Assessment 2 knuth book

# Since the subresultant prs algorithm restricts the

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Since the subresultant PRS algorithm restricts the coefficient growth to a linear Journal of the Association for Computing Machinery, Voh 18, No. 4, October 1971

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Euclid's Algorithm and Computation of Polynomial GCD' s 503 rate comparable to that which often occurs in a primitive PRS, and at the same time avoids the need for recursive applications to inflated coefficients, it appears to satisfy the most optimistic criteria that could be set for Algorithm C. Nevertheless, it may require the production of subresultants much larger than either the given polynomials or their GCD. Algorithm ~[ avoids this difficulty by taking a fundamentally different approach. The given polynomials are first projected by modular mappings into one or more simpler domains in which images of the GCD can more easily be computed. The true GCD is then constructed from these images with the aid of the Chinese re- mainder algorithm. Since the same method is used for the required GCD computa- tions in the image spaces, it is only necessary to apply Euclid's algorithm to integers and to univariate polynomials with coefficients in a finite field. This modular approach is especially well suited to the problem of GCD computa- tion, because the desired GCD is typically smaller than the given polynomials, and this limits the required number of images. In particular when the GCD is unity, a single image is usually sufficient. Of critical importance to both Algorithm C and Algorithm 5'[ is the existence of a small multiple of the GCD whose leading coefficient can be computed with negligible effort. Otherwise, it would be necessary in Algorithm 2~I to obtain enough images to build the associated subresultant, whose coefficients might be very large, and it would be necessary in both algorithms to compute the primitive part of that subresultant. A striking difference between the two algorithms is that Algorithm S views a polynomial in Z[xl, • • • , x~] as a univariate polynomial in some main variable, with polynomial coefficients, while Algorithm M treats it directly as a multivariate polynomial with integer coefficients. It is easy to see that Algorithm M profits greatly from the resulting speed of operations on the larger number of smaller co- efficients. This same idea is applied in the image space Zp [xl, • • • , x~], whose poly- nomials are viewed by Algorithm P as polynomials in the variables x~, ... , x,_~ with coefficients in Zp[x,]. In our stady of computing times, we obtained asymptotic bounds on the maximum computing times for the two algorithms, with the aid of several simplifying assump- tions which we believe to be realistic. Furthermore we showed that the maximum computing time for Algorithm ~,i is strictly dominated by the maximum computing time for the first pseudo-division in Algorithm C, in the region v _> 2. This dramat- ically illustrates the superiority of the modular approach for GCD computations in which the given polynomials are sdfficiently large and sufficiently dense.
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