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The purpose of continuation here is to consider a system with one parameter and $n$ unknowns and to determine what are the possible values of the unknowns when the parameters lie in a given range [a,b]. The values of the unknowns will be calculated for values of the parameter separated by a constant value, starting at one extremity of the range for the parameter.

The principle is to solve the system for the initial value a or b of the parameter range and then to obtain the solution for other values of the parameter using a certified Newton scheme (hence it is necessary that the equations of the system are at least $C^1$).

As soon as the initial starting point of the branches have been obtained the continuation procedure is usually very fast. The continuation will stop if the jacobian become singular. The algorithm will then increase the parameters value until we can start again a certified procedure.

Jean-Pierre Merlet 2012-12-20