The SAC-1 system: An introduction and survey

SAC-1 is a program system for performing operations on multivariate polynomials and rational functions with infinite-precision coefficients. It is programmed, with the exception of a few simple primitives, in ASA Fortran. As a result the system is extremely accessible, portable, easy to learn, and indeed has been implemented at more than 20 institutions. The SAC-1 system's range of programmed capabilities is exceptionally broad, including, besides the usual operations, polynomial greatest common divisor and resultant calculation, polynomial factorization, exact polynomial real zero calculation, partial fraction decomposition, rational function integration, and solution of systems of linear equations with polynomial coefficients. SAC-1 is also outstanding in its computing time efficiency, which is achieved partially through the use of appropriate data structures, but primarily through the use of mathematically sophisticated and analyzed algorithms, which are briefly surveyed. The efficiency gains, frequently orders of magnitude, are such that many new applications are rendered feasible.

[1]  William S. Brown,et al.  The alpak system for nonnumerical algebra on a digital computer , 1963 .

[2]  G. Hardy.  The Integration of Functions of a Single Variable , Nature.

[3]  George E. Collins,et al.  The SAC-1 Polynomial Linear Algebra System , 1972 .

[4]  E. Bareiss.  Sylvester’s identity and multistep integer-preserving Gaussian elimination , 1968 .

[5]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[6]  D. Musser.  Algorithms for polynomial factorization. , 1971 .

[7]  G. E. Collins.  Algorithmic approaches to symbolic integration and simplification: summary of the panel session sponsored by SIGSAM at the FJCC in San Francisco, December 10, 1968 , 1969, SIGS.

[8]  A. B. BASSET,et al.  Modern Algebra , 1905, Nature.

[9]  G. E. Collins.  Computing time analyses for some arithmetic and algebraic algorithms , 1969 .

[10]  E. Berlekamp.  Factoring polynomials over finite fields , 1967 .

[11]  R. Risch,et al.  Symbolic integration of elementary functions , 1969 .

[12]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

[13]  George E. Collins.  The calculation of multivariate polynomial resultants , 1971, SYMSAC '71.

[14]  George E. Collins.  PM, a system for polynomial manipulation , 1966, CACM.

[15]  L. E. Heindel,et al.  The SAC-1 Modular Arithmetic System , 1972 .

[16]  E. Horowitz.  Algorithms for partial fraction decomposition and rational function integration , 1971, SYMSAC '71.

[17]  George E. Collins.  The SAC-1 List Processing System , 1971 .

[18]  H. D. Parker,et al.  The Bell Telephone Laboratories, Inc. , 1925, Journal of the A.I.E.E..

[19]  George E. Collins,et al.  A method for overlapping and erasure of lists , 1960, CACM.

[20]  Richard Y. Kain,et al.  Symbolic factoring of polynomials in several variables , 1966, CACM.

[21]  S. Gorn,et al.  FORTRAN vs. Basic FORTRAN: a programming language for informational processing on automatic data processing systems , 1964, CACM.

[22]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[23]  George E. Collins,et al.  The Calculation of Multivariate Polynomial Resultants , 1971, JACM.

[24]  Michael Terence Mc Clellan.  Algorithms for exact solution of systems of linear equations with polynomial coefficients , 1971 .

[25]  G. E. Collins.  Computing Multiplicative Inverses in GF(p) , 1969 .

[26]  Lee E. Heindel.  Algorithms for exact polynomial root calculation , 1970, SIGS.

[27]  J. D. Lipson.  Symbolic methods for the computer solution of linear equations with applications to flowgraphs , 1969 .

[28]  Peter Henrici.  A Subroutine for Computations with Rational Numbers , 1956, JACM.