Wang, Wen-Jun; Zhu, Ju-Bo; Duan, Xiao-Jun Eigenvalue decomposition-based modified Newton algorithm. (English) Zbl 1299.65104 Math. Probl. Eng. 2013, Article ID 523895, 4 p. (2013). Summary: When the Hessian matrix is not positive, the Newton direction may not be the descending direction. A new method named eigenvalue decomposition-based modified Newton algorithm is presented, which first takes the eigenvalue decomposition of the Hessian matrix, then replaces the negative eigenvalues with their absolute values, and finally reconstructs the Hessian matrix and modifies the searching direction. The new searching direction is always the descending direction. The convergence of the algorithm is proven and the conclusion on convergence rate is presented qualitatively. Finally, a numerical experiment is given for comparing the convergence domains of the modified algorithm and the classical algorithm. MSC: 65H10 Numerical computation of solutions to systems of equations 65F15 Numerical computation of eigenvalues and eigenvectors of matrices PDF BibTeX XML Cite \textit{W.-J. Wang} et al., Math. Probl. Eng. 2013, Article ID 523895, 4 p. (2013; Zbl 1299.65104) Full Text: DOI OpenURL References: [1] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, Mass, USA, 1999. · Zbl 1040.90546 [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1250.97022 [3] A. A. Goldstein and J. F. Price, “On descent from local minima,” Mathematics of Computation, vol. 25, no. 115, pp. 569-574, 1971. · Zbl 0223.65020 [4] A. V. Fiacco and G. P. McCormick, “The sequential unconstrained minimization technique for nonlinear-a primal-dual method,” Management Science, vol. 10, no. 2, pp. 360-365, 1964. [5] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. · Zbl 0241.65046 [6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, USA, 1985. · Zbl 0576.15001 [7] M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” The Computer Journal, vol. 7, pp. 155-162, 1964. · Zbl 0132.11702 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.