Error Analysis in Numerical Integration: Learn It 1

  • Calculate how far off your numerical approximation might be from the true value
  • Use error-bound formulas to estimate the accuracy of your approximation

Absolute and Relative Error

When you use numerical approximation methods like the midpoint rule or trapezoidal rule to estimate definite integrals, it’s crucial to understand how accurate your approximation actually is. This involves calculating two types of error: absolute error and relative error.

absolute and relative error

If [latex]B[/latex] is your estimate of some quantity having an actual value of [latex]A[/latex], then:

  • Absolute error = [latex]|A - B|[/latex]
  • Relative error = [latex]\left|\frac{A - B}{A}\right| \cdot 100%[/latex]

The relative error expresses the error as a percentage of the true value.

Why Both Matter: Absolute error tells you the actual difference between your estimate and the true value. Relative error tells you how significant that difference is compared to the size of the true value. A small absolute error might still be a large relative error if the true value is very small.

Calculate the absolute and relative error in the estimate of [latex]{\displaystyle\int }_{0}^{1}{x}^{2}dx[/latex] using the midpoint rule, found in the example: Using the Midpoint Rule with [latex]{M}_{4}[/latex].

Calculate the absolute and relative error in the estimate of [latex]{\displaystyle\int }_{0}^{1}{x}^{2}dx[/latex] using the trapezoidal rule, found in the example: Using the trapezoidal rule.

Watch the following video to see the worked solutions to the example above.

You can view the transcript for “3.6.2” here (opens in new window).