Newton's Method — Question 8

Question 8

Approximate a root of the equation: $$e^x - 2x^2 = 0$$ using Newton’s Method. Start with $x_1 = 0.5$, and compute $x_2$, $x_3$, and $x_4$ to four decimal places.

Question 8 - Solution

Newton’s iteration is

$$x_{n+1}=x_n-\frac{f(x_n)}{f\prime(x_n)},\qquad f(x)=e^x-2x^2,\quad f\prime(x)=e^x-4x.$$

Keeping full precision internally gives

$$\begin{aligned}x_{2}&=0.5000000000-\frac{1.1487212707}{-0.3512787293}\approx\boxed{3.7701}\\[6pt]x_{3}&=3.7701133740-\frac{14.9574735817}{28.3045297918}\approx\boxed{3.2417}\\[6pt]x_{4}&=3.2416653238-\frac{4.5594909448}{12.6096177926}\approx\boxed{2.8801}\end{aligned}$$

These are the requested iterates; three steps do not yet give an accurate root. Continue iterating if a converged root is required.