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--- gibson:teaching:fall-2014:iam961:iam-961-hw5 [2014/12/05 09:57]
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-**Problem 1:** Implement Rayleigh quotient iteration. Apply
+**Problem 1:** Implement Rayleigh quotient iteration and demonstrate its convergence, as follows. Develop your code with a small matrix (maybe 5 x 5) and do the final demonstration with a larger matrix (maybe 100 x 100).
+. Construct a random symmetric M x M matrix A with random but known eigenvalues and eigenvectors. Hint: construct A from a random orthogonal matrix V and a known random diagonal matrix D. (4 lines of code).
+. Select a eigenvalue of A and the corresponding eigenvector (an element of D and a column of V). Verify that these are in fact an eigenvalue, eigenvector pair. (3 lines of code).
+. To get an initial guess for the Rayleigh quotient iteration, perturb the eigenvector by 10% or so and evaluate the Rayleigh quotient using the perturbed eigenvector. (5 lines of code).
+. Now do the Rayleigh quotient iteration, stopping when $\|Av-\lambda v\| < 10^{-14}$. Plot the errors in the eigenvalue and eigenvector as a function of the iteration number. The Rayleigh quotient iteration code takes about 3 lines of code, the plotting more. Don't forget to label your axes!
+Answer these questions:
+Are you amazed, or what?
+Can you confirm that the errors scale as stated on page 208 of Trefethen and Bau?