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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
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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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