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Root Finding: Bisection, Newton, Secant

What you are seeing: a function f(x)f(x) and the iterates of three classical root finders. Bisection halves the bracket each step (always convergent if ff changes sign on the bracket). Newton uses tangent lines (quadratic convergence near a simple root, but can diverge with bad initial guesses). Secant uses chord slopes (superlinear, golden-ratio order).

Figure 1. Trail of iterates from three root finders against f(x)f(x).
x0 / left1.00
x1 / right2.50
function
method

WHAT TO TRY

  • Switch methods on the same function: bisection halves the bracket every step (slow but sure), Newton rides the tangent (fast but can fly off), secant approximates the tangent from two points.
  • Give Newton a bad starting point: the tangent can overshoot, oscillate, or diverge, the price of its speed when it fails to converge.
  • Count iterations to a tolerance: Newton roughly doubles the correct digits each step while bisection adds a fixed fraction of one, the gap between quadratic and linear convergence.