Euler–Maclaurin and Gregory interpolants

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• 作者：Mohsin Javed ; Lloyd N. Trefethen
• 关键词：41A05 ; 42A15 ; 65D32 ; 65D05
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：132
• 期：1
• 页码：201-216
• 全文大小：641 KB
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• 作者单位：Mohsin Javed (1)
  Lloyd N. Trefethen (1)
  
  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

Let a sufficiently smooth function \(f\) on \([-1,1]\) be sampled at \(n+1\) equispaced points, and let \(k\ge 0\) be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree \(k\) algebraic polynomial and a degree \(n\) trigonometric polynomial, which deviates from \(f\) by \(O(n^{-k})\) and whose integral is equal to the order \(k\) Euler–Maclaurin approximation of the integral of \(f\). This interpolant makes use of the same derivatives \(f^{( j)}(\pm 1)\) as the Euler–Maclaurin formula. A variant Gregory interpolant is also defined, based on finite difference approximations to the derivatives, whose integral (for \(k\) odd) is equal to the order \(k\) Gregory approximation to the integral. Mathematics Subject Classification 41A05 42A15 65D32 65D05