Validating the Fisher approach for stage IV spectroscopic surveys
  In recent years forecasting activities have become a very important tool for
designing and optimising large scale structure surveys. To predict the
performance of such surveys, the Fisher matrix formalism is frequently used as
a fast and easy way to compute constraints on cosmological parameters. Among
them lies the study of the properties of dark energy which is one of the main
goals in modern cosmology. As so, a metric for the power of a survey to
constrain dark energy is provided by the Figure of merit (FoM). This is defined
as the inverse of the surface contour given by the joint variance of the dark
energy equation of state parameters $\{w_0,w_a\}$ in the
Chevallier-Polarski-Linder parameterisation, which can be evaluated from the
covariance matrix of the parameters. This covariance matrix is obtained as the
inverse of the Fisher matrix. Inversion of an ill-conditioned matrix can result
in large errors on the covariance coefficients if the elements of the Fisher
matrix have been estimated with insufficient precision. The conditioning number
is a metric providing a mathematical lower limit to the required precision for
a reliable inversion, but it is often too stringent in practice for Fisher
matrices with size larger than $2\times2$. In this paper we propose a general
numerical method to guarantee a certain precision on the inferred constraints,
like the FoM. It consists on randomly vibrating (perturbing) the Fisher matrix
elements with Gaussian perturbations of a given amplitude, and then evaluating
the maximum amplitude that keeps the FoM within the chosen precision. The steps
used in the numerical derivatives and integrals involved in the calculation of
the Fisher matrix elements can then be chosen accordingly in order to keep the
precision of the Fisher matrix elements below this maximum amplitude...
