1. 1) Is , in principle, oscillating convergence more efficient than monotonic?
If not particularly so, it is always better anyway, as it gives you a rough idea where your root is from the beginning

2) They say that Newton's algorithm has quadratic convergence, can one determine that a priori examining the formulas (through f''(x)) or you must take a few calculations?

If we compare the results , applying the formula for square and cube root, we see that : if the root (L) is 1,000 and we start from 10,000
a) for [(x²+a):2x] the values decrease (roughly) by 1/2: 10000, 5050, 2600, etc
b) for [(2x³+a):3x²] they decrease by 1 /3

Is convergence quadratic in both cases? can you show me how you prove that?
3) what is the correct term for these series: monoton(ic) decreasing

4)) is it difficult to change a formula so that the convergence is oscillating?

is there a general principle that makes a convergence oscillate? Can you give me some examples of oscillating (quadratic) formulas?

Thanks  2.

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