Conditioning belongs to the mathematical problem; stability belongs to the algorithm. Numerical analysis is careful about keeping these two ideas separate.
Absolute and relative condition numbers
Definition. For a differentiable map \(\Phi:X\to Y\), the absolute condition number at \(x\) is \(\|D\Phi(x)\|\). The relative condition number is \(\kappa_{\mathrm{rel}}(x)=\|D\Phi(x)\|\|x\|/\|\Phi(x)\|\), when the denominator is nonzero.
Stability
Definition. An algorithm is forward stable if its forward error is comparable to the unavoidable conditioning error. It is backward stable if its computed output is the exact output for a small perturbation of the input.
Conditioning-stability principle
Theorem. If an algorithm is backward stable and the problem has relative condition number \(\kappa\), then the relative forward error is typically bounded by a constant times \(\kappa u\), up to higher-order terms. Thus poor conditioning cannot be repaired by a stable algorithm; instability can ruin even a well-conditioned problem.
Residual versus error
Proposition. A small residual implies a small forward error only after multiplication by an appropriate condition number. For linear systems, \(\|r\|/\|b\|\) controls \(\|\widehat x-x\|/\|x\|\) through \(\kappa(A)\), not by itself.
Diagnostic interpretation
Remark. When a computation fails, the first question is whether the problem is ill conditioned or the algorithm is unstable. These require different fixes: reformulate or regularize the problem in the first case, change the algorithm in the second.
Problem sensitivity versus algorithm behavior
Conditioning belongs to the mathematical problem \(F\); stability belongs to an algorithm. A backward-stable algorithm returns \[ \widehat y=F(x+\delta x), \qquad \frac{\|\delta x\|}{\|x\|}\le C u+O(u^2), \] where \(C\) grows moderately with dimension. Combining backward stability with conditioning yields a forward-error bound of size \(C\kappa_{\mathrm{rel}}(x)u\) to first order.
Theorem (backward-to-forward transfer). If \(F\) is differentiable at \(x\), \(\widehat y=F(x+\delta x)\), and \(\|\delta x\|/\|x\|\le\eta\), then \[ \frac{\|\widehat y-F(x)\|}{\|F(x)\|} \le\kappa_{\mathrm{rel}}(x)\eta+o(\eta). \]
A large forward error from a backward-stable algorithm diagnoses an ill-conditioned problem, not an unstable implementation.
Componentwise and normwise analysis
Normwise backward error permits a perturbation bounded in one global norm. Componentwise error requires \[ |\delta x_i|\le\eta|x_i| \] and is more informative for badly scaled data. For linear systems, the componentwise condition number can differ drastically from \(\kappa_2(A)\). Scaling rows and columns can improve normwise conditioning without changing the underlying componentwise problem.
Mixed stability
Some algorithms admit a mixed statement: the computed answer is close to the exact answer for nearby data, rather than exactly equal to it. This is still useful if the extra output perturbation is small relative to the problem's scale. The precise theorem should name which quantities are perturbed and in which norm.
Composition of stable stages
Proposition. Backward stability is not automatically preserved by composing two stable algorithms. It is preserved to first order when the intermediate map is well conditioned and the second stage's backward perturbation can be pulled back through a locally stable inverse of the first map.
For example, explicitly forming \(A^{-1}\) stably and then multiplying by \(b\) is not as stable as solving \(Ax=b\): perturbations in the inverse may not correspond to a small perturbation of \(A\). Stability is attached to a problem formulation, not to isolated subroutines.
Residual-based backward error for linear systems
Proposition. For \(r=b-A\widehat x\), the smallest normwise perturbation in \(b\) alone making \(\widehat x\) exact is \(\|r\|/\|b\|\). Allowing perturbations in both \(A\) and \(b\), a common relative backward error is \[ \eta(\widehat x) =\frac{\|r\|} {\|A\|\|\widehat x\|+\|b\|}. \]
Proof. Any admissible perturbations satisfy \[ r=\Delta A\,\widehat x-\Delta b, \] so \(\|r\|\le\eta(\|A\|\|\widehat x\|+\|b\|)\). Equality can be attained in subordinate norms by rank-one perturbations aligned with the residual, giving the formula.