Complexity of Computer Computations pp 21-30 | Cite as

# Simple Proofs of Lower Bounds for Polynomial Evaluation

## Abstract

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 n^{th} 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 n^{th} degree polynomials. Eve (1964A) and others have shown that this lower bound is almost achievable: an n^{th} degree polynomial can be evaluated in [n/2] + 2 multiplications and n additions/subtractions, provided some irrational preconditioning is allowed without cost.

## Keywords

Lower Bound Simple Proof Addition Step Multiplication Step Polynomial Evaluation## Preview

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