含经验数据的奇异代数方程
Singular algebraic equations with empirical data
摘要：
具有秩亏雅可比矩阵的奇异方程在代数计算中经常出现。如本文中的案例研究所示，代数问题的直接和直观的建模往往会导致非孤立的奇异解。当需要从有限准确性的经验数据中解决问题时，挑战变得艰巨。一个新发现的低秩牛顿迭代出现作为一个有效的正则化机制，使求解奇异方程准确的误差界在同一顺序的数据错误。本文阐述了新方法在求解奇异代数方程中的应用，如奇异线性系统、多项式GCD和因子分解以及矩阵亏损本征值问题。
英文摘要：
Singular equations with rank-deficient Jacobians arise frequently in algebraic computing applications. As shown in case studies in this paper, direct and intuitive modeling of algebraic problems often results in nonisolated singular solutions. The challenges become formidable when the problems need to be solved from empirical data of limited accuracy. A newly discovered low-rank Newton's iteration emerges as an effective regularization mechanism that enables solving singular equations accurately with an error bound in the same order as the data error. This paper elaborates applications of new methods on solving singular algebraic equations such as singular linear systems, polynomial GCD and factorizations as well as matrix defective eigenvalue problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
