Derivative convergence for iterative equation solvers
Derivative convergence for iterative equation solvers
复制标题
迭代方程求解器的导数收敛
DOI：
10.1080/10556789308805549
复制
发表时间：
1993
影响因子：
2.2
通讯作者：
K. Williamson
中科院分区：
文献类型：
--
作者：
A. Griewank;C. Bischof;G. Corliss;A. Carle;K. Williamson
当非线性方程求解器应用于参数相关问题时，它们的迭代可以解释为这些可变参数的函数。这些迭代函数的导数（如果存在）可以通过自动微分的前向模式递归求值。然后，人们可能会问这些导数值是否以及以多快的速度收敛到隐式解函数的导数，这可能是参数识别、敏感性研究或设计优化所需要的。这里表明，对于一大类无记忆收缩或割线更新方法，导数收敛是通过 R 线性或可能的 R 超线性速率实现的。对于更广泛的多步收缩，我们获得了简化导数递推的 R 线性收敛，这更经济并且可以轻松推广到高阶导数。我们还基于隐函数定理制定了导数收敛的构造性准则。所有理论结果...
When nonlinear equation solvers are applied to parameter-dependent problems, their iterates can be interpreted as functions of these variable parameters. The derivatives (if they exist) of these iterated functions can be recursively evaluated by the forward mode of automatic differentiation. Then one may ask whether and how fast these derivative values converge to the derivative of the implicit solution function, which may be needed for parameter identification, sensitivity studies, or design optimization. It is shown here that derivative convergence is achieved with an R-linear or possibly R-superlinear rate for a large class of memoryless contractions or secant updating methods. For a wider class of multistep contractions, we obtain R-linear convergence of a simplified derivative recurrence, which is more economical and can be easily generalized to higher-order derivatives. We also formulate a constructive criterion for derivative convergence based on the implicit function theorem. All theoretical resul...