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Sources of Approximation in Numerical Analysis

Definition

There are many sources of approximation and inexactedness in computational science. Some approximations can occur before a computation begins.

Approximations Beyound our Control

Modelling: Some physical features of the problem may be simplified. (ie. removing air resistance, friction)

Empirical Measurements: Scientific equipment have finite precision. This accuracy can be further reduced when we have a small sample size, randomness or bias.

Previous Computation: Previous computations may have been approximate, which propagate to future computations.

Approximations we can Control

Truncation or Discretization: Some features of a mathematical model can be simplified or omitted. (Replacing derivatives by finite differences, or only taking a finite number of terms in an infinite sequence).

Rounding: Every piece of computational calculation isn't infinitely precise. There will be rounding errors no matter what.

The accuracy of the final solution is a combination of all of its preceding approximative calculations. The study of the effects of such approximations on the accuracy and stability of numerical algorithms is called error analysis.

Example

The surface area of the Earth might be computed using the formula

\[ A = 4\pi r^2 \]

for the surface area of a sphere of radius \(r\). The use of this formula for the computation involves a number of approximations:

  • The Earth is modeled as a sphere, which is an idealization of its true shape.
  • The value for the radius, r ~ 6370 km, is based on a combination of empirical measurements and previous computations.
  • The value for \(\pi\) is given by an infinite limiting process, which must be truncated at some point.
  • The numerical values for the input data, as well as the results of the arithmetic operations performed on them, are rounded in a computer or calculator.

The accuracy of the computed result depends on all of these approximations.