Development and Verification of Symbolic-Numeric Computation Suitable for Approximation, Transformation, and Interpolation Operations of Algebraic Surfaces

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K11827
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60010:Theory of informatics-related
• Research Institution: Kobe University
• Principal Investigator: Nagasaka Kosaku 神戸大学, 人間発達環境学研究科, 准教授 (70359909)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 数値・数式融合計算 / 近似GCD / 近似Groebner基底 / SLRA / 補間法 / Bernstein基底

Outline of Final Research Achievements

In practical calculations, unavoidable errors can deteriorate the properties of polynomials, making them difficult to handle. Therefore, we advanced research on the GCD and Groebner bases, which correspond to methods for finding solutions to systems of equations, as fundamental polynomial operations. As a result of our research, we proposed methods to improve resistance to errors through basis transformation, ways to improve known methods to reduce errors, and new methods that enable calculations even when the degree or the number of variables in a polynomial is large. Ultimately, we demonstrated examples of calculations for problems that were difficult to solve with previous methods.

Free Research Field

計算機代数

Academic Significance and Societal Importance of the Research Achievements

単一の代数曲面（代数曲線を含む）で複雑な形状を実現しようとすると，著しく高い次数の多項式が必要となってしまう。本研究課題では，数値・数式融合計算（厳密だが遅い計算方法と，高速だが不正確な計算方法を融合させる計算）を用いることで，様々な形で厳密な代数式を近似した形で扱えうることに着目し，実践的な計算において不可避の誤差を含んだ多項式に対して，種々の操作の基盤となる計算アルゴリズムの開発を行い，その有効性を計算例により示した。