\section{Preconditioner routines}
\label{sec:precs}

% \section{Preconditioners}
\label{sec:psprecs}
The base PSBLAS library  contains the implementation of two simple
preconditioning techniques:
\begin{itemize}
\item Diagonal Scaling
\item Block Jacobi with ILU(0) factorization
%% \item Additive Schwarz with the Restricted Additive Schwarz and
%%   Additive Schwarz with Harmonic extensions;
\end{itemize}
The supporting data type and subroutine interfaces are defined in the
module  \verb|psb_prec_mod|.

%% We also provide a companion  package of multi-level Additive
%%   Schwarz preconditioners called MD2P4; this is actually a family of 
%%   preconditioners since there is the possibility to choose between
%%   many variants, and is currently in an experimental stateIts
%%   documentation is planned to appear after stabilization of the
%%   package, which will characterize release  2.1 of our library.




\subroutine{psb\_precset}{Sets the preconditioner type}

\syntax{call psb\_precset}{prec, ptype, info}

\begin{description}
\item[\bf On Entry]
\item[prec] 
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a pronditioner data structure \precdata.
\item[ptype] the type of preconditioner. 
Scope: {\bf global} \\
Type: {\bf required}\\
Specified as: a character string, see usage notes.
%% \item[iv] integer parameters for the precondtioner. 
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: an integer array, see usage notes. 
%% \item[rs] 
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Specified as: a long precision real number.
\item[\bf On Exit]

\item[info] 
Scope: {\bf global} \\
Type: {\bf required}\\
Error code: if no error, 0 is returned.
\end{description}
\subsection*{Usage Notes}
%% The PSBLAS 2.0 contains a number of preconditioners, ranging from a
%% simple diagonal scaling to 2-level domain decomposition. These
%% preconditioners may use the SuperLU or the UMFPACK software, if
%% installed; see~\cite{SUPERLU,UMFPACK}. 
Legal inputs to this subroutine are interpreted depending on the
$ptype$ string as follows\footnote{The string is case-insensitive}:
\begin{description}
\item[NONE] No preconditioning, i.e. the preconditioner is just a copy
  operator.
\item[DIAG] Diagonal scaling; each entry of the input vector is
  multiplied by the reciprocal of the sum of the absolute values of
  the coefficients in the corresponding row of matrix  $A$;
\item[BJAC] Precondition by a  factorization of the
  block-diagonal of matrix $A$, where block boundaries are determined
  by the data allocation boundaries for each process; requires no
  communication. Only the incomplete factorization $ILU(0)$ is
  currently implemented.  
%% \item[AS] Additive Schwarz preconditioner (see~\cite{PARA04}); in this
%%   case the user may specify additional flags through the integer
%%   vector \verb|ir| as follows:
%% \begin{description}
%% \item[$iv(1)$] Number of overlap levels, an integer $novr>=0$, default
%%   $novr=1$.  
%% \item[$iv(2)$] Restriction operator, legal values: \verb|psb_halo_|,
%%   \verb|psb_none_|; default: \verb|psb_halo_|
%% \item[$iv(3)$] Prolongation operator, legal values: \verb|psb_none_|,
%%   \verb|psb_sum_|, \verb|psb_avg_|, default: \verb|psb_none_|
%% \item[$iv(4)$] Factorization type, legal values: \verb|f_ilu_n_|,
%%   \verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|.
%% \end{description}
%% Note that the default corresponds to a Restricted Additive Schwarz
%% preconditioner with $ILU(0)$ and 1 level of overlap.
%% \item[2L] Second level of a multilevel preconditioner, see below
%% \end{description}
%% If a multilevel preconditioner is desired, the user should call
%% \verb|psb_precset| twice, the first time  choosing an AS variant, and
%% a second time specifying 
%% $ptype=2L$ with the following optional parameters in $iv$ (see
%% also~\cite{APNUM06,DD2}): 
%% \begin{description}
%% \item[$iv(1)$] Type of multilevel correction, legal values: \verb|no_ml_|,
%%   \verb|add_ml_prec_|, \verb|mult_ml_prec_|,
%%   default: \verb|mult_ml_prec_|;
%% \item[$iv(2)$] Aggregation algorithm, legal values: \verb|loc_aggr_|;
%% \item[$iv(3)$] Smoother type, legal values: \verb|no_smth_|,
%%   \verb|smth_omg_|, default: \verb|smth_omg_|;
%% \item[$iv(4)$] Coarse matrix allocation, legal values:
%%   \verb|mat_distr_|, \verb|mat_repl_|, default: \verb|mat_distr_|
%% \item[$iv(5)$] Smoother position, legal values: \verb|pre_smooth_|,
%%   \verb|post_smooth_|, \verb|smooth_both_|, default:
%%   \verb|post_smooth_|
%% \item[$iv(6)$] Factorization type (for coarse matrix), legal values: \verb|f_ilu_n_|,
%%   \verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|;
%% \item[$iv(7)$] Number of Jacobi sweeps for coarse system correction,
%%   default 1.
%% \item[$rs$] Set the smoother parameter $\omega$ a user defined value;
%%   default: esitimate with the infinity norm of matrix $A$.
\end{description}
%% The 2-level preconditioners are based on the idea of building a
%% coarse-space approximation $A_C$ of the matrix $A$; given a set $W_C$
%% of coarse vertices, with size $n_C$, and a suitable restriction
%% operator $R_C \in \Re^{n_C \times n}$, $A_C$ is defined as 
%% \[
%% A_C=R_C A R_C^T .
%% \]
%% The prolongator $R_C^T$ is built with the smoothed aggregation technique,
%% in which we start from a tentative prolongator that simply maps
%% fine-level entries onto their aggregates $P_C$; if the user chooses
%% \verb|no_smth_| this is the prolongator used, otherwise it is
%% multiplied by a smoother \[ S = I - \omega D^{-1} A \], where $D$ is
%% the diagonal of $A$ and  $\omega$ may be imposed by the user or
%% estimated internally.   
%% The coarse space correction may be added to the fine level solution
%% \verb|add_ml_prec_|
%% \[
%% M_{2L-A}^{-1} = M_{C}^{-1} + M_{1L}^{-1}. 
%% \]
%% or it can be composed  in a multiplicative framework
%% (\verb|mult_ml_prec_|)as a pre-smoothed correction (\verb|pre_smooth_|)
%% \[
%% M_{2L-H1}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}, 
%% \]
%% post-smoothed correction (\verb|post_smooth_|)
%% \[
%% M_{2L-H2}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}. 
%% \]
%% or two-sided for symmetric matrices (\verb|smooth_both_|).



\subroutine{psb\_precbld}{Builds a preconditioner}

\syntax{call psb\_precbld}{a, desc\_a, prec, info, upd}

\begin{description}
\item[\bf On Entry]
\item[a] the system sparse matrix.
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a sparse matrix data structure \spdata.
\item[desc\_a] the problem communication descriptor. 
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a communication descriptor data structure \descdata.
\item[upd] 
Scope: {\bf global} \\
Type: {\bf optional}\\
Specified as: a character.
\end{description}

\begin{description}
\item[\bf On Return]
\item[prec] the preconditioner.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a precondtioner data structure \precdata\\
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
An integer value; 0 means no error has been detected. 
\end{description}



\subroutine{psb\_precaply}{Preconditioner application routine}

\syntax{call psb\_precaply}{prec,x,y,desc\_a,info,trans,work}
\syntax*{call psb\_precaply}{prec,x,desc\_a,info,trans}

\begin{description}
\item[\bf On Entry]
\item[prec] the preconditioner.
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a preconditioner data structure \precdata.
\item[x] the source vector.
Scope: {\bf local} \\
Type: {\bf require}\\
Specified as: a double precision array.
\item[desc\_a] the problem communication descriptor.
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a communication data structure \descdata.
\item[trans] 
Scope: {\bf } \\
Type: {\bf optional}\\
Specified as: a character.
\item[work] an optional work space
Scope: {\bf local} \\
Type: {\bf optional}\\
Specified as: a double precision array.
\end{description}

\begin{description}
\item[\bf On Return]
\item[y] the destination vector.
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a double precision array.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
An integer value; 0 means no error has been detected. 
\end{description}



\subroutine{psb\_prec\_descr}{Prints a description of current preconditioner}

\syntax{call psb\_prec\_descr}{prec}

\begin{description}
\item[\bf On Entry]
\item[prec] the preconditioner.
Scope: {\bf local} \\
Type: {\bf required}\\
Specified as: a preconditioner data structure \precdata.
\end{description}



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