Global Convergence and Reliable Stopping in a Safeguarded Modified Anderson--Björck Method
Abstract
The modified Anderson-Bjorck (modAB) method was first proposed in 2023 by Ganchovski and Traykov for solving nonlinear equations in structural mechanics. It was improved further in 2026 by introducing better switching criteria with account to function symmetry, and bisection fallback based on measuring the bracket reduction rate. Although the algorithm demonstrated superior performance across a great variety of functions, it lacked formal mathematical analysis and proof of convergence. In addition, some discrepancies were noted by L. Tomov in the latest version that need to be addressed: premature application of the Anderson-Bjorck (AB) correction on the first AB step, caused by retaining the endpoint-update record from the preceding bisection step, and using the corrected endpoint ordinates in symmetry factor calculation instead of the true residuals. The purpose of the present paper is to provide rigorous mathematical proof of global convergence in exact arithmetic, remove the above discrepancies, and provide floating-point safeguards for interpolation, switching and stopping, with explicit treatment of infinite and NaN values, underflow, overflow and round-off errors. These safeguards do not certify the accuracy of the supplied function values. Additional performance improvement was introduced by allowing for early bisection fallback when AB fails to make progress on reducing either the bracket width or the residual, while retaining a finite bound on consecutive exceptions to the prescribed width condition.
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BibTeX
@misc{tomov2026global,
title = {{Global Convergence and Reliable Stopping in a Safeguarded Modified Anderson--Björck Method}},
author = {Latchezar Tomov and Nedelcho Ganchovski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14193},
eprint = {2610.14193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14193},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Global Convergence and Reliable Stopping in a Safeguarded Modified Anderson--Björck Method
AU - Latchezar Tomov
AU - Nedelcho Ganchovski
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14193
UR - https://arxiv.org/abs/2610.14193
AB - The modified Anderson-Bjorck (modAB) method was first proposed in 2023 by Ganchovski and Traykov for solving nonlinear equations in structural mechanics. It was improved further in 2026 by introducing better switching criteria with account to function symmetry, and bisection fallback based on measuring the bracket reduction rate. Although the algorithm demonstrated superior performance across a great variety of functions, it lacked formal mathematical analysis and proof of convergence. In addition, some discrepancies were noted by L. Tomov in the latest version that need to be addressed: premature application of the Anderson-Bjorck (AB) correction on the first AB step, caused by retaining the endpoint-update record from the preceding bisection step, and using the corrected endpoint ordinates in symmetry factor calculation instead of the true residuals. The purpose of the present paper is to provide rigorous mathematical proof of global convergence in exact arithmetic, remove the above discrepancies, and provide floating-point safeguards for interpolation, switching and stopping, with explicit treatment of infinite and NaN values, underflow, overflow and round-off errors. These safeguards do not certify the accuracy of the supplied function values. Additional performance improvement was introduced by allowing for early bisection fallback when AB fails to make progress on reducing either the bracket width or the residual, while retaining a finite bound on consecutive exceptions to the prescribed width condition.
ER -