By JEAN-PIERRE DE SCHAMPS,

GERY JEAN ANTOINE BIOUL

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**Extra resources for (Ewiley) Synthesis Of Arithmetic Circuits--Fpga, Asic & Embedded Systems**

**Sample text**

According to the type of the function at hand, some evaluation methods may be more appropriate than others. For instance, a method well suited for a polynomial may not be the best for an exponential function. Polynomial approximation is most often recommended for function evaluation as any continuous function can be approximated in this way, and the implementation only consists of additions, multiplications, and powers. Taylor and MacLaurin series are the most classic approaches to approximate functions.

Given two polynomials g(x) and h(x), not both equal to 0, the greatest common divisor of g(x) and h(x) is the monic polynomial of greatest degree which divides both g(x) and h(x). 3. gcd(0, 0) ¼ 0. 4. A polynomial f (x) of degree at least 1 is said to be irreducible if it cannot be written as the product of two polynomials, each of positive degree. A variant of the Euclidean algorithm for polynomials (VZG2003) expresses the greatest common divider of two polynomials g(x) and h(x) in the form gcd(g, h) ¼ b(x):g(x) þ c(x):h(x): The algorithm is based on the fact that if u(x) and v(x) are two polynomials such that deg(u) ¼ m, deg(v) ¼ t and m .

The chapter is divided into three sections corresponding to natural numbers, integers, and real numbers. 1 NATURAL NUMBERS Weighted Systems Any natural number (nonnegative integer) can be represented, in a unique way, in the form of a sum of powers Bi of some natural number B greater than 1, each of Synthesis of Arithmetic Circuits: FPGA, ASIC, and Embedded Systems By Jean-Pierre Deschamps, Ge´ry J. A. Bioul, and Gustavo D. Sutter Copyright # 2006 John Wiley & Sons, Inc. 39 40 NUMBER REPRESENTATION them multiplied by a natural number smaller than B.