2.2. Error#

2.2.1. Types of Error#

Applying the trapezoidal rule and Simpson’s rule introduces the concept of error in numerical solutions.

In our work so far, we have come across two obvious kinds of error, that we’ll come back to later:

  • local truncation error, which represents how “wrong” each interval/step is compared with the exact solution; and

  • global truncation error, which is the sum of the truncation errors over the entire method.

In any numerical solution, there are five main sources of error:

  1. Error in input data: this comes from measurements, and can be systematic (for example, due to uncertainty in measurement devices) or random.

  2. Rounding errors: loss of significant digits. This comes from the fact that computers cannot represent real numbers exactly, and instead use a floating-point representation.

  3. Truncation error: due to an infinite process being broken off. For example, an infinite series or sum ending after a finite number of terms, or discretization error by using a finite step size to approximate a continuous function.

  4. Error due to simplifications in a mathematical model: “All models are wrong, but some are useful” (George E.P. Box) All models make some idealizations, or simplifying assumptions, which introduce some error with respect to reality. For example, we may assume gases are continuous, that a spring has zero mass, or that a process is frictionless.

  5. Human error and machine error: there are many potential sources of error in any code. These can come from typos, human programming errors, errors in input data, or (more rarely) a pure machine error. Even textbooks, tables, and formulas may have errors.

2.2.2. Absolute and relative error#

We can also differentiate between absolute and relative error in a quantity. If \(y\) is an exact value and \(\tilde{y}\) is an approximation to that value, then we have

  • absolute error: \(\Delta y = | \tilde{y} - y |\)

  • relative error: \(\frac{\Delta y}{y} = \left| \frac{\tilde{y} - y}{y} \right|\)

If \(y\) is a vector, then we can define error using the maximum of the elements:

(2.4)#\[\begin{equation} \max_i \frac{ |y_i - \tilde{y}_{i} |}{|y_i|} \end{equation}\]