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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?

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# Are there ill-conditioned problems in infinite precision arithmetric?

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 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?