What is the justification and intuition behind Muller's method's quadratic equation?

Question

Usually we write the quadratic formula like this:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

But Muller's is written like this: Muller's method.

Why is that?

Answer 1

Here the author is using the Citardauq Formula $x=\frac{2c}{-b\mp \sqrt{b^2-4ac}}$. It can be useful in avoiding loss of significance in the computation of one of the roots.

We can obtain the Citardauq Formula by multiplying top and bottom of the Quadratic Formula by $-b\mp\sqrt{b^2-4ac}$.

Suppose for example that $b$ is fairly large positive, and $4ac$ is quite a bit smaller. Then when we compute the root $x=\frac{-b+\sqrt{b^2-4ac}}{2a}$, there may be a serious loss of significant digits in the computation of the numerator. There is no such loss if we use the equivalent $x=\frac{2c}{-b-\sqrt{b^2-4ac}}$.

Another way of looking at the formula is that if $ax^2+bx+c=0$, where $ac\ne 0$, then $\frac{1}{x}$ is a solution of $cu^2+bu+a=0$. Solve for $u$ using the Quadratic Formula, and take the reciprocal.