Write a script in MATLAB that implements the power method for finding the larges

Question

Write a script in MATLAB that implements the power method for finding the largest eigenvalue by magnitude and compute the following for the matrices below:

a) Use the power iteration to compute an estimate for the magnitude of the largest eigenvalue of the

matrices A10, A20, A50, and A100.

b) For A100, show the current value of (k) (i.e., the current estimate of the magnitude of the largest eigenvalue) as a function of k. That is, create a graph in which you plot (k) against k for k = 1, . . . , 1000. Also state how many iterations you need until the first three significant digits of the estimate (k) do not change any more from one iteration to the next.

c) From parts (a) or (b), you can obtain an estimate for |1| of A100 if you just run the method for long enough. Since it’s not the exact value, let’s call this estimate 1. This now allows you to take the values (k) again and plot (an approximation to the) error e(k) = |(k) 1| as a function of k. State your estimate 1 and show such a plot. Because the errors will become quite small after some time, you will want to use a logarithmic scale for the y-axis.

d) Verify that the relationship of e(k) and k indeed corresponds to linear convergence where e(k) = Ce(k1) = Cke(0) with a C < 1. What C do you infer from the data you have collected?

Explanation / Answer

The matlab script:

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