
			History of CA Systems

					evans@eedsp.gatech.edu (Brian Evans)
					Georgia Institute of Technology


--> Introduction

        The roots of computer algebra apparently go all the way back
          to the 1950's.  In the 1960's, three primary groups launched into
          the field--  mathematicians, physicists, and AI researchers.  The
          first two groups created several special purpose environments.
          For example, mathematicians created Cayley for group theory and SAC
          for cylindrical decomposition, whereas physicists produced SHEEP for
          relativity and MAO for celestial mechanics.  REDUCE evolved from a
          program to do computational physics into a general computer algebra
          system.  Many AI researchers were laying the groundwork for MACSYMA
          during this time, as is evident from the work by the MathLab group
          at MIT and by The MITRE Corporation (who wrote the MATHLAB program).

        The 1960's saw other work in the field.  Bell Labs produced ALPAK
          (1964) and its successor ALTRAN (1966).  IBM produced FORMAC,
          Scratchpad, Scratchpad II, and Axiom.  (IBM recently assigned the
          copyright of Axiom, a distant relative of Scratchpad, to the
          Numerical Algorithms Group Ltd.)

        MACSYMA came together in 1970 and is the father of many popular
          computer algebra systems today.  Until the late 1970's, however,
          MACSYMA only ran on mainframe computers, although one mainframe
          (the DEC PDP-10) required a special purpose operating system.

        With the advent of personal computing in 1977, attempts were made
          to write a computer algebra program for these small computers.  The
          first commercial success for these small platforms was muMath (1979).
          In the 1980's, a host of programs for the PC were released,
          including a port of Maple to the IBM PC in the late 1980's.
          In 1982, MIT granted Symbolics Inc. the right to develop MACSYMA
          and by 1989 had ported it to personal computers (although there was
          a version running on 8 Mb 68000 Pixel Corp computer at Berkeley
          c. 1982).  Mathematica, another popular environment, was released
          in 1988 for Macintosh computers.  PowerMath, the first computer
          mathematics environment for the Macintosh, was on the market in 1985.

        Today, the computer algebra systems on the market typically run on
          several different platforms, including personal computers,
          workstations, and mainframes.


--> Summary

        Here are the dates (or approx. dates) when many of the key general
          purpose computer algebra systems were released (after Beta testing):
                           ^^^^^^^^        ^^^^

     Environment       Release     Derivative Works     e-mail ad. or phone no.
     ---------------   --------  ---------------------  -----------------------
     ALPAK             1964      ALTRAN                 (Bell Labs)
     ALTRAN            1968                             (Bell Labs)
     FORMULA (Algol).
     FORMAC                      FORMAC (PL/I)          (IBM)
     FORMAC (PL/I)                                      (IBM)
     MATHLAB (DECUS)   1968      MACSYMA                (DEC)
     CAMAL
    *REDUCE            1968                             reduce-netlib@rand.org
   **MACSYMA           1970      Symbolics Macsyma,     (See Below)
                                 VAXIMA, DOE-Macsyma,
                                 ALJABR, ParaMacs
    *SchoonShip        1971                             archive.umich.edu (FTP)
     muMath            1979      Derive
  ***VAXIMA            1980                             (312) 972-7250
 ****SMP               1982      Mathematica            NOT ON MARKET
     Symbolics MACSYMA 1983                       macsyma-service@symbolics.com
  ***DOE-Macsyma       1984      ALJABR                 gcook@llnl.gov
     Maple             1985                             wmsi@daisy.waterloo.edu
*****MathCAD           1985 (?)  Mathcad                1-800-MATHCAD
     Powermath         1985                             NOT ON MARKET
    *REDUCE/PC         1986                             reduce-netlib@rand.org
     Derive            1988                             (Soft Warehouse Inc.)
     Mathematica       1988                             info@wri.com
     Theorist          1988                     (415) 543-2252 (Prescience Corp)
     PARI              1988 (?)                         ftp to math.ucla.edu
     FORM              1989                             form@can.nl
     MACSYMA/PC        1989                       macsyma-service@symbolics.com
     ALJABR            1991                             aljabr@fpr.com
     Mathcad           1991                             1-800-MATHCAD
     SymbMath          1991                             chen@deakin.oz.au
     Axiom             1991                             (708) 971-2337
     ParaMacs          1991                             lph@paradigm.com
     SIMATH            1992                             marc@math.uni-sb.de
 
          * Some would claim that the original release of REDUCE (1968) shared
            some similarities with Ashmedai (never marketed) and SchoonSchip
            (1971) because they were begun at about the same time (1963).
            Originally, REDUCE was implemented in Lisp and SchoonSchip in CDC
            assembler, so their interfaces and command syntax are quite
            different.  Subsequently, they evolved in dramatically different
            directions.  Also, when mailing for information about REDUCE,
            entitle the e-mail message `send info-package`.  For questions
            requiring a human response, send e-mail to "reduce@rand.org".

         ** MACSYMA has certainly influenced most if not all of the symbolic
            mathematics environments released after 1971.  In fact, some
            of the implementors of MACSYMA have helped write other platforms
            (e.g., Barry Trager / Scratchpad, David Stoutemeyer / Derive,
            Bruce Char / Maple, and Steve Wolfram / Mathematica).  MIT has
            licensed a handful of companies to develop MACSYMA: Symbolics Inc.,
            Intermath, DOE, Fort Pond Research, and Paradigm Associates Inc.,
            the first being Symbolics Inc. in 1982.  This accounts for the
            many derivations.

        *** VAXIMA was ready for full release in the early 1980's but
            the exclusive licensing agreement between MIT and Symbolics Inc.
            in 1982 put the legality of this into question.  VAXIMA was
            officially released on a large scale in 1988, but full versions
            were available in universities as early as 1980.  A version of
            DOE-MACSYMA for the Vax under VMS was available in 1984.
            VAXIMA is available from the National Energy Software Center.

       **** Actually, its full release was held up after 1982 because of
            lengthy contractual negotiations between its developer (Dr.
            Wolfram), CalTech (place where it was developed), and Computer
            Mathematics Corp. (CMC, which merged later into Inference Corp.,
            was to market SMP).  The end results were that SMP was marketed 
            in 1983 and Dr. Wolfram left CalTech.

      ***** Originally, MathCAD was a numerical processing environment
            similar to MatLab but with a better user interface.  In 1991,
            MathCAD 3.0 emerged with a new spelling (Mathcad) and 
            symbolic manipulation capabilities (based on Maple).  MathCAD
            and Mathcad are trademarks of MathSoft, Inc.  Matlab is a
            trademark of The Math Works Inc.


--> About the Table

        Certainly, some key computer algebra environments (esp. early ones)
          were never "released" to the public.  For example, the 1965 version
          of Mathlab (a predecessor of Mathlab '68 and Macsyma) ran only on
          PDP-6/PDP-10 computers under MIT's ITS timesharing system.  Other
          early systems of note are Scratchpad (1966), PM/SAC/SAC-2, and
          ALDES.  Scratchpad is IBM's in-house software, with two indirect
          successors Scratchpad II (1975) and Axiom (1991).  Axiom is the
          only one that has been released.

        Similarly, it was difficult to place a date on the early systems
          that were "released".  For example, SMP and VAXIMA were caught
          up in legal disputes which delayed their large scale release.
          In these two cases, I have cited the years in which full releases
          were available to colleges and universities.

        In the above list, I have also omitted some of the special purpose
          systems like SHEEP (relativity), MAO (celestial mechanics),
          Cayley (group theory), and GAP (group theory).  SAC is considered
          special purpose because it concentrates on algebraic computation
          with polynomials especially cylindrical algebraic decompositions;
          however, developers in Austria are working on an interactive front
          end and a symbolic integrator for SAC-2.  I have also excluded
          TK!Solver (by Universal Technical Systems) from the list because
          it is a constraint-based numerical solver instead of a symbolic math
          environment.


--> References from N. Soiffer:

@INBOOK{ca-applications,
        AUTHOR = "J. A. van Hulzen and J. Calmet",
        TITLE = "{Computer Algebra Symbolic and Algebraic Computation}",
        CHAPTER = "Computer Algebra Applications",
        PUBLISHER = "Springer-Verlag",
        EDITION = "Second",
        YEAR = 1983
        }

@INPROCEEDINGS{fitch-applications,
        AUTHOR = "J. Fitch",
        TITLE = "{A Survey of Symbolic Computation in Physics}",
        BOOKTITLE = "{Symbolic and Algebraic Computation, Proc. EUROSAM '79}",
        SERIES = "Lecture Notes in Computer Science 72",
        YEAR = 1979,
        PAGES = "30--41",
        PUBLISHER = "Springer-Verlag",
        }

@INPROCEEDINGS{ng-applications,
        AUTHOR = "E. Ng",
        TITLE = "{Symbolic-Numeric Interface:  A Review}",
        BOOKTITLE = "{Symbolic and Algebraic Computation, Proc. EUROSAM '79}",
        SERIES = "Lecture Notes in Computer Science 72",
        YEAR = 1979,
        PAGES = "330-345",
        PUBLISHER = "Springer-Verlag",
        }


--> Reference from Glenn Thobe

  R.G. Tobey, R.J. Bogbrow, and  S.N. Zilles, {Proc. Amer. Soc. Info. Proc.
    Soc.}, _27_, 37. (1965),

  R.G. Tobey, {IBM Technical Report 00.1365} (1965), 

  J.E. Sammet, {Adv. Comp.}, _8_, 47 (1966), 

  W.H. Jeffreys, {Comm. Assoc. Comp. Mach.}, _14_, 538 (1971), 

  M.S. Davis, in B.D. Tapley and V. Szebehely, Eds., {Recent Advances in
    Dynamical Astronomy}, D. Reidel Publ., Dordrecht, Holland, 1973,
    p. 351 (1973),

  R. Pavelle, M. Rothstein, and J. Fitch, {Sci. Am.}, _245_, 136 (1981).


--> Other references:

  Caviness, B.  "Computer Algebra : Past and Future." EUROCAL'85,
    Buchberger, B.  (ed). Pages 1-18.

  Barton and Fitch in {Reports on Progress in Physics} vol. 35, no. 2.

  G. Gonnet and D. Gruntz.  "Algebraic Manipulation:  Systems" in the
    {Encyclopedia of Computer Science and Engineering}, 3rd ed.,
    Ralston, Reilly, and Dahlin eds., Van Nostrand Reinhold, 1991.

  D. Harper, C. Wooff, and D. Hodgkinson,  {A Guide to Computer Algebra
    Systems}, John Wiley and Sons.  ISBN 0-471-92910-7.

  D. R. Stoutemeyer, ``Crimes and misdemeanors in the computer algebra
    trade'', Notices of the AMS, Sept. 1991, pp. 701-785.


--> Contributors:

        Doug Andersen        hari@sugar.neosoft.com              (Axiom)
        Bill Beckner         beckner@math.utexas.com
        George Carrette      gjc@mitech.com                      (MACSYMA)
        Bruce Char           bchar@mcs.drexel.edu                (early sys)
        Elizabeth Chelkowska ryba@tasman.cc.utas.edu.au          (PowerMath)
        Brian Evans          evans@eedsp.gatech.edu
        Paul Evans           pdevans@sail.uwaterloo.ca           (Maple/Mathcad)
        Richard Fateman      fateman@peoplesparc.berkeley.edu    (MACSYMA etc.)
        Stuart Feldman       sif@lachesis.bellcore.com           (ALPAK/ALTRAN)
        Dan Frezza           dfrezza@vela.acs.oakland.edu        (Derive)
        Jeffrey Golden       jpg@allegheny.scrc.symbolics.com    (MACSYMA)
        Jeff Greif           jmg@inference.com                   (SMP)
        Leo Harten           lph@paradigm.com                    (ParaMacs)
        Tony Hearn           hearn@rand.org                      (REDUCE)
        Daniel Lazard        dl@posso.ibp.fr                     (Axiom)
        Michael Monagan      mbmonaga@daisy.waterloo.edu         (Maple)
        James O'Dell         jim@fpr.com                         (TK!Solver)
        Geert Oldenborgh     t19@nikhef.nl                       (SchoonSchip)
        Ue-Li Pen            upen@astro.princeton.edu
        Richard Pavelle      rp@xn.ll.mit.edu
        Andrzej Pindor       pindor@gpu.utcs.utoronto.ca         (MathCAD)
        Mohammed Rayes       rayes@mcs.kent.edu
        Richard Schroeppel   fermat!r@la.tis.com
        Joel Shapiro         shapiro@jashap.rutgers.edu
        Neil Soiffer         soiffer@wri.com                     (Mathematica)
        Glenn Thobe          getunx!thobe@uunet.uu.net           (early sys)
        Jos Vermaseren       t68@nikhef.nl                       (FORM)
        David Williams       dnw@williams.physics.lsa.umich.edu  (SchoonShip)
        Edward Wright        Edward_D_Wright@cup.portal.com


-- 
Brian L. Evans
Digital Signal Processing Laboratory
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA  30332-0250

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