Ch2.3.4 Error Propagation and Stability

• A length measurement \( l \) will necessarily include some error.
• The potential error area is shaded in figure.

• For two length measurements, assume for simplicity that

\[ \begin{align*} l_1 &= l_{1}^{'} \pm \epsilon \\ l_2 &= l_{2}^{'} \pm \epsilon \end{align*} \]

• It follows that

\[ \begin{align*} l_1 + l_2 &= ( l_{1}^{'} \pm \epsilon ) + (l_{2}^{'} \pm \epsilon) \\ &= l_{1}^{'} + l_{2}^{'} \pm 2\epsilon \end{align*} \]

• Max combined error is 2(0.5 mm) = 1 mm.

• Combined error area illustrated by shaded region.
• Error is growing linearly: two summations gives 2\( \epsilon \), while three summations would give 3 \( \epsilon \), etc.

• Other situations can amplify error more extremely.
• Imagine two measurements, length \( l \) and height \( h \), with a desire to find the surface area.
• Then

\[ \begin{align*} l \times h &= ( l' \pm \epsilon ) \times (h' \pm \epsilon) \\ &= l'h' \pm ( l'\epsilon + h'\epsilon + \epsilon^2) \end{align*} \]

• The combined error is \( l'\epsilon + h'\epsilon + \epsilon^2 \).

• Through these examples, we can see the affects of compounding errors.
• When a value is subject to machine error, through floating point representation or through some other source, the same error propagation patterns emerge.

\[ \begin{align*} l \times h &= ( l' \pm \epsilon ) \times (h' \pm \epsilon) \\ &= l'h' \pm ( l'\epsilon + h'\epsilon + \epsilon^2) \\ & \cong l'h' \end{align*} \]

• If one of source values, such as the height, had overflowed, and was now represented as Inf, then final result would be Inf.
• The same thing can happen with NaN.
• While these results are obviously unnatural, they arise just as easily through mistake as they do through actual results.
• An accidental log(0) returns -Inf, and if buried within a larger calculation, we can be left scratching our heads for a few minutes as we look for what went wrong.

log(0)

[1] -Inf

• Depending on problem, large output change for small input change may be expected.
• This happens for an exponential function, and is inherent to the problem rather than an algorithm attribute.

f <- function(x) {exp(x)}
x1 <- 10
x2 <- 10.1
c(f(x1), f(x2))

[1] 22026.47 24343.01