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Fixed points are fundamental to renormalization group theory and chaos. They are special values of nonlinear equations which may be transition points or cycles. A fixed point is obtained as an asymptotic limit, so that this technique known as the fixed point analysis is based on analysis like perhaps no other numerical methods.
The fixed-point analysis does not work if fixed points are repulsive. Recently I have found a way to make repulsive fixed points attractive. As a result, one can now apply this analysis to calculate fixed points of almost any nonlinear equations or roots of almost any polynomials of any order to any desired degree of accuracy.