Branch Point Locator
The location of BPC points in the non-generic situation (i.e. where some symmetry is present) as zeros of the test functions is numerically suspect because no local quadratic convergence can be guaranteed. This difficulty can be avoided by introducing an additional unknown
and
considering the minimally extended system:
![$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) +\beta p_1...
...{old}(t) \rangle dt+\beta p_3 & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$](img290.png) |
|
|
(50) |
where
is defined as in (84) and
is the bordering vector
in (86).
We solve this system with respect to
and
by Newton's method with initial
. A branch point
corresponds to a regular solution
of system (50)
(see [3],p. 165). We note, however that the second order partial
derivatives (Hessian) of
with respect to
and
are required.
The tangent vector
at the BPC singularity is approximated as
where
is the tangent vector in the continuation point previous to the BPC and
is the one in the next point.