Why are the round-off errors when solving the linear system $Ax = b$ of order $\varepsilon_\text{mach} x_j$?

I was reading a paper on arXiv where, in Section 2.4, the authors are discussing the error that arises in the solution of a linear system $$Ax = b,$$ or, to match up better with the paper, $$\Phi \alpha = u,$$ due to machine precision. The system is overdetermined so they solve it with least squares.

To make a long story short, they state that large coefficients will lead to severe round-off errors that hamper the convergence of a numerical method...

Now for the details. I will write the problem in a general form for clarity. Let $$M>N$$ and let $$\varepsilon_\text{mach} \approx 10^{-16}$$ represent machine precision.

They have a system matrix $$\Phi=(\phi_{i,j}) \in \mathbb{R}^{M\times N}$$, a coefficients vector $$\alpha = (\alpha_1,\dots, \alpha_N) \in \mathbb{R}^N$$ and an input vector $$u^{(N)} = (u_1,\dots, u_M)\in \mathbb{R}^M$$, giving the linear system: $$\sum_{j=1}^N \alpha_j \phi_{i,j} = u_i.$$

The vector $$u^{(N)}$$ is a numerical approximation of a continuous function $$u$$, see equation (2) in the paper. They are interested in the convergence of $$u^{(N)}$$ to $$u$$ as $$N$$ increases.

In Section 2.4 they say that because the elements of the system matrix are of $$O(1)$$, each coefficient $$\alpha_j$$ will result in round-off errors of approximately $$\varepsilon_\text{mach} \alpha_j$$ in the numerical approximation $$u^{(N)}$$. Thus the total error in $$u^{(N)}$$ due to machine precision is of order $$\varepsilon_\text{mach}||\alpha||_{l^2}$$.

They say this machine precision error places a limit on the achievable minimimum error of the numerical method. That is, as $$N$$ is increased eventually the approximation error $$t(\alpha) = ||u-u^{(N)}||$$ will decrease to approximately $$\varepsilon_\text{mach}||\alpha||_{l^2}$$ and then stop decreasing.

Thus if the norm of the coefficients $$||\alpha||_{l^2}$$ is very large this halting of convergence will happen very quickly and really impact the performance of the numerical method.

I get the overall idea but what I don't understand is how they can say that:

...because the elements of the system matrix are of $$O(1)$$, each coefficient $$\alpha_j$$ will result in round-off errors of approximately $$\varepsilon_\text{mach} \alpha_j$$ in the numerical approximation $$u^{(N)}$$.

Can this statement that the round-off errors are of size $$\varepsilon_\text{mach} \alpha_j$$ be proven? Even a rough proof? And/or, is it possible to show a simple explicit example that demonstrates that the round-off errors are indeed of this size?

I have spent several hours searching for further information on this, but although I have found many papers/books/documents, that state that large coefficients lead to round-off errors, I have not seen anything that proves or at least gives a detailed explanation that the size of the error due to each coefficient is of $$\varepsilon_\text{mach} \alpha_j$$?!

The previous and next IEEE machine numbers to $$\alpha_j$$ are at a distance $$\approx |\alpha_j| \varepsilon_{mach}$$ from each other; hence $$fl(\alpha_j)$$ (the closest machine number to $$\alpha$$) is at a distance at most $$|\alpha_j| \varepsilon_{mach}$$ from $$\alpha_j$$, but that's just an upper bound: if $$\alpha=0$$ or $$\alpha=1$$, for instance, then it is represented exactly and the error is 0.
I think that what they mean in Section 2.4 is in the reverse direction of what is usually done in error analysis: a perturbation of $$\alpha$$ to $$\alpha+f$$, with $$\|f\| \leq \varepsilon \|\alpha\|$$, gives rise to a perturbed $$u+g =\Phi (\alpha+f) = u + \Phi f$$ with $$\|g\| = \|\Phi f\| \leq \|\Phi\|\|f\| \leq O(1) \varepsilon \|\alpha\|$$, hence one expects that perturbations in $$u$$ of size smaller than that will not change the value of $$\alpha$$ further, or result only in perturbations in the last significant digit. This is confirmed by the experimental results, and it shows that it makes little sense to compute $$u$$ with very high precision.
It is not hard to construct an example where all $$\leq$$ become equalities, for instance take $$N=1$$ and $$\Phi=1$$.
It could be rephrased in the more familiar terms of condition numbers: if $$\Phi\alpha=u$$, then $$\sigma_{\min}(\Phi) \leq \|u\| / \|\alpha\|$$, and since $$\sigma_{\max}(\Phi)=O(1)$$, we get $$\kappa(\Phi) \geq O(1) \|\alpha\| / \|u\|$$. The relative error amplification in solving the linear system is then given by $$\kappa(\Phi)$$, which (if you use the classical condition number bound and convert between relative and absolute errors properly), shows that absolute errors in $$u$$ of size $$\varepsilon \|\alpha\|$$ become absolute errors in $$\alpha$$ of size at least $$\varepsilon \|\alpha\|$$, that is, relative errors of the size of roundoff precision. But that's just overcomplicating things, if you look at it in retrospect.
• Very nice answer thanks! Can you recommend and books or papers that cover this type of material, i.e., implications of machine precision on solving $Ax=b$ type systems?