgeable about the latest computational tech-niques being used for the functions in their chapter andshould therefore be able to point us to the best meansfor obtaining reliable data.

For both the still images and dynamic visualizationswe began by using available packages such as Matlaband Mathematica to plot the data so that we couldexamine the graphical representation and adjust thescaling to bring out interesting features. Also, consid-erable effort was spent determining the most informa-tive views for the still images. The still images werestored in GIF or Postscript format. To obtain dy-namic visualizations, we wrote a C program to convertthe data to VRML (Virtual Reality Modeling Lan-guage; http://www.vrml.org/) format. VRML is astandard 3D file format for describing the geometryand movement of a 3D virtual world. We chose VRMLbecause of its accessibility on the Web and its interac-tive capabilities. Also, VRML browsers for a varietyof platforms can be freely downloaded. However, it isnot a foregone conclusion that the completed version ofthe DLMF will use VRML. We have to address suchissues as whether VRML browsers will continue to bereadily available and what to do if a browser is notavailable for a specific platform. We will also examinethe feasibility of using alternatives to VRML such asJava 3D which would not require the user to down-load a browser. Currently the user is given the optionof viewing a still 3D image if a VRML browser is notavailable. Figure 3 shows a VRML display of the realpart of Airy function Ai(z). The browser controls al-low the user to rotate the figure, zoom in and out, andmove the figure in an arbitrary direction.

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Figure 3. VRML display on CosmoPlayer.

4.2 Intersection of 3D Surfaces with Cut-ting Planes

In addition to the standard controls that come withthe VRML browser, Figure 3 shows a panel labeled“Cutplane control” which gives the user additional ca-pabilities. We used VRML to create files that wouldgenerate cutting planes through a 3D surface. By ma-nipulating the panel controls, a user can study thechange in the intersection as a plane is moved througha surface. Currently, the cutting planes are limited toplanes perpendicular to the X and Y coordinate axes,but we are working on an extension to the Z direction.Future work will extend the capability to an arbitrarydirection.

When the user clicks the X button on the Cutplanecontrol panel, a bounding box appears around the fig-ure along with a cutting plane that moves perpendicu-lar to the X axis. The user moves the plane by clickingon the second row of buttons which operate like thoseof a VCR. The plane moves in sync with the projectedintersection curve, displayed on opposite faces of thebounding box as shown in Figure 4. The controls op-erate similarly in the Y direction. We are working onthe addition of a slider bar to the control panel to givethe user more flexibility and make it easier to stop theplane at desired points.

To implement the cut plane control, we used VRMLreusable components called PROTO’s. Our CutplanePROTO displays the plane and searches the surface

Figure 4. VRML display with X direction cutting

plane.

data to determine which points are closest to the spec-ified plane. Linear interpolation is then used to obtainthe coordinates for the intersection. All the surfaceswe have done to date intersect the X or Y planes incontinuous curves. If the surface contains holes, thenthe intersection curves will be disconnected at some lo-cations, so we have to be careful about how we connectthe points. Such a situation is the norm for the Z di-rection. The intersection of the Z direction plane withthe surface is the contour curve for that level which,in general, is not a single continuous curve. Therefore,we are testing various packages for contour plotting touse in determining the Z direction intersection points.In testing the effectiveness of the packages on differentmachines we are finding that an acceptable speed forthe VCR controls on one machine may actually be toofast on another. For that reason, we may decide thatthe slider bar, which gives the user more control, is abetter choice.

4.3 3D Clipping

An unexpected issue arose when some plots wererescaled to emphasize interesting features. We foundthat most of the packages we used clipped the surfacesin an unsatisfactory manner. Some packages simply re-set values above a certain height to the same constant,producing the shelf effect illustrated in the plot of Airyfunction |Bi′(z)| shown in the first graph in Figure 5

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Figure 5. Unclipped and clipped graphs.

and also seen in Thompson [9]. Others suppress theplotting of points where the function value is greaterthan a specified number, but this may produce plotswith jagged edges that are equally misleading.

Another problem is the extraction of the clippeddata and its translation to a format we can use. Inat least one package, we discovered that although theclipped surface looked fine on the screen, the output ofthe plotted data included the entire surface instead ofthe clipped surface.

We have been unable to find a package that meets allof our needs. Therefore, we are doing some work in de-veloping our own techniques. We created the smoothlyclipped second graph in Figure 5 by using techniquesfrom the field of mesh generation. First, we selectedthe height at which we wanted to clip the function,Z = 5. We then used the Z = 5 contour curve ofthe function to construct a boundary for our domain.A boundary fitted mesh was placed on the domain asshown in Figure 6. By computing the Airy functiononly at values on the mesh we obtained the smoothlyclipped surface plot in the figure. We discovered thatclipping the figure by this technique also smooths theshading, which is based on the height of the functionat the grid points. This is probably because the mesh

-4

-3

-2

-1

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3

Figure 6. Contour mesh.

lines are close to being contour curves.Creating a boundary-fitted mesh based on contour

information about the function is an ideal solution formany graphs, but it is clear that the mesh generationproblem can become quite complicated for more com-plex special functions that have features such as steepgradients, zeros, or poles. For example, a contour meshfor the gamma function would be multiply-connectedwith several holes. We may want to look at triangula-tion techniques to handle more complicated domains,although doing so may also affect the way we write thecutting plane software.

5 Numerical and Symbolic Computa-tion

The chapter on Airy functions in the prototype Website includes four subsections on the general subjectof Computations; see Table 2. The first, Methods ofComputation, gives a list of general approaches thathave been used to construct algorithms, with refer-ences to the literature. The approaches identified inthis subsection are distinguished by their generality,i.e. in principle they can be combined to achieve anydegree of precision because they start from analyticaldefinitions of the functions: series expansions, differen-tial equations, integral representations, and represen-tations in terms of other functions. Numerical con-siderations such as convergence, accuracy and stabil-ity are mentioned briefly. The second subsection, Ta-bles, gives references to published numerical tabula-tions. Such tables are of occasional use in validatingmathematical software but almost always their suit-ability for this purpose is severely limited by their in-adequate precision and range in comparison to the ca-pabilities of current software. Because of the existence

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of numerical software, tables are rarely used today fortheir original purpose of providing function values forpencil-and-paper calculations by interpolation betweenthe tabulated entries. Therefore, the DLMF will notinclude voluminous static tables, which occupied overone-half the pages of AMS 55.

The third Computations subsection is Approxima-tions. Here, references are given to papers that pro-vide fixed finite-precision approximations, usually forrestricted ranges of the independent variables. Theseare valuable when, as is often the case, their execu-tion speed is fast in comparison to other methods.They are often found at the heart of numerical sub-routines in software libraries. The fourth subsection,Software, is the one likely to be the most often con-sulted by DLMF users. Here distinctions are madeamong programs that have been constructed and pub-lished by an original author; libraries that have beenproduced by gathering programs and imposing uniformconventions with respect to documentation, style andhandling of errors; and systems that provide an interac-tive command-line interface. The subsection provides aclassification and listing of published and commerciallyavailable software, complete with the pertinent restric-tions on the ranges of the independent variables and theprecision of the computed results. It also provides im-mediate access to documentation, and even to sourcecode, via links to GAMS (http://gams.nist.gov/)(the Guide to Available Mathematical Software) andNetlib (http://www.netlib.org/); see also [3].

Eventually the DLMF will include a more interac-tive facility for computing numerical values of specialfunctions. This will allow a user to specify precisionand the ranges of independent variables quite arbitrar-ily. The actual computation may take place on ded-icated computers at NIST or, alternatively, in Javacode downloaded to the user’s local environment. Anumber of research problems remain to be solved be-fore such a facility can be put fully into place, evenfor a small selected subset of mathematical functions.The most important of these problems is the requisiteerror analysis, which is very demanding analytically,and which must be put into a computable form. Thisis essential to be able to assure that the computed re-sults are accurate to the precision specified. Neverthe-less, a prototype facility for selected functions is underconstruction that will apply within certain limits ofprecision. It will have a strong likelihood (but not aguarantee) that the precision criterion has been met.

One useful purpose for such a facility is a “softwaretest service for special functions;” see [6]. Another isto support user-driven visualizations of mathematicalfunctions 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 providea way to determine mathematical equivalence of ex-pressions, for example when a user is searching for amathematical formula which he or she expresses in aform that is different from but equivalent to a formulathat is encoded in the DLMF database.

6 Application and Educational Mod-ules

The DLMF is envisioned not only as a basic re-source for scientific professionals but also as a foun-dation for innovative, discipline-specific “applicationmodules” that can, for example, eliminate some of thevexing variations that occur in the use of mathematicalfunctions in different application areas. Some of thesevariations are merely notational but most are relatedto the fact that real-world applications involve phys-ical constants, normalization conventions, and specialconditions that have no place in a purely mathematicaltreatment. Since mathematical functions are intrinsicin so many different fields, no attempt can be made tocover all fields within the DLMF project. Thus theDLMF will contain only a very restricted set of illus-trative examples that have a connection to an applica-tion area in chapters where it is appropriate. However,one of the outstanding benefits of a richly interactiveand interlinked Web site is the opportunity it presentsto construct associated Web sites that are tailored tospecific application areas and discuss them in substan-tial detail. We expect that the DLMF will serve as therepository of core mathematical information for suchWeb sites. The DLMF project intends to provide twosuch application modules, in quantum mechanics andelectomagnetic theory, as prototypes for others to con-sider.

The DLMF provides an opportunity to constructeducational modules in exactly the same way, and aprototype in this field is being developed also. It willfocus on mathematical functions that are introduced inhigh school, elaborated in university, and used in vir-tually all engineering and scientific applications. Weexpect that it will quickly become popular as an inno-vative learning resource for well-motivated high-schoolstudents, university students and technological profes-sionals. It will provide an on-line tutorial with theability to (i) search for and retrieve formulas and otherinformation from the DLMF, (ii) generate computa-tional results on demand, (iii) generate user-controlledgraphics and visualizations on demand, (iv) check thecorrectness of exercises completed by the student, and

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(v) follow links to related Web sites.

7 Concluding Remarks

It is a truism that our world is becoming increas-ingly mathematized. This has been going on for atleast two centuries, and the pace is accelerating. Spe-cial functions is a vital branch of mathematics thathas deep and far-reaching impact in science and tech-nology. And it is not only the scientific world that ismathematized. Advanced mathematical techniques arecentral also in economics, finance, scheduling, commu-nications, risk assessment, . . . ; the list is very long.

Non-mathematicians who are faced with on-the-jobmathematical demands need ready access to reliablemathematical information that ranges across a spec-trum from critically evaluated reference data, throughscholarly synopses, to introductory tutorials. Theeventual DLMF Web site will meet needs across thisspectrum, and in a highly usable form. This is a natu-ral application area for a digital library, with its abil-ity to provide scientists, technicians and students withpowerful interactive tools.

References

[1] M. Abramowitz and I. A. Stegun, editors. Handbookof Mathematical Functions with Formulas, Graphs andMathematical Tables, volume 55 of National Bureau ofStandards Applied Mathematics Series. U. S. Govern-ment Printing Office, Washington, D. C., 1964.

[2] D. E. Amos. Algorithm 644. A portable package forBessel functions of a complex argument and nonnega-tive order. ACM Trans. Math. Software, 12:265–273,1986. for remark see same journal v. 16 (1990), p. 404.

[3] R. Boisvert, S. Browne, J. Dongarra, and E. Grosse.Digital software and data repositories for support ofscientific computing. In N. R. Adam, B. K. Bhargava,M. Halem, and Y. Yesha, editors, Advances in DigitalLibraries. Springer-Verlag, New York, 1995.

[4] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G.Tricomi. Higher Transcendental Functions, volume 1.McGraw-Hill, New York, 1953. Volumes 2 (1953)and 3 (1955) printed by same publisher. All volumesreprinted in 1981 by Krieger Publishing Company,Melbourne, Florida.

[5] E. Jahnke, F. Emde, and F. Losch. Tables of HigherFunctions. McGraw-Hill, New York, sixth edition,1960.

[6] D. W. Lozier. A proposed software test service forspecial functions. In R. F. Boisvert, editor, Quality ofNumerical Software: Assessment and Enhancement,pages 167–177. Chapman & Hall, London, 1997.

[7] F. W. J. Olver. Airy Functions, Chapter 11,Digital Library of Mathematical Functions Project.

National Institute of Standards and Technology,Gaithersburg, MD 20899-8910, U. S. A., 1998.http://math.nist.gov/DigitalMathLib/.

[8] J. Spanier and K. B. Oldham. An Atlas of Functions.Hemisphere Publishing Corporation, Washington, D.C., 1987.

[9] W. J. Thompson. Atlas for Computing Mathemati-cal Functions: An Illustrated Guide for Practitioners,with Programs in C and Mathematica. John Wiley &Sons, New York, 1997. An edition with programs inFortran exists also.

[10] S. Zhang and J. Jin. Computation of Special Func-tions. John Wiley & Sons, New York, 1996.

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