Generalized Pandrosion Residuals: Numerical Analysis of an Iterative Geometric Construction for p-th Roots

We generalize Pandrosion's geometric construction — originally designed to approximate the cube root of 2 by inserting proportional means via Thales' theorem — to any integer p ≥ 2 and any positive real x > 0. The construction produces two sequences: an internal parameter uₙ and an output vₙ converging to α = x^1/p. After normalized substitution, the iteration takes a universal form involving the geometric polynomial Sₚ (s) = 1 + s + s² +. . . + s^p-1. We prove that the fixed point is s* = x^-1/p, that convergence is linear, and we compute the exact contraction ratio λ₏, ₗ, expressed entirely in terms of the target α. We establish non-asymptotic error bounds, prove monotonicity of λ₏, ₗ in both p and x, extend the theory to x < 1 (where convergence becomes oscillatory), determine the optimal starting point, and show that Steffensen's acceleration of Pandrosion's iteration achieves quadratic convergence — bridging ancient geometric methods and modern numerical analysis.