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By Zhang G.

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Therefore it is natural to try to approximate the elementary functions by polynomials or rational functions. The questions that immediately spring to mind are: • How can we compute such polynomial or rational approximations? • What is the best way (in terms of accuracy and/or speed) to evaluate a polynomial or a rational function? , the “distance” between the function being approximated and the polynomial or rational function), and the evaluation error due to the fact that the polynomial or rational function are evaluated in finite precision floating-point arithmetic.

1. Chebyshev polynomials play a central role in approximation theory. Among their many properties, the following three are frequently used. A much more detailed presentation of the Chebyshev polynomials can be found in [36, 269]. Theorem 3 For n ≥ 0, we have n Tn (x) = 2 n/2 (−1)k k=0 (n − k − 1)! (2x)n−2k . (n − 2k)! Hence, the leading coefficient of Tn is 2n−1 . Tn has n real roots, all strictly between −1 and 1. 30 Chapter 3. 1: Graph of the polynomial T7 (x). Theorem 4 There are n + 1 points x0 , x1 , x2 , .

Digit-recurrence algorithms for division and square root [135, 270] also generate results in a “signed-digit” representation. Some of these exotic number systems allow carry-free addition. This is what we are going to investigate in this section. First, assume that we want to compute the sum s = sn sn−1 sn−2 · · · s0 of two integers x = xn−1 xn−2 · · · x0 and y = yn−1 yn−2 · · · y0 represented in the conventional binary number system. 1) ci+1 = xi yi ∨ xi ci ∨ yi ci we see that there is a dependency relation between ci , the incoming carry at position i, and ci+1 .

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A 17/10-approximation algorithm for k -bounded space on-line variable-sized bin packing by Zhang G.

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