Newton's Method

    Newton's Method is an iterative process used to approximate a root of a function; and is based on the assumption that the graph of and the tangent line at both cross the -axis at about the same point. It can be shown that a sufficient condition for the convergence of Newton's Method is



on an open interval containing zero.

Definition (Newton's Method) Let where is differentiable on an open interval containing The following method is used to calculate a root of a function and is called Newton's method.

    (i) To approximate we make an initial estimate that is close to
    
    (ii) Recursively, we can determine a new approximation using



    (iii) If is within the desired accuracy, then let serve as the final approximation. Otherwise calculate a new approximation.    

Proposition (Newton-Raphson Method) To approximate a root of the equation start with a preliminary estimate and generate a sequence using the formula

   

If on an open interval containing zero, then the sequence of numbers converges; otherwise the sequence may not have a limit.

Example (Newton's Method) Use Newton's method to approximate the zeros of the following functions.

(a) Approximate the zero of

    Solution. First we sketch a graph to get an initial estimate.



Our initial estimate is Next we form and so we compute and we have the recursively defined formula



Therefore, we estimate the zero of to be

(b) Approximate a zero of

    Solution. First we sketch a graph to get an initial estimate.



Our initial estimate is Next we form and so we compute and we have the recursively defined formula





Therefore, we estimate the zero of to be 1.1459.

(c) Approximate the zero of

    Solution. First we sketch a graph to get an initial estimate.



Our initial estimate is Next we form and so we compute and we have the recursively defined formula



Therefore, we estimate the zero of to be

(d) Approximate the zero of

    Solution. First we sketch a graph to get an initial estimate.



Our initial estimate is Next we form and so we compute and we have the recursively defined formula


Therefore, we estimate the zero of to be

• print this page
• pages linking here
• related pages

Cite this as:
The Newton Method
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-newton-method.html