1) Is , in principle, oscillating convergence more efficient than monotonic?

If not particularly so, it is always better anyway, as it gives you arough ideawhere your root is from the beginning

2) They say thatNewton'salgorithm 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