Module for Newton's Method

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    If [Graphics:Images/Newton'sMethodMod_gr_1.gif] are continuous near a root [Graphics:Images/Newton'sMethodMod_gr_2.gif], then this extra information regarding the nature of [Graphics:Images/Newton'sMethodMod_gr_3.gif] can be used to develop algorithms that will produce sequences [Graphics:Images/Newton'sMethodMod_gr_4.gif] that converge faster to [Graphics:Images/Newton'sMethodMod_gr_5.gif] than either the bisection or false position method. The Newton-Raphson (or simply Newton's) method is one of the most useful and best known algorithms that relies on the continuity of [Graphics:Images/Newton'sMethodMod_gr_6.gif].  The method is attributed to Sir Isaac Newton (1643-1727) and Joseph Raphson (1648-1715).  

Theorem (Newton-Raphson Theorem).  Assume that [Graphics:Images/Newton'sMethodMod_gr_7.gif] and there exists a number [Graphics:Images/Newton'sMethodMod_gr_8.gif], where [Graphics:Images/Newton'sMethodMod_gr_9.gif].  If   [Graphics:Images/Newton'sMethodMod_gr_10.gif], then there exists a [Graphics:Images/Newton'sMethodMod_gr_11.gif] such that the sequence [Graphics:Images/Newton'sMethodMod_gr_12.gif] defined by the iteration  

    [Graphics:Images/Newton'sMethodMod_gr_13.gif]    for    [Graphics:Images/Newton'sMethodMod_gr_14.gif]  

will converge to [Graphics:Images/Newton'sMethodMod_gr_15.gif] for any initial approximation  [Graphics:Images/Newton'sMethodMod_gr_16.gif].  

Derivation.

 

Animations (Newton's Method  Newton's Method).  Internet hyperlinks to animations.

 

Algorithm (Newton-Raphson Iteration).  To find a root of  [Graphics:Images/Newton'sMethodMod_gr_56.gif]  given an initial approximation  [Graphics:Images/Newton'sMethodMod_gr_57.gif]  using the iteration  

    [Graphics:Images/Newton'sMethodMod_gr_58.gif]   for    [Graphics:Images/Newton'sMethodMod_gr_59.gif].

 

Mathematica Subroutine (Newton-Raphson Iteration).

[Graphics:Images/Newton'sMethodMod_gr_60.gif]

Example 1.  Use Newton's method to find the three roots of the cubic polynomial  [Graphics:Images/Newton'sMethodMod_gr_61.gif].  Determine the Newton-Raphson iteration formula  [Graphics:Images/Newton'sMethodMod_gr_62.gif]  that is used.  Show details of the computations for the starting value  [Graphics:Images/Newton'sMethodMod_gr_63.gif].

Solution 1.

 

Reduce the volume of printout.

After you have debugged you program and it is working properly, delete the unnecessary print statements  
[Graphics:Images/Newton'sMethodMod_gr_136.gif]  and  
[Graphics:Images/Newton'sMethodMod_gr_137.gif]

Concise Program for the Newton-Raphson Method

[Graphics:Images/Newton'sMethodMod_gr_138.gif]

Now test the example to see if it still works. Use the last case in Example 1 given above and compare with the previous results.

[Graphics:Images/Newton'sMethodMod_gr_139.gif]
[Graphics:Images/Newton'sMethodMod_gr_140.gif]
[Graphics:Images/Newton'sMethodMod_gr_141.gif]
[Graphics:Images/Newton'sMethodMod_gr_142.gif]
[Graphics:Images/Newton'sMethodMod_gr_143.gif]
[Graphics:Images/Newton'sMethodMod_gr_144.gif]
[Graphics:Images/Newton'sMethodMod_gr_145.gif]
[Graphics:Images/Newton'sMethodMod_gr_146.gif]
[Graphics:Images/Newton'sMethodMod_gr_147.gif]
[Graphics:Images/Newton'sMethodMod_gr_148.gif]
[Graphics:Images/Newton'sMethodMod_gr_149.gif]
[Graphics:Images/Newton'sMethodMod_gr_150.gif]

Error Checking in the Newton-Raphson Method

Error checking can be added to the Newton-Raphson method.  Here we have added a third parameter  [Graphics:Images/Newton'sMethodMod_gr_151.gif]  to the subroutine which estimate the accuracy of the numerical solution.

[Graphics:Images/Newton'sMethodMod_gr_152.gif]

The following subroutine call uses a maximum of 20 iterations, just to make sure enough iterations are performed.  However, it will terminate when the difference between consecutive iterations is less than  [Graphics:Images/Newton'sMethodMod_gr_153.gif].  By interrogating  k  afterward we can see how many iterations were actually performed.

[Graphics:Images/Newton'sMethodMod_gr_154.gif]
[Graphics:Images/Newton'sMethodMod_gr_155.gif]
[Graphics:Images/Newton'sMethodMod_gr_156.gif]
[Graphics:Images/Newton'sMethodMod_gr_157.gif]
[Graphics:Images/Newton'sMethodMod_gr_158.gif]
[Graphics:Images/Newton'sMethodMod_gr_159.gif]

Old Lab Project (Newton's Method  Newton's Method).  Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

Newton-Raphson Method  Newton-Raphson Method  Internet hyperlinks to web sites and a bibliography of articles.  

  

Downloads (Newton's Method Newton's Method).  Download this Mathematica notebook.  

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2003