Bisection Method Calculator
Interactive bisection method calculator. Enter f(x), a bracket [a, b] with a sign change, tolerance, and max iterations. See each midpoint, the convergence bound, and a downloadable iteration log.
Function & Interval
Use x as the variable and Math helpers such as sin, cos, exp, log, sqrt. Powers accept ^ or **.
The method requires f(a)·f(b) < 0. If the signs match, the calculator refuses to run until you supply a proper bracket.
Accuracy & Iterations
Absolute tolerance for the midpoint value or the interval half-length.
Stops the loop early if convergence slows.
Controls how many decimals appear in the summary and table.
Set the inputs and click Calculate root to see the theoretical bound.
Stop reason will appear here once the method runs.
How to Use This Calculator
Use the bisection method to numerically approximate a root of f(x) = 0 on an interval [a, b]. The calculator checks that f(a) and f(b) have opposite signs, then iteratively halves the bracket while logging midpoints, function values, and the shrinking half-length.
Ideal for numerical analysis, engineering, or quick recon that needs a deterministic, reliable root finder. Adjust tolerance and iteration limits to balance precision against runtime, then review the iteration log to confirm convergence.
Steps at a glance
- Enter the function using x and valid Math helpers (sin, cos, exp, log, sqrt, etc.). Powers may be written with ^ or **.
- Choose a and b so that a sign change exists: f(a) · f(b) < 0.
- Set tolerance ε, cap the iterations, and select how many decimals the interface should display.
- Click Calculate root; the summary cards update and the iteration table populates when you toggle it into view.
Methodology
This implementation follows the standard fixed-point bisection algorithm. After verifying the bracket, the midpoint m = (a + b)/2 is evaluated. Depending on the sign of f(m), the algorithm keeps the subinterval that still brackets a root and repeats until the tolerance condition or iteration limit is met.
The theoretical half-length after N steps is L_N = L_0 / 2^N. Ensuring this quantity drops below ε yields the a priori bound used in the summary.
Advantages
- Guaranteed convergence whenever the function is continuous and a sign change exists.
- Simple, stable algorithm that needs no derivatives.
- Deterministic iteration count with transparent rounding.
Limitations
- Requires a bracketing interval with opposite signs.
- Slower than Newton or secant methods for smooth functions.
- Finds at most one root per bracket; other intervals are needed for additional roots.
Full original guide (expanded)
The calculator stops when |f(m)| ≤ ε, the interval half-length (b − a)/2 ≤ ε, or the max iteration cap is reached. Each row in the iteration table shows the current bracket, midpoint, function evaluations, and half-length so you can trace convergence.
Choose tolerances around ±1e-4 to ±1e-6 for teaching or quick checks. Scientific applications may push 1e-8 or tighter, but watch for floating-point limits and smoothness of f(x).