What is the bisection method?

The bisection method is a numerical algorithm for finding a root of a continuous function f(x) = 0 on an interval [a, b] where f(a) and f(b) have opposite signs. At each step, it halves the interval and selects the sub-interval where the sign of f(x) changes, guaranteeing convergence to a root under mild conditions.

When does the bisection method converge?

The bisection method converges when f is continuous on [a, b] and f(a) and f(b) have opposite signs. Under these conditions, the Intermediate Value Theorem guarantees at least one root in the interval, and the repeated halving of the interval converges to a root.

How many iterations does the bisection method need?

If the initial interval length is L0 = b − a and you want the final interval length to be at most ε, then the number of iterations N required by the bisection method is at most N ≥ log2(L0/ε). The calculator displays both the actual iteration count used and this theoretical upper bound.

What stopping criteria does this bisection calculator use?

This calculator stops when either the absolute value |f(m)| at the current midpoint m is less than or equal to the tolerance, or when the interval length (b − a)/2 falls below the tolerance. You can also impose a maximum number of iterations to avoid unnecessarily long runs.

Can the bisection method fail?

The method can fail to start if f(a) and f(b) do not have opposite signs, or if f is not continuous on [a, b]. Even when it runs, if your tolerance is unrealistically small or if the function is badly conditioned, it may reach the maximum number of iterations before meeting the stopping criteria.

Bisection Method Calculator

Use the bisection method to numerically approximate a root of \( f(x) = 0 \) on an interval \([a, b]\). This tool checks the sign-change condition, runs the iterations step-by-step, and gives you the final approximation together with an iteration table and theoretical iteration bound.

Ideal for numerical analysis courses, engineering calculations and quick checks when you need a guaranteed-convergence root-finding method.

1. Enter function and interval

Function \( f(x) \)

Use x as the variable. Supported functions include sin, cos, tan, exp, log (natural log), sqrt, and more via JavaScript’s Math object. Use ^ or ** for powers.

Left endpoint \( a \)

Right endpoint \( b \)

The bisection method requires opposite signs at the endpoints: \( f(a) \cdot f(b) < 0 \).

2. Set tolerance and iteration limit

Tolerance \( \varepsilon \)

Target accuracy for the root and interval length.

Max iterations

Safety cutoff to prevent infinite loops.

Decimals in output

Display precision in the table and results.

Deterministic & guaranteed convergence (with sign change)

Bisection method results

Root approximation

Approximate root
f(root)
Final interval
Half-length
Iterations used
Max iterations

Theoretical iteration bound

This bound assumes only the interval-length criterion \( \frac{b-a}{2^N} \le \varepsilon \). The actual stopping rule also checks \( |f(m)| \le \varepsilon \).

Iteration table

#	a	b	m	f(a)	f(m)	f(b)	(b − a)/2

Each row shows the interval \([a_k, b_k]\), its midpoint \( m_k \), the function values at the endpoints and midpoint, and the half-length of the interval. The method always keeps a sign change across the current interval.

The bisection method – theory in a nutshell

The bisection method is one of the simplest and most robust algorithms for solving nonlinear equations of the form \( f(x) = 0 \). It is based on the Intermediate Value Theorem: if \( f \) is continuous on \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one root \( \xi \in (a, b) \) such that \( f(\xi) = 0 \).

Bisection algorithm

Check that \( f(a) \cdot f(b) < 0 \). If not, the method cannot start.
Compute the midpoint \( m = \dfrac{a + b}{2} \).
Evaluate \( f(m) \).
If \( f(m) = 0 \) (or \( |f(m)| \) is within tolerance), stop: \( m \) is the root approximation.
Otherwise, choose the new interval:
- If \( f(a) \cdot f(m) < 0 \), set \( b \gets m \).
- Else, set \( a \gets m \).
Repeat from step 2 until the interval is smaller than the tolerance or you hit the iteration limit.

Convergence and error bound

Let the initial interval length be \( L_0 = b_0 - a_0 \). After \( N \) bisection steps, the interval length is

\[ L_N = \frac{L_0}{2^N}. \]

To guarantee that the interval length is at most \( \varepsilon \), we need

\[ \frac{L_0}{2^N} \le \varepsilon \quad \Rightarrow \quad N \ge \log_2 \left( \frac{L_0}{\varepsilon} \right). \]

This gives a simple a priori bound on the number of iterations needed. The method converges linearly to the root.

Advantages and limitations

Advantages

Guaranteed convergence when \( f \) is continuous and there is a sign change on \([a, b]\).
Very stable and simple to implement.
No derivatives required.

Limitations

Requires an interval with a sign change (you must bracket a root).
Converges slower than methods like Newton–Raphson or secant.
Only finds one root per bracketing interval (others require separate intervals).