We present a comprehensive computational study of a class of linear system solvers, called {\it Triangle Algorithm} (TA) and {\it Centering Triangle Algorithm} (CTA), developed by Kalantari \cite{kalantari23}. The algorithms compute an approximate solution or minimum-norm solution to $Ax=b$ or $A^TAx=A^Tb$, where $A$ is an $m \times n$ real matrix of arbitrary rank. The algorithms specialize when $A$ is symmetric positive semi-definite. Based on the description and theoretical properties of TA and CTA from \cite{kalantari23}, we give an implementation of the algorithms that is easy-to-use for practitioners, versatile for a wide range of problems, and robust in that our implementation does not necessitate any constraints on $A$. Next, we make computational comparisons of our implementation with the Matlab implementations of two state-of-the-art algorithms, GMRES and ``lsqminnorm". We consider square and rectangular matrices, for $m$ up to $10000$ and $n$ up to $1000000$, encompassing a variety of applications. These results indicate that our implementation outperforms GMRES and ``lsqminnorm" both in runtime and quality of residuals. Moreover, the relative residuals of CTA decrease considerably faster and more consistently than GMRES, and our implementation provides high precision approximation, faster than GMRES reports lack of convergence. With respect to ``lsqminnorm", our implementation runs faster, producing better solutions. Additionally, we present a theoretical study in the dynamics of iterations of residuals in CTA and complement it with revealing visualizations. Lastly, we extend TA for LP feasibility problems, handling non-negativity constraints. Computational results show that our implementation for this extension is on par with those of TA and CTA, suggesting applicability in linear programming and related problems.

翻译：我们针对由Kalantari开发的称为三角形算法（TA）和中心三角形算法（CTA）的一类线性系统求解器开展了全面的计算研究。这些算法可计算任意秩m×n实矩阵A的线性系统Ax=b或最小二乘系统A^TAx=A^Tb的近似解或最小范数解，并在A为对称半正定矩阵时具有特殊形式。基于文献《kalantari23》对TA和CTA的描述及理论性质，我们给出了便于实际应用、适用于广泛问题且具有鲁棒性的算法实现——该实现无需对A施加任何约束。随后，我们将本实现与Matlab中两种先进算法GMRES和“lsqminnorm”的实现进行计算比较。考虑m≤10000、n≤1000000的方阵和矩形矩阵，涵盖多种应用场景。结果表明，本实现无论从运行时间还是残差质量上均优于GMRES和“lsqminnorm”。此外，CTA的残差相对值比GMRES下降更快且更稳定，本实现能在GMRES报告不收敛之前提供高精度近似解；相较于“lsqminnorm”，本实现运行速度更快且能产生更优解。我们同时开展了CTA残差迭代动态的理论研究，并辅以可视化分析。最后，我们将TA扩展至处理线性规划可行性问题中的非负约束。计算结果表明，该扩展实现的性能与TA/CTA相当，这表明其在线性规划及相关问题中具有应用价值。