Example 4.7.  Trace out the next five iterates of Newton's method given an initial guess of    as a solution to the equation  ,  where  .  

Solution.  For any guess z for a solution, Newton's method gives as the next guess the number  .  Table 4.1 gives the required iterates, rounded to five decimal places.  Figure 4.2 shows the relative positions of these points on the z plane.  Note that the points are so close together that they appear to coincide, and that the value for agrees to five decimal places with the actual solution .  

            Table 4.1  The iterates of for Newton's method applied to .

            Figure 4.2  The iterates of for Newton's method applied to .

Explore Solution 4.7.

Enter the function    and find the next five iterates.

Now compute the next five iterates    and the corresponding values of  .

Plot the iterates    and observe that they are converging to  .  

Notice that the points    are co close together that they appear to coincide and that the value for    agrees to six decimal places with the actual solution  .  

(c) 2006 John H. Mathews, Russell W. Howell