## Franklin's Notes

### 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

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

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