Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizi...

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Abstract We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to infinite products. then characterize spectra involved matrices necessary for study convergence. Using these results, we discuss several examples possible or failure modes. Then, prove general theorem sequences generated by method. The key assumption proved in low-dimensional spaces up...