  Discontinuity with respect to data perturbations is common in algebraic
computation where solutions are often highly sensitive. Such problems can be
modeled as solving systems of equations at given data parameters. By appending
auxiliary equations, the models can be formulated to satisfy four easily
verifiable conditions so that the data form complex analytic manifolds on which
the solutions maintain their structures and the Lipschitz continuity. When such
a problem is given with empirical data, solving the system becomes a least
squares problem whose solution uniquely exists and enjoys Lipschitz continuity
as long as the data point is in a tubular neighborhood of the manifold. As a
result, the singular problem is regularized as a well-posed computational
problem.
