We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally numerically stable. Tangents of systems of linear equations can become instable as well as both tangents and adjoints of systems of nonlinear equations, which extends to optima of convex unconstrained objectives. Sufficient conditions for numerical stability are derived.

翻译：我们调查由于由数字求解者计算近似而导致的原始解决方案误差造成的正切和隐含功能的连接差错。线性方程式系统的连接结果在数字上是无条件稳定的。线性方程式系统的连接可能变得不稳,非线性方程式系统的切合和连接可能变得不均匀,非线性方程式系统的切合和连接也变得不均匀,因为非线性方程式系统延伸到对锥形无限制目标的优化。可以得出数字稳定性的充分条件。