3.9: Newton's Method

Newton’s Method

By sketching a graph of \(f\), we can estimate a root of \(f(x)=0\). Let’s call this estimate \(x_0\). We then draw the tangent line to \(f\) at \(x_0\). If \(f′(x_0)≠0\), this tangent line intersects the \(x\)-axis at some point \((x_1,0)\). Now let \(x_1\) be the next approximation to the actual root. Typically, \(x_1\) is closer than \(x_0\) to an actual root. Next we draw the tangent line to \(f\) at \(x_1\). If \(f′(x_1)≠0\), this tangent line also intersects the \(x\)-axis, producing another approximation, \(x_2\). We continue in this way, deriving a list of approximations: \(x_0,\, x_1,\, x_2,\, ….\) Typically, the numbers \(x_0,\, x_1,\, x_2,\, …\) quickly approach an actual root \(x^*\), as shown in the following figure.

Figure \(\PageIndex{14}\): A graph illustrating the iterative process of Newton's method. From OpenStax Calculus Volume 1, licensed under CC-BY-NC-SA.

Now, let's calculate the approximations \(x_0,\, x_1,\, x_2,\, ….\). Suppose that \(x_0\) is our first approximation. Usually, this approximation is obtained by guessing, or from a sketch of the graph, or from a graphing device or app. Then, the approximation \(x_1\) is defined by letting \((x_1,0)\) be the \(x\)-intercept of the tangent line to \(f\) at \(x_0\). The equation of this tangent line is given by

\[y=f(x_0)+f′(x_0)(x−x_0). \nonumber \]

Therefore, \((x_1,0)\) is on the line and must satisfy

\[f(x_0)+f′(x_0)(x_1−x_0)=0.\nonumber \]

If \(f'(x_0)\neq\), we can solve this equation for \(x_1\), so

\[x_1=x_0−\frac{f(x_0)}{f'(x_0)}.\nonumber \]

Similarly, the point \((x_2,0)\) is the \(x\)-intercept of the tangent line to \(f\) at \(x_1\). Therefore, \(x_2\) satisfies the equation

\[x_2=x_1−\frac{f(x_1)}{f'(x_1)}.\nonumber \]

In general, for \(n>0,x_n\) satisfies

\[x_n=x_{n−1}−\frac{f(x_{n−1})}{f'(x_{n−1})}.\label{Newton} \]