Simple Proofs of Lower Bounds for Polynomial Evaluation
The idea of establishing lower bounds on the number of arithmetic operations required to evaluate polynomials is due originally to Ostrowski (1954A). He showed that at least n multiplications and n additions/subtractions are required to evaluate nth degree polynomials for n ≦ 4. Since then, this result has been proved true for all nonnegative values of n [Belaga (1961A), Pan (1966A)]. Motzkin (1955A) introduced the idea of preconditioning; if the same polynomial is to be evaluated at many points, it may be reasonable to allow some free preprocessing of the coefficients. It has been shown [Motzkin (1955A), Belaga (1961A)] that even if this preconditioning is not counted then at least [n/2] multiplications/divisions and n additions/ subtractions are necessary to evaluate nth degree polynomials. Eve (1964A) and others have shown that this lower bound is almost achievable: an nth degree polynomial can be evaluated in [n/2] + 2 multiplications and n additions/subtractions, provided some irrational preconditioning is allowed without cost.
KeywordsLower Bound Simple Proof Addition Step Multiplication Step Polynomial Evaluation
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