We extend the ultraspherical spectral method to solving nonlinear ODE boundary value problems. We propose to use the inexact Newton-GMRES framework for which an effective preconditioner can be constructed and a fast Jacobian-vector multiplication can be effected, thanks to the structured operators of the ultraspherical spectral method. With a mixed-precision implementation, the inexact Newton-GMRES-ultraspherical framework exhibits extraordinary speed and accuracy, as we show by extensive numerical experiments.

翻译：我们扩展了超球面谱方法以求解非线性常微分方程边值问题。提出采用非精确牛顿-广义最小残差法框架，借助超球面谱方法的结构化算子特性，可构建高效预处理器并实现快速雅可比向量乘积。通过混合精度实现，数值实验表明该非精确牛顿-广义最小残差法-超球面谱方法框架展现出卓越的速度与精度。