Just do 2.4 PART A. Please, also comment on the relative merits of the different techniques suggested in 2.4.

2.4. Write a routine for estimating the condition number of a matrix A. You may use either the 1-norm or the oo-norm (or try both and compare the results). You will need to compute ||Al, which is easy, and estimate A which is more chal- lenging. As discussed in Section 2.3.3, one way to estimate ||As to choose a vector y such that the ratio | is large, where z is the solu tion to Az y. Try two different approaches to choosing y: (a) Choose y as the solution to the system A y c, where c is a vector each of whose components is +1, with the sign for each component chosen by the following heuristic. Using the factoriza- tion A-LU, the system Aly c is solved in two stages, successively solving the triangular systems UTU = C and Liy-u. At each step of the first triangular solution, choose the cor- responding component of c to be 1 or-, de- pending on which wi make the resulting com- ponent of v larger in magnitude. (You will need to write a custom triangular solution routine to implement this.) Then solve the second triangu- lar system in the usual way for y. The idea here is that any ill-conditioning in A wi be reflected in U, resulting in a relatively large v. The rel- atively well-conditioned unit triangular matrix L will then preserve this relationship, resulting in a relatively large y (b) Choose some small number, say, five, different vectors y randomly and use the one producing the largest ratio 11 /llyll. (For this you can use an or- dinary triangular solution routine.) You may use a library routine to obtain the nec- essary LU factorization of A. Test both of the approaches on each of the following matrices: 10 -7 0 5 -15 73 78 24 -80 37 10 A13 2 6 A2-1 92 66 25 How do the results using these two methods com- pare? To check the quality of your estimates, com- pute A explicitly to determine its true norm this computation can also make use of the LU factorization already computed). If you have ac- cess to linear equations software that already in- cludes a condition estimator, how do your results compare with its