  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.
