Fixed-point iteration

What the problems look like

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

1. Warm-up

   The fixed-point iteration $x_{n+1} = g(x_n)$ uses $g(x) = (ax + b)/c$ with $a = -3$, $b = 8$ and $c = 6$. Find (i) the fixed point $p$, which satisfies $p = g(p)$, and (ii) the convergence rate $|g'(p)|$. Give both as exact fractions in lowest terms.

2. Warm-up

   The fixed-point iteration $x_{n+1} = g(x_n)$ uses $g(x) = (ax + b)/c$ with $a = 4$, $b = -7$ and $c = 6$. Find (i) the fixed point $p$, which satisfies $p = g(p)$, and (ii) the convergence rate $|g'(p)|$. Give both as exact fractions in lowest terms.

3. Warm-up

   The fixed-point iteration $x_{n+1} = g(x_n)$ uses $g(x) = (ax + b)/c$ with $a = -2$, $b = -2$ and $c = 6$. Find (i) the fixed point $p$, which satisfies $p = g(p)$, and (ii) the convergence rate $|g'(p)|$. Give both as exact fractions in lowest terms.

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