【摘要】Singular algebraic equations with rank-deficient Jacobians arise in a broad range of applications but rarely mentioned in the literature of scientific computing. Solutions of such equations are considered infinitely sensitive to perturbations and thus intractable from empirical data. Textbook advice is to avoid such problems altogether. This is a glaring gap in the state of knowledge and a frontier in scientific computing. In fact, it is a misconception to consider such equations automatically hypersensitive. From a different perspective, the infinitely many solutions of a singular linear system form a unique point in an affine Grassmannian. Such a solution is of a bounded sensitivity with respect to data perturbations and can be solved accurately. Every conventional solution is actually an accurate approximation to one of the infinitely many solutions. The commonly known “error” is largely a part of the solution and not error at all. On nonlinear equations, singular non-isolated solutions can be accurately computed by a newly discovered extension of Newton’s iteration even if the data are perturbed and the exact solutions disappear. This new method enables nonlinear equations to be modeled as square or over-/under-determined with Jacobian in any ranks. There are even advantages for algebraic equations to be singular. We shall also elaborate application models in which linear and nonlinear equations are naturally singular with their solutions accurately attainable.

【报告人简介】Zhonggang Zeng (Bernard J. Brommel Distinguished Research Professor of NEIU, Professor of Mathematics) Northeastern Illinois University(homepage: arv.neiu.edu/~zzeng/)

腾讯会议ID：263 912 571