  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 vectorised, while the memory allocation demands 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 up to $\mathcal{O}(10^3)$ 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
allocations demands. The parallelisation of the algorithm is also presented in
multi-threaded and GPU accelerators' setting. The paper also comprises a
theoretical proof for the algorithmic convergence. Finally, we extend the
implementation of the proposed algorithmic rationale to feature selection
tasks.
