2.3: Secant Method
The Secant Method is second best to Newton’s Method , and is used when a faster convergence than Bisection is desired, but it is too difficult or impossible to take an analytical derivative of the function \(f(x)\) . We write in place of \(f^{\prime}\left(x_{n}\right)\) ,
\[f^{\prime}\left(x_{n}\right) \approx \frac{f\left(x_{n}\right)-f\left(x_{n-1}\right)}{x_{n}-x_{n-1}} \nonumber \]
Starting the Secant Method requires a guess for both \(x_{0}\) and \(x_{1}\) .
2.3.1. Estimate \(\sqrt{2}=1.41421356\) using Newton’s Method
The \(\sqrt{2}\) is the zero of the function \(f(x)=x^{2}-2\) . To implement Newton’s Method, we use \(f^{\prime}(x)=2 x\) . Therefore, Newton’s Method is the iteration
\[x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}} \nonumber \]
We take as our initial guess \(x_{0}=1\) . Then
\[\begin{aligned} &x_{1}=1-\frac{-1}{2}=\frac{3}{2}=1.5 \\ &x_{2}=\frac{3}{2}-\frac{\frac{9}{4}-2}{3}=\frac{17}{12}=1.416667, \\ &x_{3}=\frac{17}{12}-\frac{\frac{17^{2}}{12^{2}}-2}{\frac{17}{6}}=\frac{577}{408}=1.41426 . \end{aligned} \nonumber \]
2.3.2. Example of fractals using Newton’s Method
Consider the complex roots of the equation \(f(z)=0\) , where
\[f(z)=z^{3}-1 \nonumber \]
These roots are the three cubic roots of unity. With
\[e^{i 2 \pi n}=1, \quad n=0,1,2, \ldots \nonumber \]
the three unique cubic roots of unity are given by
\[1, \quad e^{i 2 \pi / 3}, \quad e^{i 4 \pi / 3} \nonumber \]
With
\[e^{i \theta}=\cos \theta+i \sin \theta, \nonumber \]
and \(\cos (2 \pi / 3)=-1 / 2, \sin (2 \pi / 3)=\sqrt{3} / 2\) , the three cubic roots of unity are
\[r_{1}=1, \quad r_{2}=-\frac{1}{2}+\frac{\sqrt{3}}{2} i, \quad r_{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2} i \nonumber \]
The interesting idea here is to determine which initial values of \(z_{0}\) in the complex plane converge to which of the three cubic roots of unity.
Newton’s method is
\[z_{n+1}=z_{n}-\frac{z_{n}^{3}-1}{3 z_{n}^{2}} \nonumber \]
If the iteration converges to \(r_{1}\) , we color \(z_{0}\) red; \(r_{2}\) , blue; \(r_{3}\) , green. The result will be shown in lecture.