Numerical solution of equations

What you’ll learn

• Locate roots using sign changes and interval methods.
• Apply linear interpolation and Newton–Raphson iteration.

Understand the idea

For a continuous function, opposite signs at the ends of an interval guarantee at least one root inside it. Bisection repeatedly halves this bracket; linear interpolation uses the straight line joining the two endpoint values to estimate where the curve crosses the axis.

Newton–Raphson uses the tangent at a current estimate. It can converge quickly near a simple root, but a zero or very small derivative, a poor starting point, or a discontinuity can make the iteration fail. Retain extra digits until the final answer and check the requested accuracy with a bracket when appropriate.

Key formulae

$$f(a)f(b)<0$$

$$x_{\mathrm{mid}}=\frac{a+b}{2}$$

$$x_{\mathrm{lin}}=a-\frac{f(a)(b-a)}{f(b)-f(a)}$$

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

Worked example

Use one Newton–Raphson step for $x^3-5=0$ starting at $x_0=2$.

1. Let $f(x)=x^3-5$, so $f'(x)=3x^2$.
2. At $x_0=2$, $f(2)=3$ and $f'(2)=12$.
3. Compute $x_1=2-3/12$.

Answer: $x_1=1.75$.