This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be expensive, we compute directional derivatives with Taylor-mode automatic differentiation. We first implement Householder's method with arbitrary order for one variable, and investigate the trade-off between computational cost and convergence order. We find that the second-order variant, i.e., Halley's method, to be the most valuable, and further generalize Halley's method to systems of nonlinear equations and demonstrate that it can scale efficiently to large-scale problems. We further apply Halley's method on solving large-scale ill-conditioned nonlinear problems, as well as solving nonlinear equations inside stiff ODE solvers, and demonstrate that it could outperform Newton's method.

翻译：本文提出并实现了一系列具有高阶收敛性的非线性求解器新方法，其核心在于高效计算高阶导数。为避免计算成本高昂的完整导数，我们采用泰勒模式自动微分技术计算方向导数。我们首先实现了任意阶单变量Householder方法，并深入研究了计算成本与收敛阶数之间的权衡关系。研究发现二阶变体（即Halley方法）最具实用价值，进而将Halley方法推广至非线性方程组求解，并证明其能高效扩展至大规模问题。我们进一步将Halley方法应用于大规模病态非线性问题求解，以及在刚性常微分方程求解器中处理非线性方程组，实验证明其性能可超越传统牛顿法。