Jesus Cernades Gomez

Although interior-point methods (IPMs) have transformed mathematical programming since 1984, the original projective Karmarkar algorithm is rarely documented step by step on reproducible engineering examples that combine algorithmic transparency with real resource allocation constraints. This article therefore does not propose a new variant of Karmarkar’s algorithm; rather, its scientific contribution is the reproducible MATLAB implementation, canonical-form conversion, and comparative validation of the original projective method against the revised Simplex method and Barnes’ affine scaling variant in two engineering settings. The case studies are (i) the minimum-weight plastic design of a rigid frame with seven candidate plastic hinge locations and six collapse mechanisms and (ii) the optimal allocation of crop patterns in the Caplina Valley (Tacna, Southern Peru), an arid irrigated system with an irrigated command area of 1253 ha, monthly labor availability of 22,239 jornales, and water availability derived from Caplina River discharges at 75% persistence. For Case I, the algorithm reached F = 1.001 in the normalized dual space, which corresponds to F = 4.251 in the original structural objective after applying the scaling factor 17/4; relative to the analytical optimum F* = 4.25, this gives |4.251 − 4.25|/4.25 = 2.4 × 10⁻⁴ after 20 iterations. For Case II, the model yielded the maximum net production value of USD 703,135.92, allocating 948.47 ha among 12 crops while satisfying water, labor, market, and land constraints. The double validation confirms the algorithm’s strictly interior trajectory, polynomial-time rationale, and transparent internal parameters (α = 0.7968, ε = 10⁻⁸), making the implementation a reproducible benchmark for educational use and for future AI–operations research hybrid solvers in regions with limited access to commercial optimization software.

This paper studies various numerical methods for computing the square root of a matrix, a problem of interest in several areas of applied mathematics, such as the solution of differential equations and the analysis of dynamical systems. The work analyzes the conditions under which a matrix has a square root, and explores different techniques for its approximation, including Newton’s method, the Denman-Beavers iteration, and Pade based methods. Stability, convergence and computational efficiency are discussed for each approach. To validate the algorithms, numerical experiments implemented in MATLAB are presented using specific test matrices. The results show that some algorithms, such as the scaled Pade method, offer greater stability and faster convergence, particularly for symmetric positive definite matrices. Finally, the methods are compared in terms of the number of iterations, the accuracy achieved, and their applicability depending on the spectral properties of the matrix.