Condition numbers, cont.

\[ \kappa_f(x) = \lim_{\epsilon \rightarrow 0}\frac{|f(x) - f(x(1+\epsilon))|}{|\epsilon f(x)|} = \left\vert \frac{xf'(x)}{f(x)} \right\vert \]

• relative error of output to relative error of input
• computer/algorithm agnostic

Condition number of simple addition \(f(x) = x - c\):

\[ \kappa_f(x) = \left\vert \frac{x}{x-c} \right\vert \]

• Condition number of constant multiplication is 1 \(\leftarrow\) no potential for cancellation error
• Ill-conditioned: problem with large rel. condition number
• Rel. condition number < 1 is no different than equivalent to 1 in practice
• see text for table