Errors And Approximations In Numerical Computations

1. Core Concept Overview & Intuition

Numerical computation uses approximate inputs and finite-precision arithmetic, so a computed value can differ from the exact value. Error analysis identifies that difference, measures its size, and tracks how it changes through arithmetic operations. This lesson focuses on round-off methods, absolute and relative error, significant figures, and propagated error.

2. Mathematical Formalization & Core Definitions

Let $x$ be the exact value and $x_r$ its approximation. The absolute error is

$$
e_{abs}=|x-x_r|.
$$

The relative error compares the absolute error with the size of the exact value:

$$
e_r=\frac{|x-x_r|}{|x|},\qquad x\ne 0.
$$

Percentage error is $100e_r\%$. If $x=0$, relative error is undefined; report absolute error instead.

A value rounded to $n$ decimal places has maximum absolute rounding error

$$
e_{abs}\leq \frac{1}{2}10^{-n}.
$$

The source treats finite machine representation as the reason round-off occurs. It describes chopping (also called truncating in this context) and symmetric rounding as ways to handle the finite digits. Do not confuse this use of truncating with a separate definition of truncation error unless that definition is supplied by the course material.

For exact inputs $x,y$ and approximations $x_r,y_r$, write $x=x_r+\epsilon$ and $y=y_r+\eta$. The propagated error is the error in the result of an operation caused by the input errors. The accumulated error includes propagated error and any new error introduced at the current computation step. For a differentiable function $f$ with input error bounds $e_i$, the source gives the first-order maximum-error estimate

$$
e_f\approx \sum_i\left|\frac{\partial f}{\partial x_i}\right|e_i.
$$