A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions | Zendy

Mohsin Javed | Zendy; Lloyd N. Trefethen | Zendy

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A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions

Author(s) -

Mohsin Javed,

Lloyd N. Trefethen

Publication year - 2013

Publication title -

proceedings of the royal society a mathematical physical and engineering sciences

Language(s) - English

Resource type - Journals

eISSN - 1471-2946

pISSN - 1364-5021

DOI - 10.1098/rspa.2013.0571

Subject(s) - midpoint method , trapezoidal rule , mathematics , discretization , midpoint , euler's formula , mathematical analysis , boundary (topology) , quadrature (astronomy) , limit (mathematics) , upper and lower bounds , convergence (economics) , analytic function , numerical integration , geometry , economic growth , electrical engineering , economics , engineering

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another. Similar results are also given for the midpoint rule.

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