SICP Workout (1.45) Resolution summary

SICP 1.45 is a summary of some of the previous exercises on fixed points.

The question recalls the fixed-point search method we used in the 1.3.3 section. When looking for the fixed point of X/Y, the transform itself does not converge. Need to do an average damping talent enough.

For Y-x/(y^2) This transformation can also be convergent by an average damping.

Just once the average damping is not enough for the four-time equation, that is, for Y-and x/(y^3) This transformation, an average damping is not enough to make it converge, need to do two times the average damping.

The topic follows the abstract principles that have always been. Ask us to do a few more tests to find out that the change of Y--X/(Y^N) requires several average damping.

Let's look at the rules we know right now,

Y-x/(y^1) requires 1 times average damping

Y-x/(y^2) requires 1 times average damping

Y-x/(y^3) requires 2 times average damping

Would it be necessary to n/2 the average damping for a simple guess?

By guessing, of course not. We need to test it a few times.

To facilitate the test, I wrote the following scenario:

(Define (N-RT x n try-average-time)  (Fixed-point (Repeat Average-damp try-average-time) (Lambda (y)  (/x (Fast-expt y (-N 1))))))

This allows you to arbitrarily specify the n-th equation and the corresponding average damping times. Start with the 5-time equation and see if the test results are consistent with my predictions.

The test found that my push was too unreliable. It is found that the 4,5,6,7 quadratic equation can be convergent by 2-times average damping.

Continue to guess (LG N) times, to tell the truth, my math sensitivity is not yet a bit to (LG N) to guess the extent, see their own very many test results, combined with some of the online students of the problem solving process only to locate (LG N) on.

Of course. This is the right time to guess.

Finally I wrote a method such as the following:

(Define (Final-n-root x N)  (Define (nth-root N)    (N-rt x N (lg N)))  (nth-root N))

The above method calls the previously defined N-RT process for testing, and simply uses (LG N) The average number of damping to be calculated.

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SICP Workout (1.45) Resolution summary