Example 1.  Use Newton's method to find the three roots of the cubic polynomial  [Graphics:Images/Newton'sMethodMod_gr_22.gif].  
Determine the Newton-Raphson iteration formula  [Graphics:Images/Newton'sMethodMod_gr_23.gif]  that is used.  Show details of the computations for the starting value  [Graphics:Images/Newton'sMethodMod_gr_24.gif].

Solution 1.

[Graphics:../Images/Newton'sMethodMod_gr_25.gif]
[Graphics:../Images/Newton'sMethodMod_gr_26.gif]

Graph the function.

[Graphics:../Images/Newton'sMethodMod_gr_27.gif]

[Graphics:../Images/Newton'sMethodMod_gr_28.gif]

[Graphics:../Images/Newton'sMethodMod_gr_29.gif]

How many real roots are there ?  Really !

[Graphics:../Images/Newton'sMethodMod_gr_30.gif]

[Graphics:../Images/Newton'sMethodMod_gr_31.gif]

[Graphics:../Images/Newton'sMethodMod_gr_32.gif]

The Newton-Raphson iteration formula  g[x]  is

[Graphics:../Images/Newton'sMethodMod_gr_33.gif]

[Graphics:../Images/Newton'sMethodMod_gr_34.gif]
[Graphics:../Images/Newton'sMethodMod_gr_35.gif]

Starting with  [Graphics:../Images/Newton'sMethodMod_gr_36.gif], Use the Newton-Raphson method to find a numerical approximation to the root.  First, do the iteration one step at a time.  Type each of the following commands in a separate cell and execute them one at a time.

[Graphics:../Images/Newton'sMethodMod_gr_37.gif]
[Graphics:../Images/Newton'sMethodMod_gr_38.gif]

[Graphics:../Images/Newton'sMethodMod_gr_39.gif]
[Graphics:../Images/Newton'sMethodMod_gr_40.gif]

[Graphics:../Images/Newton'sMethodMod_gr_41.gif]
[Graphics:../Images/Newton'sMethodMod_gr_42.gif]

[Graphics:../Images/Newton'sMethodMod_gr_43.gif]
[Graphics:../Images/Newton'sMethodMod_gr_44.gif]

[Graphics:../Images/Newton'sMethodMod_gr_45.gif]
[Graphics:../Images/Newton'sMethodMod_gr_46.gif]

[Graphics:../Images/Newton'sMethodMod_gr_47.gif]
[Graphics:../Images/Newton'sMethodMod_gr_48.gif]

[Graphics:../Images/Newton'sMethodMod_gr_49.gif]
[Graphics:../Images/Newton'sMethodMod_gr_50.gif]


[Graphics:../Images/Newton'sMethodMod_gr_51.gif]

[Graphics:../Images/Newton'sMethodMod_gr_52.gif]
[Graphics:../Images/Newton'sMethodMod_gr_53.gif]
[Graphics:../Images/Newton'sMethodMod_gr_54.gif]
[Graphics:../Images/Newton'sMethodMod_gr_55.gif]
[Graphics:../Images/Newton'sMethodMod_gr_56.gif]
[Graphics:../Images/Newton'sMethodMod_gr_57.gif]
[Graphics:../Images/Newton'sMethodMod_gr_58.gif]
[Graphics:../Images/Newton'sMethodMod_gr_59.gif]
[Graphics:../Images/Newton'sMethodMod_gr_60.gif]
[Graphics:../Images/Newton'sMethodMod_gr_61.gif]
[Graphics:../Images/Newton'sMethodMod_gr_62.gif]

From the second graph we see that there are two other real roots, use the starting values  0.0  and  1.4  to find them.
First, use the starting value [Graphics:../Images/Newton'sMethodMod_gr_63.gif].  

[Graphics:../Images/Newton'sMethodMod_gr_64.gif]

[Graphics:../Images/Newton'sMethodMod_gr_65.gif]
[Graphics:../Images/Newton'sMethodMod_gr_66.gif]
[Graphics:../Images/Newton'sMethodMod_gr_67.gif]
[Graphics:../Images/Newton'sMethodMod_gr_68.gif]
[Graphics:../Images/Newton'sMethodMod_gr_69.gif]
[Graphics:../Images/Newton'sMethodMod_gr_70.gif]
[Graphics:../Images/Newton'sMethodMod_gr_71.gif]
[Graphics:../Images/Newton'sMethodMod_gr_72.gif]
[Graphics:../Images/Newton'sMethodMod_gr_73.gif]
[Graphics:../Images/Newton'sMethodMod_gr_74.gif]
[Graphics:../Images/Newton'sMethodMod_gr_75.gif]
[Graphics:../Images/Newton'sMethodMod_gr_76.gif]

Then use the starting value [Graphics:../Images/Newton'sMethodMod_gr_77.gif].  

[Graphics:../Images/Newton'sMethodMod_gr_78.gif]

[Graphics:../Images/Newton'sMethodMod_gr_79.gif]
[Graphics:../Images/Newton'sMethodMod_gr_80.gif]
[Graphics:../Images/Newton'sMethodMod_gr_81.gif]
[Graphics:../Images/Newton'sMethodMod_gr_82.gif]
[Graphics:../Images/Newton'sMethodMod_gr_83.gif]
[Graphics:../Images/Newton'sMethodMod_gr_84.gif]
[Graphics:../Images/Newton'sMethodMod_gr_85.gif]
[Graphics:../Images/Newton'sMethodMod_gr_86.gif]
[Graphics:../Images/Newton'sMethodMod_gr_87.gif]

Compare our result with Mathematica's built in numerical root finder.

[Graphics:../Images/Newton'sMethodMod_gr_88.gif]

[Graphics:../Images/Newton'sMethodMod_gr_89.gif]

[Graphics:../Images/Newton'sMethodMod_gr_90.gif]

[Graphics:../Images/Newton'sMethodMod_gr_91.gif]

 

This can also be done with Mathematica's built in symbolic solve procedure.

[Graphics:../Images/Newton'sMethodMod_gr_92.gif]

[Graphics:../Images/Newton'sMethodMod_gr_93.gif]

[Graphics:../Images/Newton'sMethodMod_gr_94.gif]

[Graphics:../Images/Newton'sMethodMod_gr_95.gif]

[Graphics:../Images/Newton'sMethodMod_gr_96.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004