Module for Newton's Method

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If are continuous near a root , then this extra information regarding the nature of can be used to develop algorithms that will produce sequences that converge faster to 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 .  The method is attributed to Sir Isaac Newton (1643-1727) and Joseph Raphson (1648-1715).

Theorem (Newton-Raphson Theorem).  Assume that and there exists a number , where .  If , then there exists a such that the sequence defined by the iteration for will converge to for any initial approximation .

Derivation.

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

Algorithm (Newton-Raphson Iteration).  To find a root of given an initial approximation using the iteration for .

Mathematica Subroutine (Newton-Raphson Iteration).

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Example 1.  Use Newton's method to find the three roots of the cubic polynomial .  Determine the Newton-Raphson iteration formula that is used.  Show details of the computations for the starting value .

Solution 1.

Reduce the volume of printout.

After you have debugged you program and it is working properly, delete the unnecessary print statements and Concise Program for the Newton-Raphson Method

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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.

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Error Checking in the Newton-Raphson Method

Error checking can be added to the Newton-Raphson method.  Here we have added a third parameter to the subroutine which estimate the accuracy of the numerical solution.

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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 .  By interrogating  k  afterward we can see how many iterations were actually performed.

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

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

(c) John H. Mathews 2003