Lecture 36

# Lecture 36 - Chapter 17 Numerical Integration of Functions...

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Chapter 17 Numerical Integration of Functions

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Numerical Integration Tabulated data – the accuracy of the integral is limited by the number of data points Continuous function – we can generate as many f(x) as required to attain the required accuracy Richardson extrapolation and Romberg integration Gauss Quadratures
n I n E n = |I n -I exact | Ratio 4 1.10384615384615 0.003302564 8 1.10631663171835 0.000832086 3.969017 16 1.10694046254292 0.000208255 3.995511 32 1.1070966393429 5.20785E-05 3.998876 2 2 0 1 1 I dx x = + I exact = tan -1 (2) = 1.10714871779409… Evaluate using trapezoidal rule with n=4, 8, 16, 32 intervals ≈4

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Round-off errors may limit the precision of lower- order Newton-Cotes composite integration formula Use Romberg Integration for efficient integration 1 2 1 4 How do we take advantage of this error behavior? Error
Romberg integration - uses Richardson extrapolation Idea behind Richardson extrapolation - improve the estimate by eliminating the leading term of truncation error at coarser grid levels Romberg Integration

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The exact integral can be represented as This is true for any h = (
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Lecture 36 - Chapter 17 Numerical Integration of Functions...

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