hwk5 - worth it becomes This famous example is due to Carl...

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Homework # 5 Due Thursday, 02/14 1. The two important proofs in “Polynomial Interpolation” are the proof of existence and uniquness of the interpolating polynomial the error estimate (a) Make sure you understand these two proofs. Read them one more time. Make sure you understand that finding an interpolating polynomial is equivalent to solving a matrix equation where the unknown are the coefficient of the polynomial. (b) Read the proof of the error estimate which is in your book (page 108): this is the same proof than we did in class, but the wording and notations are slighlty different. (It is always good to have two points of view.) 2. Use the mean value theorem to prove that if G ( x i ) = G ( x i +1 ) = 0, then G ( x ) has a zero between x i and x i +1 . (We did use this fact in class while proving the error estimate). (Also you need to assume that G is differentiable) 3. Runge Phenomenon In this problem, we will see that polynomial interpolation doesn’t work well to interpolate the function f ( x ) = 1 1 + x 2 on the interval [-5,5]. To be more precise, we will see that the more node points you put in [-5,5], the
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Unformatted text preview: worth it becomes. This famous example is due to Carl Runge. Let x , x 1 , . . . , x n be n + 1 equally spaced points between-5 and 5 (with x =-5 and x n = 5). Write a matlab code to compute and plot the unique polynomial p n ( x ) of degree ≤ n which interpo-lates f ( x ) on these n + 1 points. You will need to use the divided difference algorithm and the nested evaluation for polynomial that you have seen during the discussion. Do it for n = 2 , n = 4 , n = 6 , n = 10 , n = 14 , n = 30 You should provide one matlab code (not 6) and 6 plots. On each plot, I want to see both f ( x ) and p n ( x ). Also the node points ( x i , f ( x i )) should be marked by a cross. (see next page for an example of what your plot should look like. Do ’help plot’ if you don’t know how to do it). 1-5-4-3-2-1 1 2 3 4 5-0.5 0.5 1 1.5 2 inter. poly. of degree 10 f(x) node points 2...
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