Newton's Method — Question 3

Question 3

Use Newton’s Method to approximate a solution to the equation: $$\cos(x) = x$$ Start with an initial guess of $x_1 = 0.5$, and compute the next three iterations: $x_2$, $x_3$, and $x_4$, rounding answers to four decimal places.

Question 3 - Solution

Newton’s iteration is

$$x_{n+1}=x_n-\frac{f(x_n)}{f\prime(x_n)},\qquad f(x)=\cos x-x,\quad f\prime(x)=-\sin x-1.$$

Keeping full precision internally gives

$$\begin{aligned}x_{2}&=0.5000000000-\frac{0.3775825619}{-1.4794255386}\approx\boxed{0.7552}\\[6pt]x_{3}&=0.7552224171-\frac{-0.0271033119}{-1.6854506318}\approx\boxed{0.7391}\\[6pt]x_{4}&=0.7391416661-\frac{-0.0000946154}{-1.6736538108}\approx\boxed{0.7391}\end{aligned}$$