B. Buchberger. Grobner Bases in Mathematica: Enthusiasm and Frustration. In P.W. Gaffney, E.N. Houstis, editors, Proc. IFIP TC2/WG2.5 Working Conference on Programming Environments for High-Level Scientific Problem Solving, Karlsruhe, Germany, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm, some more advanced than others. For example, the basic algorithm implemented in Mathematica (version 2.0 and later) is poor in the sense that ffl it only works for lexicographic ordering and for rational coefficients; ffl apparently works fast enough only for small problems. cf. [10]) A little bit more useful but still not too impressive is the grobner package in Maple (release 4.0 and later) ffl it allows both pure lexicographic ordering and total degree inverse lexicographic ordering; ffl coefficients may be polynomials with rational coefficients; ffl some algorithmic ....

B. Buchberger. Grobner Bases in Mathematica: Enthusiasm and Frustration. In P.W. Gaffney, E.N. Houstis, editors, Proc. IFIP TC2/WG2.5 Working Conference on Programming Environments for High-Level Scientific Problem Solving, Karlsruhe, Germany, 1991.

....in the arbitrary precision library discussed in [64] are 50 to 100 times slower than floating point operations, even when no additional precision is required. Certain application programs that use computer algebra systems are 1,000 to 3,000 times slower than equivalent programs written in C [16]. To overcome the speed limitation of existing software tools, direct hardware support is required. To improve the accuracy and performance of vector and matrix operations, coprocessors which support exact dot products have been designed [43] 29] 39] To facilitate the use of interval ....

....finding roots of polynomials, solving systems of equations, and performing vector and matrix operations [31] Although computer algebra systems provide exact or extremely accurate solutions, they are often too slow for numerically intensive computations. According to a study done in [16], Maple programs are more than 1,000 times slower than equivalent programs written in C, and Mathematica programs are more than 3,000 times slower. For several applications, the computation time required by computer algebra systems is prohibitive. 8 Chapter 1 2.5 Problem Solving Environments ....

B. Buchberger. Grobner Bases in MATHEMATICA: Enthusiasm and Frustration. In Programming Environments for High-level Scientific Problem Solving, pages 80--91, 1991.

....functioning is hidden for proprietary reasons. Even if you are willing to pay the substantial penalty in performance, you cannot tell how much functionality must be recreated. An attempt to make extensive use of Mathematica rules in defining a system using abstract data types is reported by Buchberger (1991) who found it resulted in frustratingly slow computation. Two other shortcomings in Contexts are worth noting: Mentioning a name before reading in the package defining it shields the name in the package from the global environment, effectively disabling the package. Debugging, never easy in ....

....any computer algebra system. Perhaps the major factor is that most systems, including Mathematica, are interpreter based. Consequently, a simple program that can just as easily be expressed in a compiled language (e.g. C) runs orders of magnitude slower than if programmed in C. see, for example, Buchberger (1991), where basic list structure operations are shown to execute 3000 8000 times slower) One trick is to write such programs in C, and call them from the interpreter. There are other factors: ffl Generality. Even with a compiler (introduced in version 2.0) it is unreasonable to expect ....

Buchberger, B. (1991). Grobner bases in Mathematica: Enthusiasm and Frustration. RISC-LINZ Report 3-3, J. Kepler Univ., A-4040 Linz, Austria.

....one can choose a particular variant of the Grobner bases algorithm. The global variables and their possible values are listed in Table 1, the exact meaning of the possible values of the global variables is explained in Section 2.5. An evaluation of the performance of the package is given in [Buchberger 91] Calling the function GB with one parameter, a set of polynomials, causes the system to call the version of the Grobner bases algorithm that is stored in GroebnerBasisPN. The result is the reduced Grobner basis with respect to the ordering GreaterEL. The normal form algorithm used during ....

....Floating Point numbers with discrete zero (tag: FP) Complex Numbers (built in Mathematica Complex Numbers, tag: Complex) Finite Fields (tag: FF) Rational Functions (built in Mathematica Rational Functions; tag: MRF) # Nests, Lists 2. 5 Implementation Details As already mentioned in [Buchberger 91] and Section 2.1, Mathematica lists do not have the complexity behavior of real lists because of their internal representation by arrays. For example, computing Rest[ l] comparable to cdr in LISP) Prepend[ l, a] comparable to cons in LISP) where l is an arbitrary list and a is any ....

[Article contains additional citation context not shown here]

B.Buchberger. Grobner bases in Mathematica: Enthusiasm and Frustration. Technical Report, RISC-Linz, University of Linz, Austria, 1991, submitted to the IFIP Workshop on Scientific Computation, Karlsruhe, September 1991.

.... of the first author s diploma thesis, Windsteiger, 1992 ] and is modeled after earlier implementations by the second author in various symbolic computation languages, notably in Mathematica, see [ Buchberger, 1990 ] The design reasons for using C in the present implementation are described in [ Buchberger, 1991 ] Summarizing, a good implementation of an algebraic algorithm should combine both speed and polymorphism . By using C and a systematic renaming mechanism, which is based on a proposal by the second author and is described in the present document, we believe that, in fact, we can achieve C ....

B. Buchberger. Grobner Bases in Mathematica: Enthusiasm and Frustration. In P.W. Gaffney and E.N. Houstis, editors, Proc. IFIP TC2/WG2.5 Working Conference on Programming Environments for High-Level Scientific Problem Solving, Karlsruhe, Germany, 1991.

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Bruno Buchberger. Grobner Bases in Mathematica: Enthusiasm and Frustation. In Proceedings IFIP TC2/WG2.5 Working Conference on Programming Environments for High-Level Scientific Problem Solving, Karlsruhe, Germany, September 23--27, 1991. Also Technical Report 91-11, RISC-Linz, Johannes Kepler University, Linz, Austria, 1991.

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