Krylov-Simplex method that minimizes the residual in ℓ_1-norm or ℓ_∞-norm
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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_∞, 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.
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