Algorithms used in calculators to obtain roots of numbers by numerical iteration:

SQUARE ROOT

For instance: (= 3.16227766)

Xo=3 (reasonable guess)

X1=1/2 (3 +10/3) = 3.166666667

X2=1/2 (3.166666667+10/3.166666667) = 3.162280702

X3=1/2 (3.162280702+10/3.162280702) = 3.16227766

Pretty fast convergence!!!

Xo=1 (pretty poor guess)

X1=1/2 (1+10/1) = 5.5

X2=1/2 (5.5+10/5.5) = 3.659090909

X3=1/2 (3.659090909+10/3.659090909) = 3.196005082

X4=1/2 (3.196005082+10/3.196005082) = 3.162455623

X5=1/2 (3.162455623+10/3.162455623) = 3.162227766

Still pretty fast!

CUBIC ROOT

As an exercise, calculate : (= 2.15443469)

Also notice that another possible algorithm could be:

But this has a much slower convergence (check it!)