Mathematical Software with High Accuracy Based on the Fast Automatic Differentiation

Research Project

Project/Area Number	63580023
Research Category	Grant-in-Aid for General Scientific Research (C)
Allocation Type	Single-year Grants
Research Field	Informatics
Research Institution	Faculty of Engineering, Fukui University
Principal Investigator	HASEGAWA Takemitsu Fukui University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (70023314)
Project Period (FY)	1988 – 1989
Project Status	Completed (Fiscal Year 1989)
Budget Amount *help	¥1,000,000 (Direct Cost: ¥1,000,000) Fiscal Year 1989: ¥100,000 (Direct Cost: ¥100,000) Fiscal Year 1988: ¥900,000 (Direct Cost: ¥900,000)
Keywords	Automatic Differentiation / Mathematical Software / Preprocessor / Error Analysis / Interval Analysis / Numerical Integration / 数学ソフトウェア / 区間解析
Research Abstract	In scientific computation it is frequently required to evaluate the partial derivatives of a function as well as the function value itself. Recently a fast automatic differentiation scheme based on the formula manipulation has been established. In this project: 1. A preprocessor system for implementing the automatic differentiation on a computer was developed. 2. Software for solving numerically system of nonlinear equations by using the Newton Method is developed. Automatic differentiation scheme is incorporated into the software to evaluate the partial derivatives needed in the Newton method. 3. Combining the automatic differentiation scheme and the method of interval analysis estimates the bounds of error incurred inevitably and accumulated in the process of calculating a given function value. 4. An efficient automatic integration scheme was developed for evaluating the oscillatory integral on finite integration interval. The integrand is approximated by the Hermite interpolating polynomial using the first derivative of function, which is computed by the automatic differentiation scheme.