1. (Alternative version of power methods) In class, we derived the following algorithm, Set q =...

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1. (Alternative version of power methods) In class, we derived the following algorithm, Set q = qo Until convergence, do z + Aq q+2/22 The second step A+q'Aq end do q+z/||2||2 is the rescaling step to prevent underflow and overflow. There is an alternative way to rescale. In this problem we examine the following algorithm, Set q=90 Until convergence, do z + Aq end do end do where is the element in z with the largest absolute value. For example, if z = [134], then=-4. Here the step qz/u is the rescaling step. The step approximate the dominant eigenvalue. is used to a. Perform two iterations of power methods using both algorithms to the matrix Take initial guess q = [11]. You should obtain q2 and A2 for both methods. b. For the second algorithm, explain that the largest number in q is always 1. 2 (DUE IN CLASS MARCH 24) c. For the second algorithm, assume qev as koo. Show that A as k . Here A is the dominant eigenvalue and v is the associated eigenvector with largest component 1. Hint. For part [c.], from the first and third step of the second algorithm, we know qk+1= Aq/k. Then use the definition of eigenvalues and part [b.] to show the result. 1. (Alternative version of power methods) In class, we derived the following algorithm, Set q = qo Until convergence, do z + Aq q+2/22 The second step A+q'Aq end do q+z/||2||2 is the rescaling step to prevent underflow and overflow. There is an alternative way to rescale. In this problem we examine the following algorithm, Set q=90 Until convergence, do z + Aq end do end do where is the element in z with the largest absolute value. For example, if z = [134], then=-4. Here the step qz/u is the rescaling step. The step approximate the dominant eigenvalue. is used to a. Perform two iterations of power methods using both algorithms to the matrix Take initial guess q = [11]. You should obtain q2 and A2 for both methods. b. For the second algorithm, explain that the largest number in q is always 1. 2 (DUE IN CLASS MARCH 24) c. For the second algorithm, assume qev as koo. Show that A as k . Here A is the dominant eigenvalue and v is the associated eigenvector with largest component 1. Hint. For part [c.], from the first and third step of the second algorithm, we know qk+1= Aq/k. Then use the definition of eigenvalues and part [b.] to show the result.