It is hoped that in the future with the advent of quantum computing that fundamental operations on a computer will have arbitrarily high precision. Moreover, that even with such high precision, computation times with be realistic. An important concept in Numerical Analysis is ill-conditioning. 


For instance it is common to call the following problem, an ill-conditioned problem:


Finding the roots of a quadratic polynomial. The example is from 
Datta, BN Numerical Linear Algebra and Applications
SIAM, Second edition (2010)


$
z^2-2z+1=0 \rightarrow z^2-2.0001z+1=0
$


since a relatively small change in the polynomial coefficients can cause a much larger perturbation in the polynomial roots. 


However the ill-conditioning doesn't seem to be part of the problem because in arbitrarily high precision we have the quadratic formula allowing us to compute the roots. The ill-conditioning seems to be caused by working in finite precision which is not inherent to the problem.


Researchers often talk about the difference between an ill-conditioned problem and an ill-conditioned method of solving. This distinction seems to be entirely blurred to me. 


To help clarify the situation I have two closely related questions:


1. Is ill-conditioning an issue in arbitrarily high precision?
2. Is there an ill-conditioned problem which remains ill-conditioned even in infinite precision computing?



Are there ill-conditioned problems in infinite precision arithmetric?
Ill-conditioning isn't a concept that depends on the precision that you use to compute the solution. "A small change in the data turns into a large change of the solution" isn't a concept that involves actual computation. Think of it as "the map data $\mapsto$ solution has a large Lipschitz constant".


The error on the input data does not come only from finite precision representation:
most data you put into a computer come from physical measurements or approximation processes, and often the error implied by these measurement/processes is way larger than one part in $10^{-16}$. How many physical constants do we know up to that precision, for instance? Ill-conditioning says that you need to provide your input data with a large precision, otherwise any solution you get will be rubbish, even in infinite precision arithmetic.


Ill-conditioning *per se* refers only on the sensitivity to input data and is independent of the actual method you use to compute a solution. The error introduced by using finite precision along the computation (highly depending on the algorithm, including order of summation and parenthesization) is often referred to as *algorithmic error* or *stability* of the algorithm.


