Numerical Methods

Created

For many problems, we cannot obtain analytical solutions. Numerical methods yield approximate results.

  • Accuracy: how closely a computed/measured value agrees with the true value
  • Precision: how closely individual computed or measured values agree with each other.

True Value = Approximation + Error True Error = True value - Approximation (+/-) True Fractional Relative Error = (true error)/(true value)

However, we usually do not know the true values.

Newton's method has an iterative approach ϵa=current  approximation−previous  approximationcurrent  approximation×100\epsilon_{a}=\dfrac{current\;approximation - previous\;approximation}{current\;approximation}\times100%

So, you must compute until ∣ϵa∣<ϵs|\epsilon_{a}| < \epsilon_{s} where ϵs\epsilon_{s} is a pre-specified % tolerance.

  • Example: to make sure that your result is correct to at least n signifcant figure, you can set ϵs=(0.5×10(2−n))\epsilon_{s} = (0.5\times10^{(2-n)})%

Common types of errors:

  • Truncation error: results from ignoring all but a finite number of terms in an infinite series.
  • Round-off error: the difference between an approximation of a number used in computation and its exact (correct) value.
  • Blunders: errors caused by malfunctions of the computer or human imperfection.
  • Model errors: errors resulting from incomplete mathematical models
  • Data uncertainty: errors resulting from the accuracy and/or precision of the data.

Linked from Computational Modeling