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How to stabilize a fixed point

While reading the thesis on 'Renormalization in area preserving maps' by Robert Mackay (Princeton 1982), I noticed an intriguing method proposed for computing an unstable fixed point of a nonlinear operator, in the case where we know approximately the eigenvalue of the linearized operator. Mackay attributed this method to his advisor Martin Kruskal. Although simple and practical, I had not seen this idea before, and since an error in the thesis makes it unnecessarily difficult to follow, I thought this presentation would be of interest. All credit is due to Mackay and Kruskal.