In the case of singular (and possibly even nonisolated) solutions of nonlinear equations, while superlinear convergence of the Newton method cannot be guaranteed, local linear convergence from large domains of starting points still holds under certain reasonable assumptions. We consider a linesearch globalization of the Newton method, combined with extrapolation and over-relaxation accelerating techniques, aiming at a speed up of convergence to critical solutions (a certain class of singular solutions). Numerical results indicate that an acceleration is observed indeed.

中文翻译：

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加速全局牛顿法收敛到非线性方程的临界解

在非线性方程的奇异（甚至可能是非孤立的）解的情况下，虽然不能保证牛顿法的超线性收敛，但在某些合理的假设下，从大范围出发点的局部线性收敛仍然成立。我们考虑牛顿法的线性搜索全球化，结合外推法和过度松弛加速技术，旨在加速收敛到关键解（一类奇异解）。数值结果表明确实观察到了加速度。