摘要
求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望.
Solving systems of algebraic equations is one of the most fundamental problems in computational algebraic geometry. It is ubiquitous and widely applied across the engineering and sciences, such as in robotics,computer vision, machine learning, artificial intelligence, cryptography, optimization, control theory and etc. One main challenge is to compute isolated singular solutions, which plays an important rule in geometric modelings.Based on recent research results of the authors and their collaborators, a survey for symbolic-numeric methods on computing isolated singular solutions of algebraic systems is conducted, especially for refining and certifying approximate solutions. Some directions for future studies on the topic are discussed as well.
作者
李楠
支丽红
Nan Li;Lihong Zhi
出处
《中国科学:数学》
CSCD
北大核心
2021年第1期17-42,共26页
Scientia Sinica:Mathematica
基金
国家重点研发计划(批准号:2018YFA0306702)
国家自然科学基金(批准号:11601378和11571350)资助项目。
关键词
代数方程组
孤立奇异解
符号数值方法
近似解的精化
近似解的验证
system of algebraic equations
isolated singular solution
symbolic-numeric method
refining approximate solutions
certifying approximate solutions