Arithmetic Complexity of Computations (CBMS-NSF Regional 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 interested by computing a process of bilinear kinds.

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

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Similarly we can use (u — oo)r instead of just (u — oo). A point of explanation as to the meaning of (u — oo)r may be in order. We used the term multiplication modulo u — oo to indicate the construction of the Case 2 of Theorem 1 of IVb; that is, using the identity: Q(u} = Q(u] modp(u) + xmynp(u), where deg Q(M) = deg P(u). That is, we computed Q(«) modulo p(u] and thereby modified some of the coefficients by multiples of xmyn, the highest coefficient of Q(u), and then "corrected" this modification.

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 follows from Theorem 1 of § Ilia that /A(ZO, • • • , zm+fl)^ m + n + l.

This requires 2 x 4 = 8 additions. The final part of the algorithm computes and uses four additions. Example 2. In this example we will discuss a 32-tap filter with a 2:1 decimation. More specifically we will derive an algorithm for F(24, 32; 2) = 2F(24, 16). Algorithms for F(24, 16) were derived in Example 1 of § Va; we will follow this derivation keeping track of the number of input additions and output additions. We partition the matrix of F(24,16) into 8x8 blocks, and denote them by z0, z\, z2,23.

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