Abstract
This paper presents efficient simplification techniques tailored for sign semi-definite conditions (SsDCs). The SsDCs for a polynomial \(f\in \mathbb {R}[y]\) with parametric coefficients are written as \(\underset{\begin{array}{c} y\\ L \le y \le U \end{array}}{\forall }\ f(y) \ge 0\) and \(\underset{\begin{array}{c} y\\ L \le y \le U \end{array}}{\forall }\ f(y) \le 0\). We give sufficient conditions for the simplification techniques to be sound for linear and quadratic polynomials. We show their effectiveness compared to state of the art quantifier elimination tools for input formulae occurring in the optimal numerical algorithms synthesis problem by an implementation on top of Reduce command of Mathematica.
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Notes
- 1.
We will not consider here the sign definite conditions but they can be treated in a similar fashion.
- 2.
(2) was motivated by the desire for simplifying the bounds of \(\sqrt{x}\). (3) and (4) are motivated by (1) and (2) and by the initial definitions of \(L^{\prime }\) and \(U^{\prime }\):
$$\begin{aligned} L^{\prime }&=L+\frac{y^{2}+(p_{0}+p_{1}+p_{2})L^{2}+(p_{1}+2p_{2} )LW+p_{2}W^{2}}{(p_{3}+p_{4})L+p_{4}W}\\ U^{\prime }&=L+W+\frac{y^{2}+(q_{0}+q_{1}+q_{2})L^{2}+(2q_{0} +q_{1})LW+q_{0}W^{2}}{(q_{3}+q_{4})L+q_{3}W} \end{aligned}$$.
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The author thanks Hoon Hong for providing feedback on an earlier draft of this paper and to anonymous referees.
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Eraşcu, M. (2016). Efficient Simplification Techniques for Special Real Quantifier Elimination with Applications to the Synthesis of Optimal Numerical Algorithms. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_13
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