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I am trying to determine the numerical condition of the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ As I understand it, the quadratic formula can be interpreted as a mapping $$\phi:\mathbb{R}^3\to\mathbb{R}, (a,b,c)\mapsto \frac{-b \pm \sqrt{b^2-4ac}}{2a} \text{, for } a \ne 0.$$

My notes define the condition number for some $\phi$ as follows: $\kappa(\phi):= \frac{\lVert x\rVert\lVert D\phi(x)\rVert_M}{\lVert \phi(x)\rVert}$ for some compatible matrix norm. Which simplifies to $\kappa(f)= \frac{\lvert xf'(x)\rvert}{\lvert f(x)\rvert}$ in the one-dimensional case (which isn't the case anyway).

What I have tried so far was yet another method of computing the condition via the formula: $$\kappa_{ij}=\frac{x_j}{\phi_i(x)}\cdot\frac{\partial\phi_i(x)}{\partial x_j}\quad \text{Here: } i=1, j\in\{1,2,3\}$$

I am not really sure if the given formula(s) is/are the best approach to solving this problem or if there is some other preferably easier method. Looking at the discussion on this post about the numerical stability of the quadratic formula I am hoping to find some intuition to solving this problem without these long winded calculations.

The Tralfamadorian
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  • There is no alternative to long calculations. The alternative is abstract definition of condition numbers that your notes have omitted. One of them, the normwise relative condition number, reduces to the formula that you have be given when $\phi$ is differentiable. It is possible to say substantially more about this. Some of the words can be found here and in the paper cited there. The abstract definition provided insight into the nature of condition numbers, but they are not useful for practical calculations. – Carl Christian Feb 27 '22 at 01:57
  • Thankyou, I just looked at your other answer too. Compared to problems about stability, condition does not seem to be approachable by means of some simple heuristic. – The Tralfamadorian Feb 27 '22 at 09:29
  • You are very welcome. Calculating the condition numbers of a function is nontrivial, but straightforward, it is mainly a question of differentiation. Determining the stability of an algorithm can require considerable ingenuity and knowledge of finite precision arithmetic. Condition numbers are important because they allow us to identify the problems that can be solved with the available hardware. Stability analysis is important because it allows to classify algorithms as good or bad. – Carl Christian Feb 27 '22 at 17:21

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