02 Mar 2026

Roots of Functions

Numerical meaning of roots, bracketing, bisection, Newton-Raphson method, and convergence checks.

msc semester-i numerical-methods roots

A root of a function is a value $x=\alpha$ for which

\[f(\alpha)=0.\]

Root-finding is common in physics: equilibrium positions, turning points, resonance conditions, and eigenvalue equations often reduce to finding zeros of a function.

Bracketing idea

If $f(a)$ and $f(b)$ have opposite signs, then a continuous function has at least one root between $a$ and $b$.

\[f(a)f(b)<0.\]

This is the basis of the bisection method.

Bisection method

The midpoint is

\[c=\frac{a+b}{2}.\]

If $f(a)f(c)<0$, the root lies in $[a,c]$; otherwise it lies in $[c,b]$. Repeating this process gives a narrower interval.

Newton-Raphson method

Newton’s method uses the tangent line at the present approximation:

\[x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.\]

It is usually faster than bisection, but it needs a derivative and a good starting value.

Stopping criteria

An iteration may be stopped when either

\[|x_{n+1}-x_n|<\epsilon\]

or

\[|f(x_n)|<\epsilon.\]

Here $\epsilon$ is the chosen tolerance.

Key points

• Bisection is slow but reliable.
• Newton-Raphson is fast but can fail for poor starting values.
• Always check the final value by substituting it into $f(x)$.