1. In this part you will find the unique degree 20 polynomial that interpolates the 21...

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1. In this part you will find the unique degree 20 polynomial that interpolates the 21 points given above, using Newton's divided differences algorithm. Then you will evaluate, and plot, your polynomial. Note that if you plot a polynomial using just a few points with a plotting program that connects points with straight lines, you will not get a smooth graph (think about a parabola graphed with just three points). Expect to use a lot of points to get a smooth graph of your degree 20 polynomial. Follow these steps: (a) Evaluate the divided differences of order 0-20 for the data given in the table above, using the Matlab code divdif.m. To be sure you understand how to use the program, first try to reproduce the results of Table 4.1 on p. 128 of our textbook: input the ri's and cosi's given in the table (as "x_nodes" and "y_values", respectively) and see if you get the Di's (as divdif_y"). (b) Now calculate Newton's form of the interpolation polynomial using the Matlab code in- terp.m. You will need the divided differences you computed in (a) as inputs to interp.m. You get to choose the r-values at which to compute the value of the polynomial - they go in the input vector "x_eval." Now you want to plot your results, so go on to part (c). (c) Start by evaluating your degree 20 polynomial at a small number of points in an interval which includes the duck, say [0.9, 13.3] (you could also start a little ahead of and end a little behind the duck, if you want). If you choose 30 equally spaced x-values between 0.9 and 13.3, like I did, then you'll get a plot that looks something like this (the orange curve is the top profile of the duck; the purple curve is some representation of P20(x)): Interpolating polynomial, 30 points evaluated Does it look like 30 equally-spaced points is enough to visualize your polynomial well? Probably not a polynomial is smooth, and this looks choppy. You need to plot more points to do a good job of representing your polynomial. Start off by evaluating P20(x) at, say, 50 equally spaced points, and keep on increasing the number of points until the graph is smooth and stops changing. (It will take quite a few points.) The plot that you turn in with your report should show, together on the same plot, the duck with a marker at each of its 21 points and a smooth graph of your interpolating polynomial, each curve clearly labeled. If the graph of your polynomial looks choppy anywhere, then you haven't evaluated it at enough points. d) Comment on what you observe (reading the remarks at the end of this Project may help). Would you expect better results if you used Chebyshev nodes? If so, why? Would it be easy to use Chebyshev nodes in this probem? 1. In this part you will find the unique degree 20 polynomial that interpolates the 21 points given above, using Newton's divided differences algorithm. Then you will evaluate, and plot, your polynomial. Note that if you plot a polynomial using just a few points with a plotting program that connects points with straight lines, you will not get a smooth graph (think about a parabola graphed with just three points). Expect to use a lot of points to get a smooth graph of your degree 20 polynomial. Follow these steps: (a) Evaluate the divided differences of order 0-20 for the data given in the table above, using the Matlab code divdif.m. To be sure you understand how to use the program, first try to reproduce the results of Table 4.1 on p. 128 of our textbook: input the ri's and cosi's given in the table (as "x_nodes" and "y_values", respectively) and see if you get the Di's (as divdif_y"). (b) Now calculate Newton's form of the interpolation polynomial using the Matlab code in- terp.m. You will need the divided differences you computed in (a) as inputs to interp.m. You get to choose the r-values at which to compute the value of the polynomial - they go in the input vector "x_eval." Now you want to plot your results, so go on to part (c). (c) Start by evaluating your degree 20 polynomial at a small number of points in an interval which includes the duck, say [0.9, 13.3] (you could also start a little ahead of and end a little behind the duck, if you want). If you choose 30 equally spaced x-values between 0.9 and 13.3, like I did, then you'll get a plot that looks something like this (the orange curve is the top profile of the duck; the purple curve is some representation of P20(x)): Interpolating polynomial, 30 points evaluated Does it look like 30 equally-spaced points is enough to visualize your polynomial well? Probably not a polynomial is smooth, and this looks choppy. You need to plot more points to do a good job of representing your polynomial. Start off by evaluating P20(x) at, say, 50 equally spaced points, and keep on increasing the number of points until the graph is smooth and stops changing. (It will take quite a few points.) The plot that you turn in with your report should show, together on the same plot, the duck with a marker at each of its 21 points and a smooth graph of your interpolating polynomial, each curve clearly labeled. If the graph of your polynomial looks choppy anywhere, then you haven't evaluated it at enough points. d) Comment on what you observe (reading the remarks at the end of this Project may help). Would you expect better results if you used Chebyshev nodes? If so, why? Would it be easy to use Chebyshev nodes in this probem?