By Georgiev P., Pardalos P., Theis F.

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**Example text**

This is easier to solve than the one in the previous paragraph. We immediately have I14(2~) = n and, again, it turns out that M(N) z 1gN. Of course, it’s not always possible to get by with such trivial manipulations. For a slightly more difficult example, consider an algorithm of the type described in the previous paragraph which must somehow examine each element before or after the recursive step. The running time of such an algorithm is described by the recurrence M(N) = M(N/2) + N. Substituting N = 2n and applying the same recurrence to itself n times now gives This must be evaluated to get the result I~f(2~) = 2n+1 - 1 which translates to M(N) z 2N for general N.

This method requires recomputation of the powers of x; an alternate method, which requires extra storage, would save the powers of x as they are computed. A simple method which avoids recomputation and uses no extra space is known as Homer’s rule: by alternat:ing the multiplication and addition operations appropriately, a degree-N polynomial can be evaluated using only 45 CHAPTER 4 46 N - 1 multiplications and N additions. The parenthesization P(X) = x(x(x(x + 3) - 6) + 2) + 1 makes the order of computation obvious: Y:=PN; for i:=N-I downto 0 do y:=x*y+p[i]; This program (and the others in this section) assume the array representation for polynomials that we discussed in Chapter 2.

The remaining problem is equivalent to multiplying 2-by-2 matrices. Just as we were able to reduce the number of multiplications required from four to three by combining terms in the polynomial multiplication problem, Strassen was able to find a way to combine terms to reduce the number of multiplications required for the 2-by-2 matrix multiplication problem from 8 to 7. The rearrangement and the terms required are quite complicated. 81. This result was quite surprising when it first appeared, since it had previously been thought that N3 multiplications were absolutely necessary for matrix multiplication.

### A bilinear algorithm for sparse representations by Georgiev P., Pardalos P., Theis F.

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