Download PDF by Shmuel Winograd: Arithmetic Complexity of Computations (CBMS-NSF Regional

By Shmuel Winograd

ISBN-10: 0898711630

ISBN-13: 9780898711639

Makes a speciality of discovering the minimal variety of mathematics operations had to practice the computation and on discovering a greater set of rules while development is feasible. the writer concentrates on that type of difficulties excited by computing a method of bilinear types.

Results that result in purposes within the quarter of sign processing are emphasised, for the reason that (1) even a modest relief within the execution time of sign processing difficulties can have sensible value; (2) leads to this sector are rather new and are scattered in magazine articles; and (3) this emphasis exhibits the flavour of complexity of computation.

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Assume also that (PI(U),p2(u)) = 1. We will now construct an algorithm A which computes the coefficients of (£"=o *;w')(Z"=o y,M') mod p ( u ) , where p(u)=pi(u) • p2(u] and n =degp(u) = nl + n2. The algorithm A has three parts. -w' modp 2 (w). Since the coefficients of p\(u] andp 2 (w)arein G, this part of A uses no m/d steps. The second part of A consists of using algorithm AI to compute the coefficients of ti(u) = r\(u] • s\(u) mod p\(u] and the algorithm A2 to compute the coefficients of t2(u) = r2(u) • s2(u) mod p2(u}.

For convenience of notation we will denote the identity underlying the algorithm of Case 2 as Q(u) = Q(u) mod [(u -oo) n^V ("-&)]• This way we can unify Cases 1 and 2 of Theorem 1 by saying that they stem from the identity Q(u) = O(u] mod [O/l'o" (u ~#i)]» where at most one of the a,-'s is oo and all the rest are in G. Example. We will illustrate the construction of the preceding discussion by deriving the algorithm used in § Ha. Let R(u) be the polynomial R(u) = x0 + xiU, and let S(u) be the polynomial S(u) = yo + yi".

Closer inspection shows that only four additions are required. The quantity h0 + hi is needed for computing z0 and zi, but also for computing z2 and z3, z4 and z5, and so on. We can compute this quantity only once and use it whenever needed. As a matter of fact ho + hi, can be computed when the program is written, and does not have to be computed when it is executed. , two additions per output. We will therefore say that this algorithm uses (2M; 2A) per output. A 2-tap filter is not very common in practice.

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Arithmetic Complexity of Computations (CBMS-NSF Regional Conference Series in Applied Mathematics) by Shmuel Winograd


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