The L(,(INFIN))-norm has been widely studied as a criterion of curve fitting problems. We are interested in the best L(,(INFIN))-approximation to a given finite array of number A = (a(,ij))(,mXn), i.e.;A natural iterated polishing (Mid-Range Polish) algorithm is shown, and its convergence in the L(,(INFIN))-norm is proved. Since the conver- gence of the Mid-Range Polish algorithm may take infinitely many iterations, we developed a new algorithm which converges in a finite number of steps to an optimal matrix of residual A* = (a(,ij)-r(,i)*-s(,j)*)(,mXn), whose L(,(INFIN))-norm u is the minimum one.;Several conditions are obtained, each of which is necessary and sufficient for a given matrix of residual to be optimal. For instance, a matrix of residual is optimal if and only if the set of entries, which equal to the L(,(INFIN))-norm of the matrix u in absolute value, contain a loop L alternating the value of u and -u. This criterion leads to an elegant and efficient finite algorithm for calculating the best L(,(INFIN))-approximation. Examples and results of the computational experience with a computer code version of some of the algorithms are presented.