We expect that an iterative method, such as Jacobi or Gauss-Seidel, will produce a sequence of approximations that get closer and closer to the true solution. In this problem we consider the question of whether we ever reach the true solution exactly. Use Jacobi’s Method to solve the system
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Since the true solution is x = (1, 1), let us center the viewing window around that point, by changing the minimum and maximum boundaries for both x1 and x2 to −4 and 6 (bottom left part of the applet — be sure to press Enter after entering the new values). For this problem, use an initial guess of x(0) = (4, −3). Also, for this problem do not write down the results of your iterations.
- Do 10 iterations. On the graph in the applet, does it appear that the approximations have already reached the true solution? Now zoom in about 10 times by clicking on the Zoom in button, and answer the same question.
- Do 10 more iterations, for a total of 20, and answer the same question as in (a). As in (a), zoom in about 10 more times and answer the same question again.
- Does it appear that we will ever reach the solution exactly? Although it would be nice to have the true solution exactly, is an approximation actually good enough? (Note: if you attempt to continue to iterate and zoom in, you will eventually, perhaps quickly, exhaust the precision of your computer, and it may produce strange results — your computer can zoom in only so far.)