Computing the square root of complex number in a stable manner

Solution 1:

You must first study the concept of catastrophic cancellation. This occurs when we subtract approximations of two real numbers that are relatively close to each other. This is a basic but extremely important point in numerical analysis. You will find very brief summary and analysis in this answer

Returning to the problem of complex square roots. The expression $$u = \sqrt{\frac{\sqrt{x^2+y^2}+x}{2}} $$ will not experience catastrophic cancellation as long as $x>0$. Afterwards, you can compute $v$ using $v = \frac{y}{2u}$.

However, the expression for $u$ will (almost certainly) experience catastrophic cancellation when $x<0$ and $y$ is nearly zero. If $x<0$, then you first compute $$v =\sqrt{\frac{\sqrt{x^2+y^2}-x}{2}}$$ and then $$u = \frac{y}{2v}.$$