Abstract
The modified Anderson–Björck (modAB) method was first proposed in 2023 by Ganchovski and Traykov for solving nonlinear equations in structural mechanics. It was improved in 2026 with better switching criteria that account for function symmetry and a bisection fallback based on the bracket reduction rate. Although the algorithm performed well in benchmarks, it lacked formal mathematical analysis and proof of convergence. Later analysis by L. Tomov showed some inconsistencies like premature endpoint corrections on the first AB step, and the use of corrected ordinates in the symmetry factor calculation instead of the true residuals. This paper removes the above inconsistencies, provides a proof of global convergence in exact arithmetic, and adds floating-point safeguards for interpolation, method switching and convergence checks, with explicit treatment of infinite and NaN values, underflow, overflow and round-off errors. As a result, the reliability of the algorithm is improved. Performance is also enhanced by an adaptive bisection fallback when AB fails to reduce either the bracket width or the residual, keeping a finite bound on residual reducing steps. Numerical experiments were conducted for an extended set of 100 functions.