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

By Shmuel Winograd

Specializes in 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 category of difficulties inquisitive about computing a procedure of bilinear varieties.

Results that result in functions within the zone of sign processing are emphasised, on the grounds that (1) even a modest relief within the execution time of sign processing difficulties can have useful value; (2) ends up in this region are rather new and are scattered in magazine articles; and (3) this emphasis shows the flavour of complexity of computation.

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Extra resources for Arithmetic Complexity of Computations (CBMS-NSF Regional Conference Series in Applied Mathematics)

Example text

Recall that we use B = G(J{x0, • • •, xm}(J {yo,' • • ,yn}, and that the linear independence is that of the z,'s as elements of H' = H/LG(B)). It follows from Theorem 1 of § Ilia that /A(ZO, • • • , zm+fl)^ m + n + l. We will exhibit an algorithm for computing the z,'s using m + n +1 m/d steps in the case that G has m + n + l elements, thus showing that jLt(z 0 , • • • , z m + n ) = m + n +1. ) We will assume in the rest of these notes that |G| is infinite, and even that G is of characteristic 0.

We will now combine these algorithms to obtain an algorithm for F(216, 81). We can view this problem as that of F(108, 81) when each of the entries is a pair. We will obtain the algorithm by iterating the algorithm for F(4, 3) and the algorithm for F(3, 3) three times. The number of m/d steps of this combined algorithm is 6x5x5x6 = 750. But since each step is multiplication modulo u2 +1 we have 3x750 = 2,250 multiplications and 3x750 = 2,250 additions. 3 mod u2 +1 which requires only two multiplications and no additions, and only the remaining 642 m/d steps require three multiplications and three additions each.

Let R(u) be the polynomial R(u) = x0 + xiU, and let S(u) be the polynomial S(u) = yo + yi". We wish to compute the coefficients of the polynomial O(u) — z0 + ziu+z2u2 where Q(u) = R(u) • S(u). 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.

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