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Nonlinear successive over-relaxation. (English) Zbl 0566.65045
Let \(\Phi\) be a real strictly convex functional defined and twice continuously differentiable on a convex domain in \({\mathbb{R}}^ n\). To find its minimum the authors solve a system of nonlinear equations \(F(x)=0\) in \({\mathbb{R}}^ n\), where F denotes grad \(\Phi\). They present two theorems giving sufficient conditions of convergence for: (i) a nonlinear analogue of the Gauss-Seidel method for positive-definite matrices, (ii) a nonlinear successive overrelaxation method, used for such a system of equations. The theorems are parallel to results of Schechter (1962, 1968) but give more general sufficient conditions and may be applied for a more general class of functionals whose Hessian matrix may be singular.
Reviewer: S.Ząbek

MSC:
65K05 Numerical mathematical programming methods
90C25 Convex programming
65H10 Numerical computation of solutions to systems of equations
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References:
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