Biopolymeric films based on chitosan and starch offer an ecological alternative for food protection. Nevertheless, their practical application is often limited by their low mechanical properties and high solubility in aqueous solutions, due to weak interactions between the chains of the biopolymers. One approach to resolve this problem is to obtain biopolymeric films based on (bio)polyelectrolyte complex ((bio)PEC). These films exhibit stronger electrostatic interactions and homogeneous biopolymeric structure. In this study, films based on (bio)PEC were obtained by the casting method, using chitosan and carboxymethyl cassava starch with different degrees of substitution with a biopolymer concentration of 2.5 wt.% at pH = 6. The obtained films were analyzed using the optical and scanning microscopy, color method, ATR-FTIR spectroscopy, thermogravimetry, mechanical analysis under tension, solubility in water, simulated gastric fluid (SGF), and phosphate-buffered saline (PBS) solutions, and contact angle of water. The results demonstrated that the tensile strength and Young’s modulus of films based on (bio)PEC increased by 2–4 times, and the elongation at break by 20% compared to films based on a mixture chitosan and starch. This is due to the increase in the attraction between oppositely charged polyelectrolytes in (bio)PEC films. Additionally, the solubility of (bio)PEC films was reduced by ~40%, 35% and 70% in water, SGF and PBS solutions, respectively, when the carboxymethyl starch with highest degree of substitution was used, and z was near to 1.

We develop a Laurent-series framework for the analytical characterization of singularities in any three-step frozen-Jacobian iterative operator parametrized by $(\lambda ,\alpha ,\beta )$ (where $\lambda$ controls the predictor step and $(\alpha ,\beta )$ 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 ($\beta \ne 0$) 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 $\lambda$ and $\alpha$; when $\beta =0$ the order reduces to $m=d+1$, recovering the classical Jarratt-class result. Second, for each fixed $\lambda$ there exists a unique parameter pair $({\alpha }^{*}\left(\lambda \right),{\beta }^{*}\left(\lambda \right))$ that minimises the Laurent residue $|a_{-1}|$ subject to fifth-order convergence; for the NJN ($\lambda =2/3$) the explicit solution ${\alpha }^{*}=\frac{1}{4}$, ${\beta }^{*}=\frac{3}{8}$ provides the first analytical justification for these design values. Third, the Laurent Validity Radius ${\rho }_L={\inf }_{z_s\in S\setminus \left\{z_c\right\}}|z_c-z_s|$ is the exact boundary of the domain in which the single-pole representation is valid; starting points satisfying $|z_0-z_c|\ll \left|\kappa \right|$ experience pole-dominated first-iterate growth $|G\left(z_0\right){|\sim |\kappa |}^m{\delta }^{-m}\to +\infty$. All results are validated on five test functions (including polynomial, transcendental, meromorphic, and non-symmetric cases) and all three, quantities can be computed from $f\left(z_c\right)$, $f^{\prime }\left(z_c\right)$, $f^{″}\left(z_c\right)$ alone.