Typically, Newton’s method is used to find roots fairly quickly. However, things can go wrong. Some reasons why Newton’s method might fail include the following:
At one of the approximations xn, the derivative f′ is zero at xn, but f(xn)≠0. As a result, the tangent line of f at xn does not intersect the x-axis. Therefore, we cannot continue the iterative process.
The approximations x0,x1,x2,⋯ may approach a different root. If the function f has more than one root, it is possible that our approximations do not approach the one for which we are looking, but approach a different root (see Figure 4). This event most often occurs when we do not choose the approximation x0 close enough to the desired root.
The approximations may fail to approach a root entirely. In the example below, we provide an example of a function and an initial guess x0 such that the successive approximations never approach a root because the successive approximations continue to alternate back and forth between two values.
Figure 4. If the initial guess x0 is too far from the root sought, it may lead to approximations that approach a different root.
Consider the function f(x)=x3−2x+2. Let x0=0. Show that the sequence x1,x2,⋯ fails to approach a root of f.
For f(x)=x3−2x+2, the derivative is f′(x)=3x2−2. Therefore,
x1=x0−f(x0)f′(x0)=0−f(0)f′(0)=−2−2=1.
In the next step,
x2=x1−f(x1)f′(x1)=1−f(1)f′(1)=1−11=0.
Consequently, the numbers x0,x1,x2,⋯ continue to bounce back and forth between 0 and 1 and never get closer to the root of f which is over the interval [−2,−1] (see Figure 5). Fortunately, if we choose an initial approximation x0 closer to the actual root, we can avoid this situation.
Figure 5. The approximations continue to alternate between 0 and 1 and never approach the root of f.
Watch the following video to see the worked solution to this example.
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From the example above, we see that Newton’s method does not always work. However, when it does work, the sequence of approximations approaches the root very quickly. Discussions of how quickly the sequence of approximations approach a root found using Newton’s method are included in texts on numerical analysis.