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Arithmetic complexity of computations by Shmuel Winograd

By Shmuel Winograd

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

Results that result in functions within the sector of sign processing are emphasised, in view that (1) even a modest aid within the execution time of sign processing difficulties can have functional value; (2) leads to this sector are really new and are scattered in magazine articles; and (3) this emphasis shows the flavour of complexity of computation.

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Proof. Again, we examine the transpose problem of computing the coefficients of Q(u) = R(u)x S(u). The coefficients of the polynomial S(u) = £"=0 /i,«' satisfy hi = —hn-i-i, hi = 0 so S(u) = (1 — u2)S'(u) where S'(«) is a symmetric polynomial of degree n— 3. The coefficients h\ of S"(w) are h'o = h0, h(=hi, h\ = h'i-2+hi(i^2). Therefore Q(u) = (i-u2)R(u)S'(u). Transposing the problem back we obtain Fs(ra, n — 2) in terms of tap values h\, and signal values x\ = Xi + Xt-2- The rest of the assertion of the theorem is immediate.

Let G be a field, R a commutative ring without zero divisors which includes G; p ( u ) , p\(u), p2(u)& G[«] be polynomials such that (p\(u), p2(u)) = 1 and p(u) = p\(u] • p2(u). Then R[u]/(p(u)) = R[u]/(pi(u))xR[u]/(p2(u)). The isomorphism s is given by the following: For every r(u)<=R[u]/(p(u)), s(r(u)) = (si(r(u}), s2(r(u})} = (r(u) mod pi(u], r(u}modp2(u)}, and for every (r^u), r2(u}}zR^Kp^u}}xR[u]/(p2(u)) s~\(ri(u),r2(u))) = (rl(u)Xp2(u)xq2(u) + r2(u)Xpi(u)xqi(u))modp(u) where qi(u] andq2(u) satisfy p i ( u } q i ( u ) + p2(u)q2(u) = 1.

The polynomial Q(u) is quadratic, so we have to choose three distinct constraints in G U {00} (we assume that G is the field of rational numbers). Let these constants be a0 = 0, a\ = —l, a 2 = °°- We thus obtain that To compute Q'(u) we have to compute By the Chinese remainder theorem: And as we have Equating coefficients we obtain the algorithm of § Ha. IVc. Heuristic algorithms. We saw in § IVa that the use of interpolation to obtain an algorithm for computing convolution leads to algorithms with large constants.

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