8.5. HORNER: Evaluating a Polynomial
The evaluation of polynomials is essential to implementations of trigonometric functions and other transcendental functions. In practice, simple power series expressions for any F(x) are replaced by Chebyshev polynomial expansions in order to attain the smoothest possible distribution of discrepancies between the approximation and the exact function across the whole range of x (see Markstein).
Consider the operations required to evaluate a third-order polynomial involving real numbers:
P(x) = 1.0 + 0.5 x + 0.25 x2 + 0.125 x3
that we might directly transcribe from algebra into pseudocode:
sum = 1.0 power = x tmp = 0.5 * power sum = sum + tmp power = power * x tmp = 0.25 * power sum = sum + tmp power = power ...
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