Numerical solution of equations — FAQ

First, use the change-of-sign rule to find a small interval $[a, b]$ that contains the root. A good choice for $x_0$ is the midpoint of this interval, $(a+b)/2$. Alternatively, sketch the graph to see which end of the interval looks closer to the root.

The iterative method $x_{n+1} = g(x_n)$ works by finding a fixed point, where the input equals the output ($x = g(x)$). This is graphically where the curve $y=g(x)$ intersects the line $y=x$. This intersection point is also a root of the original equation $f(x)=0$. Simply iterating with $f(x)$ (e.g., $x_{n+1} = f(x_n)$) doesn't have this property and won't converge to a root of $f(x)=0$.

Both show how an iteration converges. A staircase diagram happens when $0 < g'(x) < 1$, and the approximations approach the root from one side. A cobweb diagram happens when $-1 < g'(x) < 0$, and the approximations oscillate around the root, spiralling inwards.

Not always. Newton-Raphson usually converges much faster (it has 'quadratic convergence'), but it is more sensitive to the starting value and can fail spectacularly if you start near a turning point. The standard $x=g(x)$ iteration is often more robust, even if it's slower.

Examiners are testing your understanding of the numerical process, not just your ability to use a calculator's 'solve' function. Marks are awarded for showing the correct method: stating the formula, showing the values of the first few iterations to sufficient precision, and performing a final sign-change check to justify your rounded answer.