### 3.2 Recovery of primitive variables

On the other hand, as for the case of the GRHD equations discussed before, iterative (root-finding)
algorithms are also required for the GRMHD equations to recover the primitive variables from the
state vector . Not surprisingly, the recovery procedure is in the GRMHD case more involved than for
unmagnetized flows. A number of approaches are available in the literature. In particular, we direct the
interested reader to [289] for a comparison of six different methods for performing such inversions from
conserved to primitives variables for the case in which the matter thermodynamics is described by an
ideal gas EOS. In [289] not only the accuracy of each of the methods is assessed but also its
run-time.
For illustrative purposes, we discuss next with some detail the specific approach followed by [24], which
is an extension to full general relativity of that developed by [199] in the special relativistic
case. The procedure relies on the fact that it is not necessary to solve the system given by the
definition of the conserved variables in terms of the primitives (see Equation (57)) for the three
components of the momentum, but instead for its modulus . By eliminating the
components of through Equations (59) and after some algebra, it is possible to write as

where . In addition, the equation for the total energy can be worked out in a similar way
Equations (68) and (69), together with the definitions of and , form a system for the unknowns
, and , assuming the function is provided. The roots of this system can be found
using a two-dimensional scheme (e.g. bisection or Newton–Raphson).