Before revision:  In this note we analyze in detail how the EIM at each iteration attempts to choose the interpolation nodes so as to make the related Vandermonde-type matrix " as invertible as possible " as well as attempting to optimize the conditioning of its inversion and minimizing a Lebesgue-type constantfor accuracy. We then compare through numerical experiments the EIM performance with fully optimized nested variations.
After revision:  In this note we analyze in detail how the EIM at each iteration attempts to choose the interpolation nodes so as to make the related Vandermonde-type matrix " as invertible as possible ; not necessarily optimizing its conditioning or accuracy of the interpolant as is sometimes thought. In fact, we introduce two new variations of the EIM, nested as well, which do optimize with respect to conditioning and the Lebesgue constant, respectively, and compare them through numerical experiments the EIM performance with fully optimized nested variations.