  To find consistent initial data points for a system of differential-algebraic
equations, requires the identification of its missing constraints. An efficient
class of structural methods exploiting a dependency graph for this task was
initiated by Pantiledes. More complete methods rely on differential-algebraic
geometry but suffer from other issues (e.g. high complexity). In this paper we
give a new class of efficient structural methods combined with new tools from
numerical real algebraic geometry that has much improved completeness
properties. Existing structural methods may fail for a system of
differential-algebraic equations if its Jacobian matrix after differentiation
is still singular due to symbolic cancellation or numerical degeneration.
Existing structural methods can only handle degenerated cases caused by
symbolic cancellation. However, if a system has parameters, then its parametric
Jacobian matrix may be still singular after application of the structural
method for certain values of the parameters. This case is called numerical
degeneration.
  For polynomially nonlinear systems of differential-algebraic equations,
numerical methods are given to solve both degenerated cases using numerical
real algebraic geometry. First, we introduce a witness point method, which
produces at least one witness point on every constraint component. This can
help to ensure constant rank and detection of degeneration on all components of
such systems. Secondly, we present a Constant Rank Embedding Lemma, and based
on it propose an Index Reduction by Embedding (IRE) method which can construct
an equivalent system with a full rank Jacobian matrix. Thirdly, IRE leads to a
global structural differentiation method, to solve degenerated
differential-algebraic equations on all components numerically. Application
examples from circuits, mechanics, are used to demonstrate our method.
