Successive column correction algorithms for solving sparse nonlinear systems of equations (Q1122315)

The paper deals with a system of nonlinear equations \(F(x)=0\) where F: \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\) is continuously differentiable on an open convex set \(D\subset {\mathbb{R}}^ n\), and the Jacobian matrix \(F'(x)\) is sparse. Two algorithms are proposed: (CM)-successive column correction algorithm and a modified CM-successive column correction algorithm. The former is based on the algorithm by \textit{T. F. Coleman} and \textit{J. J. Moré} [SIAM J. Numer. Anal. 20, 187-209 (1983; Zbl 0527.65033)] and the column-update algorithm. The later is a combination of the CM-successive column correction algorithm and the sparse Broyden (SB)-algorithm [cf. \textit{C. G. Broyden}, Math. Comput. 25, 285-294 (1971; Zbl 0227.65038)]. Both algorithms require only two function values at each iterative step. The numerical results show that the CM-successive column correction algorithms, especially the modified one, are competitive with the algorithm by \textit{A. R. Curtis}, \textit{M. J. D. Powell}, and \textit{J. K Reid} [J. Inst. Math. Appl. 13, 117-119 (1974; Zbl 0273.65036)] and the (SB)-algorithm.