HW7 (1)

EGM 3344 Summer 2023 Introduction to Numerical Methods of Engineering Analysis University of Florida Mechanical and Aerospace Engineering Homework 7: I’m Looking for a Great r 2 . Oh, Great r 2 . r 2 Not Make One Great Due: Friday, July 28, 2023, 11:59pm. Instructions: 1. Submit your solutions as a PDF document on Canvas. 2. Include any and all plots in the PDF. 3. Also submit your individual .m files. 4. Do not submit a zipped file. 5. You may use MATLAB’s left division to solve systems of linear equations as long as they are square. 6. Remember to use a superfluous amount of comments! Problem 1: Below are the floor-routine scores and the total scores for the University of Florida Gymnastics team 1 for each meet in 2023. Write a MATLAB script called hw7_p1.m to fit a line to the data, to fit a parabola to the data, and to fit an exponential function to the data using linear least squares. Plot the line, the parabola 2 , the exponential, and the data in the same plot. (Remember that parabolas and exponential functions are smooth and that data points are points 3 .) Floor Total 49.4 197.75 49.625 197.825 49.3 197.325 49.425 197.9 49.25 197.875 49.725 198.35 49.575 197.975 49.6 198.15 49.175 197.7 49.6 198.175 49.675 198.425 49.4 197.875 1 Go Gators!! 2 MATLAB will give you a warning for the parabola. You can ignore it. 3 This should be obvious, but you would be surprised, and by “you,” I mean one of y’all’s’ll roll into office hours wondering why your parabola/exponential is a line. . . That is okay—that is the reason we have office hours. Jonathan Brooks 1

EGM 3344 Summer 2023 Discuss your results: Which fit best models the 2023 gymnastics team and why? Is it a good model? As engineers, what should we advise Coach Rowland? Jenkins 4 , analysis! Problem 2: The file hw7_p2.mat contains US daily coronavirus cases for the months of March, April, and May of 2020. Write a MATLAB script called hw7_p2.m to compute the cubic polynomial of best fit. Use this “cubic model” to predict the number of daily cases in August 2020. Plot the fitted curve through July 2020, along with the data. Discuss your results: Based on your (probably) limited knowledge of epidemiology, what are the merits and/or critiques of this “cubic model?” Problem 3: On its mission, our rover has collected several soil samples and found various concen- trations of the radioactive isotope, nickel-59. The samples are from different eras in the Martian past. The file, hw7_p3.mat , contains the measurements made by the rover (time in years, where 0 denotes the present; and percent concentration of nickel-59). Write a MATLAB script called hw7_p3.m to use linear least squares to find the line of best fit, parabola of best fit 5 , and exponential function of best fit for the data. Plot the data points and each of your best-fit curves in the same plot. Discuss your results: What are the arguments for or against each model? Caveat: Make sure you do an “apples-to-apples” for the exponential fit. 67 10 bonus points: Use MATLAB’s fminsearch function to find the exponential func- tion of best fit using nonlinear least squares (5 points), and explain why the answer is different (5 points) 8 . Problem 4: This problem uses hw7_p4.mat , which contains the results of a food study. 100 partici- pants tasted 10 cookies made with different recipes using varying amounts of 5 different ingredients and rated their enjoyment levels; each participant tests the same set of 10 cookies. We want to find the equation of the best-fit hyperplane so that we may deter- mine the ultimate cookie recipe! (Because there are 5 ingredients for each cookie and a participant rating, that means there are 6 numbers associated with each data point—5 x values and 1 y value.) Treat each rating from each participant as a separate data point, and write a MAT- LAB script called hw7_p4.m to fit a hyperplane that relates the ingredients to overall enjoyment. Discuss your results: What do these results mean for our cookie recipe? Suggestion: Instead of officially using multiple linear regression, use general linear least squares 9 . Hint: Before you employ least squares, you will need to do a little preprocessing on 4 For the purposes of this sentence, your name is Jenkins. 5 MATLAB will give you a warning for the parabola. You can ignore it. 6 Cryptic! 7 This also applies to problem 1, too, but it is especially important for this problem. 8 Recall I mentioned this in class once before. . . It didn’t scare you enough then. It does now. 9 Or don’t. I’m not your mom. Jonathan Brooks 2