混合积分  如果将切线性模式和伴随模式相结合，往往可以避免梯度向量运算中的诸多冗余计算。例如，ADJIFOR系统在求解雅可比矩阵时，在语句级微分实现中首先用伴随方法求得所有偏导数，然后做梯度向量积分；其计算时间代价与 和模式的语句数目有关，而其存储代价为 。具体讨论可参考文献[7]。

5．结论

        切线性模式在无截断误差意义上计算函数的方向导数、梯度或雅可比矩阵，以及在模式的可预测性及参数敏感性分析、伴随模式构造等相关问题中有着广泛应用。DFT系统主要用于求解FORTRAN 77语言编写的切线性模式，具有很强的全局数据相关分析能力。此外，DFT系统还具有其它几个重要特色，如结构化的微分实现、自动生成微分测试程序以及基于语句级的微分代码优化。本文简单给出了DFT系统在求解数值和符号导数和微分、稀疏雅可比矩阵中的应用。为评价一类自动微分系统，本文初步提出了统计准确率的概念。
  
参考文献

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[5]   Mu Mu, et al，The predictability problem of weather and climate prediction, Progress in Nature Science, accepted.

[6]  Giering R. et al. Recipes for Adjoint Code Construction. ACM Trans. On Math. Software. 1998，24（4）：
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[7]  C. Bischof, A. Carle, P. Khademi, and G. Pusch. "Automatic Differentiation: Obtaining Fast and Reliable
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