Abstract
An algorithm for the numerical solution of general systems of complex linear equations in the $l_\infty $, or Chebyshev, norm is presented. The objective is to find complex values for the unknowns so that the maximum magnitude residual of the system is a minimum. The unknowns are required to satisfy certain convex constraints; in particular, bounds on the magnitudes of the unknowns are imposed. In the algorithm presented here, this problem is replaced by a linear program generated in such a way that the relative error between its solution and a solution of the original problem can be estimated. The maximum relative error can easily be made as small as desired by selecting an appropriate linear program. Order of magnitude improvements in both computation time and computer storage requirements in an implementation of the simplex algorithm to this linear program are presented. Three numerical examples are included, one of which is a complex function approximation problem.
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