What is a relative condition number of a sum of positive values?

Question

We want to compute the relative condition number of:

$$x_1+x_2+x_3+\cdots$$

We assume all values are positive, and we will do a limit of a large $x_1=10^{8}$, and smaller values for all the other numbers $x_2=x_3=\cdots=10^{-3}$, as a worse-case scenario:

1 term

For

$$f(x_1) = x_1$$

the relative condition number is:

$$\kappa={\|J\|\|x\|\over \|f(x)\|}={1\cdot x \over x} = 1$$

2 terms

For

$$f(x_1, x_2) = x_1+x_2$$

we get in the infinity norm:

$$\kappa={\|J\|_\infty \|x\|_\infty\over \|f(x)\|}={2\cdot \max(x_1, x_2) \over x_1 + x_2} \rightarrow 2$$

Which for $x_1=10^{8}$ and $x_2=10^{-3}$ approaches 2.

The exact value is:

In [1]: x1 = 1e8; x2 = 1e-3; 2*max(x1, x2)/(x1+x2)
Out[1]: 1.99999999998

3 terms

For

$$f(x_1, x_2, x_3) = x_1+x_2+x_3$$

we get in the infinity norm:

$$\kappa={\|J\|_\infty \|x\|_\infty\over \|f(x)\|}={3\cdot \max(x_1, x_2, x_3) \over x_1 + x_2 + x_3} \rightarrow 3$$

Which for $x_1=10^{8}$ and $x_2=x_3=10^{-3}$ approaches 3. The exact value is:

In [2]: x1 = 1e8; x2 = 1e-3; x3 = 1e-3; 3*max(x1, x2, x3)/(x1+x2+x3)
Out[2]: 2.99999999994

n terms

The relative condition number for summing n-terms of the type:

$$10^8 + 10^{-3} + 10^{-3} + \cdots$$

is

$$\kappa=n$$

Is that correct? So for large $n$, the sum of such values becomes ill-conditioned?

And the only way it becomes well-conditioned is if the magnitude of the numbers is similar, since then the condition number seems to stay low even for large $n$, e.g.:

In [3]: x1 = 1e-3; x2 = 2e-3; x3 = 3e-3; x4 = 4e-3; 4*max(x1, x2, x3, x4)/(x1+x2+x3+x4)
Out[3]: 1.6

Answer 1

The mistake I made is that the Jacobian in the infinity norm is equal to 1, not $n$. As such, the relative condition number of the sum is:

$$\kappa=1$$

and as such is always well-conditioned, no matter what magnitudes the values are.

Answer 2

I don't think your own answer is correct, though my method somewhat differs in so far as I use a different norm than you - the Euclidean norm. My result:

The relative condition for the sum of two numbers $x+y$ is $$\kappa_R = \dfrac{\sqrt{2x^2+2y^2}}{|x+y|}.$$ The absolute condition is $$\kappa_A = \sqrt 2\approx 1.41.$$

Hence the sum is very poorly conditioned if $x\approx -y$, which is somewhat to be expected.

$\textbf{Proof:}$ Consider the function $f:\mathbb{R}\to\mathbb{R}:(x,y)\mapsto x+y$. Then $\nabla f = (1,1)$. Hence $$\kappa_A = ||\nabla f|| = \sqrt 2,$$ where we use the Euclidean norm. Then we can calculate the relative condition number as $$\kappa_R = \dfrac{||\nabla f||\cdot||(x,y)||}{||f(x,y)||}= \dfrac{\sqrt{2x^2+2y^2}}{|x+y|}.$$

The conclusion of the elementary operation $x+y$ is that though it is a $\textit{stable}$ numerical problem (Indeed, the only $\textbf{new}$ errors made by $+$ if we consider no input-errors are bounded by $\varepsilon_{\text{machine}}$), it is not a well-conditioned operation for relative errors i.e it propagates existing errors by making them bigger.

Also note that from https://en.wikipedia.org/wiki/Condition_number#One_variable you can already see that your answer $\kappa_R = 1$ cannnot be right. If we exchange $y$ for $a\in\mathbb{R}$ your answer does not change, while mine indeed conforms to the condition number (then $\kappa_A = 1$)$$\kappa_R=\dfrac{|x|}{|x+a|}.$$