Absolute and relative forward error for an approximate value 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), let $f:X\to Y$ be a map, and let $x$ be an element of the domain of $f$. Let $\widehat{y}\in Y$ be an approximate value of $f$ at $x$: an element of the target space meant to stand for $f(x)$.

The absolute forward error of $\widehat{y}$ at $x$ is the norm

$\parallel \widehat{y}-f(x)\parallel ,$

and, provided $f(x)\mathrm{\ne }0$, the relative forward error of $\widehat{y}$ at $x$ is

$\frac{\parallel \widehat{y}-f(x)\parallel }{\parallel f(x)\parallel }\mathrm{.}$

The exact value $f(x)$ is called the exact answer, and $\widehat{y}$ the computed (or approximate) answer. Forward error is the error in the answer itself; it is measured in the target space $Y$ and knows nothing about how $\widehat{y}$ was produced.

• Why the relative version needs $f(x)\mathrm{\ne }0$. Relative error divides by $\parallel f(x)\parallel$; when $f(x)=0$ the division is not defined, and the only sensible quantity is the absolute error. Every statement on this page that writes a relative forward error carries the hypothesis $f(x)\mathrm{\ne }0$.

• Approximate values are plain elements of $Y$. No machine arithmetic is assumed here: $\widehat{y}$ can come from hand calculation, a rounded formula, or an algorithm. The floating-point model is introduced separately in The standard relative floating-point model and unit roundoff, and nothing in this definition depends on it.