  The paper presents two variants of a Krylov-Simplex iterative method that
combines Krylov and simplex iterations to minimize the residual $r = b-Ax$. The
first method minimizes $\|r\|_\infty$, i.e. maximum of the absolute residuals.
The second minimizes $\|r\|_1$, and finds the solution with the least absolute
residuals. Both methods search for an optimal solution $x_k$ in a Krylov
subspace which results in a small linear programming problem. A specialized
simplex algorithm solves this projected problem and finds the optimal linear
combination of Krylov basis vectors to approximate the solution. The resulting
simplex algorithm requires the solution of a series of small dense linear
systems that only differ by rank-one updates. The $QR$ factorization of these
matrices is updated each iteration. We demonstrate the effectiveness of the
methods with numerical experiments.
