The above is a general way to detect and locate singularities depending on one test function.
However, it may happen that it is not possible to represent a singularity with only one test
function.
Suppose we have a singularity
which depends on
test functions.
Also assume we have found two consecutive points
and
and all
test functions change sign:
![\begin{displaymath}
\forall j\in [1,n_t]: \phi_j(x_i)\phi_j(x_{i+1}) < 0
\end{displaymath}](img85.png) |
(27) |
Also assume we have found, using a one-dimensional secant method, all zeros
of the test functions. In the ideal (exact) case all these zeros will
coincide:
![\begin{displaymath}
\forall j\in [1,n_t]: x^*=x^*_j \quad\mbox{and}\quad \phi_j(x^*_j) = 0
\end{displaymath}](img87.png) |
(28) |
Since the continuation is not exact but numerical, we cannot assume this.
However, the locations of
probably will be clustered around some center point
.
In this case we will glue the points
to
.
A cluster will be detected if
for some
small value
. In this case we define
as the mean of all located zeroes:
 |
(29) |