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More 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 nt test functions. Also assume we have found two consecutive points xi and xi+1 and all nt test functions change sign:

"j Î [1,nt]: fj(xi)fj(xi+1) < 0
(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:

"j Î [1,nt]: x*=x*j    and    fj(x*j) = 0
(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:

x* = 1

nt

nt
å
j=1 
x*j
(29)