romberg_gauss_intro_2010

# romberg_gauss_intro_2010 - Revisiting Numerical Integration...

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Revisiting Numerical Integration: Getting More from Fewer Points Part II March 2010 Bruce Cohen Lowell High School, SFUSD http://www.cgl.ucsf.edu/home/bic David Sklar San Francisco State University Ver. 5.00

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Plan Romberg integration: getting more with fewer points Gaussian quadrature: getting even more with even fewer points Bibliography
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 ( 29 2 3 4 1 2 3 4 E h c h c h c h c h = + + + + L ( 29 ( 29 ( 29 , b a I f x dx T h E h = = + ( 29 2 4 6 8 2 4 6 8 E h c h c h c h c h = + + + + L We’ll illustrate Richardson’s technique by applying it to the trapezoidal rule. ( 29 ( 29 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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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 ( 29 ( 29 ( 29 ( 29 , 2 , 4 , 8 . T h T h T h T h ( 29 ( 29 ( 29 2 4 6 8 2 4 6 8 I T h E h T h c h c h c h c h = + = + + + + + L 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 = + = + + + + + L ( 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 - = - + + + + L 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 2 2 h h h h h I T T T h c c c - = - + - + + + + L We can eliminate the h 2 term by multiplying the second by 2 2 and subtracting the first giving
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romberg_gauss_intro_2010 - Revisiting Numerical Integration...

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