  This paper proposes a generalization of the conjugate gradient (CG) method
used to solve the equation $Ax=b$ for a symmetric positive definite matrix $A$
of large size $n$. The generalization consists of permitting the scalar control
parameters (= stepsizes in gradient and conjugate gradient directions) to be
replaced by matrices, so that multiple descent and conjugate directions are
updated simultaneously. Implementation involves the use of multiple agents or
threads and is referred to as cooperative CG (cCG), in which the cooperation
between agents resides in the fact that the calculation of each entry of the
control parameter matrix now involves information that comes from the other
agents. For a sufficiently large dimension $n$, the use of an optimal number of
cores gives the result that the multithread implementation has worst case
complexity $O(n^{2+1/3})$ in exact arithmetic. Numerical experiments, that
illustrate the interest of theoretical results, are carried out on a multicore
computer.
