Applications of Fast Differentiation to Numerical Solution of Ordinary Differential Equations, Numerical Integration and Optimization.

(1)自动微分在常微分方程数值解中的应用。考虑了使用函数二阶导数的显式龙格-库塔方法。结果表明，只存在嵌入三阶方法的两阶四阶方法。此外，我们还提出了五阶和六阶方法，这是关于局部截断误差最有希望的方法。同时考虑了在欧拉-梯形规则中的应用，即采用牛顿法一次迭代的Rosenbrock法计算修正量。我们现在比较普通的Rosenbrock法和自动微分法。(2)自动微分在数值积分中的应用。我们还考虑了Romberg积分的新变体，它对于没有奇异点的函数是有效的。只在两个端点处求导。结果表明，在这些变量中，使用一阶导数的方法就计算工作量而言是最有希望的方法之一，因为它与半步长的标准Romberg积分达到相同的精度。(3)自动微分在光线追踪中的应用。自动微分可用于解决计算机图形学中的各种各样的问题。提出了一种利用自动微分法对隐曲面进行光线追踪的算法。并研究了这些方法对其他问题的有效性。

(1) Application of Automatic Differentiation to Numerical Solution of Ordinary Differential Equations. Explicit Runge-Kutta methods using the second derivatives of function are considered. It is shown that there exist only two stage fourth-order method in which the third-order method is embedded. Furthermore, we presented the fifth-and sixth-order methods which are the most promising methods with respect to the local truncation error. The application to Euler-Trapezoidal rule is also considered, namely, Rosenbrock method using one iteration of Newton method in the computation of the corrector. We are now comparing the ordinary Rosenbrock method with that using automatic differentiation.(2) Application of Automatic Differentiation to Numerical Integration. We also considered the new variants of the Romberg integration which is efficient for the functions having no singularities. They use the derivatives only at both end points. It is shown that among these variants the method using the first derivatives is one of the most promising with respect to the amount of computational work, because it achieves the same accuracy as the standard Romberg integration with half stepsize.(3) Application of Automatic Differentiation to Ray Tracing. Automatic differentiation can be used to solve a wide variety of problems in computer graphics. We presented an algorithm for ray tracing implicit surfaces using automatic differentiation method. and investigated the usefulness of such methods for other problems.