Survey of formula manipulation

ABSTRACT

The field of formula manipulation is surveyed, with particular attention to the specific capabilities of differentiation, integration and the supporting capabilities of simplification, displays and input/output editing, and precision arithmetic. General systems—both batch and online—are described. Finally, some programs to solve specific applications are discussed.

REFERENCES

• AED00650: AED-0 user kit. MIT, Cambridge, Mass.
• ALPIR650: ALPIAR, R. The ALGAN-I manual (User's handbook for the algebraic FORTRAN programme). Eidg. Institut fur Reaktorforschung Wurenlingen Schweiz Eir-Bericht Nr. 76, March 1965, 69 pp.
• BALLW660: William E. Ball , Robert I. Berns, AUTOMAST: automatic mathematical analysis and symbolic translation, Communications of the ACM, v.9 n.8, p.626-633, Aug. 1966 [doi>10.1145/365758.365797]
• BARNM640: BARNETT, M. P., WACTLAR, H. D. A septuple precision rational fraction package. Tech. Note No. 22, Cooperative Computing Laboratory, MIT, Cambridge, Mass., 1964, pp. 1-7.
• BELLR650: R. E. Bellman , H. Kagiwada , R. E. Kalaba, Wengert's numerical method for partial derivatives, orbit determination and quasilinearization, Communications of the ACM, v.8 n.4, p.231-232, April 1965 [doi>10.1145/363831.364886]
• BENDB660: BENDER, B. MANIP-a computer system for algebra and analytic differentiation. Inst. for Environmental Research, Environmental Science Services Administration, March 1966, 26 pp.; Comm. ACM 9, 8 (Aug. 1966), 552, summary
• BERNM610: BERNICE, M. D., CALLENDER, E. D., SANFORD, J. R. ALGY-an algebraic manipulation program. 1961 Western Joint Comput. Conf., May 1961, pp. 389-392.
• BLACF630: BLACKWELL, F. W. ALMS-analytic language manipulation system. ACM 18th Nat. Conf., Aug. 1963, 6 pp. (summary)
• BLEIL650: BLEIWEISS, L., BOND, E., CUNDALL, P., HIRSCHKOP R. A time-shared algebraic desk calculator version of FOR- MAC. Tech. Rep. No. TR00.1415, IBM, Sys. Develop. Div., Poughkeepsie, N.Y., March 1966, 19 pp.
• BONDE640: E. Bond , M. Auslander , S. Grisoff , R. Kenney , M. Myszewski , J. Sammet , R. Tobey , S. Zilles, FORMAC an experimental formula manipulation Compiler, Proceedings of the 1964 19th ACM national conference, p.112.101-112.1019, January 1964 [doi>10.1145/800257.808916]
• BOYSS560: BOYS, S. F., CooK, G. B., REEVES, C. M., SHAVITT, I. Automatic fundamental calculations of molecular structure. Nature 178, 4544 (Dec. 1956), 1207-1209.
• BROWW630: BROWN, W. S. The ALPAK system for nonnumerical algebra on a digital computer-I. Polynomials in several variables and truncated power series with polynomial coefficients. Bell Sys. Tech. J. 42, 5 (Sept. 1963), 2081-2119.
• BROWW642: ----, HYDE, J. P., TAGUE, B. A. The ALPAK system for non-numerical algebra on a digital computer-II. Rational functions of several variables and truncated power series with rational-function coefficients. Bell Sys. Tech. J. 42, 2 (Mar. 1964), 785-804.
• BROWW643: ----, LEAGUS, D. C. VPRPAK-a computer programming system for variable precision rational arithmetic. Bell Telephone Laboratories, Murray Hill, N. J., 1964, 21 pp. (U)
• BROWW653: ---- A language and system for symbolic algebra on a digital computer. Bell Telephone Laboratories, Murray Hill, N. J., Oct. 1965, 20 pp. Proc. IBM Scientific Computing Syrup. on Computer-Aided Experimentation, Oct. 1965, in press; Comm. ACM 9, 8 (Aug. 1966), 548, summary.
• CLAPL630: CLAPP, L. C., KAIN, R. Y. A computer aid for symbolic mathematics. Proc. AFIPS 1963 Fall Joint Comput. Conf., Vol. 24, Nov. 1963, pp. 509-517.
• COLLG652: COLLINS, G.E. REFCO III, a reference count list processing system for the IBM 7094. Res. Rep. RC-1436, IBM Watson Research Center, Yorktown Heights, N. Y., May 1965, 33 pp.
• COLLG663: G. E. Collins, PM, a system for polynomial manipulation, Communications of the ACM, v.9 n.8, p.578-589, Aug. 1966 [doi>10.1145/365758.365770]
• CORBF630: F. J. Corbató, The Compatible Time-Sharing System: A Programmer's Guide, The MIT Press, 1963
• CUTHE650: CUTHILL, E., VOIGT, S., ULLOM, S. Use of computers in the solution of boundary value and initial value problems. Annual Prog. Rep. SR011-01-01 TASK 0401, AML Problem 821-911, David Taylor Model Basin, Dept. of the Navy, Carderock, Md., June 1965, 89pp.
• CUTHE661: ---- . A FORMAC program for the solution of linear boundary and initial value problems. David Taylor Model Basin, Dept. of the Navy, Carderock, Md., March 1966, 44pp.; Comm. ACM 9, 8 (Aug. 1966), 550 abstract only.
• DANBJ660: DANBY, J. M., DEPRIT, A., ROM, A.R. The symbolic manipulation of Poisson series. Yale University Observatory, Boeing Scientific Research Laboratories, March 1966, 33pp.; Comm. ACM 9, 8 (Aug. 1966), 550, abstract only.
• DARTM590: Symbolic work on high speed computers. Proj. Rep. No. 4, Dartmouth Mathematics Project, Dartmouth College, Hanover, N. H., June 1959, 69pp.
• EDWAD590: EDWARDS, D. Symbolic circuit analysis with the 704 electronic computer. B.S. Thesis, Electrical Eng. Dept., MIT, Cambridge, Mass., 1959, 48pp.
• ENGEC650: ENGELMAN, C. MATI-ILAB-a program for on-line machine assistance in symbolic computations. Proe. AFIPS 1965 Fall Joint Comput. Conf., Vol. 27, Pt. I, Nov. 1965, pp. 413-422.
• ENGEM651: ENGELI, M. Formal manipulation of algebraic expressions within an algorithmic language. Federal Institute of Technology, Zurich, Switzerland, May 1965, 23pp. (U)
• FELDJ660: Jerome A. Feldman, A formal semantics for computer languages and its application in a compiler-compiler, Communications of the ACM, v.9 n.1, p.3-9, Jan. 1966 [doi>10.1145/365153.365156]
• FLETJ660: FLETCHER, J. GRAD assistant-aprogram for symbolic algebraic manipulation and differentiation. (University of California, Lawrence Radiation Laboratory, March 1966, 23pp. Comm. ACM 9, 8 (Aug. 1966), 552, summary.
• FLETR630: FLETCHER, R., REEVES, C.M. A mechanization of algebraic differentiation and the automatic generation of formulae for molecular integrals of Gaussian orbitals. Comput. J . 6, 3 (1963), 287-292.
• FORMI650: FORMAC (operating and user's preliminaryreference manual). No. 7090 R2 IBM 0016 IBM Program Inf. Dept., Hawthorne, N. Y., Aug. 1965, 230pp.
• FOWLM640: FOWLER, M. E., MACMASTERS, J. A. A FORTRAN program for polynomial multiplication. Rep. No. TR-24.012, IBM, Data Processing Div., Kingston, N. Y., March 1964, 10pp.
• GAWLH630: H. J. Gawlik, MIRFAC: a compiler based on standard mathematical notation and plain English, Communications of the ACM, v.6 n.9, p.545-547, Sept. 1963 [doi>10.1145/367593.367618]
• GERAJ650: J. M. Gerard , M. P. Barnett , I. G. Izsak, Mechanization of tedious algebra: the Newcomb operators of planetary theory, Communications of the ACM, v.8 n.1, p.27-32, Jan. 1965 [doi>10.1145/363707.363710]
• GOLDB590: GOLDBERG, S. H. Solution of an electrical network using a digital computer. M.S. thesis, Electrical Engineering Dep't., MIT, Cambridge, Mass., 1959, 203pp.
• GOTLC660: GOTLIEB, C. C. NOVAK, R. J. ALGEM-an algebraic manipulator. University of Toronto, March 1966, 10 pp. Comm. ACM 9, 8 (Aug. 1966), 549, abstract only.
• HANSJ621: James W. Hanson , Jane Shearin Caviness , Camilla Joseph, Analytic differentiation by computer, Communications of the ACM, v.5 n.6, p.349-355, June 1962 [doi>10.1145/367766.368195]
• HARTK640: Kenneth Hartt, Some Analytical Procedures for Computers and Their Applications to a Class of Multidimensional Integrals, Journal of the ACM (JACM), v.11 n.4, p.416-421, Oct. 1964 [doi>10.1145/321239.321242]
• HARTT610: HART, T. Simplify. Memo 27, Artificial Intelligence Proj. MIT, Cambridge, Mass. 1961. (U)
• HEARA660: A. C. Hearn, Computation of algebraic properties of elementary particle reactions using a digital computer, Communications of the ACM, v.9 n.8, p.573-577, Aug. 1966 [doi>10.1145/365758.365766]
• HELLL590: HELLERMAN, L. A computer analytic method for solving differential equations. 1959 Proe. Eastern Joint Comput. Conf., Boston, Mass., Dec. 1959, pp. 238-243.
• HENRP560: Peter Henrici, A Subroutine for Computations with Rational Numbers, Journal of the ACM (JACM), v.3 n.1, p.6-9, Jan. 1956 [doi>10.1145/320815.320818]
• HERGP590: HERGET, P., MUSEN, P. The calculation of literal expansions. Astron. J. 64, 1266, (Feb. 1959), 11-20.
• HYDEJ640: HYDE, J .P . The ALPAK system for non-numerical algebra on a digital computer-III. Systems of linear equations and a class of side relations. Bell Sys. Tech. J. 43, 4, pt. 2 (July 1964), 1547-1562.
• IZSAI640: IzsAK, I. G., GERARD, J. M., EFIMBA, R., BARNETT, M. P. Construction of Newcomb operators on a digital computer. Special Rep. No. 140, Smithsonian Astrophysical Observatory, Cambridge, Mass. Jan. 1964, 103pp.
• JIRAD640: JIRAUCH, D. H., WESTERWICK, R. A. A use of digital computers for algebraic computation. Rep. No. RTD-TDR-63-4200, General Dynamics/Astronautics, San Diego, Calif., March 1964, 129pp.
• JORDD660: Dale E. Jordan , Lewis C. Clapp , Richard Y. Kain, Symbolic factoring of polynomials in several variables, Communications of the ACM, v.9 n.8, p.638-643, Aug. 1966 [doi>10.1145/365758.365809]
• KAHRH530: KAHRIMANIAN, H. G. Analytical differentiation by a digital computer. M.A. Thesis, Temple U., Philadelphia, Pa. May 1953, 48pp.
• KELLK630: KELLEY, K. R., WACTLAR, H.D. An integer and rational fraction polynomial manipulation package and a full word binary integer coefficient polymonial manipulation package. Tech. Note No. 21, Cooperative Computing Lab., MIT, Cambridge, Mass., 1963, 14pp.
• KITSA650: KITSELMAN, A.L. CALPAK and CALTRAN. La Jolla, Calif., 1965. (U)
• KLERM640: Melvin Klerer , Jack May, An experiment in a user-oriented computer system, Communications of the ACM, v.7 n.5, p.290-294, May 1964 [doi>10.1145/364099.364266]
• KORSK661: KORSVOLD, K. An on line program for non-numerical algebra. Norwegian Defence Research Establishment, March 1966, 15pp. Comm. ACM 9, 8 (Aug. 1966), 553, abstract only.
• LAPIA651: Arnold Lapidus , Max Goldstein, Some experiments in algebraic manipulation by computer, Communications of the ACM, v.8 n.8, p.501-508, Aug. 1965 [doi>10.1145/365474.365515]
• LEDED650: LEDERMAN, D. Application of FORMAC to generating functions. Rep. 9851-331, TRW Space Technology Labs., June 1965, 6 pp.
• LEDED651: ---. An application of FORMAC to partial differentiation. Rep. 9851-330, TRW Space Technology Labs., June 1965, 5pp.
• MALIK590: K. Maling, The LISP Differentiation Demonstration Program, Massachusetts Institute of Technology, Cambridge, MA, 1959
• MARTW640: William Martin , Tim Hart, Hash-Coding Functions of a Complex Variable, Massachusetts Institute of Technology, Cambridge, MA, 1964
• MARTW651: ----. PDP-6 LISP input-output for the display. Memo. MAC-M-242, Artificial Intelligence Proj., Project MAC, MIT, Cambridge, Mass. June 1965, 6pp. (U)
• MARTW652: William Martin, Syntax and Display of Mathematical Expressions, Massachusetts Institute of Technology, Cambridge, MA, 1965
• MCCAJ600: McCARTHY, J., BRAYTON, R., EDWARDS, D., Fox, P., HODES, L., LUCKHAM, D., MALING, K., PARK, D., RUSSELL, S., LISP I programmer's manual. Comput. Center and Res. Lab. Electronics, MIT, Cambridge, Mass., March 1960, 156pp.
• MINSM630: Marvin Minsky, MATHSCOPE Part I: A Proposal for a Mathematical Manipulation- Display System, Massachusetts Institute of Technology, Cambridge, MA, 1963
• MOSEJ660: Joel Moses, Solutions of systems of polynomial equations by elimination, Communications of the ACM, v.9 n.8, p.634-637, Aug. 1966 [doi>10.1145/365758.365802]
• NOLAJ530: NOLAN, J. Analytical differentiation on a digital computer. M.A. Thesis, Math. Dept., MIT, Cambridge, Mass., May 1953, 71pp.
• NOVAR650: NovAx, R .J . Algorithms for algebraic manipulations. Proj. Rep. Inst. Comput. Sci., U. of Toronto, Toronto, Canada, Sept. 1965, 17pp. (U)
• PERLA640: A. J. Perlis , Renato Iturriaga, An extension to ALGOL for manipulating formulae, Communications of the ACM, v.7 n.2, p.127-130, Feb. 1964 [doi>10.1145/363921.363943]
• PERLA652: ----,----., STANDISH, T.A. A preliminary sketch of Formula ALGOL. Carnegie Inst. of Technology, Pittsburgh, Pa., April 1965, 52pp.
• PERLA663: ----,----,----, A definition of Formula ALGOL. Carnegie Institute of Technology, March 1966, 52pp. Comm. ACM 9, 8 (Aug. 1966), 549, abstract only.
• REYNJ650: REYNOLDS, J .C . COGENT programming manual, ANL-7022, Argonne Natl. Lab., Argonne, Ill. March 1965, 116pp.
• RICHR571: RICHTMYER, R.D. Detached-shock calculations by power series, I. NYO-7973, Inst. Math. Sci., New York U., New York, Oct. 1957, 49pp.
• RICHR600: ----. Power series solution, by machine, of a nonlinear problem in two-dimensional fluid flow. Ann. New York Acad. Sci. 86, Art. 3 (May 1960), 828-843.
• RIETR650: "VAN DE RIET, R. P. Algebraic operations in ALGOL 60 (a second order problem). TW 96, Math. Center, Amsterdam, Netherlands, March 1965, 40pp.
• RIETR651: ----. Algebraic operations in ALGOL 60, the Cauchy Problem I. TW 97, Mathematical Center, Amsterdam, Netherlands, 1965, 38pp.
• ROMAR610: Arnold R. M. Rom, Manipulation of algebraic expressions, Communications of the ACM, v.4 n.9, p.396-398, Sept. 1961 [doi>10.1145/366696.366769]
• ROSSD640: Ross, D. AED JR.-an experimental language processor. Rep. ESL-TM-211, Electron. Sys. Lab., MIT, Cambridge, Mass., Sept. 1964.
• SAMMJ640: SAMMET, J. E., BOND, E. Introduction to FOR- MAC. IEEE Trans. Electron. Computers EC-IS, 4, (Aug. 1964), 386-394.
• SAMMJ651: ----. An overall view of FORMAC. Tech. Rep. No. TR00.1367, IBM Sys. Develop. Div., Poughkeepsie, N. Y., Nov. 1965, 19pp.
• SAMMJ652: ----. Formula manipulation by computer. Advances in Computers, Vol. 8. McGraw Hill, New York, in press; Preliminary version as Tech. Rep. No. TR00.1363 IBM Sys. Develop. Div., Poughkeepsie, N. Y., Nov. 1965, 44pp.
• SAMMJ664: Jean E. Sammet, Survey of formula manipulation, Communications of the ACM, v.9 n.8, p.555-569, Aug. 1966 [doi>10.1145/365758.365762]
• SAMMJ665: ----. An annotated descriptor based bibliography on the use of computers for non-numerical mathematics. Computing Reviews 7, 4 (July-Aug. 1966), B1-B31.
• SCHOH650: SCHORR, H. Analytic differentiation using a syntaxdirected compiler. Comput. J. 7, 4 (Jan. 1965), 290-298.
• SCONP650: SCONZO, P., LESCHAeK, A. R., TOBEY, R.G. Symbolic computation of f and g series by computer. Tech. Rep. No. TR00.1262, IBM, Data Sys. Div. Poughkeepsie, N. Y., March 1965, 30pp.
• SHAPM640: SHAPIRO, M. NYU DIF-Analytic differentiation FORTRAN and FAP coded 7090. Writeup No. 0-640227, AEC Computing and Appl. Math. Center, New York U., New York, Feb. 1964, 3pp. (U)
• SHURV610: SHURYGIN, V. A., YANENXO, N.N. On the realization of algebraic-differential algorithms on electronic computers. Probl. Kiberneliki No. 6 (1961).
• SLAGJ631: James R. Slagle, A Heuristic Program that Solves Symbolic Integration Problems in Freshman Calculus, Journal of the ACM (JACM), v.10 n.4, p.507-520, Oct. 1963 [doi>10.1145/321186.321193]
• SPIEK610: Kurt Spielberg, Representation of Power Series in Terms of Polynomials, Rational Approximations and Continued Fractions, Journal of the ACM (JACM), v.8 n.4, p.613-627, Oct. 1961 [doi>10.1145/321088.321099]
• TAKAL630: TAKACS, L. A single-server queue with feedback. Bell Sys. Tech. J . 42, 2 (March 1963), 505-519.
• TOBER650: TOBEY, R. G., BORROW, R. J., ZILLES, S. Automatic simplification in FORMAC. Proc. AFIPS 1965 Fall Joint Comput. Conf., Vol. 27, Ft. 1, Nov. 1965, pp. 37-52.
• TOBER651: ----. Eliminating monotonous mathematics with FORMAC. Tech. Rep. No. TR00.1365, IBM, Sys. Develop. Div., Poughkeepsie, N. Y., Nov. 1965, 32pp. Comm. ACM to be published.
• TOBER662: R. G. Tobey, Experience with FORMAC algorithm design, Communications of the ACM, v.9 n.8, p.589-597, Aug. 1966 [doi>10.1145/365758.365773]
• WACTH640: H. D. Wactlar , M. P. Barnett, Mechanization of tedious algebra—the e coefficients of theoretical chemistry, Communications of the ACM, v.7 n.12, p.704-710, Dec. 1964 [doi>10.1145/355588.365113]
• WALTJ650: WALTON, J .J . Formula manipulation code for the calculation of Christoffel symbols. UCRL-14177, U. of California, Lawrence Rad. Lab., Livermore, Calif., May 1965, 40pp.
• WELLM610: Mark B. Wells, MADCAP: a scientific compiler for a displayed formula textbook language, Communications of the ACM, v.4 n.1, p.31-36, Jan. 1961 [doi>10.1145/366062.366080]
• WENGR640: R. E. Wengert, A simple automatic derivative evaluation program, Communications of the ACM, v.7 n.8, p.463-464, Aug. 1964 [doi>10.1145/355586.364791]
• WILKR640: R. D. Wilkins, Investigation of a new analytical method for numerical derivative evaluation, Communications of the ACM, v.7 n.8, p.465-471, Aug. 1964 [doi>10.1145/355586.364792]
• WILLL620: Leland H. Williams, Algebra of Polynomials in Several Variables for a Digital Computer, Journal of the ACM (JACM), v.9 n.1, p.29-40, Jan. 1962 [doi>10.1145/321105.321109]
• WOLMB650: WOLMAN, B.L. Operators for manipulating verbal and graphical language structures. M.S. thesis, Electrical Eng. Dep't., MIT, Cambridge, Mass., June 1965, ll0pp.
• WOLMB661: ----. Operators for manipulating language structures. Electronic Systems Laboratory, MIT, March 1966, 25pp. Comm. ACM 9, 8 (Aug. 1966), 553, abstract only.
• WOOLD630: WOOLDRIDGE, D. An algebraic simplify program in LISP. Memo. No. 11, Artificial Intelligence Proj., Stanford U., Stanford, Calif. Dec. 1963, 26pp. (U)