Abstract
By extending the classical Newton’s irrational method defined by an iteration x n + 1 = x n - f ( x n ) f ′ ( x n ) 2 - f ( x n ) f ″ ( x n ) , we present a high-order k-fold pseudo-Newton’s irrational method for locating a simple zero of a nonlinear equation. Its order of convergence is proven to be at least k + 3 and the convergence behavior of the asymptotic error constant is investigated near the corresponding simple zero. A root-finding algorithm is described as well as the introduction on the convergence of the fixed-point iterative method. Various numerical examples have successfully demonstrated a good agreement with the theory presented here.
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