Hello, please do it perfectly and soon

(a) Here we consider the QR algorithm when combined with the reduction to tridiagonal form (for symmetric matrices). Show that the steps of the QR algorithm preserve the tridiagonal structure. (b) When computing the QR factorisation for a tridiagonal matrix, it is a waste to use a Householder rotation for the entire below diagonal component of the column as all but one of the entries in that component are already zero. Propose a less wasteful approach that only uses 2 x 2 Householder rotations on the component of the column containing non-zero entries. Provide a brief explanation of the difference in operation count between the two approaches. (c) Implement your proposed approach as a Python function mr.factor.tri, supported by appropriate tests that you should add. It should avoid multiplication by, or addition of, values that are known to be zero. Further, it should not compute the mxm matrix Q, but just return all of the 2-dimensional vectors v used to generate the Householder reflections. (d) Write a new function gr.alg.tri implementing the unshifted QR algorithm applied to tridiagonal ma- trices, using your code implemented in the last step. It should avoid multiplication by, or addition of values that are known to be zero. It should not explicitly form the Q matrix, but instead work with the Householder reflection generators that are produced in qr.factor.tri. Your function should stop when the m, m - 1 element of the m x m tridiagonal matrix 7 satisfies Tm.m-11