Mathematics / Free calculator

Bisection root finder

Find a real root within a sign-changing interval and inspect each bracket update.

01 Your inputs

Use x, + − * / ^, sin, cos, exp, ln, sqrt. Angles are in radians. Use explicit multiplication for clarity.

Entered function
f(x)=x2−2

1e-14 to 0.1. Arrows adjust by a factor of 10; you can also type any value in range. A residual tolerance is not a guaranteed bound on root error.

1–80 whole iterations.

Calculated on your device. Inputs are not transmitted or stored.

02 Your result

Enter your values and select Calculate.

The method

xmid=a+b2

A worked example

For f(x) = x² − 2 on [0, 2], the root is √2 ≈ 1.4142135624. Each iteration halves the enclosing interval.

Assumptions & limits

Real scalar functions with the restricted expression parser. Tolerances are absolute. Stopping by residual does not guarantee a small root error for flat or poorly conditioned functions. Discontinuities, poles, and convergence failures require independent review. Requires a < b and opposite endpoint signs unless an endpoint is exactly zero. Continuity on the entire bracket is essential; a sign change across a pole does not establish a root. Full double precision between operations; displayed values use up to 10 decimal places. Numerical outputs above 10¹⁰⁰ and nonfinite function evaluations are rejected.

Calculation source: Math Workbench: bisection method.

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