Root Finding With Binary Search

In a Siemens NX geometry workflow, a parameterized calculation may require finding where a monotonic function reaches zero. Implement a binary-search, or bisection, method to approximate the unique root of a cubic polynomial.

The polynomial is defined as f(x) = a*x^3 + b*x + c, where a > 0 and b >= 0. Given coefficients [a, b, c], an interval [lo, hi], an absolute tolerance, and a maximum iteration count, return an approximation to the root. The input guarantees that f(lo) <= 0 <= f(hi), so the root is within the interval. Return the result rounded to 6 decimal places.

Formal Specification

Implement root_binary_search(coefficients, lo, hi, tolerance, max_iterations).

• coefficients is a list [a, b, c] of numbers.
• lo, hi, and tolerance are floating-point numbers.
• max_iterations is a positive integer.
• Return a floating-point approximation to the unique x such that f(x) = 0.

At each iteration, evaluate the midpoint and retain the half-interval that still contains the sign change.