Arizona State University



Curve Fitting and Applications of DiagonalizationDirections in green on this worksheet tell you what exactly to turn in. Do not turn in a MATLAB diary. Create an OpenOffice text document and write/copy final answers into this document. You may submit a hardcopy or electronic copy by email. Your document cannot have more than three pages.Find all square roots of A=90-41, i.e. all matrices B with B2=A. Do “help eig” in MATLAB to find out how to compute eigenvalues and eigenvectors of a matrix.Turn in: the list of matrices.A Vandermonde matrix is a square matrix whose columns contain the powers 0, 1, 2, .. (n-1) for a list of n numbers. Compute vander([1,2,3,4])in MATLAB for an example. Observe that the MATLAB version of the Vandermonde matrix starts goes from higher to lower power. You can use fliplr to reverse the order of the columns.Turn in : nothing.A science experiment produced the following set of points:(1, 3.0975) (2, 6.2785) (3, 9.5469) (4, 12.9575) (5, 15.9649) (6, 18.1576)Compute the 5th degree polynomial that interpolates (exactly goes through) the points using a Vandermonde matrix. Observe that this part of the question is NOT asking for a least squares fit. Now compute the best straight line fit using least squares. Do not set up a new matrix; extract the needed columns from the Vandermonde matrix. Recall from our first lab that the left division operator (“\”) will automatically find the least squares solution of an overdetermined system for you.Turn in: a plot containing the 5th degree polynomial (in RED), the straight line (in GREEN) and the original points (as point-only plot with dots representing points.)The linear fit of the data you created in the previous problem is only an approximation. Yet, a working scientist may prefer this fit to the exact 5th degree solution and not just for the simple reason that a linear function is mathematically more expedient to work with. Turn in: an explanation in one or two sentences.Use the following points x = 0.1:0.1:1y= [3.2641 2.8281 5.0391 3.9653 5.4878 6.3570 4.3012 10.8792 10.0212 11.7558]and compute the 9th degree polynomial that interpolates them, using the same technique as in problem 3. Plot the points, and the polynomial on the domain [0.1, 1]. What do you observe?Now create a best quadratic fit and add it to the plot. What is your conclusion?Turn in: a plot containing the 9th degree polynomial, the quadratic and the original points, on the domain [0.1, 1], and a one-sentence conclusion you draw from this experiment.Earlier during the course, we examined a Markov process that modeled the brand loyalty of Windows, Mac and Linux users. We came up with the following model:xnynzn=0.60.050.20.10.940.050.30.010.75Anx0y0z0We found the steady state distribution of the system by observing that the transition matrix had a limit as n goes to infinity, and we approximated that limit A∞ by raising A to a large power.Now that we know eigenvalue theory, we can do better.Diagonalize the transition matrix and use that to compute the steady state without approximation. Call the change of basis matrix from the eigenbasis to the standard basis V. Turn in: a formula that gives A∞ in terms of V, V-1 and a diagonal matrix. Do not substitute a numerical approximation for V and V-1. Inspired by the example of the previous problem, describe a condition involving the eigenvectors of a diagonalizable matrix that guarantees that An has a nonzero limit as n goes to infinity.Turn in: a condition, and a brief explanation. ................
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