We investigate the possibilities of reconstructing the cosmic equation of state (EoS) for high redshift. In order to obtain general results, we use two model-independent approaches. The first reconstructs the EoS using comoving distance and the second makes use of the Hubble parameter data. To implement the first method, we use a recent set of Gamma-Ray Bursts (GRBs) measures. To implement the second method, we generate simulated data using the Sandage-Loeb ($SL$) effect; for the fiducial model, we use the $Lambda CDM$ model. In both cases, the statistical analysis is conducted through the Gaussian processes (non-parametric). In general, we demonstrate that this methodology for reconstructing the EoS using a non-parametric method plus a model-independent approach works appropriately due to the feasibility of calculation and the ease of introducing a priori information ($H_ {0}$ and $Omega_{m0}$). In the near future, following this methodology with a higher number of high quality data will help obtain strong restrictions for the EoS.
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