Numerical method&analysis.

Taking A to be a 10 by 10 matrix, try the following:

(a) What information does Gerschgorins theorem give you about the eigenvalues of this matrix?

(b) Implement the power method to compute an approximation to the eigenvalue of largest absolute value and its corresponding eigenvector. [Note: Use a random initial vector. If you choose something special, like the vector of all 1s, it may turn out to be orthogonal to the eigenvector you are looking for.] Turn in a listing of your code together with the eigenvalue/eigenvector pair that you computed. Once you have a good approximate eigenvalue, look at the error in previous approximations and comment on the rate of convergence of the power method.

(c) Implement the QR algorithm to take A to diagonal form. You may use MATLAB routine [Q,R] = qr(A); to perform the necessary QR decompositions. Comment on the rate at which the off-diagonal entries in A are reduced.

(d) Take one of the eigenvalues computed in your QR algorithm and, using it as a shift in inverse iteration, compute the corresponding eigenvector.

5. Consider the matrix -1 2 A= Taking A to be a 10 by 10 matrix, try the following: (a) What information does Gerschgorin's theorem give you about the eigenvalues of this (b) Implement the power method to compute an approximation to the eigenvalue of matrix? largest absolute value and its corresponding eigenvector. [Note: Use a random initial vector. If you choose something special, like the vector of all 1s, it may turn out to be orthogonal to the eigenvector you are looking for.] Turn in a listing of your code together with the eigenvalue/eigenvector pair that you computed. Once you have a good approximate eigenvalue, look at the error in previous approximations and com- ment on the rate of convergence of the power method. (c) Implement the QR algorithm to take A to diagonal form. You may use MATLAB rou- tine to,R] q(A) to perform the necessary QR decompositions. Comment on the rate at which the off-diagonal entries in A are reduced. (d) Take one of the eigenvalues computed in your QR algorithm and, using it as a shift in inverse iteration, compute the corresponding eigenvector