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 nt test functions.
Also assume we have found two consecutive points xi and xi+1 and all nt
test functions change sign:
| (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:
| (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 xc. In this case we will glue the points x*j to x* = xc.
A cluster will be detected if "i,j Î [1,nt]: ||x*i-x*j|| £ e for some
small value e. In this case we define x* as the mean of all located zeroes:
| (29) |