This is a project that professor ask us to do. But I have no idea how to do it. Thank you very much!

If we think of this method as a train taking us to the unique solution of our linear system located at the bottom of a valley, its track is taking us on the fastest route to the bottom from our current starting point by stopping along the way and changing directions if there is a faster route now available from the new stop. The next iterate x1 gives us the rst stop we make on this route from our starting point 3:0 to the solution x'. Using the minimum found in part (b) of Exercise 1.3 for t5 , our rst stop is the terminal point of the position vector in: x1=xo+t3 do. We repeat the above process to get (11 = VF (x1) = (Ax1 1;) and ti' to move to the next stopping point x2: X2 = XI + t; d1. After It: stops, we are at the point xk = Xkl + $2-1 dkl: where ,. _ Ildk_1|l2 _1 a dkl ' (Adkll and . dkl = (Axk1 bl- With every stop, we get closer to the minimum of the function F, which is the unique solution to our linear system. This process does not necessarily terminate in a nite number of steps; to implement the steepest descent algorithm, we need to specify a stopping rule, or a tolerance for how \"bad\" our nal iterate is allowed to be. One way to set this \"badness\" tolerance is to use the magnitude ||dk||. This magnitude measures how well xk solves the linear system. So if E is our prescribed tolerance (could be relatively large for a hard problem or as small as machine epsilon when you need to be precise), then the algorithm for the method of steepest descent can be described as follows: Prescribe a tolerance 6. Start with any initial itemte x0. Compute . (1) dkl = (Axk1 bl, Ildklll2 2 t*_ =, U k 1 dbl-(Adan (3) xk = xk_1 + t;_1 dk_1. Compute ||dk||. If ||dk|| 2 E, repeat computations (1)(3) for the neat itemte. Else if ||dk||