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

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(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.