The problem of optimal scaling of matrices with respect to the ℓ2 condition number is an important one in numerical analysis of linear systems. The optimization problem corresponding to this is nondifferentiable unless the largest and smallest singular values are simple at a solution. In this paper, an algorithm is given which is capable of converging to the minimum at a second-order rate independently of the multiplicity of these singular values. It is based mainly on recent work of Overton (1988) on the problem of minimizing the largest eigenvalue of a symmetric matrix.