  Tomographic wave-front reconstruction is the main computational bottleneck to
realize real-time correction for turbulence-induced wave-front aberrations in
future laser-assisted tomographic adaptive-optics (AO) systems for ground-based
Giant Segmented Mirror Telescopes (GSMT), because of its unprecedented number
of degrees of freedom, $N$, i.e. the number of measurements from wave-front
sensors (WFS). In this paper, we provide an efficient implementation of the
minimum-mean-square error (MMSE) tomographic wave-front reconstruction mainly
useful for some classes of AO systems not requiring a multi-conjugation, such
as laser-tomographic AO (LTAO), multi-object AO (MOAO) and ground-layer AO
(GLAO) systems, but also applicable to multi-conjugate AO (MCAO) systems. This
work expands that by R. Conan [ProcSPIE, 9148, 91480R (2014)] to the
multi-wave-front, tomographic case using natural and laser guide stars. The new
implementation exploits the Toeplitz structure of covariance matrices used in a
MMSE reconstructor, which leads to an overall $O(N\log N)$ real-time complexity
compared to $O(N^2)$ of the original implementation using straight
vector-matrix multiplication. We show that the Toeplitz-based algorithm leads
to 60\,nm rms wave-front error improvement for the European Extremely Large
Telescope Laser-Tomography AO system over a well-known sparse-based tomographic
reconstruction, but the number of iterations required for suitable performance
is still beyond what a real-time system can accommodate to keep up with the
time-varying turbulence
