We consider Broyden's method and some accelerated schemes for nonlinear equations having a strongly regular singularity of first order with a one-dimensional nullspace. Our two main results are as follows. First, we show that the use of a preceding Newton-like step ensures convergence for starting points in a starlike domain with density 1. This extends the domain of convergence of these methods significantly. Second, we establish that the matrix updates of Broyden's method converge q-linearly with the same asymptotic factor as the iterates. This contributes to the long-standing question whether the Broyden matrices converge by showing that this is indeed the case for the setting at hand. Furthermore, we prove that the Broyden directions violate uniform linear independence, which implies that existing results for convergence of the Broyden matrices cannot be applied. Numerical experiments of high precision confirm the enlarged domain of convergence, the q-linear convergence of the matrix updates, and the lack of uniform linear independence. In addition, they suggest that these results can be extended to singularities of higher order and that Broyden's method can converge r-linearly without converging q-linearly. The underlying code is freely available.
翻译:我们考虑布洛伊登的方法和一些非线性方程式的加速方案,这些方法与一维空格高度正常的一阶一阶一阶一等。我们的两个主要结果如下:首先,我们证明前牛顿式步骤的使用确保了类似恒星域与密度的起点的趋同性。首先,我们表明,前牛顿式步骤的使用确保了类似恒星域与密度的起点的趋同性。这大大扩展了这些方法的趋同性领域。第二,我们确定布洛伊登法的矩阵更新使q线性系数与迭代相趋同。这促使长期存在的问题,即布洛伊登矩阵是否通过显示这确实属于手动设置的情况而趋同。此外,我们证明,布洛伊登方向违反了统一的线性独立性,这意味着无法应用现有布洛伊登式矩阵趋同性结果。高精确度的数值实验证实了趋同性范围扩大,矩阵更新的q线性趋同性系数,以及缺乏统一的线性独立性。此外,它们表明这些结果可以扩大到更高顺序的奇异性,而布洛伊登法则可以自由趋同正弦基。