Primal revised simplex with Bland's rule

Input the exact standard-form data of Revised simplex standard-form linear program with full-row-rank $A$ and a primal-feasible basis $B$ as in Basis dictionaries and reduced costs for revised simplex. Use exact arithmetic. At each iteration compute $x_B=A_B^{-1}b$, $y=A_B^{-\mathsf{T}}{c}_B$, and all nonbasic reduced costs ${\bar{c}}_j=c_j-a_j^{\mathsf{T}}y$.

1. If every nonbasic ${\bar{c}}_j\ge 0$, return optimal with $x(B)$ and $y$.
2. Otherwise choose the least index $j\mathrm{\notin }B$ with ${\bar{c}}_j<0$ and solve $A_{B}d=a_j$.
3. If no component $d_i$ is positive, return unbounded with the ray whose $j$th component is $1$, whose basic components are $-d$, and whose other nonbasic components are $0$.
4. Otherwise compute $\theta ={\min }_{d_i>0}{x}_{B_i}\mathrm{/}{d}_i$. Among minimizing rows, leave the basic variable of least index. Replace its basis column by $a_j$ and repeat.

The algorithm assumes an exact initial feasible basis. It makes no iteration bound polynomial in $m$ or $n$.