Backward Error Analysis

See Figure 12 Forward error propagation: difficult and usually leads to overestimates (pessimistic).

Figure 12: Schematic for the forward and backward error analysis.

Backward error propagation: How much error in input would be required to explain all output error?

Assumes that approximate solution to problem is good IF IT IS THE exact solution to a ``nearby'' problem.

Example Want to approximate $f(x)=e^x$. We evaluate its accuracy at $x=1$.

$$\begin{eqnarray*} f(x)=e^x=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3\cdots\\ \widehat f(x)=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3 \end{eqnarray*}$$

Backward Error: 1) Find $\widehat x$ such that $f(\widehat x)=\widehat f(x)$

$$% latex2html id marker 8344f(\widehat x)=e^{\widehat x}\q... ...\quad \widehat x=\log (f({\widehat x})) = \log (\widehat f(x))$$

$$\begin{eqnarray*} f(x=1)=2.718282\quad \widehat f(x)=2.666667 \mbox {to seven place.}\\ \widehat x=\log(2.666667)=0.980829 \end{eqnarray*}$$

The following are different and cannot be compared:

$$\begin{eqnarray*} &&\mbox {Forward error }\widehat f(x)-f(x)=-0.051615\quad \m... ...{Ok, because output is correct for a slightly perturbed input.}) \end{eqnarray*}$$

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