The square root algorithm

I was taught that to get the square root of a number $a$ you iterate $x\mapsto \frac{x+\frac ax}2$ until you're happy with the result. More generally, to scale a vector $v$ to have quadrance $a$ you iterate $v\mapsto \frac{1+\frac a{v\cdot v}}2v$. That's a direct analog. With negative quadrances you get chaotic behavior.