On Cline’s Direct Method for Solving Overdetermined Linear Systems in the $L_\infty $ Sense

An algorithm is presented for computing a vector x which satisfies a given $m \times n(m > n \geqq 2)$ linear system in the sense that the $L_\infty $ norm of the residual vector is minimized. That is, letting $a_1 , \cdots ,a_m $ be the columns of a matrix A, each column being of length n, and letting $\beta _1 , \cdots ,\beta _m $ be the components of a vector b, we wish to find a vector x which minimizes \[\phi (x) = ||A^T x - b||_\infty = \mathop {\max }\limits_i |a_i^T x - \beta _i |.\]The proposed algorithm is a direct (i.e. descent) method which minimizes the function $\phi $ in a finite number of steps. It is closely related to Cline’s algorithm for the above problem:The algorithm represents an improvement over others which proceed by directly minimizing the function $\phi $ in that it employs recent advances in the use of fast Givens transformations to update orthogonal matrix factorizations. This permits the work per cycle in the algorithm to be reduced without sacrificing numerical stability. F...