This paper develops and analyzes new high-order iterative schemes for the effective evaluation of the matrix square root (MSR). By leveraging connections between the matrix sign function and the MSR, we design stable algorithms that exhibit fourth-order convergence under mild spectral conditions. Detailed error bounds and convergence analyses are provided, ensuring both theoretical rigor and numerical reliability. A comprehensive set of numerical experiments, conducted across structured and large-scale test matrices, demonstrates the superior performance of the proposed methods compared to classical approaches, both in terms of computational efficiency and accuracy. The results confirm that the proposed iterative strategies provide robust and scalable tools for practical applications requiring repeated computation of matrix square roots.

Clustered proportion responses often exhibit bounded support, skewness, heterogeneous dispersion, and within-cluster dependence. We propose a correlated mean–precision random-effects beta regression model that jointly represents cluster-level heterogeneity in the conditional mean and conditional precision. Its main innovation is to treat the cross-submodel random-effect correlation as a scientific estimand. The same frequentist Beta mixed model estimates and tests this correlation while allowing nonlinear adjustment and cluster-level interpretation. Penalized B-splines allow nonlinear effects in both submodels, and estimation is performed by maximizing a Laplace-approximated penalized marginal likelihood. The fitted model provides likelihood-based inference for the mean–stability association and empirical Bayes estimates for cluster ranking and quadrant classification. Among M1–M5, M5 gives the lowest average errors for conditional-mean and conditional-precision recovery and the best average AIC, cluster-level BIC, and full-data NLPD across the Monte Carlo settings considered. It is also the only model compared here that estimates and tests the latent association while retaining paired cluster effects for interpretation. All M5 fits succeeded in the enlarged 30-replication stress suite, and the five aggregate mean M5–M4 criteria favored M5 across 299 successful pairs; quadrature checks indicate where safeguards are needed under weak information. In both CDC PLACES and the representative World Bank panel of 128 eligible countries, M5 has the largest marginal likelihood, the smallest AIC and cluster-level BIC, a significant M4–M5 likelihood-ratio test, and the lowest application-specific point-prediction errors. For the World Bank data, $\widehat{\rho }=-0.500$ (profile 95% interval $-0.621$ to $-0.295$), the likelihood-ratio statistic is $17.370$ ($p=3.08\times {10}^{-5}$), and M5 has the lowest rolling-origin MSPE, RMSE, and MAE. On the combined evidence from fit, prediction, and correlation inference, M5 is the best overall model evaluated in both applications.