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Home › Math & Conversions › Core Math & Algebra › Bisection Method Calculator 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. Calculate root Clear table 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 Copy as CSV # 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). Bisection method – FAQ What happens if f(a) and f(b) have the same sign? + If \( f(a) \) and \( f(b) \) do not have opposite signs, the bisection method cannot guarantee a root between a and b. This calculator will show an error and refuse to start. In practice, you should scan the function or plot it to locate intervals where the sign changes. How does this compare to Newton’s method? + Newton’s method typically converges much faster (quadratically near the root) but requires a good initial guess and the derivative \( f'(x) \). It can diverge if the guess is poor. The bisection method is slower but much more robust, provided you can bracket a root with a sign change. What tolerance should I choose? + It depends on your application. For teaching examples and quick checks, \( 10^{-4} \) or \( 10^{-6} \) is usually enough. For high-precision scientific work you may want \( 10^{-8} \) or smaller, but keep in mind machine precision limits and that the function itself might not be known that accurately. Can this method find complex roots? + No. The classic bisection method works on real-valued functions on real intervals. To locate complex roots you need different techniques (for example, complex Newton methods or contour integral methods). Core Math & Algebra tools G-test (likelihood-ratio test) FOIL Method Calculator Graph Adjacency Matrix Speed, Distance, Time Friedman Test Traveling Salesman Problem (TSP) Stem and Leaf Plot Generator Bisection Method – you are here Monte Carlo Simulation Lottery Odds Dot Product Matrix Determinant Partial Fraction Decomposition Trapezoid Area Combination and Permutation More numerical methods Newton–Raphson Method Secant Method Euler’s Method (ODE) Taylor Series Expansion
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