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Numerical Integration

  • Vilmos Komornik
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

For his celestial mechanics Newton had to evaluate complicated integrals. He and his followers generalized the trapezoidal rule \(\int _{a}^{b}f(x)\ dx \approx \frac{f(a)+f(b)} {2} (b - a)\) and Simpson’s rule \(\int _{a}^{b}f(x)\ dx \approx \frac{f(a)+4f(\frac{a+b} {2} )+f(b)} {6} (b - a)\) by seeking approximations of the form \(\int _{I}f(x)\ dx \approx A_{1}f(x_{1}) + \cdots + A_{n}\,f(x_{n})\) where the points x k and coefficients A k do not depend on the particular choice of the function f.

Keywords

Asymptotic Expansion Orthogonal Polynomial Trapezoidal Rule Celestial Mechanic Compact Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  • Vilmos Komornik
    • 1
  1. 1.Department of MathematicsUniversity of StrasbourgStrasbourgFrance

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