Multistep High-Order Methods for Nonlinear Equations Using Padé-Like Approximants

We present new high-order optimal iterative methods for solving a nonlinear equation, , by using Padé-like approximants. We compose optimal methods of order 4 with Newton's step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.

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