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Test functions

The idea to detect singularities is to define smooth scalar functions which have regular zeros at the singularity points. These functions are called test functions. Suppose we have a singularity S which is detectable by a test function f:Rn+1®R. Also assume we have found two consecutive points xi and xi+1 on the curve

F(x)=0,    F: Rn+1® Rn .
(23)

The singularity S will then be detected if

f(xi)f(xi+1) < 0 .
(24)

Having found two points xi and xi+1 one may want to locate the point x* where f(x) vanishes. A logical solution is to solve the following system

F(x)
=
0
(25)
f(x)
=
0
(26)

using Newton iterations starting at xi. However, to use this method, one should be able to compute the derivatives of f(x) with respect to x, which is not always easy. To avoid this difficulty we implemented by default a one-dimensional secant method to locate f(x)=0 along the curve. Notice that this involves Newton corrections at each intermediate point.