https://doi.org/10.65770/BNRC9106
ABSTRACT
This paper proposes a modified hybrid Steffensen method for solving nonlinear equations, in which the fixed convex combination weight of the standard hybrid scheme is replaced by an adaptive residual-based weight. At each step, the forward and backward Steffensen candidates are computed and combined through a weight that automatically concentrates on whichever candidate has the smaller residual. The convergence analysis establishes that the proposed scheme retains second-order convergence and, under the condition that the derivative of the function at the simple root lies strictly between zero and one, achieves a smaller asymptotic error constant than the equal-weight combination. Numerical experiments on twenty test functions of polynomial, trigonometric, exponential, logarithmic, and mixed type confirm that the adaptive variant requires fewer or equal iterations than the constant-weight method in every case considered, with the largest reduction observed where the fixed weight is locally inefficient. The number of function evaluations and the computational order of convergence are reported alongside the iteration counts to give a balanced view of the cost of the adaptive scheme.
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