Simplex ratio test ($\theta ={\min }_{i:d_i>0}{x}_i/d_i$)

In a feasible simplex basis, increasing a nonbasic variable by $\theta$ changes basic coordinates to $x-\theta d$. Nonnegativity limits $\theta$ by the smallest ratio $x_i/d_i$ with $d_i>0$; a minimizing coordinate leaves the basis. Ties can leave zero-valued basic variables. This pivot test is distinct from the analytic ratio test for series convergence.