new

[book_gpu.git] / BookGPU / Chapters / chapter12 / ch12.tex

diff --git a/BookGPU/Chapters/chapter12/ch12.tex b/BookGPU/Chapters/chapter12/ch12.tex
index a514438a7ecac26fe29d7b80526eb12ee933b892..0269fd28a3957ccd5faaf8dec9ee15c92eee3703 100755 (executable)
--- a/BookGPU/Chapters/chapter12/ch12.tex
+++ b/BookGPU/Chapters/chapter12/ch12.tex
@@ -217,7 +217,7 @@ r_k \bot A \mathcal{K}_k(A, v_1).
 \end{array}
 \label{ch12:eq:13}
 \end{equation}
 \end{array}
 \label{ch12:eq:13}
 \end{equation}

-GMRES uses the Arnoldi process~\cite{ch12:ref5}\index{iterative method!Arnoldi process} to construct an
+GMRES uses the Arnoldi iterations~\cite{ch12:ref5}\index{iterative method!Arnoldi iterations} to construct an

 orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{Hessenberg matrix}
 $\bar{H}_k$ of order $(k+1)\times k$:
 \begin{equation}
 orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{Hessenberg matrix}
 $\bar{H}_k$ of order $(k+1)\times k$:
 \begin{equation}


@@ -313,7 +313,7 @@ $V$ to $m$ orthogonal vectors.
 Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
 It solves the left-preconditioned\index{sparse linear system!preconditioned} sparse linear
 system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration
 Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
 It solves the left-preconditioned\index{sparse linear system!preconditioned} sparse linear
 system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration

-$k$, GMRES uses the Arnoldi process\index{iterative method!Arnoldi process} (defined from
+$k$, GMRES uses the Arnoldi iterations\index{iterative method!Arnoldi iterations} (defined from

 line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
 Hessenberg matrix\index{Hessenberg matrix} $\bar{H}_m$ of size $(m+1)\times m$. Then, it
 solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$
 line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
 Hessenberg matrix\index{Hessenberg matrix} $\bar{H}_m$ of size $(m+1)\times m$. Then, it
 solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$


@@ -526,7 +526,7 @@ is managed by one MPI process and is composed of one CPU core and one GPU card.
 All tests are made on double-precision floating point operations. The parameters of both linear
 solvers are initialized as follows: the residual tolerance threshold $\varepsilon=10^{-12}$, the
 maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$, and the
 All tests are made on double-precision floating point operations. The parameters of both linear
 solvers are initialized as follows: the residual tolerance threshold $\varepsilon=10^{-12}$, the
 maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$, and the

-initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi process\index{iterative method!Arnoldi process}
+initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi iterations\index{iterative method!Arnoldi iterations}

 used in the GMRES method to $16$ iterations ($m=16$). For the sake of simplicity, we have chosen
 the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows us to easily
 compute the required inverse matrix $M^{-1}$, and it provides a relatively good preconditioning for
 used in the GMRES method to $16$ iterations ($m=16$). For the sake of simplicity, we have chosen
 the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows us to easily
 compute the required inverse matrix $M^{-1}$, and it provides a relatively good preconditioning for