relaxed components to be used in the computational process may be guided by any criterion,
and in particular, a natural criterion is to pickup the most recently available
values of the components computed by the processors. Furthermore, the asynchronous
-iterations are implemented by means of nonblocking MPI communication subroutines\index{MPI subroutines!nonblocking}
+iterations are implemented by means of nonblocking MPI communication subroutines\index{MPI!nonblocking}
(asynchronous communications).
The important property ensuring the convergence of the parallel projected Richardson
and \verb+cublasGetVectorAsync()+ in the asynchronous algorithm. Moreover, we
use the communication routines of the MPI library to carry out the data exchanges
between the neighboring nodes. We use the following communication routines: \verb+MPI_Isend()+
-and \verb+MPI_Irecv()+ to perform nonblocking\index{MPI subroutines!nonblocking}
+and \verb+MPI_Irecv()+ to perform nonblocking\index{MPI!nonblocking}
sends and receives, respectively. For the synchronous algorithm, we use the MPI
routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node in
blocking status until all data exchanges with neighboring nodes (sends and receives)
are completed. In contrast, for the asynchronous algorithms, we use the MPI routine
\verb+MPI_Test()+ which tests the completion of a data exchange (send or receives)
-without putting the MPI process in blocking status\index{MPI subroutines!blocking}.
+without putting the MPI process in blocking status\index{MPI!blocking}.
The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{ch13:alg:02})
computes, at each iteration, the new elements of the iterate vector $U$. Its general code
conv \leftarrow true;
\end{array}
$$
-where the function $AllReduce()$ uses the MPI global reduction subroutine\index{MPI subroutines!global}
+where the function $AllReduce()$ uses the MPI global reduction subroutine\index{MPI!global}
\verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the local
absolute errors, $error$, of all computing nodes, and $p$ (in Algorithm~\ref{ch13:alg:02})
is used as a counter of the local relaxations carried out by a computing node. In
