TAOCP 4.6.4 Exercise 21
The reviewer's criticism is directed at the previous writeup, but it does not resolve the underlying problem: the exercise as presented here is missing the two definitions on which the computation dep...
Section 4.6.4: Evaluation of Polynomials
Exercise 21. [**] [20] Find two additional ways to evaluate the polynomial $x^8 + 13x^5 + 49x^4 + 33x^3 - 61x^2 - 37x + 3$ by scheme (12), using the two roots of (15) that were not considered in the text.
Verified: no
Solve time: 15m43s
The reviewer's criticism is directed at the previous writeup, but it does not resolve the underlying problem: the exercise as presented here is missing the two definitions on which the computation depends.
Exercise 4.6.4.21 explicitly requires:
- evaluation by scheme (12);
- use of the two roots of equation (15) that were not treated in the text.
Neither scheme (12) nor equation (15) appears in the material provided. Without them, there is no mathematically well-defined problem to solve.
The reviewer states that a correct solution should:
- identify the two unused roots of (15);
- substitute them into scheme (12);
- derive the corresponding evaluation formulas;
- verify that the formulas evaluate
$$ x^8+13x^5+49x^4+33x^3-61x^2-37x+3. $$
All four of those steps require the actual content of (12) and (15). There is no way to determine the required roots or the resulting evaluation schemes from the polynomial alone.
Therefore the correct mathematical conclusion is:
The exercise cannot be solved from the information supplied in the prompt. A complete solution requires the explicit formulas labeled (12) and (15) in TAOCP §4.6.4.
Once those formulas are provided, one can carry out the requested derivation and produce the two additional evaluation schemes. In their absence, any purported "solution" would necessarily be a guess and could not be justified rigorously.