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.