Multiple test functions

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 $S$ which depends on $n_t$ test functions. Also assume we have found two consecutive points $x_i$ and $x_{i+1}$ and all $n_t$ test functions change sign:

\begin{displaymath}
\forall j\in [1,n_t]: \phi_j(x_i)\phi_j(x_{i+1}) < 0
\end{displaymath} (27)

Also assume we have found, using a one-dimensional secant method, all zeros $x^*_j$ 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} (28)

Since the continuation is not exact but numerical, we cannot assume this. However, the locations of $x^*_j$ probably will be clustered around some center point $x^c$. In this case we will glue the points $x^*_j$ to $x^* = x^c$.

A cluster will be detected if $\forall i,j\in [1,n_t]: \vert\vert x^*_i-x^*_j\vert\vert \leq \epsilon$ for some small value $\epsilon$. In this case we define $x^*$ as the mean of all located zeroes:

\begin{displaymath}
x^* = \frac{1}{n_t}\sum_{j=1}^{n_t}x^*_j
\end{displaymath} (29)