Trapezoidal rule and its error analysis for the Grünwald-Letnikov operator

Matthew Harker, Paul O'Leary

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)


In this paper, the trapezoidal rule for the Grünwald-Letnikov operator is derived. It is a trapezoidal rule in the sense that the formula yields the exact Grünwald-Letnikov derivative/integral of a piecewise linear function. Firstly, the formula for evenly spaced points is derived, and is used as a basis to derive the equivalent formula for arbitrary abscissae. Further, an analytic bound on the residual error is derived, which depends on a bound on the second derivative of the function. The derived trapezoidal rule can therefore be used to compute fractional integrals and derivatives to within a given error tolerance. Through numerical testing it is shown that the new formula yields results that are orders-of-magnitude more accurate than the classical formula, even for arbitrary functions. A simple adaptive algorithm is proposed for computing the result of applying the Grünwald-Letnikov operator to a function to within a desired accuracy.
Original languageEnglish
Pages (from-to)18-29
Number of pages12
JournalInternational Journal of Dynamics and Control
Issue numberMarch
Early online date11 Mar 2016
Publication statusPublished - 1 Mar 2017


  • Error analysis
  • Grünwald-Letnikov operator
  • Trapezoidal rule

Cite this