are bigger than the absolute value of the off-diagonals, but the magni eo Exercise 3 Determine a set of test points and charge locations for the 2 square problem that generates an A matrix where the magnitude of the diagon 8.1.3 Computational Results SLIDE 14 Circle w⊥ th char Potentials on the n=20 n=40 ote ll It is possible to construct a numerical scheme for solving exterior Laplace prob- lems by adding progressively more point charges so as to match more boundary onditions. In the slide above, we show an example of using such a method to compute the potential exterior to a circle of radius 10, where the potential or the circle is given to be unity. In the example, charges are placed uniformly on a circle of radius 9.5, and test points are placed uniformly on the radius 10 circle. If 20 point charges are placed in a circle of radius 9.5, then the potential produced will be exactly one only at the 20 test points on the radius 10 circle. The potential produced by the twenty point charges on the radius 10 circle is plotted in the lower left corner of the slide above. As might be expected, the potential produced on the radius 10 circle is exactly one at the 20 test points but then oscillates between l and 1.2 on the radius 10 circle. If 40 charges and test points are used, the situation improves. The potential on the circle still oscillates, as shown in the lower right hand corner, but now the amplitude is only between 1 and 1.004 Exercise 3 Determine a set of test points and charge locations for the 2-D square problem that generates an A matrix where the magnitude of the diagonals are bigger than the absolute value of the off-diagonals, but the magnitude of the diagonal is smaller than the absolute sum of the off-diagonals. 8.1.3Computational Results Slide 14 R=10 r Circle with Charges r=9.5 n=20 n=40 Potentials on the Circle Note 11 results It is possible to construct a numerical scheme for solving exterior Laplace prob￾lems by adding progressively more point charges so as to match more boundary conditions. In the slide above, we show an example of using such a method to compute the potential exterior to a circle of radius 10, where the potential on the circle is given to be unity. In the example, charges are placed uniformly on a circle of radius 9.5, and test points are placed uniformly on the radius 10 circle. If 20 point charges are placed in a circle of radius 9.5, then the potential produced will be exactly one only at the 20 test points on the radius 10 circle. The potential produced by the twenty point charges on the radius 10 circle is plotted in the lower left corner of the slide above. As might be expected, the potential produced on the radius 10 circle is exactly one at the 20 test points, but then oscillates between 1 and 1.2 on the radius 10 circle. If 40 charges and test points are used, the situation improves. The potential on the circle still oscillates, as shown in the lower right hand corner, but now the amplitude is only between 1 and 1.004. 10