  The adiabatic temperature change ($\Delta T_{ad}$) of a magnetic refrigerant
can be indirectly estimated through field ($H$) and temperature ($T$) dependent
magnetization ($M$) and specific heat ($C_p$) measurements. A direct
integration approach for this estimation is frequently reported, which is an
approximation to a rigorous mathematical approach. In this work, we propose an
iterative method in small $H$ steps, to estimate $\Delta T_{ad}$ from indirect
measurements. We show that this approach is able to reproduce the reversibility
of the magnetocaloric effect, and provides a more accurate estimation of
$\Delta T_{ad}$, up to 10\% when considering a detailed $M(H,T)$ and $Cp(H,T)$
dataset that reproduces the magnetothermal properties of gadolinium, a
benchmark room-temperature magnetic refrigerant.
