Root finding by bisection in Java

Run this example

Complete Getting Started, then run from the repository root:

mvn exec:java -Dexec.mainClass=dev.nm.examples.RootFindingExample

Problem and mathematics

Find a root in [0.4, 1.0] of

f(x)=x^4-\sin(x)\cos(x).

Bisection repeatedly halves a bracket. For a continuous function, opposite signs at the endpoints establish that at least one root lies inside. The chosen interval matters: this example is not a search for every root of the equation.

Walk through the API

Create an anonymous AbstractUnivariateRealFunction and implement evaluate(double). Construct BisectionRoot with tolerance 1e-10 and a maximum of 10,000 iterations, matching the source test. Call solve(f, lower, upper).

The example checks the endpoint signs before calling the solver. It leaves NoRootFoundException visible to the caller. A production application should report convergence failure with the input interval and model context.

Expected result and validation

x\approx0.8396570801734923.

The executable checks the distance to this reference and the function residual abs(f(root)), both to 1e-10. These checks answer different questions: a small residual does not always mean a small error in x, especially near a flat root.

Assumptions and mistakes to avoid

Opposite signs across a discontinuity do not establish a root. A bracket whose endpoints have the same sign may contain roots but is unsuitable for this bisection setup. In particular, [0.4, 0.8] does not bracket the root used here; the upstream tests cover that invalid interval.

Use function units and variable scale when choosing tolerances. Avoid requesting precision beyond what double arithmetic and the model can support.

Try it

Replace the function with x*x - 2 on [1, 2] and compare against Math.sqrt(2). Then deliberately choose [2, 3] and observe the bracket validation.

Source and API reference

Adapted from BisectionRootTest.java, test_solve_0010. See source provenance for the exact NM Dev commit and test path. The full runnable Java source linked above includes imports, construction steps and checks. Browse the NM Dev Javadoc for the classes used here.