message: Updated documentation of ParIlu
diff: @@ -53,12 +53,27 @@ namespace factorization {
  /**
  - * An incomplete LU (ILU) factorization is a sparse approximation of the LU
  - * factorization, which is often used as a preconditioner.
  + * ParILU is an incomplete LU factorization which is computed in parallel.
  *
  - * $L$ is a lower unitriangular, while $U$ is an upper triangular matrix, which
  - * approximate a given matrix $A$ with $A \approx LU$. Here, $L$ and $U$ have
  - * the same sparsity pattern as $A$, which is also called ILU(0).
  + * The ParILU algorithm generates the incomplete factors iteratively, using a
  + * fixed-point iteration of the form
  + * $F(L, U) = \begin{cases}\frac{1}{u_{jj}}\left(a_{ij}-\sum_{k=1}^{j-1}
  + * l_{ik}u_{kj}\right), \quad &i>j\\
  + * a_{ij}-\sum_{k=1}^{i-1} l_{ik}u_{kj},\quad &i\leq j \end{cases}*$
  + *
  + * In general, the entries of $L$ and $U$ can be iterated in parallel and in
  + * asynchronous fashion, the algorithm asymptotically converges to the
  + * incomplete factors $L$ and $U$ fulfilling $\left(R = A - L \cdot
  + * U\right)\vert_\mathcal{S} = 0\vert_\mathcal{S}$ where $\mathcal{S}$ is the
  + * pre-defined sparsity pattern (in case of ILU(0) the sparsity pattern of the
  + * system matrix $A$). The number of ParILU sweeps needed for convergence
  + * depends on the parallelism level: For sequential execution, a single sweep
  + * is sufficient, for fine-grained parallelism, 3 sweeps are typically
  + * generating a good approximation.
  + *
  + * The ParILU algorithm in Ginkgo follows the design of E. Chow and A. Patel,
  + * Fine-grained Parallel Incomplete LU Factorization, SIAM Journal on Scientific
  + * Computing, 37, C169-C193 (2015).
  *
  * @tparam ValueType Type of the values of all matrices used in this class
  * @tparam IndexType Type of the indices of all matrices used in this class
  @@ -145,8 +160,8 @@ protected:
  * @note: system_matrix must be convertable to a Csr
  * Matrix, otherwise, an exception is thrown.
  * @param skip_sorting if set to `true`, the sorting will be skipped.
  - * @note: If the matrix was not sorted, the
  - * factorization might be wrong.
  + * @note: If the matrix is not sorted, the
  + * factorization fails.
  * @return A Composition, containing the incomplete LU factors for the
  * given system_matrix (first element is L, then U)
  */
  