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 desirous about computing a process of bilinear kinds.

Results that bring about purposes within the zone of sign processing are emphasised, on account that (1) even a modest relief within the execution time of sign processing difficulties may have functional importance; (2) ends up in this region are quite new and are scattered in magazine articles; and (3) this emphasis exhibits the flavour of complexity of computation.

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0, 1, • • • , ra + «, we can compute the z/c's with no additional m/d step. («,-) = Zflo^/ai and S(a,-) = £/"=o y/aj, / = 0, 1, • • • , m + n. (a,) and 5(a,-) without any m/d step. ,•) using only one m/d step; and thus we can compute all the O(a,)'s using m + n + 1 m/d steps. That means that we can compute all the z fc 's, k =0, 1, • • • , m+n, with an algorithm which has only m + n +1 m/d steps. We have thus shown that if z = x * y then IJL(ZO,ZI, • • • ,zm+n) = m + n + l. This method of constructing the algorithm is due to Toom.

We will use the second algorithm, in conjunction with the algorithm for F(2, 2), to obtain a new sequence of algorithms for a 16-tap filter. More precisely, we will obtain a sequence of algorithms for F(24, 16). ALGORITHM I'. This is the straightforward algorithm, which is a (16M, 15A) algorithm. ALGORITHM 2'. We partition the matrix of F(24, 16) to 8x88blocks. The addition corresponding to (ZQ — z2) requires 15 additions, that of (z\ — z$) eight more additions, and that of (z\-\- z2) and (z2 — Zi) 15 additions each.

THEOREM 1. LetM(x)y be a system of bilinear forms. This system is of minimal complexity if and only if the following two conditions hold: 1. No linear combination of rows ofM(x) (with coefficients in G) yields a row all of whose entries are 0. 2. There exists a nonsingular matrix C~l (with entries in G) such that for every row c 0/C~V(dW(jt)y) = i. For every C~l there exists an algorithm A = A(C~l) computing M(x)y, satisfying n(A) = /tt(M(jt)y) such that its /cth m/d step mk is given by mk = (£/=i «ifc*«)(Z/=i j3/fcy/), where the a,fc's and /S/fc's are in G.

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