A Laurent-Series Framework for Analytical Characterisation of Singularities in Frozen-Jacobian Multi-Step Iterative Solvers: Application to the Fifth-Order Newton–Jarratt–Newton Method

We develop a Laurent-series framework for the analytical characterization of singularities in any three-step frozen-Jacobian iterative operator parametrized by (λ,α,β) (where λ controls the predictor step and (α,β) are the weight-function coefficients) and apply it to the fifth-order Newton–Jarratt–Newton (NJN) method. Three principal results are established. First, any operator with a quadratic weight function (β≠0) possesses a pole of order m=d(d+1)+1 at every simple critical point of f, where d is the asymptotic polynomial degree, independently of λ and α; when β=0 the order reduces to m=d+1, recovering the classical Jarratt-class result. Second, for each fixed λ there exists a unique parameter pair (α*(λ),β*(λ)) that minimises the Laurent residue |a−1| subject to fifth-order convergence; for the NJN (λ=2/3) the explicit solution α*=14, β*=38 provides the first analytical justification for these design values. Third, the Laurent Validity Radius ρL=infzs∈S∖{zc}|zc−zs| is the exact boundary of the domain in which the single-pole representation is valid; starting points satisfying |z0−zc|≪|κ| experience pole-dominated first-iterate growth |G(z0)|∼|κ|mδ−m→+∞. All results are validated on five test functions (including polynomial, transcendental, meromorphic, and non-symmetric cases) and all three, quantities can be computed from f(zc), f′(zc), f″(zc) alone.