Let A be a symmetric, tridiagonal matrix. You learned that the matrices A defined by the...

 

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Let A be a symmetric, tridiagonal matrix. You learned that the matrices A defined by the QR-algorithm converge to a diagonal matrix that is similar to (and thus has the same eigenvalues as) A. The convergence speed depends on the absolute value of the ratio of consecutive eigenvalues. Let r = (0, 1) and A = [(} }]) (a) Calculate the eigenvalues of A as a function of r (by hand). (b) Implement the QR-algorithm using MATLAB's (or Python's) implementation of the QR-factorization, qr(). Your code should run for a quadratic matrix of any size. (c) Now define a tolerance, e.g., T = 10-10. Introduce a stopping criterion in your code, causing it to stop when the maximal difference between the true eigenvalues of A and the diagonal entries of A is smaller than T. (d) Use your code with the matrix given for at least five values of r (0, 1) and make a plot with r versus the number of iterations needed to achieve the given tolerance. Explain your findings by examining the ratio between the eigenvalues of A using (a). Let A be a symmetric, tridiagonal matrix. You learned that the matrices A defined by the QR-algorithm converge to a diagonal matrix that is similar to (and thus has the same eigenvalues as) A. The convergence speed depends on the absolute value of the ratio of consecutive eigenvalues. Let r = (0, 1) and A = [(} }]) (a) Calculate the eigenvalues of A as a function of r (by hand). (b) Implement the QR-algorithm using MATLAB's (or Python's) implementation of the QR-factorization, qr(). Your code should run for a quadratic matrix of any size. (c) Now define a tolerance, e.g., T = 10-10. Introduce a stopping criterion in your code, causing it to stop when the maximal difference between the true eigenvalues of A and the diagonal entries of A is smaller than T. (d) Use your code with the matrix given for at least five values of r (0, 1) and make a plot with r versus the number of iterations needed to achieve the given tolerance. Explain your findings by examining the ratio between the eigenvalues of A using (a).