Cline/Moler/Stewart/Wilkinson gave a strategy for estimating the condition number \(\mu:=\| A\|_ 1\| A^{-1}\|\) of an \(n\times n\)-matrix A, which is incorporated in LINPACK. It needs the solution of two systems with A and \(A^ T\), but underestimates \(\mu\) by a factor of 0.55 in average, and this factor is getting worse with growing n. The author develops a new idea by considering the relative maxima of the convex function \(f(x):=\| Bx\|_ 1\). He achieves an average factor of 0.97, which obviously is not depending on n and needs 4.2 system solutions with A and \(A^ T\) in average. By splitting this method in cycles in certain subspaces it is possible to reach factors 0.991 and 0.997 by doubling and trebling the effort resp. The worst estimate for 200 random matrices of different orders are 0.32, 0.44 and 0.7 for one, two and three cycles resp.