About the Applet
Newton's Method is an algorithm used to approximate the roots of a real valued function. The procedure begins with choosing an initial 'guess,' denoted \(x_0\). The next approximation \(x_1\) is obtained using the formula
\[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \]
The process repeats until a satisfactory approximation to the root has been found, where the \(n\)-th approximation is found from the previous one by
\[ x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})} \]
Graphically, at each approximation \(x_n\), the value of \(x_{n+1}\) is the \(x\)-intercept of the tangent line to \(f\) at \(x_n\).
Generally, Newton's method will have quadratic convergence, meaning the difference between the root and the approximation is squared at each step. However, there are times when the method:
- converges slowly. This can happen when the derivative of the function is 0 at the root, as in \(f(x) = x^2\).
- converges to the 'wrong' root. The method may find a root of the function in this case, but if the function has multiple distinct roots, the root found by Newton's method may not be the closest one to the initial starting point. Choosing a new value of \(x_0\) can have different results. An example is \(f(x) = \sin x\). An initial guess of \(x_0 = 1.5\) will not converge to 0 or even \(\pi\), as may be expected.
- fails to converge at all. There are a few reasons this may happen. Some initial guesses may cause the algorithm to 'loop', with each approximation alternating between two values. If a function is not continuously differentiable near the root, the method may also diverge. An example is \(f(x) = \sqrt[3]{x}\). Any initial guess will result in failure to converge.
Using the Applet
Choose a function from the drop down menu. Set an initial guess \(x_0\) by using the input field or sliding the red circle on the graph, then click the 'Set \(x_0\)' button. Click the 'Find Next Iteration Point' to calculate the next step in the algorithm. The values of each approximation will be listed.
