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Tracing the equation of state and the density of the cosmological constant along z

We investigate the equation of state w(z) in a non-parametric form using the latest compilations of the luminosity distance from SNe Ia at high z. We combine the inverse problem approach with a Monte Carlo method to scan the space of priors. In the light of the latest high redshift supernova data sets, we reconstruct w(z). A comparison between a sample including the latest results at z>1 and a sample without those results shows the improvement achieved through observations of very high z supernovae. We present the prospects for measuring the variation of dark energy density along z by this method.