x = g(x) Iteration

Iteration is a repetitive process where the output of one step is used as the input for the next to achieve a more accurate result.

An iterative formula is expressed as a recurrence relation: $x_{n+1} = g(x_n)$, where $x_n$ is the current approximation and $x_{n+1}$ is the next.

The goal is to find a fixed point where $x = g(x)$, which corresponds to a root of the original equation $f(x) = 0$.

To use this method, an equation $f(x) = 0$ must be algebraically rearranged into the form $x = g(x)$. There are often multiple ways to rearrange a single equation, and not all will lead to a successful solution.

The sequence $x_0, x_1, x_2, ...$ converges to a root $\alpha$ if the values get progressively closer to $\alpha$ as $n$ increases.

Mathematically, convergence depends on the gradient of the function $g(x)$ near the root. If $|g'(\alpha)| < 1$, the sequence will converge; if $|g'(\alpha)| > 1$, the sequence will diverge away from the root.

Staircase Diagrams occur when $g'(x)$ is positive ($0 < g'(x) < 1$). The sequence of approximations approaches the root from one side, creating a step-like visual pattern between the curve $y = g(x)$ and the line $y = x$.

Cobweb Diagrams occur when $g'(x)$ is negative ($-1 < g'(x) < 0$). The approximations alternate above and below the root, spiraling inward toward the intersection point.

If the gradient is too steep ($|g'(x)| > 1$), these diagrams will show the sequence 'spiraling out' or stepping away, indicating divergence.

Step 1: Rearrangement. Transform $f(x) = 0$ into $x = g(x)$. For example, $x^2 - x - 1 = 0$ could become $x = \sqrt{x+1}$ or $x = x^2 - 1$.

Step 2: Initial Value. Choose a starting value $x_0$ close to the suspected root.

Step 3: Iteration. Substitute $x_0$ into $g(x)$ to find $x_1$, then substitute $x_1$ to find $x_2$, and so on.

Step 4: Termination. Stop the process when the values of $x_n$ and $x_{n+1}$ are identical to the required degree of accuracy (e.g., 3 decimal places).

Unlike the Change of Sign method, which only identifies an interval, iteration provides a specific numerical approximation.

Iteration is more sensitive to the choice of $x_0$ and the form of $g(x)$ than methods like the Trapezium Rule.

Calculator Efficiency: Use the 'ANS' key. Type your $x_0$ value and press '=', then type the formula for $g(x)$ using the 'ANS' button instead of $x$. Repeatedly pressing '=' will generate $x_1, x_2, x_3$ rapidly.

Show Your Working: Always write down the first few iterations ($x_1, x_2, x_3$) even if using a calculator shortcut to demonstrate the process.

Verification: If asked to prove a root is correct to $k$ decimal places, use the Change of Sign method on the upper and lower bounds of that value (e.g., for $1.26$, check $1.255$ and $1.265$).