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Solutions of systems of polynomial equations by elimination

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Joel MosesAuthors Info & Claims

Communications of the ACM, Volume 9, Issue 8

Pages 634 - 637

https://doi.org/10.1145/365758.365802

Published: 01 August 1966 Publication History

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Abstract

The elimination procedure as described by Williams has been coded in LISP and FORMAC and used in solving systems of polynomial equations. It is found that the method is very effective in the case of small systems, where it yields all solutions without the need for initial estimates. The method, by itself, appears inappropriate, however, in the solution of large systems of equations due to the explosive growth in the intermediate equations and the hazards which arise when the coefficients are truncated. A comparison is made with difficulties found in other problems in non-numerical mathematics such as symbolic integration and simplification.

References

[1]

WILLIAMS, L .H. Algebra of polynomials in several variables for a digital computer. J. ACM 9 (Jan. 1962), 29-40.

Digital Library

Google Scholar

[2]

LISP 1.5 programmer's manual. Res. Lab. of Electronics, MIT, Cambridge, Mass., 1962.

Digital Library

Google Scholar

[3]

BOND, E. R. ET AL., FORMAC-An experimental formula manipulation compiler. Proc. ACM 19th Nat. Conf., Philadelphia, 1964, Paper K2.1-1.

Digital Library

Google Scholar

[4]

MULLER, D. A method for solving algebraic equations using a digital computer. Math. Tables Other Aids Comput. 10 (1956), 208-215.

Crossref

Google Scholar

[5]

BAREISS, E.H. Resultant procedure and the mechanization of the Graeffe process. J. ACM 7 (Oct. 1960), 346-386.

Digital Library

Google Scholar

[6]

RICHARDSON, D. Doc. Thesis, U. of Bristol, Bristol, England.

Google Scholar

[7]

SLAGLE, J. R. A heuristic program that solves symbolic integration problems. Doc. Thesis, MIT, Cambridge, Mass., 1961.

Google Scholar

[8]

MARTIN, W.A. Hash-coding functions of a complex variable. Artificial Intelligence Project Memo 70, MIT, Cambridge, Mass., 1964.

Digital Library

Google Scholar

[9]

COLLINS, G. PM, a system for polynomial manipulation. J. ACM 9 (Aug. 1966), 578-589.

Digital Library

Google Scholar

[10]

KUTTA., W. Beitrag zum naherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46 (1901), 435-453.

Google Scholar

[11]

ENGLEMAN, C. Mathlab: a program for on-line machine assistance in symbolic computation. Rept. MTP-18, MITRE Corp., Bedford, Mass., 1965.

Google Scholar

[12]

HARDY, G .H. The Integration of Funclions of a Single Variable. 2nd. ed., Cambridge U. Press, Cambridge, England, 1916.

Google Scholar

[13]

BROWN, W. S., HYDE, J. P., AND TAGUE, B.A. The ALPAK system for nonnumerical algebra on a digital computer-II. Bell Sys. Tech. J. 43 (March 1964), 785-804.

Crossref

Google Scholar

[14]

MARTIN, W. A. Syntax and display for mathematical expressions. Artificial Intelligence Project Memo 85, MIT, Cambridge, Mass., 1965.

Digital Library

Google Scholar

Cited By

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Pal J(2015)Compensator Design for Dynamic Optimisation of Multivariable Systems with Prescribed Pole LocationsIETE Journal of Research10.1080/03772063.1985.1143649831:3(81-84)Online publication date: 2-Jun-2015
https://doi.org/10.1080/03772063.1985.11436498
Schwarz F(2005)A REDUCE package for determining first integrals of autonomous systems of ordinary differential equationsEUROCAL '8510.1007/3-540-15984-3_338(601-602)Online publication date: 8-Jun-2005
https://doi.org/10.1007/3-540-15984-3_338
Hulzen J(2005)A note on methods for solving systems of polynomial equations with floating point coefficientsSymbolic and Algebraic Computation10.1007/3-540-09519-5_86(346-357)Online publication date: 24-May-2005
https://doi.org/10.1007/3-540-09519-5_86
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Solutions of systems of polynomial equations by elimination
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis

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Published In

Communications of the ACM Volume 9, Issue 8

Aug. 1966

114 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/365758

Editor:
Gerard Salton
Cornell Univ., Ithaca, NY

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Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 August 1966

Published in CACM Volume 9, Issue 8

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Pal J(2015)Compensator Design for Dynamic Optimisation of Multivariable Systems with Prescribed Pole LocationsIETE Journal of Research10.1080/03772063.1985.1143649831:3(81-84)Online publication date: 2-Jun-2015
https://doi.org/10.1080/03772063.1985.11436498
Schwarz F(2005)A REDUCE package for determining first integrals of autonomous systems of ordinary differential equationsEUROCAL '8510.1007/3-540-15984-3_338(601-602)Online publication date: 8-Jun-2005
https://doi.org/10.1007/3-540-15984-3_338
Hulzen J(2005)A note on methods for solving systems of polynomial equations with floating point coefficientsSymbolic and Algebraic Computation10.1007/3-540-09519-5_86(346-357)Online publication date: 24-May-2005
https://doi.org/10.1007/3-540-09519-5_86
Gruntz DGander WHřebíček J(2004)Symbolic Computation of Explicit Runge-Kutta FormulasSolving Problems in Scientific Computing Using Maple and MATLAB®10.1007/978-3-642-18873-2_19(281-297)Online publication date: 2004
https://doi.org/10.1007/978-3-642-18873-2_19
Barton DFitch J(2002)Applications of algebraic manipulation programs in physicsReports on Progress in Physics10.1088/0034-4885/35/1/30535:1(235-314)Online publication date: 5-Aug-2002
https://doi.org/10.1088/0034-4885/35/1/305
Gruntz DGander WHřebíček J(1997)Symbolic Computation of Explicit Runge-Kutta FormulasSolving Problems in Scientific Computing Using Maple and MATLAB®10.1007/978-3-642-97953-8_19(281-296)Online publication date: 1997
https://doi.org/10.1007/978-3-642-97953-8_19
Gruntz DGander WHřebíček J(1995)Symbolic Computation of Explicit Runge-Kutta FormulasSolving Problems in Scientific Computing Using Maple and MATLAB®10.1007/978-3-642-97619-3_19(267-283)Online publication date: 1995
https://doi.org/10.1007/978-3-642-97619-3_19
Shackell J(1993)Zero-equivalence in function fields defined by algebraic differential equationsTransactions of the American Mathematical Society10.1090/S0002-9947-1993-1088022-2336:1(151-171)Online publication date: 1993
https://doi.org/10.1090/S0002-9947-1993-1088022-2
Gruntz DGander WHřebíček J(1993)Symbolic Computation of Explicit Runge-Kutta FormulasSolving Problems in Scientific Computing Using Maple and Matlab ®10.1007/978-3-642-97533-2_19(251-266)Online publication date: 1993
https://doi.org/10.1007/978-3-642-97533-2_19
Ganesan SShieh LMehio M(1991)Sequential design of linear quadratic state regulators with prescribed eigenvalues and specified relative stabilityComputers & Mathematics with Applications10.1016/0898-1221(91)90241-U21:4(1-10)Online publication date: 1991
https://doi.org/10.1016/0898-1221(91)90241-U
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Abstract

References

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Recurrence equations and their classical orthogonal polynomial solutions

Chebyshev polynomial solutions of systems of linear integral equations

Polynomial solutions of differential-difference equations

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