  In this paper we describe in detail the implementation and main properties of
a new inversion code for the polarized radiative transfer equation (VFISV: Very
Fast inversion of the Stokes vector). VFISV will routinely analyze pipeline
data from the Helioseismic and Magnetic Imager (HMI) on-board of the Solar
Dynamics Observatory (SDO). It will provide full-disk maps (4096$\times$4096
pixels) of the magnetic field vector on the Solar Photosphere every 10 minutes.
For this reason VFISV is optimized to achieve an inversion speed that will
allow it to invert 16 million pixels every 10 minutes with a modest number
(approx. 50) of CPUs. Here we focus on describing a number of important
details, simplifications and tweaks that have allowed us to significantly speed
up the inversion process. We also give details on tests performed with data
from the spectropolarimeter on-board of the Hinode spacecraft.
