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\begin{document}
Consider the matrix problem $ A x = b$, where $A = L + U + D$.

\section*{Notes on SOR Implementation}


Symmetric successive over-relaxation as a simple iterative solver can be written as the two-step process
\[
x_i^{n+1/2} =  x_i^n + \omega A_{ii}^{-1}( b_i - \sum_{j < i} A_{ij} x_j^{n+1/2} - \sum_{j \ge i} A_{ij} x_j^{n}) = (1 - \omega) x_i^n + \omega A_{ii}^{-1}( b_i - \sum_{j < i} A_{ij} x_j^{n+1/2} - \sum_{j > i} A_{ij} x_j^{n})
\]
for $ i=1,2,...n$. Followed by
\[
x_i^{n+1} = x_i^{n+1/2} + \omega A_{ii}^{-1}( b_i - \sum_{j \le i} A_{ij} x_j^{n+1/2}  - \sum_{j > i} A_{ij} x_j^{n+1}) = (1 - \omega) x_i^{n+1/2} + \omega A_{ii}^{-1}( b_i - \sum_{j < i} A_{ij} x_j^{n+1/2}  - \sum_{j > i} A_{ij} x_j^{n+1})
\]
for $ i=n,n-1,....1$. It is called over-relaxation because generally $ \omega $ is greater than one, though on occasion underrelaxation with $ \omega < 1$  has the fastest convergence.

To use this as a preconditioner, just start with $x^0 = $ to obtain
\[
x_i^{1/2} =  \omega A_{ii}^{-1}( b_i - \sum_{j < i} A_{ij} x_j^{1/2})
\]
for $ i=1,2,...n$. Followed by
\[
x_i = (1 - \omega) x_i^{1/2} + \omega A_{ii}^{-1}( b_i - \sum_{j < i} A_{ij} x_j^{1/2} - \sum_{j > i} A_{ij} x_j)
\]
for $ i=n,n-1,....1$.

Rewriting in matrix form
\[
x^{1/2} = \omega (L + D)^{-1} b
\]
\[
x = (1 - \omega) x^{1/2} + \omega (U + D)^{-1}(b - L x^{1/2}) = x^{1/2} + \omega (U+D)^{-1}(b - A x^{1/2}).
\]

For the SBAIJ matrix format
\begin{verbatim}
v  = aa + 1;
      vj = aj + 1;
      for (i=0; i<m; i++){
        nz = ai[i+1] - ai[i] - 1;
        tmp = - (x[i] = omega*t[i]*aidiag[i]);
        for (j=0; j<nz; j++) {
          t[vj[j]] += tmp*v[j];
        }
        v  += nz + 1;
        vj += nz + 1;
      }
\end{verbatim}
the array $t$ starts with the value of $b $ and is updated a column of the matrix at a time to contain the value of $ (b - L x^{1/2})$ that
are then needed in the upper triangular solve
\begin{verbatim}
      v  = aa + ai[m-1] + 1;
      vj = aj + ai[m-1] + 1;
      nz = 0;
      for (i=m-1; i>=0; i--){
        sum = 0.0;
        nz2 = ai[i] - ai[i-1] - 1;
        PETSC_Prefetch(v-nz2-1,0,PETSC_PREFETCH_HINT_NTA);
        PETSC_Prefetch(vj-nz2-1,0,PETSC_PREFETCH_HINT_NTA);
        PetscSparseDensePlusDot(sum,x,v,vj,nz);
        sum = t[i] - sum;
        x[i] =   (1-omega)*x[i] + omega*sum*aidiag[i];
        nz  = nz2;
        v  -= nz + 1;
        vj -= nz + 1;
      }
\end{verbatim}
Since the values in $ aa[]$ and $ aj[]$ are visited ``backwards'', the prefetch is used to load the needed previous row of matrix values and column indices into cache before they are needed.

For the AIJ format $t$ is updated a row at a time to contain $ (b - Lx^{1/2}).$


\section*{Notes on Sequential Eisenstat Implementation}


\[
  x = \omega (L + D)^{-1}b
\]
is the same as
\[
   x_i = \omega D_{ii}^{-1}(b_i - \sum_{j<i} A_{ij} x_j)
\]
\[
   x_i = (D_{ii}/\omega)^{-1}(b_i - \sum_{j<i} A_{ij} x_j)
\]
resulting in
\[
  x =  (L + D/\omega)^{-1}b
\]

Rather than applying the left preconditioner obtained by apply the two step process $ (L + D/\omega)^{-1} $ and then $ (U + D/\omega)^{-1} $
one can apply the two ``halves'' of the preconditioner symmetrically to the system resulting in
\[
 (L + D/\omega)^{-1} A (U + D/\omega)^{-1} y = (L + D/\omega)^{-1} b.
\]
Then after this system is solved, $ x = (U + D/\omega)^{-1} y$. If an initial guess that is nonzero is supplied then the
initial guess for $ y$ must be computed via $ y = (U + D/\omega) x$.
\begin{eqnarray*}
 (L + D/\omega)^{-1} A (U + D/\omega)^{-1} & =  & (L + D/\omega)^{-1} (L + D + U) (U + D/\omega)^{-1} \\
  & = &  (L + D/\omega)^{-1} (L + D/\omega + U + D/\omega + D - 2D/\omega) (U + D/\omega)^{-1} \\
  & = &  (U + D/\omega)^{-1} + (L+D/\omega)^{-1}(I + \frac{\omega - 2}{\omega}D(U + D/\omega)^{-1}).
\end{eqnarray*}

\end{document}