  Characterization of nonstabilizerness is fruitful due to its application in
gate synthesis and classical simulation. In particular, the resource monotone
called the $\textit{stabilizer extent}$ is indispensable to estimate the
simulation cost using the rank-based simulators, one of the state-of-the-art
simulators of Clifford+$T$ circuits. In this work, we propose fast numerical
algorithms to compute the stabilizer extent. Our algorithm utilizes the Column
Generation method, which iteratively updates the subset of pure stabilizer
states used for calculation. This subset is selected based on the overlaps
between all stabilizer states and a target state. Upon updating the subset, we
make use of a newly proposed subroutine for calculating the $\textit{stabilizer
fidelity}$ that (i) achieves the linear time complexity with respect to the
number of stabilizer states, (ii) super-exponentially reduces the space
complexity by in-place calculation, and (iii) prunes unnecessary states for the
computation. As a result, our algorithm can compute the stabilizer fidelity and
the stabilizer extent for Haar random pure states up to $n=9$ qubits, which
naively requires a memory of 305 EiB. We further show that our algorithm runs
even faster when the target state vector is real. The optimization problem size
is reduced so that we can compute the case of $n=10$ qubits in 4.7 hours.
