Locating roots by change of sign, iterative methods of the form x equals g of x, the Newton-Raphson method, and the conditions under which these numerical methods fail.

What this dot point is asking

OCR wants you to locate a root of an equation by showing a change of sign over an interval, solve an equation by a fixed-point iteration $x_{n+1} = g(x_n)$ (and illustrate convergence with a staircase or cobweb diagram), use the Newton-Raphson method, and explain when each method fails (for example a sign change missed because two roots lie close together, an iteration that diverges, or Newton-Raphson at a turning point).

The answer

Locating a root by change of sign

If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, then $f$ has at least one root between $a$ and $b$. You must state continuity for the argument to be valid.

Fixed-point iteration

Rearrange $f(x) = 0$ into the form $x = g(x)$, then iterate $x_{n+1} = g(x_n)$ from a starting value. If the iteration converges, the limit is a root. Different rearrangements converge at different speeds, and some diverge.

A staircase diagram (when $g'(x) > 0$) or a cobweb diagram (when $g'(x) < 0$) drawn between $y = g(x)$ and $y = x$ shows the successive iterates closing in on, or spiralling away from, the root.

The Newton-Raphson method

Newton-Raphson uses the tangent at the current estimate to leap to a better one. Geometrically, you follow the tangent line at $x_n$ down to the $x$-axis, and that crossing is the next estimate. When it works it roughly doubles the number of correct digits each step, so it usually converges very fast from a good starting value.

Examples in context

Using an iteration to a required accuracy

To show a root is correct to a number of decimal places, find a sign change of $f$ over an interval whose endpoints round to that value (for example $[1.8595, 1.8605]$ for $1.860$ to three decimal places).

Establishing an error bound

Once an iteration appears to have settled, you confirm the accuracy by bracketing: evaluate $f$ at the two ends of a small interval centred on the rounded value. A sign change over that interval, with continuity, proves the true root lies inside it, so the rounded value is correct to the stated accuracy. This sign-change check is the standard way to justify a final answer to a required number of decimal places.

When the methods fail

Each method has a failure mode you may be asked to explain. A change-of-sign search misses a repeated root or two close roots where $f$ does not change sign. An iteration diverges when $|g'(x)| > 1$ near the root, spiralling or staircasing away. Newton-Raphson fails or jumps far away if $f'(x_n)$ is zero or very small (a turning point near the estimate), or if the starting value is poor.

Try this

Q1. Show that $x^3 + x - 5 = 0$ has a root between $1$ and $2$. [2 marks]

• Cue. $f(1) = -3 < 0$, $f(2) = 5 > 0$, and $f$ is continuous, so a root lies in $(1, 2)$.

Q2. With $f(x) = x^2 - 5$ and $x_0 = 2$, apply Newton-Raphson once. [3 marks]

• Cue. $f'(x) = 2x$, so $x_1 = 2 - \dfrac{4 - 5}{4} = 2 + 0.25 = 2.25$.