Conditioning and stability

Objective. Separate problem conditioning from algorithm stability.

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Learn: the key idea

Conditioning and stability separate two different sources of error in numerical computation: conditioning describes how sensitive the exact mathematical problem is to small perturbations in input data, while stability describes whether a particular algorithm amplifies rounding errors during computation. A problem can be well-conditioned yet solved by an unstable algorithm, producing inaccurate results, or ill-conditioned so that even a stable algorithm cannot avoid large output errors. For solving a linear system Ax = b, a large condition number of A signals ill-conditioning regardless of which solution method is used.

Worked example

A linear system has condition number κ(A) = 10^6. Explain the implication of this conditioning for numerical solution accuracy.

1 $Recall condition number measures sensitivity of the solution to input errors$
2 $A value of 10^6 is very large, indicating high sensitivity$
3 $Small input perturbations can be amplified by this factor in the solution$
4 $State the conclusion about the problem's conditioning$

Answer: The problem is ill-conditioned, so small input errors cause large solution errors

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Common mistake. Students often blame an unstable algorithm whenever results are inaccurate, without checking whether the underlying problem itself is ill-conditioned, in which case no algorithm could do better. Always assess the problem's conditioning separately from the method's stability.

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