Absolute and relative condition numbers

Given a multivariate function $f:\mathbb C^m\to\mathbb C^n$, and given a value of $x\in\mathbb C^m$ we use $\delta f$ to denote the quantity $f(x+\delta x)-f(x)$. Using this, the absolute condition number $\hat\kappa$ of computing $f$ at $x$ is defined as

$$\hat\kappa(x) = \lim_{\delta\to 0}\sup_{\lVert\delta x\rVert \leq \delta} \frac{\lVert\delta f\rVert}{\lVert \delta x\rVert}=\lVert J(x)\rVert$$

where $J$ is the Jacobian. Further, the relative condition number $\kappa$ of computing $f$ and $x$ is defined as

$$\kappa(x) = \lim_{\delta\to 0}\sup_{\lVert\delta x\rVert \leq \delta} \frac{\lVert\delta f\rVert/\lVert f\rVert}{\lVert \delta x\rVert/\lVert x\rVert}=\frac{\lVert J(x)\rVert}{\lVert f(x)\rVert/\lVert x\rVert}$$

Notice that, when $f$ is differentiable, we have that

$$\kappa = \frac{\hat\kappa}{\lVert f(x)\rVert/\lVert x\rVert}$$

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