By B. Buchberger (auth.), Dr. Alfonso Miola, Dr. Marco Temperini (eds.)
New methodological points with regards to layout and implementation of symbolic computation structures are thought of during this quantity aiming at integrating such features right into a homogeneous software program atmosphere for clinical computation. The proposed method relies on a mixture of other recommendations: algebraic specification via modular method and of completion algorithms, approximated and precise algebraic computing equipment, object-oriented programming paradigm, computerized theorem proving via tools ? l. a. Hilbert and techniques of usual deduction. specifically the proposed remedy of mathematical gadgets, through suggestions for approach abstraction, buildings category, and particular illustration, the programming technique which helps the layout and implementation matters, and reasoning features supported by means of the complete framework are defined.
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Additional info for Advances in the Design of Symbolic Computation Systems
5 Working Conference on Programming Environments for High Level Scientific Problem Solving, Karlsruhe, Germany, Sept. 23-27, 1991. NorthHolland, Amsterdam, pp. 217-226. Lipson, J. D. (1976): Newton's method: a great algebraic algorithm. In: Proceedings ACM Symposium on Symbolic and Algebraic Computation, Symsac '76. Association for Computing Machinery, New York, pp. 260-270. , Miola, A. (1986): On the integration on numeric and algebraic computations. , Clausen, M. ): Applied algebra, algebraic algorithms and error-correcting codes.
ArCf») -= 0 mod If , ajCf) = a·J - mod I ' i = 1, ... , n . 1 , ... ,r. J= Proof. The proof is given by induction on t. See Lauer (1983). o The Hensel lemma is a specialization of that theorem. The lemma is specialized by assuming n = 1, r = 2, R = Z[x] and p = I, p being a prime number. Thus, assume AI, A2, C E Z[x]. Rename Al and A2 as G and H, respectively, 41 Uniform representation of mathematical objects and consider C = F. Also suppose that G·H== Fmodp, and that G and F are relatively prime mod p.
Following a procedure like in the previous example the computation of F (x) = x 2 - 5 is briefly shown: input: F(x); G = 0; H = 0; P = 11; Go = 0; Ho = 0; Xl = 7; r = 3; C. Limongelli and M. Temperini 50 output: = ~xo· pO + ~X1 = 8059. y'5 in the p-adic arithmetic. It is simple to verify that, in this arithmetic, (7606, 0) x (7606, 0) = (5000,0), which is the representation of the integer number 5. The importance of dealing with exact results is well known. y'5 represented by (7606,0). (1000,0) F3=(7606,0)· [( 0000,0) + (7606,0»)3 _ (0000,0) - (7606, 0»)3J (2000,0) (2000,0) F = (1000,0) .
Advances in the Design of Symbolic Computation Systems by B. Buchberger (auth.), Dr. Alfonso Miola, Dr. Marco Temperini (eds.)