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.

中文翻译：

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隐函数的切线和伴随的数值稳定性

我们研究由于数值求解器计算的近似值而导致原始解中的错误导致的隐函数的切线和伴随错误。线性方程组的伴随结果证明是无条件数值稳定的。线性方程组的切线以及非线性方程组的切线和伴随都可能变得不稳定，这扩展到凸无约束目标的优化。推导出数值稳定性的充分条件。