\section{Computational routines}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%      DENSE MATRIX SUM
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage\subsection*{psb\_geaxpby --- General Dense Matrix Sum}
\addcontentsline{toc}{subsection}{psb\_geaxpby}

This subroutine is an interface to the computational kernel for
dense matrix sum:
\[ y \leftarrow  \alpha\> x+ \beta y \] 
%% where:
%% \begin{description}
%% \item[$x$] represents the global dense submatrix $x_{:, :1}$
%% \item[$y$] represents the global dense submatrix $y_{:, :}$
%% \end{description}

\begin{verbatim}
call psb_geaxpby(alpha, x, beta, y, desc_a, info)
\end{verbatim}
%% \syntax*{call psb\_geaxpby}{alpha, x, beta, y, desc\_a, info, n, jx, jy}

%( calculating y <- alpha*x+beta*y )
\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$x$, $y$, $\alpha$, $\beta$ & {\bf Subroutine}\\
\hline
Short Precision Real & psb\_geaxpby \\
Long Precision Real & psb\_geaxpby \\
Short Precision Complex & psb\_geaxpby \\
Long Precision Complex & psb\_geaxpby \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90axpby}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[alpha] the scalar $\alpha$.\\ Scope: {\bf global} \\ Type: {\bf
required} \\ Intent: {\bf in}.\\ Specified as: a number of the data
type indicated in Table~\ref{tab:f90axpby}.
\item[x] the local portion of global dense matrix
$x$.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a rank one or two array or an object of type \vdata\ 
containing numbers of type 
specified in Table~\ref{tab:f90axpby}.  The rank of $x$ must be the same of $y$. 
\item[beta] the scalar $\beta$.\\
Scope: {\bf global} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a number of the data type indicated in Table~\ref{tab:f90axpby}.
\item[y] the local portion of the global dense matrix
$y$. \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as:  a rank one or two array or an object of type \vdata\  containing numbers of the type 
indicated in Table~\ref{tab:f90axpby}.  The rank of $y$ must be the same of $x$. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
%% \item[n] number of columns in dense submatrices $x$ and $y$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ and $y$ are of rank 2.\\
%% Default: \verb|min(size(x,2),size(y,2))|.\\
%% Specified as: an integer variable $n\ge 0$.
%% \item[jx]  the column index of the global dense matrix $x$,
%% identifying the first column of the submatrix $x$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ and $y$ are of rank 2.\\
%% Default: $jx = 1$.\\
%% Specified as: an integer variable $jx\ge 1$. 
%% \item[jy]  the column index of the global dense matrix $y$,
%% identifying the first column of the submatrix $y$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ and $y$ are of rank 2.\\
%% Default: $jy = 1$.\\
%% Specified as: an integer variable $jy\ge 1$. 

\end{description}

\begin{description}
\item[\bf On Return]
\item[y] the local portion of result submatrix $y$.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as: a rank one or two array or an object of type \vdata\ containing numbers of the type
indicated in Table~\ref{tab:f90axpby}.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       F90DOT PRODUCT
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage\subsection*{psb\_gedot --- Dot Product}
\addcontentsline{toc}{subsection}{psb\_gedot}

This function computes dot product between two vectors $x$ and
$y$.\\
If $x$ and $y$ are real vectors
it computes dot-product as:
\[dot \leftarrow x^T y\]
Else if $x$ and $y$ are complex vectors then it computes dot-product as:
\[dot \leftarrow x^H y\]
%% where:
%% \begin{description}
%% \item[$x$] represents the global subvector $x_{:,jx}$
%% \item[$y$] represents the global subvector $y_{:,jy}$
%% \end{description}

\begin{verbatim}
psb_gedot(x, y, desc_a, info [,global])
\end{verbatim}
%% \syntax*{psb\_gedot}{x, y, desc\_a, info, jx, jy}
\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$dot$, $x$, $y$ & {\bf Function}\\
\hline
Short Precision Real & psb\_gedot \\
Long Precision Real & psb\_gedot \\
Short Precision Complex & psb\_gedot \\	
Long Precision Complex & psb\_gedot \\	
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90dot}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$.\\
%%  This function computes the location of the first element of
%% local subarray used, based on $jx$ and the field $matrix\_data$ of $desc\_a$ . \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90dot}. The rank of $x$ must be the same of $y$. 
\item[y] the local portion of global dense matrix
$y$. \\
%% This function computes the location of the first element of
%% local subarray used, based on $iy, jy$ and the field $matrix\_data$ of $desc\_a$ . \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90dot}. The rank of $y$ must be the same of $x$. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[global]  Specifies whether the computation should include the
  global reduction across all processes.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
Specified as: a logical scalar.
Default: \verb|global=.true.|\\
%% \item[jx]  the column index of global dense matrix $x$,
%% identifying the column of subvector $x$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ and $y$ are of rank 2.\\
%% Default: $jx = 1$.\\

%% \item[jy]  the column index of global dense matrix $y$,
%% identifying the column of subvector $y$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ and $y$ are of rank 2.\\
%% Default: $jy = 1$.\\
%% Specified as: an integer variable $jy\ge 1$. 
\item[\bf On Return] 
\item[Function value] is the dot product of subvectors $x$ and $y$.\\
Scope: {\bf global}  unless the optional variable
\verb|global=.false.| as been specified\\
Specified as: a number of the data type indicated in Table~\ref{tab:f90dot}.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

{\par\noindent\large\bfseries Notes}
\begin{enumerate}
\item The computation of a global result requires a global
  communication, which entails a significant overhead. It may be
  necessary and/or advisable to compute multiple dot products at the same 
  time; in this case, it is possible to improve the runtime efficiency
  by using the following scheme:
  \begin{lstlisting}
    vres(1) = psb_gedot(x1,y1,desc_a,info,global=.false.)
    vres(2) = psb_gedot(x2,y2,desc_a,info,global=.false.)
    vres(3) = psb_gedot(x3,y3,desc_a,info,global=.false.)
    call psb_sum(ictxt,vres(1:3))
  \end{lstlisting}
  In this way the global communication, which for small sizes is a
  latency-bound operation, is invoked only once.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
%       F90DOT PRODUCT
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage\subsection*{psb\_gedots --- Generalized Dot Product}
\addcontentsline{toc}{subsection}{psb\_gedots}

This subroutine computes a series of  dot products among the columns of
two dense matrices  $x$ and $y$: 
\[ res(i) \leftarrow x(:,i)^T y(:,i)\]
If the matrices are complex, then the
usual convention applies, i.e. the conjugate transpose of $x$ is
used. If $x$ and $y$ are of rank one, then $res$ is a scalar, else it
is a rank one array. 

\begin{verbatim}
call psb_gedots(res, x, y, desc_a, info)
\end{verbatim}
\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$res$, $x$, $y$ & {\bf Subroutine}\\
\hline
Short Precision Real & psb\_gedots \\
Long Precision Real & psb\_gedots \\
Short Precision Complex & psb\_gedots \\	
Long Precision Complex & psb\_gedots \\	
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90mdot}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90mdot}. The rank of $x$ must be the same of $y$. 
\item[y] the local portion of global dense matrix
$y$. \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90mdot}. The rank of $y$ must be the same of $x$. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[\bf On Return] 
\item[res] is the dot product of subvectors $x$ and $y$.\\
Scope: {\bf global} \\
Intent: {\bf out}.\\
Specified as: a number or a rank-one array  of the data type indicated
in Table~\ref{tab:f90dot}. 
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       VECTOR INFINITY-NORM 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_normi --- Infinity-Norm of Vector}
\addcontentsline{toc}{subsection}{psb\_normi}    

This function computes 
 the infinity-norm of a vector $x$.\\
If $x$ is a real  vector
it computes infinity norm as:
\[ amax \leftarrow \max_i |x_i|\]
else if $x$ is a complex vector then it computes the infinity-norm  as:
\[ amax \leftarrow \max_i {(|re(x_i)| + |im(x_i)|)}\]
%% where:
%% \begin{description}
%% \item[$x$] represents the global subvector $x_{:,jx}$
%% \end{description}

\begin{verbatim}
psb_geamax(x, desc_a, info [,global])
psb_normi(x, desc_a, info [,global])
\end{verbatim}
%% \syntax*{psb\_geamax}{x, desc\_a, info, jx}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$amax$ & $x$ & {\bf Function}\\
\hline
Short Precision Real& Short Precision Real & psb\_geamax \\
Long Precision Real&Long Precision Real & psb\_geamax \\
Short Precision Real&Short Precision Complex & psb\_geamax \\
Long Precision Real&Long Precision Complex & psb\_geamax \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90amax}}
\end{table}


\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. %% This function computes the location of the first element of
%% local subarray used, based on $jx$ and the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\  
containing numbers of type specified in
Table~\ref{tab:f90amax}.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[global]  Specifies whether the computation should include the
  global reduction across all processes.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
Specified as: a logical scalar.
Default: \verb|global=.true.|\\%% \item[jx]  the column index of global dense matrix $x$,
%% identifying the column of subvector $x$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ is of rank 2.\\	
%% Default: $jx = 1$\\	
%% Specified as: an integer variable $jx\ge 1$. 

\item[\bf On Return] 
\item[Function value] is the infinity norm of subvector $x$.\\
Scope: {\bf global} unless the optional variable
\verb|global=.false.| as been specified\\
Specified as: a long precision real number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

{\par\noindent\large\bfseries Notes}
\begin{enumerate}
\item The computation of a global result requires a global
  communication, which entails a significant overhead. It may be
  necessary and/or advisable to compute multiple norms at the same 
  time; in this case, it is possible to improve the runtime efficiency
  by using the following scheme:
  \begin{lstlisting}
    vres(1) = psb_geamax(x1,desc_a,info,global=.false.)
    vres(2) = psb_geamax(x2,desc_a,info,global=.false.)
    vres(3) = psb_geamax(x3,desc_a,info,global=.false.)
    call psb_amx(ictxt,vres(1:3))
  \end{lstlisting}
  In this way the global communication, which for small sizes is a
  latency-bound operation, is invoked only once.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
%       Infinity norm
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage\subsection*{psb\_geamaxs --- Generalized Infinity Norm}
\addcontentsline{toc}{subsection}{psb\_geamaxs}

This subroutine computes a series of  infinity norms on the columns of
a  dense matrix  $x$: 
\[ res(i) \leftarrow \max_k |x(k,i)| \]

\begin{verbatim}
call psb_geamaxs(res, x, desc_a, info)
\end{verbatim}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$res$&  $x$& {\bf Subroutine}\\
\hline
Short Precision Real    &Short Precision Real    & psb\_geamaxs\\
Long Precision Real    &Long Precision Real    & psb\_geamaxs\\
Short Precision Real &Short Precision Complex & psb\_geamaxs\\	
Long Precision Real &Long Precision Complex & psb\_geamaxs\\	
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90mamax}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. \\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90mamax}. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[\bf On Return] 
\item[res] is the infinity norm of the columns of $x$.\\
Scope: {\bf global} \\
Intent: {\bf out}.\\
Specified as: a number or a rank-one array  of long precision real numbers. 
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       1-NORM OF A VECTOR
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_norm1 --- 1-Norm of Vector}
\addcontentsline{toc}{subsection}{psb\_norm1}    

This function computes the 1-norm of a vector $x$.\\
If $x$ is a real vector
it computes 1-norm as:
\[ asum \leftarrow  \|x_i\|\]
else if $x$ is a complex vector then it computes 1-norm  as:
\[ asum \leftarrow \|re(x)\|_1 + \|im(x)\|_1\]


\begin{verbatim}
psb_geasum(x, desc_a, info [,global])
psb_norm1(x, desc_a, info [,global])
\end{verbatim}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$asum$ & $x$ & {\bf Function}\\
\hline
Short Precision Real&Short Precision Real & psb\_geasum \\
Long Precision Real&Long Precision Real & psb\_geasum \\
Short Precision Real&Short Precision Complex & psb\_geasum \\
Long Precision Real&Long Precision Complex & psb\_geasum \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90asum}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. %% This function computes the location of the first element of 
%% local subarray used, based on the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90asum}.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[global]  Specifies whether the computation should include the
  global reduction across all processes.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
Specified as: a logical scalar.
Default: \verb|global=.true.|\\

\item[\bf On Return] 
\item[Function value] is the 1-norm of vector $x$.\\
Scope: {\bf global} unless the optional variable
\verb|global=.false.| as been specified\\
Specified as: a long precision real  number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

{\par\noindent\large\bfseries Notes}
\begin{enumerate}
\item The computation of a global result requires a global
  communication, which entails a significant overhead. It may be
  necessary and/or advisable to compute multiple norms at the same 
  time; in this case, it is possible to improve the runtime efficiency
  by using the following scheme:
  \begin{lstlisting}
    vres(1) = psb_geasum(x1,desc_a,info,global=.false.)
    vres(2) = psb_geasum(x2,desc_a,info,global=.false.)
    vres(3) = psb_geasum(x3,desc_a,info,global=.false.)
    call psb_sum(ictxt,vres(1:3))
  \end{lstlisting}
  In this way the global communication, which for small sizes is a
  latency-bound operation, is invoked only once.
\end{enumerate}


\clearpage\subsection*{psb\_geasums --- Generalized 1-Norm of Vector}
\addcontentsline{toc}{subsection}{psb\_geasums}    

This subroutine computes a series of  1-norms on the columns of
a  dense matrix  $x$: 
\[ res(i) \leftarrow \max_k |x(k,i)| \]
This function computes the 1-norm of a vector $x$.\\
If $x$ is a real vector 
it computes 1-norm as:
\[ res(i) \leftarrow  \|x_i\|\]
else if $x$ is a complex vector then it computes 1-norm  as:
\[ res(i) \leftarrow \|re(x)\|_1 + \|im(x)\|_1\]


\begin{verbatim}
call psb_geasums(res, x, desc_a, info)
\end{verbatim}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$res$ & $x$ & {\bf Subroutine}\\
\hline
Short Precision Real&Short Precision Real & psb\_geasums \\
Long Precision Real&Long Precision Real & psb\_geasums \\
Short Precision Real&Short Precision Complex & psb\_geasums \\
Long Precision Real&Long Precision Complex & psb\_geasums \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90asums}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. %% This function computes the location of the first element of 
%% local subarray used, based on the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90asums}.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.

\item[\bf On Return] 
\item[res] contains the 1-norm of (the columns of) $x$.\\
Scope: {\bf global} \\
Intent: {\bf out}.\\
Short as: a long precision real  number.
Specified as: a long precision real  number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       2-NORM OF A VECTOR 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_norm2 --- 2-Norm of Vector}
\addcontentsline{toc}{subsection}{psb\_norm2}    

This function computes the 2-norm of a vector $x$.\\
If $x$ is a  real  vector
it computes 2-norm as:
\[ nrm2 \leftarrow \sqrt{x^T x}\]
else if $x$ is a complex vector then it computes 2-norm  as:
\[ nrm2 \leftarrow \sqrt{x^H x}\]
%% where:
%% \begin{description}
%% \item[$x$] represents the global subvector $x_{:,jx}$
%% \end{description}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$nrm2$ & $x$ & {\bf Function}\\
\hline
Short Precision Real&Short Precision Real & psb\_genrm2 \\
Long Precision Real&Long Precision Real & psb\_genrm2 \\
Short Precision Real&Short Precision Complex & psb\_genrm2 \\
Long Precision Real&Long Precision Complex & psb\_genrm2 \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90nrm2}}
\end{table}

\begin{verbatim}
psb_genrm2(x, desc_a, info [,global])
psb_norm2(x, desc_a, info [,global])
\end{verbatim}
%% \syntax*{psb\_genrm2}{x, desc\_a, info, jx}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$.%%  This function computes the location of the first element of
%% local subarray used, based on $jx$ and the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90nrm2}.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[global]  Specifies whether the computation should include the
  global reduction across all processes.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
Specified as: a logical scalar.
Default: \verb|global=.true.|\\%% \item[jx]  the column index of global dense matrix $x$,
%% identifying the column of subvector $x$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ is of rank 2.\\	
%% Default: $jx = 1$\\	
%% Specified as: an integer variable $jx\ge 1$. 

\item[\bf On Return] 
\item[Function Value] is the 2-norm of subvector $x$.\\
Scope: {\bf global} unless the optional variable
\verb|global=.false.| as been specified\\
Type: {\bf required} \\
Specified as: a long precision real number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

{\par\noindent\large\bfseries Notes}
\begin{enumerate}
\item The computation of a global result requires a global
  communication, which entails a significant overhead. It may be
  necessary and/or advisable to compute multiple norms at the same 
  time; in this case, it is possible to improve the runtime efficiency
  by using the following scheme:
  \begin{lstlisting}
    vres(1) = psb_genrm2(x1,desc_a,info,global=.false.)
    vres(2) = psb_genrm2(x2,desc_a,info,global=.false.)
    vres(3) = psb_genrm2(x3,desc_a,info,global=.false.)
    call psb_nrm2(ictxt,vres(1:3))
  \end{lstlisting}
  In this way the global communication, which for small sizes is a
  latency-bound operation, is invoked only once.
\end{enumerate}



\clearpage\subsection*{psb\_genrm2s --- Generalized 2-Norm of Vector}
\addcontentsline{toc}{subsection}{psb\_genrm2s}    

This subroutine computes a series of  2-norms on the columns of
a  dense matrix  $x$: 
\[ res(i) \leftarrow \|x(:,i)\|_2 \]


\begin{verbatim}
call psb_genrm2s(res, x, desc_a, info)
\end{verbatim}

\begin{table}[h]
\begin{center}
\begin{tabular}{lll}
\hline
$res$ & $x$ & {\bf Subroutine}\\
\hline
Short Precision Real&Short Precision Real & psb\_genrm2s \\
Long Precision Real&Long Precision Real & psb\_genrm2s \\
Short Precision Real&Short Precision Complex & psb\_genrm2s \\
Long Precision Real&Long Precision Complex & psb\_genrm2s \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90nrm2s}}
\end{table}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[x] the local portion of global dense matrix
$x$. %% This function computes the location of the first element of 
%% local subarray used, based on the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90nrm2s}.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.

\item[\bf On Return] 
\item[res] contains the 1-norm of (the columns of) $x$.\\
Scope: {\bf global} \\
Intent: {\bf out}.\\
Specified as: a long precision real  number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       1-NORM OF A MATRIX 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_norm1 --- 1-Norm of Sparse Matrix}
\addcontentsline{toc}{subsection}{psb\_norm1}    

This function computes the 1-norm of a matrix $A$:\\

\[ nrm1 \leftarrow \|A\|_1 \]
where:
\begin{description}
\item[$A$] represents the global matrix $A$
\end{description}

\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$A$ & {\bf Function}\\
\hline
Short Precision Real & psb\_spnrm1 \\
Long Precision Real & psb\_spnrm1 \\
Short Precision Complex & psb\_spnrm1 \\
Long Precision Complex & psb\_spnrm1 \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90nrm1}}
\end{table}

\begin{verbatim}
psb_spnrm1(A, desc_a, info)
psb_norm1(A, desc_a, info)
\end{verbatim}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[a] the local  portion of the global sparse matrix
$A$. \\   
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \spdata.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[\bf On Return] 
\item[Function value] is the 1-norm of sparse submatrix $A$.\\
Scope: {\bf global} \\
Specified as: a long precision real number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       INFINITY-NORM OF A MATRIX 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_normi --- Infinity Norm of Sparse Matrix}
\addcontentsline{toc}{subsection}{psb\_normi}    

This function computes the infinity-norm of a matrix $A$:\\

\[ nrmi \leftarrow \|A\|_\infty \]
where:
\begin{description}
\item[$A$] represents the global matrix $A$
\end{description}

\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$A$ & {\bf Function}\\
\hline
Short Precision Real & psb\_spnrmi \\
Long Precision Real & psb\_spnrmi \\
Short Precision Complex & psb\_spnrmi \\
Long Precision Complex & psb\_spnrmi \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90nrmi}}
\end{table}

\begin{verbatim}
psb_spnrmi(A, desc_a, info)
psb_normi(A, desc_a, info)
\end{verbatim}

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[a] the local  portion of the global sparse matrix
$A$. \\   
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \spdata.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[\bf On Return] 
\item[Function value] is the infinity-norm of sparse submatrix $A$.\\
Scope: {\bf global} \\
Specified as: a long precision real number.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       SPARSE MATRIX by DENSE MATRIX PRODUCT
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_spmm --- Sparse Matrix by Dense Matrix
  Product}
\addcontentsline{toc}{subsection}{psb\_spmm}   

This subroutine computes the Sparse Matrix by Dense Matrix Product:

\begin{equation}
y \leftarrow \alpha  A  x + \beta y
\label{eq:f90spmm_no_tra}
\end{equation}
\begin{equation}
y \leftarrow \alpha  A^T x + \beta y
\label{eq:f90spmm_tra}
\end{equation}
\begin{equation}
y \leftarrow \alpha  A^H  x + \beta y
\label{eq:f90spmm_con}
\end{equation}

where:
\begin{description}
\item[$x$] is the global dense matrix $x_{:, :}$
\item[$y$] is the global dense matrix $y_{:, :}$
\item[$A$] is the global sparse matrix $A$
\end{description}

\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$A$, $x$, $y$, $\alpha$, $\beta$ & {\bf Subroutine}\\
\hline
Short Precision Real & psb\_spmm \\
Long Precision Real & psb\_spmm \\
Short Precision Complex & psb\_spmm \\
Long Precision Complex & psb\_spmm \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90spmm}}
\end{table}

\begin{verbatim}
call psb_spmm(alpha, a, x, beta, y, desc_a, info)
call psb_spmm(alpha, a, x, beta, y,desc_a, info, &
             & trans, work)
\end{verbatim} 

\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[alpha] the scalar $\alpha$.\\
Scope: {\bf global} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a number of the data type indicated in
Table~\ref{tab:f90spmm}. 
\item[a] the local portion of the sparse matrix
$A$. \\ 
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \spdata.
\item[x] the local portion of global dense matrix
$x$. %% This subroutine computes the location of the first element of
%% local subarray used, based on $jx$ and the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90spmm}.  The rank of $x$ must be the same of $y$. 
\item[beta] the scalar $\beta$.\\
Scope: {\bf global} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a number of the data type indicated in Table~\ref{tab:f90spmm}.
\item[y] the local portion of global dense matrix
$y$. %% This subroutine computes the location of the first element of
%% local subarray used, based on $jy$ and the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90spmm}. The rank of $y$ must be the same of $x$. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[trans] indicates what kind of operation to perform.
\begin{description}
\item[trans = N] the operation is specified by equation \ref{eq:f90spmm_no_tra}
\item[trans = T] the operation is specified by equation
\ref{eq:f90spmm_tra}
\item[trans = C] the operation is specified by equation
\ref{eq:f90spmm_con}
\end{description}
Scope: {\bf global} \\
Type: {\bf optional}\\	
Intent: {\bf in}.\\
Default: $trans = N$\\	
Specified as: a character variable.
%% \item[k] number of columns in dense submatrices $x$ and $y$. \\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\	
%% Default: \verb|min(size(x,2)-jx+1,size(y,2)-jy+1)|\\	
%% Specified as: an integer variable $ k \ge 1$.
%% \item[jx]  the column index of global dense matrix $x$,
%% identifying the column of subvector $x$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $x$ is of rank 2.\\	
%% Default: $iy = 1$\\	
%% Specified as: an integer variable $jx\ge 1$. 
%% \item[jy]  the column index of global dense matrix $y$,
%% identifying the column of subvector $y$.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}; can only be present if $y$ is of rank 2.\\	
%% Default: $jy = 1$\\	
%% Specified as: an integer variable $jy\ge 1$. 

\item[work]  work array.\\
Scope: {\bf local} \\
Type: {\bf optional}\\	
Intent: {\bf inout}.\\
Specified as: a rank one array of the same type of $x$ and $y$ with
the TARGET attribute. 

\item[\bf On Return]
\item[y] the local portion of result matrix $y$.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as: an array of rank one or two
containing numbers of type specified in
Table~\ref{tab:f90spmm}.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%       TRIANGULAR SYSTEM SOLVE
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage\subsection*{psb\_spsm --- Triangular System Solve}
\addcontentsline{toc}{subsection}{psb\_spsm}   

This subroutine computes the Triangular System Solve:

\begin{eqnarray*}
y &\leftarrow& \alpha  T^{-1}  x + \beta y\\
y &\leftarrow& \alpha D  T^{-1}  x + \beta y\\
y &\leftarrow& \alpha  T^{-1}  D x + \beta y\\
y &\leftarrow& \alpha  T^{-T}  x + \beta y\\
y &\leftarrow& \alpha D  T^{-T}  x + \beta y\\
y &\leftarrow& \alpha  T^{-T}  D x + \beta y\\
y &\leftarrow& \alpha  T^{-H}  x + \beta y\\
y &\leftarrow& \alpha D  T^{-H}  x + \beta y\\
y &\leftarrow& \alpha  T^{-H}  D x + \beta y\\
\end{eqnarray*}


where:
\begin{description}
\item[$x$] is the global dense matrix $x_{:, :}$
\item[$y$] is the global dense matrix $y_{:, :}$
\item[$T$] is the global sparse block triangular submatrix $T$
\item[$D$] is the scaling diagonal matrix.
\end{description}

\begin{verbatim}
call psb_spsm(alpha, t, x, beta, y, desc_a, info)
call psb_spsm(alpha, t, x, beta, y, desc_a, info,&
             & trans, unit, choice, diag, work)
\end{verbatim} 

\begin{table}[h]
\begin{center}
\begin{tabular}{ll}
\hline
$T$, $x$, $y$, $D$, $\alpha$, $\beta$ & {\bf Subroutine}\\
\hline
Short Precision Real & psb\_spsm \\
Long Precision Real & psb\_spsm \\
Short Precision Complex & psb\_spsm \\
Long Precision Complex & psb\_spsm \\
\hline
\end{tabular}
\end{center}
\caption{Data types\label{tab:f90spsm}}
\end{table}



\begin{description}
\item[Type:] Synchronous.
\item[\bf On Entry]
\item[alpha] the scalar $\alpha$.\\
Scope: {\bf global} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a number of the data type indicated in
Table~\ref{tab:f90spsm}.
\item[t] the global portion of the sparse matrix
$T$.  \\ 
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object type specified in
\S~\ref{sec:datastruct}.
\item[x] the local portion of global dense matrix
$x$. %% This subroutine computes the location of the first element of
%% local subarray used, based on $jx$ and the field $matrix\_data$ of $desc\_a$ .
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90spsm}.  The rank of $x$ must be the same of $y$. 
\item[beta] the scalar $\beta$.\\
Scope: {\bf global} \\
Type: {\bf required} \\
Intent: {\bf in}.\\
Specified as: a number of the data type indicated in Table~\ref{tab:f90spsm}.
\item[y] the local portion of global dense matrix
$y$. %% This subroutine computes the location of the first element of
%% local subarray used, based on $jy$ and the field $matrix\_data$ of $desc\_a$ .
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as:  a rank one or two array or an object of type \vdata\ 
containing numbers of type specified in
Table~\ref{tab:f90spsm}. The rank of $y$ must be the same of $x$. 
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: an object of type \descdata.
\item[trans] specify with {\em unitd} the operation to perform.
\begin{description}
\item[trans = 'N'] the operation is with no transposed matrix
\item[trans = 'T'] the operation is with transposed matrix.
\item[trans = 'C'] the operation is with conjugate transposed matrix.
\end{description}
Scope: {\bf global} \\
Type: {\bf optional}\\	
Intent: {\bf in}.\\
Default: $trans = N$\\	
Specified as: a character variable.
\item[unitd] specify with {\em trans} the operation to perform.
\begin{description}
\item[unitd = 'U'] the operation is with no scaling
\item[unitd = 'L'] the operation is with left scaling
\item[unitd = 'R'] the operation is with right scaling.
\end{description}
Scope: {\bf global} \\
Type: {\bf optional}\\	
Intent: {\bf in}.\\
Default: $unitd = U$\\	
Specified as: a character variable.
\item[choice] specifies the update of overlap elements to be performed
  on exit:
\begin{description}
\item \verb|psb_none_|
\item \verb|psb_sum_|
\item \verb|psb_avg_|
\item \verb|psb_square_root_|
\end{description}
Scope: {\bf global} \\
Type: {\bf optional}\\	
Intent: {\bf in}.\\
Default: \verb|psb_avg_|\\	
Specified as: an integer variable.
\item[diag] the diagonal scaling matrix.\\
Scope: {\bf local} \\
Type: {\bf optional}\\	
Intent: {\bf in}.\\
Default: $diag(1) = 1 (no scaling)$\\	
Specified as: a rank one  array containing numbers of the type
indicated in Table~\ref{tab:f90spsm}.
\item[work] a work array. \\
Scope: {\bf local} \\
Type: {\bf optional}\\	
Intent: {\bf inout}.\\
Specified as: a rank one array of the same type of $x$ with the
TARGET attribute. 

\item[\bf On Return] 
\item[y] the local portion of global dense matrix
$y$. %% This subroutine computes the location of the first element of
%% local subarray used, based on $jy$ and the field $matrix\_data$ of $desc\_a$ . 
\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf inout}.\\
Specified as: an array of rank one or two
containing numbers of type specified in
Table~\ref{tab:f90spsm}.
\item[info] Error code.\\
Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected. 
\end{description}




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