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 occupied with computing a process of bilinear kinds.

Results that result in purposes within the quarter of sign processing are emphasised, due to the fact (1) even a modest aid within the execution time of sign processing difficulties can have functional importance; (2) ends up in this sector are quite new and are scattered in magazine articles; and (3) this emphasis exhibits the flavour of complexity of computation.

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The second concept we have to introduce is that of a symmetric pair of filters. Consider two n -tap filters z, = £"=o xi+jhj and z\ = Z"=o x'i+jh'j, where the jc, terms and x't terms stand for disjoint of indeterminates, and h\ =/i n _i-,. A symmetric pair of filters is the computation of z" = z,• + z \. We denote the computation of m outputs of a symmetric pair of filters by FF(m, n)_. THEOREM 6. /u(FF(w, n)) = n(Fs(m, n)) + /u,(Fs(m, n)). More specifically, for every algorithm, A, for computingFs(m, n)+Fs(m, n), there exists an algorithm for computingFF(m, n) having the same number ofm/d steps as A, and 2(m + n — 1 additions more than A.

That is, we know how to derive a (5M, 15|A) algorithm for computing F5(8,15; 2). The regular algorithm is an (8M, 14A) algorithm. In order to continue our investigation of the algorithms for symmetric filters with decimation, we need two new concepts: that of a skew-symmetric filter, and that of a symmetric pair of filters. An n-tap filter with tap values of h0, hi,- - • ,hn-\ is said to be skew symmetric if for all i = 0,1, • • • , n — 1, hf = —hn-\-i. Of course if the number of taps n = 21 +1 is odd then hi = 0.

Each of the 12 F(4,4)'s uses 12 output additions for the total of 144. Each of the four F(2, 2) algorithms (on 4 x 4 blocks) we used requires 2 x 4 = 8 output additions for a total of 32 additions. Finally the F(3, 2) algorithm (on 8 x 8 blocks) uses 8x4 = 32 additions. So altogether this algorithm has 144 + 32 + 32 = 208 output additions. To summarize, the F(24, 16) algorithm we use has 192 multiplications, 94 input additions, and 208 output additions. Consequently the F(24, 32; 2) algorithm has 2 x 192 = 384 multiplications and 2 x 94 +192 + 208 = 588 additions.

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Arithmetic complexity of computations by Shmuel Winograd
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