Lecture37

# Lecture37 - Numerical Integration Tabulated data the...

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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

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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
More efficient methods to achieve better accuracy have been developed 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 = ( b - a )/ n Use trapezoidal rule as an example ( 29 ( 29 h E h I I + = ( 29 ( 29 ( 29 ( 29 2 2 1 1 h E h I h E h I I + = + = ( 29 ( 29 ( 29 2 2 2 1 2 1 2 h h h E h E f h 12 a b E 2245 - - = ) ( ξ Richardson Extrapolation
Composite Trapezoidal Rule Evaluate the integral dx xe I 4 0 x 2 = [ ] [ 4, (0) 2 (1) 2 (2) 2 2 (3) (4) 7288.79 8, (0) 2 (0.5) 2 (1) 2 2 ( 39.71% 1.5) 1 0. 2 5 h n I f f f f f h n I f h f f f h ε = = + + + + = = = + = - = + = + + ] [ (2) 2 (2.5) 2 (3) 2 (3.5) (4) 5764.76 16, (0) 2 (0.25) 2 (0.5) 2 10.5 2 (3.5) 2 ( 0.25 0% h f f f f f h n I f f f f f + + + + = = = = + + + + + - = L ] 3.75) (4) 5355.95 .6 % 2 6 f - + = = Error decreases by a factor of 4 when h is reduced by a factor of 2 because ( 29 ( 29 ( 29 2 1 1 2 2 2 2 2 2 ( ) 12 4 ( / 2) b a E h h h E f E h h h h ξ - ′′ = - 2245 = =

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Truncation error for trapezoidal rule Substitute into the exact integral Which leads to ( 29 ( 29 2 2 1 2 1 h h h E h E 2245 ) ( ) ( ) ( ) ( 2 2 2 2 1 2 1 h E h I h h h E h I I + 2245 + = ( 29 2 2 1 2 1 2 h h 1 h I h I h E ) / ( ) ( ) ( - - 2245 Richardson Extrapolation

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Lecture37 - Numerical Integration Tabulated data the...

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