Apply the Newton-Raphson method to find a root of an equation numerically and discuss its convergence and failure cases

What this dot point is asking

SEAB wants you to apply the Newton-Raphson method to find a root of an equation $\mathrm{f}(x) = 0$ numerically, to understand its geometric basis as the intercept of the tangent line, to perform iterations to a stated accuracy, and to discuss its convergence (typically fast) and the circumstances in which it fails.

The answer

The iterative formula

Starting from an estimate $x_0$, the Newton-Raphson iteration is

Each step uses both the function value and its derivative at the current estimate.

The geometric meaning

At the point $(x_n, \mathrm{f}(x_n))$ draw the tangent to the curve. The next estimate $x_{n+1}$ is where that tangent crosses the $x$-axis. Because a smooth curve is well approximated by its tangent near a root, the estimates close in on the root rapidly.

Choosing a starting value

A good $x_0$ is one near the root, ideally with $\mathrm{f}$ and the iteration well behaved nearby. Locating a sign change ($\mathrm{f}(a)$ and $\mathrm{f}(b)$ of opposite sign) brackets a root and suggests a sensible start between $a$ and $b$.

Convergence

When it works, Newton-Raphson converges quadratically: the number of correct decimal places roughly doubles each step. This is much faster than simple iteration, which is why it is the workhorse root-finder. Stop when successive estimates agree to the required accuracy.

When the method fails

The method can fail or behave badly when:

• $\mathrm{f}'(x_n)$ is zero or near zero (a nearly horizontal tangent sends the estimate far away);
• the starting value is too far from the root (it may converge to a different root or diverge);
• the iteration falls into a cycle (oscillating between values without converging);
• there is an inflection or symmetry that causes the estimates to overshoot.

Examples in context

Example 1. Square roots inside a calculator. The iteration $x_{n+1} = \tfrac{x_n}{2} + \tfrac{a}{2x_n}$ for $\sqrt{a}$ is Newton-Raphson applied to $x^2 - a = 0$, and is essentially the algorithm hardware uses to compute square roots quickly.

Example 2. Solving implicit equations in engineering. Equations such as the Colebrook equation for pipe friction have no closed-form solution and are solved by Newton-Raphson, which is why the method is built into engineering and scientific software.

Try this

Q1. Write the Newton-Raphson iteration formula. [1 mark]

• Cue. $x_{n+1} = x_n - \dfrac{\mathrm{f}(x_n)}{\mathrm{f}'(x_n)}$.

Q2. What does $x_{n+1}$ represent geometrically? [1 mark]

• Cue. The point where the tangent to the curve at $(x_n, \mathrm{f}(x_n))$ crosses the $x$-axis.

Q3. Why should the starting value avoid a turning point of $\mathrm{f}$? [2 marks]

• Cue. There $\mathrm{f}'(x_n) \approx 0$, so the correction $\tfrac{\mathrm{f}}{\mathrm{f}'}$ is huge and throws the next estimate far from the root.