The gamut of a colour space is defined by a number of extreme points. The best inks to achieve an accurate spectral reproduction of a given target are those which span the target's spectral gamut. Using a modified non-negative matrix factorization (NMF) algorithm we derive m colorants and their spectral curves such that they are the extreme points of the targets gamut. Using the spectral Neugebauer printing model where eight colorants are assumed we compare our new method with existing techniques. Comparison with a set of optimal rotated principal vectors as well as the classical NMF clearly shows that the performance of the new method is superior.