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romberg_gauss_intro_2010[1]

# romberg_gauss_intro_2010[1] - A Short Introduction to...

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A Short Introduction to Romberg Integration and Gaussian Quadrature: Getting More from Fewer Points April 2010 David Sklar San Francisco State University [email protected] Ver. 1.00

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Romberg Integration The method arises from a technique called Richardson extrapolation which can be used whenever the error E ( h ) can be expanded in a series of the form ( 2 3 4 1 2 3 4 E h c h c h c h c h = + + + + ( 29 ( 29 ( 29 , b a I f x dx T h E h = = + ( 2 4 6 8 2 4 6 8 E h c h c h c h c h = + + + + We’ll illustrate Richardson’s technique by applying it to the trapezoidal rule. ( ( 2 with E h O h = In fact we can show that if f can be expanded in a Taylor series on each subinterval then E can be expanded in a series of the form
Romberg Integration To implement the method we don’t need to know the coefficients, we need only know that they exist. Assume that we have computed trapezoidal estimates for I for h, h/2, h/4, h/8, so we have Just looking at the first two we have ( ( ( ( , 2 , 4 , 8 . T h T h T h T h ( ( ( 2 4 6 8 2 4 6 8 I T h E h T h c h c h c h c h = + = + + + + + 2 4 6 8 2 4 6 8 2 4 6 8 and 2 2 2 2 2 2 2 h h h h h h h I T E T c c c c = + = + + + + + ( 29 ( 29 4 6 8 2 2 4 6 8 4 6 8 2 1 2 2 2 2 2 h h h h I T T h c c c - = - + + + + Which we can rewrite as ( 29 ( 29 ( 29 4 6 8 2 2 4 6 8 4 6 8 2 1 2 1 2 2 2

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