Absolute and relative local condition numbers of a problem map

Let $X$ and $Y$ be normed spaces over $\mathbb{R}$ (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms) and let $f:X\to Y$ be a map. For $x\in X$ and $\delta >0$ write

$P_{f,x}(\delta ):=\{h\in X:0<\parallel h\parallel <\delta ,x+h\in \mathrm{dom}f\}\mathrm{.}$

Define the worst amplification over perturbations of size at most $\delta$ by

$S_{f,x}(\delta ):=\{\begin{array}{ll}0,& P_{f,x}(\delta )=\mathrm{\varnothing },\\ \sup \{{\displaystyle \frac{\parallel f(x+h)-f(x)\parallel }{\parallel h\parallel }}:h\in P_{f,x}(\delta )\},& P_{f,x}(\delta )\mathrm{\ne }\mathrm{\varnothing }\mathrm{.}\end{array}$

so $S_{f,x}(\delta )\in [0,+\mathrm{\infty }]$ in every case.

For each fixed $x$ the map $\delta \mapsto S_{f,x}(\delta )$ is nondecreasing as $\delta$ increases, so the infimum over all $\delta >0$ exists in $[0,+\mathrm{\infty }]$. The absolute (local) condition number of $f$ at $x$ is

${\kappa }_{\mathrm{a}\mathrm{b}\mathrm{s}}(f,x):={\inf }_{\delta >0}{S}_{f,x}(\delta )={\lim }_{\delta \to 0^{+}}{S}_{f,x}(\delta )\in [0,+\mathrm{\infty }]\mathrm{.}$

If $x\mathrm{\ne }0$ and $f(x)\mathrm{\ne }0$, the relative (local) condition number of $f$ at $x$ is

${\kappa }_{\mathrm{r}\mathrm{e}\mathrm{l}}(f,x):={\kappa }_{\mathrm{a}\mathrm{b}\mathrm{s}}(f,x)\cdot \frac{\parallel x\parallel }{\parallel f(x)\parallel }\in [0,+\mathrm{\infty }]\mathrm{.}$

For the computed value $\widehat{y}=f(x+h)$ of Absolute and relative forward error for an approximate value of a problem map, the quantity $\parallel h\parallel$ is the absolute backward error. When $x\mathrm{\ne }0$, the quotient $\parallel h\parallel \mathrm{/}\parallel x\parallel$ is the relative backward error: the amount by which the data must be moved, relative to its size, for the computed answer to be exactly correct. At $x=0$ only the absolute backward error is defined. The condition number converts backward error into a bound on forward error; that conversion is Local conditioning times backward error controls forward error to first order.

• The infimum form is the limit of the supremum form. Because $S_{f,x}$ is monotone in $\delta$, the infimum over $\delta >0$ equals the one-sided limit ${\lim }_{\delta \to 0^{+}}{S}_{f,x}(\delta )$, which is the familiar definition of a condition number as the worst relative amplification of infinitesimal perturbations. The infimum formulation is used below because it is an honest value of the library's ordered arithmetic with no separate limit apparatus.

• The value is allowed to be $+\mathrm{\infty }$. An ill-posed (discontinuous) map has infinite condition number at a point of discontinuity; ${\kappa }_{\mathrm{a}\mathrm{b}\mathrm{s}}=+\mathrm{\infty }$ is a legitimate value and no arithmetic is performed with it. The first-order conversion theorem only yields a finite bound when ${\kappa }_{\mathrm{a}\mathrm{b}\mathrm{s}}<+\mathrm{\infty }$.

• Conditioning belongs to the problem, not to the algorithm. The condition number depends only on $f$, the point $x$, and the two norms; no algorithm enters its definition. The separate notion of stability belongs to an algorithm under an arithmetic model and is defined in Forward and backward stability for a problem family under an arithmetic model.