Assessment 2 knuth book

# Which differs very little from the corresponding

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which differs very little from the corresponding primitive PRS (12), except at the last step. Although it is not known whether this behavior is typical, it is known that there is a broad spectrum of possibilities. On the one hand, there are subresultant PRS's in which all polynomials except the last are primitive. On the other hand [1, p. 377], let F~, F2, • - - , Fk be a normal subresultant PRS, and let G be a primitive poly- nomial with leading coefficient g. Then it is easy to show that the subresultant PRS for GFi and GF2 is GF1, GF~ , g~+~GF3 , g~l+3GF4 , • • • , g6~+2k 5GFk , which diverges linearly from the primitive PRS GF~ , GF2 , • • • , GFk . 3.7 FURTHER IMPROVEMENTS. Having used the subresultant PRS algorithm to compute Fi = Ti [see (16)] for i = 3, .-. , j, it may happen that a divisor rj of cont(Ti) is available with little or no extra work. For example, we know that 0 = gcd (f~, f2) divides Ti for j = 3, -.. , k, and it occasionally happens that f~ divides T1 for some i < j. When such a rj is available, we would like to incorporate it into/3j, so that Fj = TJrj. However, if we do so, we cannot necessarily complete the PRS by direct application of (24). Fortunately, it is possible to modify (24) so as to complete the PRS and furthermore to guarantee that Fi I Ti for i = j + 1, • - • , k. Let F1, F2, T3 .... , Tk be a subresultant PRS, and let F1, F~, F3,..., F~ be an im- proved PRS with Fi = Ti/ri for i = 3, • • • , k. Let Ti = F1, T2 = F2, and rl = r2 = 1, and let t~ denote the leading coefficient of T~. Then T~ = riF~, (27) where i = 1, • • • , k. Now from (9), (24), (25), and (27) it is easy to show that the improved PRS is defined by /~3/r3 = (-- 1)~'+t, ~i/r, = --f,-2¢~ i-~rT-~ -2-', where Given F~, i = 4, ...,k, (28) @3 -~ --1, (29) ~b~ = (--r,_2fi_2)~-~b~-~ '-~, i = 4, ..., k. • ", F~_i, we first compute prem(Fi_2, Fi-1) and divide by ([~i/rl) from (28) to obtain the subresultant Ti = riFi. We then choose ri, and divide by it to obtain Fi. Clearly, there are many possible variations on this theme. To illustrate the theme, let us return to the example (4), for which the primitive PRS is (12) and the sub- resultant PRS is (26). The reader should imagine the problem to be enough harder that he could not easily discover the contents of the subresultants T~. Choosing ri = fi-l, whenever fi-1 I T~, and rl = 1 otherwise, the improved PRS is Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971

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488 w.s. BROWN 1, 0, 1, 0, -3, -3, 8, 2, -5 3, 0, 5, 0, -4, -9, 21 5,0, -1,0,3 13, 25, --49 9326, - 12300 260708. (30) By this simple strategy, we have succeeded in making F3 and F4 primitive, but we have failed to remove the factor of 2 from Fs. 3.8 COMPARISON. Let us now compare the foregoing algorithms. It is clear that the Euclidean PRS algorithm suffers so severely from coefficient growth that it does not merit further consideration. We also exclude the reduced 1)RS algorithm be- cause it is never better than the subrcsultant 1)RS algorithm, and it can be extremely inefficient in certain abnormal cases.
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