A Cholesky-Factor-Based Levenberg-Marquardt Method for Solving Nonlinear Systems of Equations
DOI:
https://doi.org/10.33003/Keywords:
Levenberg--Marquardt method, Nonlinear systems, Cholesky factorization, Numerical analysis.tAbstract
We present a Levenberg-Marquardt (LM) algorithm for solving nonlinear system of equations.Nonlinear systems of equations are central to numerous scientific and engineering applications,
yet solving them efficiently remains computationally challenging. To address limitations in existing
Levenberg--Marquardt (LM) parameter selection strategies, this paper presents an efficient LM algorithm
(ACLMM) designed to reduce computational cost. In the proposed scheme, the regularization parameter was
derived analytically using established result in combination with the maximum diagonal entry of the Cholesky
factor matrix. The proposed method was evaluated on seven standard benchmark problems and compared against
two existing LM methods the Corrected Levenberg-Marquardt (CLM) and Algebraic Rule for Levenberg-Marquardt
Parameter (ARCLM) methods using iteration counts and CPU time, across varying problem dimensions. Numerical
results demonstrated that the ACLMM algorithm generally exhibited competitive or improved performance across
the tested problems. Notably, for large-scale instances such as Problem 1 ($n=1000$), ACLMM reduced the iteration
count to 8 compared to 23 for CLM and 14 for ARCLM while maintaining small residual norms. Overall, the
incorporation of Cholesky factor diagonal entries into the regularization parameter provides an effective
and numerically stable alternative for solving nonlinear systems.
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Copyright (c) 2026 Ya"u Balarabe Musa, Muhammad Yusuf Waziri, Hassan Muhammad
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