  A new domain decomposition preconditioner is introduced for efficiently
solving linear systems Ax = b with a symmetric positive definite matrix A. The
particularity of the new preconditioner is that it is not necessary to have
access to the so-called Neumann matrices (i.e.: the matrices that result from
assembling the variational problem underlying A restricted to each subdomain).
All the components in the preconditioner can be computed with the knowledge
only of A (and this is the meaning given here to the word algebraic). The new
preconditioner relies on the GenEO coarse space for a matrix that is a low-rank
modification of A and on the Woodbury matrix identity. The idea underlying the
new preconditioner is introduced here for the first time with a first version
of the preconditioner. Some numerical illustrations are presented. A more
extensive presentation including some improved variants of the new
preconditioner can be found in [7]
(https://hal.archives-ouvertes.fr/hal-03258644).
