The author considers the problem of minimizing a sum of Euclidean norms \[ F(x)=\sum^{m}_{i=1}r_i(x) \] where the residuals \(r_i(x)\) are affine functions from \(R^n\) to \(R^l\) (\(n\geq l\geq 2,\quad m\geq 2)\). This arises in a number of applications, including single- and multi-facility location problems. The function \(F\) is, in general, not differentiable at \(x\) if at least one \(r_i(x)\) is zero. Computational methods described previously in the literature generally converge quite slowly if the solution is at such a point. A new method is described which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along which \(F\) is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method given recently by \textit{P. H. Calamai} and \textit{A. R. Conn} [Numerical analysis, Proc. 9th bienn. Conf., Dundee/Scotl. 1981, Lect. Notes 912, 1--25 (1982; Zbl 0485.65013)]. The new method has quadratic convergence to a solution \(x\) under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at \(x\), then \(F\) is differentiable at \(x\) and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, say \(r_i(x)\), is zero, then, as the iteration proceeds, the Hessian of \(F\) becomes extremely ill-conditioned. It is proved that this ill-conditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold \(\{x \mid r_i(x)=0\}\). If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence of Newton's method applies. If several residuals are zero at \(x\), several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results. The proof of the convergence rate in the nondifferentiable case involves the development of estimates for the change in eigenvalues and eigenvectors of a symmetric matrix with multiple eigenvalues which undergoes a symmetric perturbation.