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.