Iteration

The Iterative Process

1. Rearrange the equation into the form $x = g(x)$.
2. Start with an initial value $x_0$.
3. Apply: $x_{n+1} = g(x_n)$.
4. Repeat until values settle (converge).

Example

Solve $x^3 + x - 3 = 0$ using $x_{n+1} = \sqrt[3]{3 - x_n}$ with $x_0 = 1$.

$x_1 = \sqrt[3]{3-1} = \sqrt[3]{2} \approx 1.2599$

$x_2 = \sqrt[3]{3-1.2599} \approx 1.1784$

$x_3 \approx 1.2042$

$x_4 \approx 1.1959$

Values converge to approximately $1.2$.

Showing a Root Exists

Use the sign change method: if $f(a) < 0$ and $f(b) > 0$ (or vice versa), there's a root between $a$ and $b$.

$f(1) = 1 + 1 - 3 = -1 < 0$ $f(2) = 8 + 2 - 3 = 7 > 0$

Sign change → root between 1 and 2.