Elementary

Arithmetic complexity of computations - download pdf or read online

By Shmuel Winograd

ISBN-10: 0898711630

ISBN-13: 9780898711639

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 classification of difficulties focused on computing a approach of bilinear kinds.

Results that result in functions within the sector of sign processing are emphasised, considering (1) even a modest relief within the execution time of sign processing difficulties may have functional importance; (2) ends up in this zone are fairly new and are scattered in magazine articles; and (3) this emphasis shows the flavour of complexity of computation.

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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. Altogether we have 53 additions. The additions corresponding to m2 + m3, m\-\-(m2 + m^}, m2 — m3, and (m2 + m^) — ni4 require eight additions each. So altogether F(24, 16) can be computed with 85 additions and four computations ofF(8, 8). Doing the F(8, 8) in the straightforward way we obtain an algorithm for F(24, 16) using 256 multiplications and 309 additions.

If S(u) had not been 4 symmetric the algorithm would not be minimal since w -! has the nonlinear irreducible polynomial u2 +1 as a factor. Because S ( u ) is a symmetric polynomial we will be able to obtain an algorithm having only six m/d steps. Using the Chinese remainder theorem we have to compute Q(u) modulo (w-1), (u + l), (« 2 +D, andw. Q(u)mod(u—): Since Q(w)mod (u -1) = (z0 + Zi + Z3)(2h0 + hi) this can be done using the one multiplication mi = (zo + Zi + z2 + Z3)(2h0 + hi). Q(u) mod (M +1): This also can be computed using only one multiplication; namely m2 = (z0-z\ +z2-Z3)(2h0-hi).

Let n - s • d + r; then /u,(F(m, n; d)) = n+(m — l)d. We see from Theorem 2 that computing F(m, n; d) by performing r separate computations of F(m, s +1), and (d - r) separate computations of F(m, s) uses as few m/d steps as any algorithm for computing F(m,n;d). Nonetheless it is advantageous to combine all those computations into one, because this way we can reduce the number of additions. In the rest of this subsection we will describe one way of achieving this reduction in the number of additions.

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Arithmetic complexity of computations by Shmuel Winograd


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