Implementation and Development of Application of Fast Automatic Differentiation

快速自动微分(FAD)是由Masao IRI于1983年提出的。本研究的目的是开发一个软件来实现FAD及其在实际问题中的应用，并将FAD建立为一种数值计算技术。本研究的结果如下：(1)FAD前处理程序的改进：在已有的FORTRAN前处理程序的基础上，利用C语言开发了一种操作符重载技术。这两种实现被改进为一个用于数值实验的软件系统。它们还可以计算函数和梯度的舍入误差估计，并可移植到各种类型的计算机上。(2)严格的舍入误差估计理论及其在数值算法中的应用：从理论上和实验上证明，使用FAD可以得到更严格和更精确的误差估计。指出了如何改进计算图的构造方法。将FAD应用于求解非线性方程组和几何/地理优化问题，证明了FAD的有效性。事实上，结果的准确性、计算速度、收敛准则等都得到了显著的改善。(3)FAD与公式操纵的混合：由于(1)和(2)中所述的研究比预期的时间长，这一点尚未完成。上述研究成果已在国内外研究会议上公布，并在世界上领先于这类研究。

FAST AUTOMATIC DIFFERENTIATION (FAD) was proposed by Masao IRI in 1983. The objectives of this research are to develop a software to realize FAD and its application to real-world problems and to establish FAD as a technique of numerical computation.The results of this research are as follows :(1) Improvement of FAD preprocessor : In addition to the already available FORTRAN preprocessor, an operator-overloading technique by means of c++ was developed. Those two types of implementations were improved to a software system for numerical experiment. They are capable also of computing the rounding error estimates of functions and gradients and are portable to various types of computers.(2) Rigorous theory of rounding error estimation and use of the estimation in numerical algorithms : It was theoretically proved and experimentally evidenced that, using FAD, the error estimation can be rigorous and sharper. It was pointed out how to improve the method of constructing computational graphs. Application of FAD to the solution of nonlinear equation systems and geometrical/geographical optimization problems proved the effectiveness of FAD. In fact the accuracy of the results, the computational speed, the convergence criterion, etc. Were remarkably ameliorated.(3) Hybridization of FAD and formula manipulation : This has not yet been completed because it took longer than expected for the research described in (1) and (2).The above-described research results have been publicized at research meetings at home and abroad, and turned out to lead this kind of research in the world.