A strongly polynomial-time algorithm for the strict homogeneous linear-inequality feasibility problem

A strongly polynomial-time algorithm is proposed for the strict homogeneous linear-inequality feasibility problem in the positive orthant, that is, to obtain $$x\in \mathbb {R}^n$$x∈Rn, such that $$Ax > 0$$Ax>0, $$x> 0$$x>0, for an $$m\times n$$m×n matrix $$A$$A, $$m\ge n$$m≥n. This algorithm requires $$O(p)$$O(p) iterations and $$O(m^2(n+p))$$O(m2(n+p)) arithmetical operations to ensure that the distance between the solution and the iteration is $$10^{-p}$$10-p. No matrix inversion is needed. An extension to the non-homogeneous linear feasibility problem is presented.