The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Evaluation of polynomials with super-preconditioning
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Complexity measures and hierarchies for the evaluation of integers, polynomials, and n-linear forms
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
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It is well known from the work of Motzkin [55], Belaga [58] and Pan [66], that “most” nth degree polynomials p &egr; R[x] require about n/2 ×, ÷ ops and n ± ops and that these bounds can always be achieved within the framework of preconditioned evaluation (1). More precisely, if p can be computed using less than [equation] ×, ÷ or less than n ± ops, then the coefficients of p are algebraically dependent. The situation when counting ± ops with the potential of unlimited * ops, is not as clear. While the arguments based on algebraic dependence provide us with our best lower bounds thus far, a different approach of independent interest is taken in section IV. Namely, we are able to show that the number of ± ops required to compute any p &egr; R[x] is bounded below by a function of the number of distinct real zeros of p. The potential (e.g., for producing non linear lower bounds) and limitations of this approach will be discussed.