Newton's method

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Newton's method applied to $f(x) = \dfrac{1}{x} - 3$ gives the division-free iteration $x_{n+1} = x_n(2 - 3x_n)$ for computing $1/3$. Starting from $x_0 = 1/2$, find (i) the next iterate $x_1$ and (ii) its error term $e_1 = 1 - 3x_1$. (The errors $e_n = 1 - 3x_n$ satisfy $e_{n+1} = e_n^{2}$.)

2. Warm-up

   Newton's method applied to $f(x) = \dfrac{1}{x} - 6$ gives the division-free iteration $x_{n+1} = x_n(2 - 6x_n)$ for computing $1/6$. Starting from $x_0 = 1/4$, find (i) the next iterate $x_1$ and (ii) its error term $e_1 = 1 - 6x_1$. (The errors $e_n = 1 - 6x_n$ satisfy $e_{n+1} = e_n^{2}$.)

3. Warm-up

   Newton's method applied to $f(x) = \dfrac{1}{x} - 2$ gives the division-free iteration $x_{n+1} = x_n(2 - 2x_n)$ for computing $1/2$. Starting from $x_0 = 1/5$, find (i) the next iterate $x_1$ and (ii) its error term $e_1 = 1 - 2x_1$. (The errors $e_n = 1 - 2x_n$ satisfy $e_{n+1} = e_n^{2}$.)

Problems shown under their original terms: Generated (template, solver-computed).