Arithmetic complexity of computations 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 whilst development is feasible. the writer concentrates on that type of difficulties curious about computing a process of bilinear kinds.

Results that result in purposes within the zone of sign processing are emphasised, due to the fact (1) even a modest aid within the execution time of sign processing difficulties may have useful importance; (2) leads to this zone are rather new and are scattered in magazine articles; and (3) this emphasis exhibits the flavour of complexity of computation.

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So we can compute F(m, n;d)by first computing pi, p2, • • • , pd (which uses da\ additions), then summing these d vectors (which uses (d — \}l additions where / is the size of the p/s), and finally multiplying by A (which uses a 2 additions). Altogether this algorithm uses d(a\ + l) + a2 — l additions. Whenever a2>l — m the second method uses (d — I)(a2 + m — /) fewer additions. Example 1. We will derive an algorithm for computing F(3, 6; 3). As we saw, F(3,6;3) = 3F(3,2). where m x = (z 0 -z 2 )y 0 , ™2 = (zl + z2}((yo + yi)/2), m3 = (z 2 - Zi)((y 0 + yi)/2), and m4 = (zi — z 3 )yi.

We will therefore devote this section to exploring efficient algorithms for computing convolutions. IVa. Minimal algorithms. It is easily established that no linear combination of the m + n +1 quantities z, = £/=o x/y,-/, / = 0, 1, • • • , m + n, with coefficients in a field G yields an element of LG(1, x0, • • • , xm, y0, • • • , yn). That means that the z,'s are linearly independent. (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 is clear that we can always compute Fs(m,n] using an algorithm for F(m, n). That is /ji(Fs(m, «))g/Lt(F(m, n)) = m + n — 1. Using Theorem 2 of § Illb we obtain the following: THEOREM 1. For n =21 + 1 an odd number n(Fs(m, n)) = m + n — l. For n =21 an even number ^(Fs(m, n)) = m + n—2. Proof. The first part of the theorem is immediate from Theorem 2 of § Illb (which implies that /u(F s (m, n))^m +n — 1), and from the observation that /x(F s (m, n))^/x(F(m, n}) = m + n -1. As for the second part of the theorem, Theorem 2 of § Illb implies that n(Fs(m, 2/))^m + 2/-2.

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