Algorithmic Solution for Systems of Linear Equations, in $\mathcal{O}(mn)$ time
This provides a faster and more efficient solution for computational tasks involving linear systems, though it appears incremental as it builds on existing algorithmic paradigms.
The paper tackles solving systems of linear equations with a novel algorithm that achieves over 100x speed-up compared to state-of-the-art methods, while maintaining high accuracy and low memory demands, especially for non-square systems.
We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and, by definition, vectorized, while the memory allocation demands are trivial, because, for each iteration, only one dimension of the given input matrix $\mathbf X$ is utilized. The execution time is very short compared with state-of-the-art methods, exhibiting $> \times 10^2$ speed-up and low memory allocation demands, especially for non-square Systems of Linear Equations, with ratio of equations versus features high (tall systems), or low (wide systems) accordingly. The accuracy is high and straightforwardly controlled, and the numerical results highlight the efficiency of the proposed algorithm, in terms of computation time, solution accuracy and memory demands. The paper also comprises a theoretical proof for the algorithmic convergence, and we extend the implementation of the proposed algorithmic rationale to feature selection tasks.