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# Newton-Raphson Method

The method can be derived in a variety of different ways. Here we use a graphical approcah. We start with an initial guess for the root . Call this initial estimate . Next we construct the tangent to the curve at . Then, we calculate where this tangent crosses the -axis. This provides the next estimate, , for the root . This is illustrated in Figure 1.3. The equation of the tangent at is where Hence, (1.9)

The tangent intersects the -axis at and this value of gives us our next estimate of the root. Thus, (1.10)

Using (1.10) we obtain . By replacing by in (1.10) generates , which could be a better estimate for the root than . Repeating this process generates a sequence of iterates which possibly converges to the root . Thus, Knowing we can generate using (1.10) or specifically (1.11)

This is the Newton-Raphson method for obtaining a root of the function . This is the most widely used practical technique for solving algebraic equations. It has fast convergence and this will be discussed later.

Example 1. .5Consider Then using the product rule for differentiating a product with and , the derivative of is This is positive if is positive. Now since there is only one root in the interval . Then the Newton-Raphson formula is Take as our intial guess . Probably a better guess would be to take the midpoint of the interval, i.e. . However, if the method converges, the initial guess is not too crucial. The results of the iteration are presented in the table.    1 1.71828 5.43656 0.68394 0.68394 0.35534 3.33701 0.57745 0.57445 0.02872 2.810211 0.56723 0.56723 2.763615 0.567143296 0.567143296 2.763223 0.567143289

When do we stop the iteration procedure?

Subsections   Next: Stopping Criterion Up: Solutions of Equations Previous: Approximate Numerical Methods
Prof. Alan Hood
2000-02-01