The Newton-Raphson Method - PDF

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					                                   The Newton-Raphson Method

The Newton-Raphson method is a simple, efficient method for estimating a root. The method
has the form,


                                             x n+1 = x n −
                                                                  ( )
                                                                f xn
                                                                  ( )
                                                                f ′ xn
                                                                       .                              (1)


Here, x n is a guess for the value of x at iteration n, which is used to compute a new guess, x n+1 .
The difference between the two calculated as the absolute error, ε a ,

                                               ε a = x n +1 − x n .                                   (2)

If the difference is larger than a specified convergence criterion, C a , x n+1 is used as the next
guess (replacing the original value of x n ), and Equation (1) is solved again. This procedure is
repeated until ε a becomes smaller than C a .

        Example 1. Derive an algorithm for the function, f ( x ) = x p − q = 0 , where p is the
        power and q is a constant.

        Substitute f ( x ) and f ′( x ) into Equation (1), yielding:


                                         x   n +1
                                                    =x   n
                                                             −
                                                               (x ) − q .
                                                                 n p
                                                                                                      (3)
                                                                p (x )
                                                                   n p −1




        The algorithm for solving Equation (3) and finding the root is as follows:

        1. Ask the user to enter the values of q and p in x = p q (how was this equation
            obtained?). Ask also for a starting guess, x n=0 , and how many digits of precision are
            required, C a .

        2. Find the root
           a. Calculate x n+1 from Equation (3).
           b. Calculate ε a from Equation (2).
           c. Check for convergence (to see if the root is found).
               i. If ε a < C a , the root is found; otherwise
                ii. Set x n = x n+1 (which makes x n=1 the new guess) and return to Step (a).

        3. Display the root and the number of times Step (2) was repeated (the number of
           iterations).


D. Haugli, Lecturer                         Aer E 161                          Aerospace Engineering
2/18/2005                       The Newton-Raphson Method, Page 1               Iowa State University
       Example 2. Apply the algorithm from Example 1 to find the cube root of 8. The actual
       root is 2, but to demonstrate the method, make an initial guess of x n=0 = 3 .

        Iteration, n Guess, x n     x n+1 (Eq. 3)   Difference, ε a (Eq. 2)
             1         3.000000      2.296296              0.703704
             2         2.296296      2.036587              0.259709
             3         2.036587      2.000653              0.035934
             4         2.000653      2.000000              0.000653
             5         2.000000      2.000000              0.000000

       In this example, the root, x = 2 , is found through the sixth decimal place after only five
       iterations.




D. Haugli, Lecturer                      Aer E 161                            Aerospace Engineering
2/18/2005                    The Newton-Raphson Method, Page 2                 Iowa State University