short and expensive. Consequently, computer centre managers were not easily persuaded to install such sys Computer Algebra tems. This is another reason why many specialized CAS have been developed in many different institutions. From the Jacques Calmet, Grenoble * very beginning it was felt that the (UFIA / INPG) breakthrough of Computer Algebra is linked to the availability of adequate computers: several megabytes of main As soon as computers became avai handling of very large pieces of algebra memory and very large capacity disks. lable the wish to manipulate symbols which would be hopeless without com Since the advent of the VAX's and the rather than numbers was considered. puter aid, the possibility to get insight personal workstations, this era is open This resulted in what is called symbolic into a calculation, to experiment ideas ed and indeed the use of CAS is computing. The sub-field of symbolic interactively and to save time for less spreading very quickly. computing which deals with the sym technical problems. When compared to What are the main CAS of possible bolic solution of mathematical problems numerical computing, it is also much interest to physicists? A fairly balanced is known today as Computer Algebra. closer to human reasoning. answer is to quote MACSYMA, RE The discipline was rapidly organized, in DUCE, SMP, MAPLE among the general 1962, by a special interest group on Computer Algebra Systems purpose ones and SCHOONSHIP, symbolic and algebraic manipulation, Computer Algebra systems may be SHEEP and CAYLEY among the specia SIGSAM, of the Association for Com classified into two different categories: lized ones. MACSYMA is the largest of puting Machinery. Soon after, the first general purpose systems and specializ all CAS. This means that its library of major Computer Algebra Systems (CAS) ed packages. The latter are generally procedures, methods and techniques is were designed and available. Simulta very efficient at solving some well defin the most complete. This originates in the neously, several specialized packages ed classes of problem. Most often, their fact that for many years it was only ac were successful and greatly contributed architecture reflects their goal and it is cessible at MIT through networking and to the fame of the field. For instance, seldom possible to transport them to dif all relevant programs written by users SCHOONSHIP was used at CERN to ferent makes of computer. General pur were added to its library. Today it is distri solve problems in high energy physics pose systems are expected to be less ef buted by Symbolics Inc., except for the from the early 1970s. ficient for very specialized applications, US Department of Energy version which A main characteristic of the computer but easily portable and they offer a large is subject to possible restrictions. It is algebra community was its diversity. In library of algorithms. Their architectures Lisp-based and available on many diffe deed, it gathered mathematicians most share many common features: rent computers starting from the SUN's ly interested in designing algebraic algo (i) a programming language which is and VAX's. REDUCE, also Lisp-based, is rithms, computer scientists motivated Pascal or Algol-like with a strong flavour less powerful but very clean, well by the will to manipulate symbols and, of Lisp, debugged and well documented. Over surprisingly enough, many users who (ii) a kernel constituted by the basic 1000 copies are distribued worldwide as designed their own CAS. Despite the functions performing the generic opera of today and it is probably the most used fact that the discipline is now well esta tions on mathematical expressions such of all CAS. SMP and MAPLE are written blished in computer science depart as input, output, representations, simpli in C. SMP is announced to be as power ments, these three classes of practi fication, substitutions, term generation, ful as MACSYMA but this remains to be tioners still remain active. The field is still flow control, storage and special com fully proved and the debugging comple lacking good textbooks. The only excep mands, ted. Eventually, it ought to be very well tion [1] was planned as a collection of (iii) one or several modules of algebraic suited to the physicist's needs. MAPLE the basic material needed to set up algorithms embedding the mathemati is listed here since it is the system to courses at all levels of the curriculum. cal knowledge of the system. select for a classroom environment. In Also illustrative are the proceedings of Specialized CAS have always origina deed, it enables many jobs to be run effi the annual conferences usually publish ted from research needs and stick to ciently simultaneously. Both REDUCE ed in Lecture Notes in Computer Science them. Although general purpose CAS and MAPLE are available on machines of Springer-Verlag and the SIGSAM Bul have sometimes a similar origin, they ranging from the MacPIus (MAPLE) and letin. Since 1985, the Journal of Sym have evolved to become "consumers" IBM PC's (REDUCE) to the Cray's. Ac bolic Computing covers the research ac oriented products and, in fact, some cording to the opinion of Computer tivities of the field and is eagerly looking have become commercial products Algebra specialists, SCHOONSHIP is a for contributions to its "Letters on Ap which is a common trend in computer rather crude system but it is perfectly plications" section. science. Nevertheless, it must be noted tailored to the needs of high energy phy Before outlining the tools of Computer that most of the very important applica sicists and still in use. It is written in Algebra, what it can bring to physicists tions have been worked out usinq gene- machine code for the CDC's and now for and what the new frontiers of the field ralCAS. the Atari micro-computer. CAYLEY is are, it is relevant to point out that a CAS Over the years, more than 70 CAS specialized to group theory and SHEEP is presently a very powerful technical have been designed [1]. A major reason to tensors and thus to general relativity. assistant to perform algebraic computa for such a diversity of systems lies with A frequent question from potential tions. Its benefits are the accuracy, the technology of computers. Twenty users is which CAS is best suited to their speed and reliability of calculations, the years ago, a general purpose CAS was problem and if it appears that the re already a large piece of software at a quired methods are not immediately * Now with the University of Karlsruhe time when computer resources were available they may be tempted to design 28 Simple Example of the Use of a Computer Algebra System (Reduce) in a Combined Algebraic- numerical Problem. RisØ National Laboratory Problem: Find the minimum value of Danish Research Academy 0 (x + y)5 cos(x) dx 1st TopsØe Summerschool The following REDUCE session solves the algebraic part: on (x + y) 5*cos(x); enter function into ws (work space) Superconductivity int(ws.x); integrate with respect to x sub(x = pi,ws) -sub(x = 0,ws); definite integral from 0 to June 20-24, 1988 on fort; turn on Fortran generation held at ws; generate Fortran for the definite integral df(ws,y); and for its derivative with respect to y RisØ, Roskilde, Denmark List of lecturers includes: REDUCE generates the following ouput: G. Aeppli, AT&T Bell Laboratories K. Bechgaard, Copenhagen University J. Clarke, Berkeley FC = 5.*(2.*Y**4 +4.*Y**3*PI + 6.*Y**2*PI**2 — 24.*Y**2 + A. Davidson, IBM Yorktown . 4.*Y*PI**3—24.*Y*PI + PI**4 — 12.*PI**2 + 48.) L.M. Falicov, Berkeley M. Garber, Brookhaven National Lab. H. R. Ott, ETH Zürich GC = 20.*(2.*Y**3 + 3.*Y**2*PI + 3.*Y*PI**2 — 12.*Y +PI**3 — N. Falsig Pedersen, Tech. Univ., Lyngby . 6.*PI) B. Raveau, University of Caen P.H. Wu, Nanjing University The above statements can then be used with a numerical minimization Experimental programme organized by: package which requires the function and its first derivative. J. Als-Nielsen, RisØ National Lab. N. Hessel Andersen, RisØ National Lab. M.T. Levinsen, Copenhagen Univ. their own system. It is necessary to rational fractions and of polynomials de M. Nielsen, RisØ National Lab. point out that this means implementing fined over diverse fields, including their Scope all the general features which are re arithmetics, factorization, gcd, zeros de The first TopsØe summerschool will give a broad coverage of superconducting phenomenology, quired in symbolic manipulation. Conse termination... Linear algebra including the quantum mechanical background and the quently it is much better to select a CAS matrix with symbolic elements opera relations to crystal structures and magnetism. The Josephson junction and its application in and to add their own techniques. Indeed, tions such as inverse, determinant, pro microdevices will illustrate technical aspects a CAS is also a programming language duct... Solutions of equations and of along with lectures on high-current application. so that this approach is always more systems of equations, both linear and Level efficient than starting from scratch. algebraic, are now well mastered. To The school is intended for graduate students and Examples of running 10 of the most im handle power series is easy. Non com researchers with a general background in solid state physics, chemistry and/or materials science portant CAS are reproduced in [2] which mutative algebra is sometimes possible. but who have not necessarily taken any special thus gives a flavour of what program To differentiate is trivial. To evaluate in courses or have other experience in supercon ming with CAS is. definite integrals is a much more difficult ductivity. task but, nowadays symbolic integrators Workshop Algebraic Algorithms are very performant and more reliable Wednesday June 22 will be devoted to a The heart and power of CAS are their than tables. The Rish's theory of integra Workshop on Ceramic Superconductors with review lectures that summarize the evolu library of algebraic algorithms. In such a tion in finite terms [1] is a highlight of the tion and current understanding and with presen short survey it is not possible to specify research activity in the field. tations of the most recent experimental and theoretical advances. Poster stands for presenta each system. It is enough to know that Among the capabilities which are only tion of the participants own work will be avai the following capabilities are available partially implemented are the evaluation lable. within one or another CAS and are very of definite integrals for which the com Format general: Arithmetics on integers inclu putation of limits is still a difficult pro The school will consist of lectures in the morn ings and practical experiments in the afternoons. ding arbitrarily long ones, on rational and blem, the solution of ordinary and partial The experiments will cover synthesis, electron- (partially) algebraic numbers, p-adic and differential equations, which are difficult microscopy, neutron diffraction, solid reactions, transport properties, magnetic properties, and modular ones as well. Manipulation of mathematical problems anyhow, and tunnelling. Lecture notes will be provided during the school. EUROPEAN SYNCHROTRON RADIATION FACILITY Accommodation, price and grants The participation is limited to 40 graduate stu Grenoble, FRANCE dents. Accommodation at Hotel Søfryd, located 12 km north of Risø at the scenic Roskilde Bay. The ESRF is a state-of-the-art international synchrotron Bus transportation will be provided. The all radiation source to be built in Grenoble, France, to meet inclusive price for participation will be DKR the needs of the European scientific community for 3000.-. Grants will be available to assist Danish x-rays. We invite applications for a post of as well as some foreign graduate students. Head of organizing committee: Chartered Engineer for Insertion Device Group N. Hessel Andersen, Risø National Lab. with broad band experience in: — coil magnets and permanent magnets, Information, application and registration: — physical measurements, Secretariat, Physics Department — mechanical engineering, Risø National Laboratory P. O. Box 49, DK-4000 Roskilde, Denmark to be responsible for the construction of insertion devices. Phone + 45 2 37 12 12, Telex 43 116, Please send your curriculum vitae, date of availability and names of three referees Telefax + 45 2 36 06 09 by 30 March 1988 to: ESRF - Personnel Office - Ref/FP02.88 BP 220 - 38043 GRENOBLE CEDEX FRANCE Deadline for registration: 1 May, 1988
29 the handling of special functions. The of a CAS was always acknowledged in A Delegate Enquires capabilities available in present systems published papers (several hundreds of do not always reflect the state of know them) nowadays, they are often regard Dear Sir ledge in algebraic algorithm design. For ed as casual tools and no longer refe May I use your columns to write to the Indi instance new powerful methods to solve renced. vidual Ordinary Members of the EPS ? I am differential equations in closed forms a recently-elected delegate of the lOM's to have been found but are not yet fully New Frontiers the Council of the EPS. I believe I could do implemented. Despite the many successful applica this job more effectively (and this goes for other Council members too) if any IOM tions of CAS in many fields of science, with strong views about any aspect of Prototypical Applications they have obviously some drawbacks. how the EPS runs its affairs — conferen CAS have been applied to many diffe Apart from technical ones which are ces, journals, general structure (including rent fields of physics [1]. In high energy irrelevant for this brief survey, the major financial matters), or indeed anything else physics, a breakthrough application weakness is probably that, surprisingly — would write to me about it. I can then has been to perform the tedious algebra enough, they know very little mathema form a better idea of what EPS members' arising from the calculation of Feynman tics. To bypass this limitation, two diffe attitudes are on the issues they find impor diagrams in quantum electrodynamics rent research directions are possible and tant. The next Council meeting is on 24- and to determine the theoretical values investigated. The first is to concentrate 25 March 1988, so it would be especially of the anomalous magnetic moments of on enlarging the library of available alge helpful to have any comments before then. the electron and muon and of the Lamb- braic algorithms and procedures. As out Such comments need not, of course, be shift effect at high orders of the pertur lined previously, several breakthroughs restricted to remarks (favourable or unfa bative expansion. Also noticeable are have been achieved in the design of vourable) about current practices; ideas several diagram calculations in quantum algebraic algorithms and there is no for the future development of the EPS and chromodynamics and the two loops reason to believe that this trend will stop. views on broader issues would also be computation in the standard SU(2) x This ensures that existing CAS will re valuable. I should mention further that I am U(1) model. main alive and expand in the foreseeable not intending in the least to suggest that In celestial mechanics an outstanding future. An alternative approach is to con adequate channels of communication do result is the duplication in 1970 by sider that CAS are not only suitable to not exist already in the EPS; it is simply Deprit and co-workers, of the 1867 work manipulate formulae as they exclusively that I personally feel I could play a more of Delaunay on the calculation of the do today, but that they must also accom useful role as a delegate if I were better in orbit of the moon with corrections up to modate the concept of mathematical formed. order 9. To get an error free result took objects and their associated properties. Yours sincerely, The IBM project SCRATCHPAD, which is them only 20 hours on a mini-computer Derek N. Stacey instead of 20 years of Delaunay's life. In well under way, is the first CAS of this this example, the capability to imple generation. But, it is possible further to Clarendon Laboratory, ment a Poisson series representation in a enlarge the concepts upon which com Oxford 0X1 3PU, England CAS was the key to their success. The puter algebra is based by stating that its same type of computation is applicable ultimate goal is to represent and manipu to the orbit of satellites for instance. late mathematical knowledge. Then, ar Optics Elections In general relativity specialized tificial intelligence techniques become Elections will be held shortly of the 12 (CLAM, SHEEP,...) as well as general pur compulsory. Unfortunately, mathema members of the Board of the Optics pose systems such as MACSYMA are tics is a very complex domain and the Division. Candidates proposed by the used to solve the basic equations star techniques developed for the usual do Provisional Board are listed below, but ting with the definition of the covariant mains of artificial intelligence such as additional nominations are invited. metric tensor which determines the expert systems or robotics are not ela These should be supported by three length of a line element in space. Then, borated enough for its needs. This latter members of the Division coming from the Christoffel's symbol and Rieman's, trend [3] is thus a longer term research at least two countries and the candi Ricci's and Einstein's tensors are obtain track which aims to design completely dates agreement to stand should be ap ed and the 10 Einstein's field equations new types of mathematical systems but pended. solved. CAS are useful to point out the not to replace present CAS. J. Bescos, Optics Institute, Madrid possible equivalence of some metrics. Further information and references K. Biedermann, Royal Inst, of Technology, Many other applications have been re are available from the author. Stockholm ported on. J.J.M. Braat, Philips Research Lab., Since CAS are devoted to symbolic Eindhoven mathematics, they are obviously used J. Bulabois, Franche-Comté Univ., Besançon REFERENCES R. Dändliker, Inst, of Microtechniques, by mathematicians as well. It is worth Neuchâtel while to notice that although they en [1] Computer Algebra: Symbolic and Alge J.C. Dainty, Imperial College, London compass only a little part of the whole braic Computation 2nd ed., Eds. B. Buch- H.A. Ferwerda, University, Groningen knowledge of mathematicians, they can berger et al. (Springer-Verlag) 1983. J.-P. Huignard, Thomson-CSF, Orsay be used to solve, from scratch, some [2] Yun D.Y.Y. and Stoutmyer D.R., 'Symbolic O.J. Løkberg, Norway of the "open problems" in the relevant Mathematical Computation', Encyclopedia E.R. Pike, Kings College, London of Computer Science and Technology, Eds. A. Podmaniczky, Technical University, section of the American Mathematical Beizer, Holzman & Kent, 15 (1980) 235-310. Budapest Monthly. [3] Calmet J., "Intelligent Computer Algebra V. Russo, IROE, Florence A general feature of applications is Systems: Myth, Fancy or Reality?' To ap O.D.D. Soares, University, Porto that most of them involve simple and pear in Proc. of the Trends in Computer Alge Nominations should be sent to the straightforward but long and tedious bra, Conference. Bad Neuenahr, FRG. May EPS Secretariat by the end of March, algebra. While many years ago the use 1987, LNCS (Springer-Verlag) 1988. 1988. 30