Assess learners' ability to analyze and compare the convergence behavior of the Newton–Raphson and Secant methods for root finding: test theoretical local convergence orders and conditions, effects of multiple roots and near-zero derivatives, asymptotic error constants, basins of attraction and sensitivity to initial guesses, iteration counts versus function/derivative evaluation cost, common failure modes, and selection of robust stopping criteria; include proof sketches, quantitative error estimates, and short numerical experiments on representative functions to interpret and justify conclusions.