Newton's Method — Question 7

Question 7

Use Newton’s Method to approximate a solution to the equation: $$\ln(x) + x^2 = 3$$ Start with an initial guess of $x_1 = 1.5$, and compute $x_2$, $x_3$, and $x_4$. Round each to four decimal places.

Question 7 - Solution

Newton’s iteration is

$$x_{n+1}=x_n-\frac{f(x_n)}{f\prime(x_n)},\qquad f(x)=\ln x+x^2-3,\quad f\prime(x)=1/x+2x.$$

Keeping full precision internally gives

$$\begin{aligned}x_{2}&=1.5000000000-\frac{-0.3445348919}{3.6666666667}\approx\boxed{1.5940}\\[6pt]x_{3}&=1.5939640614-\frac{0.0069454631}{3.8152948397}\approx\boxed{1.5921}\\[6pt]x_{4}&=1.5921436351-\frac{0.0000026613}{3.8123713061}\approx\boxed{1.5921}\end{aligned}$$