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Unformatted text preview: MAE 107 – Computational Methods Summer Session II 2009 Final Exam Friday, September 4 5 Problems + 6 Short questions – 50 points Guidelines : This is a closed book and closed notes 3h exam. No calculator or computer allowed . To obtain full credit to a question, you need to clearly justify your answers . Detail all necessary intermediate steps and clearly identify your final answers (e.g. box them in). Do not forget to put your name and student ID on your test and staple/attach all the pages together. Problem 1 – 12 points 1. In this section, we consider the second-order polynomial p 2 ( x ) interpolating three equispaced data points ( x 1 ,f ( x 1 )), ( x 2 ,f ( x 2 )) and ( x 3 ,f ( x 3 )). (a) Using Newton’s formula, write down the expression for p 2 ( x ) in terms of x , x 1 , x 2 , f ( x 1 ), f [ x 1 ,x 2 ] and f [ x 1 ,x 2 ,x 3 ]. Also, give the definition of f [ x 1 ,x 2 ] and f [ x 1 ,x 2 ,x 3 ]. (b) Assuming h = x 3- x 2 = x 2- x 1 (equispaced data points), simplify the expression obtained in the previous question to show that: p 2 ( x ) = f ( x 1 ) + f ( x 2 )- f ( x 1 ) h ( x- x 1 ) +...
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- Summer '08
- Numerical Analysis, Romberg's method