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Graphical iteration It is perhaps more instructive to follow through the iteration steps of table 1 in graphical form. To do this, we first plot the function f(x) of equation (2) by a red round curve in figure 2, and then draw a 45° diagonal in blue, which is the function x. It should be noted that the intersection of the round curve and diagonal would correspond to x = f(x), which occurs at x = 0.5 in figure 2. Let us begin by marking initial population x0 = 0.3 on the x-axis of figure 2. From x0 we draw a vertical line to the red round curve; the intersection corresponds to the first iterate x1 = 0.42. Next, we extend the line horizontally to the diagonal and again vertically to the red round curve; the intersection is the second iterate x2 = 0.4872. Similarly, by a horizontal and vertical line extension the third iterate is located, and so on. However, as the iteration proceeds further, you see in figure 2 that the horizontal and vertical line segments get trapped in the wedge of round curve and diagonal. Hence, they all pile up on the intersection x = 0.5, which is the fixed point of x = f(x) that we are seeking.
Figure 2. Graphical iteration starting from |
