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\title{ScaLAPACK Users' Guide}
\author{L. S. Blackford, J. Choi, A. Cleary, E. D'Azevedo, J. Demmel, \\ 
I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, \\ 
K. Stanley, D. Walker, and R. C. Whaley}
\date{PRELIMINARY DRAFT:  9 May 1997}

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The royalties from the sales of this book are being placed in a fund to
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This fund is administered by SIAM and qualified individuals are
encouraged to write directly to SIAM for guidelines. }}}\hbox{}\vfil

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\label{tabdrivelineq}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||	} \hline
Type of matrix & Operation & \multicolumn{2}{c|}{Single precision} 
& \multicolumn{2}{c||}{Double precision} \\  \cline{3-6}
and storage scheme &       & Real & Complex & Real & Complex \\  \hline
general    
& simple driver              & PSGESV\index{PSGESV}  &
PCGESV\index{PCGESV}  & PDGESV\index{PDGESV} & PZGESV\index{PZGESV} \\ 
(partial pivoting)
& expert driver              & PSGESVX\index{PSGESVX} &
PCGESVX\index{PCGESVX} & PDGESVX\index{PDGESVX} & PZGESVX\index{PZGESVX} \\ 
\hline
general band
& simple driver              & PSGBSV\index{PSGBSV} & PCGBSV\index{PCGBSV} & PDGBSV\index{PDGBSV} & PZGBSV\index{PZGBSV} \\ 
(partial pivoting) &   &  &  &  &  \\ 
\hline
general band
& simple driver              & PSDBSV\index{PSDBSV} & PCDBSV\index{PCDBSV} & PDDBSV\index{PDDBSV} & PZDBSV\index{PZDBSV} \\ 
(no pivoting) &   &  &  &  &  \\ 
\hline
general tridiagonal
& simple driver              & PSDTSV\index{PSDTSV}  & PCDTSV\index{PCDTSV} & PDDTSV\index{PDDTSV} & PZDTSV\index{PZDTSV}  \\ 
(no pivoting)
&  &  &  &  &  \\ 
\hline
symmetric/Hermitian
& simple driver              & PSPOSV\index{PSPOSV}  & PCPOSV\index{PCPOSV} & PDPOSV\index{PDPOSV} & PZPOSV\index{PZPOSV} \\ 
positive definite
& expert driver              & PSPOSVX\index{PSPOSVX} & PCPOSVX\index{PCPOSVX} & PDPOSVX\index{PDPOSVX} & PZPOSVX\index{PZPOSVX} \\ 
\hline
symmetric/Hermitian
& simple driver              & PSPBSV\index{PSPBSV}  & PCPBSV\index{PCPBSV}  & PDPBSV\index{PDPBSV}  & PZPBSV\index{PZPBSV}  \\ 
positive definite band
& & & & &  \\ 
\hline
symmetric/Hermitian
& simple driver              & PSPTSV\index{PSPTSV}  & PCPTSV\index{PCPTSV}  & PDPTSV\index{PDPTSV}  & PZPTSV\index{PZPTSV}  \\ 
positive definite &  &  &  &  &  \\ 
tridiagonal &  &  &  &  &  \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{equation}\label{llsq}
\mathop{\mbox{minimize }}_{x} \| b - A x {\|}_2 ,
\end{equation}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[htb]

\label{tabdrivellsq}
\begin{center}
\begin{tabular}{||l|l|l|l|l||} \hline
Operation & \multicolumn{2}{c|}{Single Precision} 
& \multicolumn{2}{c||}{Double Precision} \\  \cline{2-5}
                                        & Real & Complex & Real & Complex \\ 
\hline
solve LLS using $QR$ or $LQ$ factorization
 & PSGELS\index{PSGELS}  & PCGELS\index{PCGELS}  & PDGELS\index{PDGELS}  & PZGELS\index{PZGELS}  \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}Az = \lambda z, \quad A = A^T, \mbox{  where } A \mbox{ is real}.
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}Az = \lambda z, \quad A = A^H.
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = Z \Lambda Z^T ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
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\stepcounter{subsubsection}
{\newpage\clearpage\samepage
\begin{displaymath}A = U \Sigma V ^T, \quad (A=U\Sigma V ^H \quad \mbox{in the complex case}) ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
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\begin{displaymath}\sigma_1 \ge \sigma_2 \ge \ldots \sigma_{\min (m,n)} \ge 0 .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A v_i = \sigma_i u_i \quad \mbox{and} \quad
A^T u_i = \sigma_i v_i \quad ({\rm or} \quad 
A^H u_i = \sigma_i v_i ) ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{table}[htb]

\label{tabdriveseig}
\begin{center}
\begin{tabular}{||c|l|l|l|l|l||} \hline
Type of & Function & \multicolumn{2}{c|}{Single Precision} 
& \multicolumn{2}{c||}{Double Precision} \\  \cline{3-6}
Problem & 
& Real & Complex & Real & Complex \\  
\hline
SEP
& simple driver              
& PSSYEV\index{PSSYEV}  &  & PDSYEV\index{PDSYEV}  &  \\ 
& expert driver            
& PSSYEVX\index{PSSYEVX} & PCHEEVX\index{PCHEEVX} & PDSYEVX\index{PDSYEVX} & PZHEEVX\index{PZHEEVX} \\ 
\hline
SVD
& singular values/vectors
& PSGESVD\index{PSGESVD} &  & PDGESVD\index{PDGESVD} & \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[htb]

\label{tabdrivegeig}
\begin{center}
\begin{tabular}{||c|l|l|l|l|l||} \hline
Type of & Function& \multicolumn{2}{c|}{Single precision} 
& \multicolumn{2}{c||}{Double precision} \\  \cline{3-6}
problem & 
& Real & Complex & Real & Complex \\  
\hline
GSEP
& expert driver              
& PSSYGVX\index{PSSYGVX}  & PCHEGVX\index{PCHEGVX}  & PDSYGVX\index{PDSYGVX}  & PZHEGVX\index{PZHEGVX}  \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

\stepcounter{section}
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{\newpage\clearpage\samepage
\begin{equation}A x = b ,        \label{Axb1}
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}A X = B        \label{AXB}
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = PLU , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = U^{T}U \; \; {\rm or}\; \; A = LL^{T} \mbox{(in the symmetric case)}\end{displaymath}}\hbox{}\vfil

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\begin{displaymath}A = U^{H}U \; \; {\rm or}\; \; A = LL^{H} \mbox{(in the Hermitian case)} , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = PLUQ , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = PLUP^{T} , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = PU^{T}UP^{T} \; \; {\rm or}\; \; A = PLL^{T}P^{T} \mbox{(in the symmetric case)}\end{displaymath}}\hbox{}\vfil

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\begin{displaymath}A = P U^{T} D U P^{T} \; \; {\rm or}\; \; A = P L D L^{T} P^{T} \mbox{(in the symmetric case)}\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{table}[htb]

\label{tabcomplineq1}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Matrix Type & Operation & \multicolumn{2}{c|}{Single Precision} 
& \multicolumn{2}{c||}{Double Precision} \\  \cline{3-6}
and Storage &       & Real & Complex & Real & Complex \\  \hline
general    
& factorize                  & PSGETRF\index{PSGETRF} & PCGETRF\index{PCGETRF} & PDGETRF\index{PDGETRF} & PZGETRF\index{PZGETRF} \\ 
partial pivoting
& solve using fact. & PSGETRS\index{PSGETRS} & PCGETRS\index{PCGETRS} & PDGETRS\index{PDGETRS} & PZGETRS\index{PZGETRS} \\ 
& est. cond. number  & PSGECON\index{PSGECON} & PCGECON\index{PCGECON} & PDGECON\index{PDGECON} & PZGECON\index{PZGECON} \\ 
& error bds. for soln. & PSGERFS\index{PSGERFS} & PCGERFS\index{PCGERFS} & PDGERFS\index{PDGERFS} & PZGERFS\index{PZGERFS} \\ 
& invert using fact. & PSGETRI\index{PSGETRI} & PCGETRI\index{PCGETRI} & PDGETRI\index{PDGETRI} & PZGETRI\index{PZGETRI} \\ 
& equilibrate                & PSGEEQU\index{PSGEEQU} & PCGEEQU\index{PCGEEQU} & PDGEEQU\index{PDGEEQU} & PZGEEQU\index{PZGEEQU} \\ 
\hline
general band
& factorize                  & PSGBTRF\index{PSGBTRF} & PCGBTRF\index{PCGBTRF} & PDGBTRF\index{PDGBTRF} & PZGBTRF\index{PZGBTRF} \\ 
partial pivoting
& solve using fact.  & PSGBTRS\index{PSGBTRS} & PCGBTRS\index{PCGBTRS} & PDGBTRS\index{PDGBTRS} & PZGBTRS\index{PZGBTRS} \\ 
\hline
general band
& factorize                  & PSDBTRF\index{PSDBTRF} & PCDBTRF\index{PCDBTRF} & PDDBTRF\index{PDDBTRF} & PZDBTRF\index{PZDBTRF} \\ 
no pivoting
& solve using fact. & PSDBTRS\index{PSDBTRS} & PCDBTRS\index{PCDBTRS} & PDDBTRS\index{PDDBTRS} & PZDBTRS\index{PZDBTRS} \\ 
\hline
general
& factorize                  & PSDTTRF\index{PSDTTRF} & PCDTTRF\index{PCDTTRF} & PDDTTRF\index{PDDTTRF} & PZDTTRF\index{PZDTTRF} \\ 
tridiagonal & & & & & \\ 
no pivoting
& solve using fact. & PSDTTRS\index{PSDTTRS} & PCDTTRS\index{PCDTTRS} & PDDTTRS\index{PDDTTRS} & PZDTTRS\index{PZDTTRS} \\ 
\hline
symmetric/
& factorize                  & PSPOTRF\index{PSPOTRF} & PCPOTRF\index{PCPOTRF} & PDPOTRF\index{PDPOTRF} & PZPOTRF\index{PZPOTRF} \\ 
Hermitian
& solve using fact.  & PSPOTRS\index{PSPOTRS} & PCPOTRS\index{PCPOTRS} & PDPOTRS\index{PDPOTRS} & PZPOTRS\index{PZPOTRS} \\ 
positive definite
& est. cond. number  & PSPOCON\index{PSPOCON} & PCPOCON\index{PCPOCON} & PDPOCON\index{PDPOCON} & PZPOCON\index{PZPOCON} \\ 
& error bds. for soln.  & PSPORFS\index{PSPORFS} & PCPORFS\index{PCPORFS} & PDPORFS\index{PDPORFS} & PZPORFS\index{PZPORFS} \\ 
& invert using fact. & PSPOTRI\index{PSPOTRI} & PCPOTRI\index{PCPOTRI} & PDPOTRI\index{PDPOTRI} & PZPOTRI\index{PZPOTRI} \\ 
& equilibrate                & PSPOEQU\index{PSPOEQU} & PCPOEQU\index{PCPOEQU} & PDPOEQU\index{PDPOEQU} & PZPOEQU\index{PZPOEQU} \\ 
\hline
symmetric/
& factorize                  & PSPBTRF\index{PSPBTRF} & PCPBTRF\index{PCPBTRF} & PDPBTRF\index{PDPBTRF} & PZPBTRF\index{PZPBTRF} \\ 
Hermitian
& solve using fact. & PSPBTRS\index{PSPBTRS} & PCPBTRS\index{PCPBTRS} & PDPBTRS\index{PDPBTRS} & PZPBTRS\index{PZPBTRS} \\ 
positive definite & & & & & \\ 
band & & & & & \\ 
\hline
symmetric/
& factorize                  & PSPTTRF\index{PSPTTRF} & PCPTTRF\index{PCPTTRF} & PDPTTRF\index{PDPTTRF} & PZPTTRF\index{PZPTTRF} \\ 
Hermitian
& solve using fact. & PSPTTRS\index{PSPTTRS} & PCPTTRS\index{PCPTTRS} & PDPTTRS\index{PDPTTRS} & PZPTTRS\index{PZPTTRS} \\ 
positive definite & & & & & \\ 
tridiagonal & & & & & \\ 
\hline
triangular
& solve                      & PSTRTRS\index{PSTRTRS} & PCTRTRS\index{PCTRTRS} & PDTRTRS\index{PDTRTRS} & PZTRTRS\index{PZTRTRS} \\ 
& est. cond. number  & PSTRCON\index{PSTRCON} & PCTRCON\index{PCTRCON} & PDTRCON\index{PDTRCON} & PZTRCON\index{PZTRCON} \\ 
& error bds. for soln. & PSTRRFS\index{PSTRRFS} & PCTRRFS\index{PCTRRFS} & PDTRRFS\index{PDTRRFS} & PZTRRFS\index{PZTRRFS} \\ 
& invert                     & PSTRTRI\index{PSTRTRI} & PCTRTRI\index{PCTRTRI} & PDTRTRI\index{PDTRTRI} & PZTRTRI\index{PZTRTRI} \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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\begin{displaymath}A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
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\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
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A^T = Q^T \left( \begin{array}{c} L^T \\ 0\end{array}\right) .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}x = Q^T \left( \begin{array}{c} L^{-1} b \\  0 \end{array} \right)
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{\newpage\clearpage\samepage
\begin{displaymath}|r_{11}| \ge |r_{22}| \ge \ldots \ge |r_{nn}|
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}x = P \left( \begin{array}{c} R_{11}^{-1} \hat{c}_1 \\  0 \end{array} \right),
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{\newpage\clearpage\samepage
\begin{displaymath}\left( \begin{array}{cc}R_{11} & R_{12}\end{array}\right) Z = 
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{\newpage\clearpage\samepage
\begin{displaymath}A P = Q \left( 
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{cc} 0 & R \end{array} \right) Q, 
\quad \mbox{if $m \leq n$.}
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{\newpage\clearpage\samepage
\begin{table}[htb]

\label{tabcompof}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Type of Fact. & Operation & \multicolumn{2}{c|}{Single Precision} 
& \multicolumn{2}{c||}{Double Precision} \\  \cline{3-6}
and Matrix             &       & Real & Complex & Real & Complex \\  \hline

$QR$, general
& fact. with pivoting    & PSGEQPF\index{PSGEQPF} & PCGEQPF\index{PCGEQPF} & PDGEQPF\index{PDGEQPF} & PZGEQPF\index{PZGEQPF} \\     
& fact., no pivoting     & PSGEQRF\index{PSGEQRF} & PCGEQRF\index{PCGEQRF} & PDGEQRF\index{PDGEQRF} & PZGEQRF\index{PZGEQRF} \\ 
& generate $Q$               & PSORGQR\index{PSORGQR} & PCUNGQR\index{PCUNGQR} & PDORGQR\index{PDORGQR} & PZUNGQR\index{PZUNGQR} \\ 
& multiply by $Q$     & PSORMQR\index{PSORMQR} & PCUNMQR\index{PCUNMQR} & PDORMQR\index{PDORMQR} & PZUNMQR\index{PZUNMQR} \\  
\hline
$LQ$, general
& fact., no pivoting     & PSGELQF\index{PSGELQF} & PCGELQF\index{PCGELQF} & PDGELQF\index{PDGELQF} & PZGELQF\index{PZGELQF} \\ 
& generate $Q$               & PSORGLQ\index{PSORGLQ} & PCUNGLQ\index{PCUNGLQ} & PDORGLQ\index{PDORGLQ} & PZUNGLQ\index{PZUNGLQ} \\ 
& multiply by $Q$     & PSORMLQ\index{PSORMLQ} & PCUNMLQ\index{PCUNMLQ} & PDORMLQ\index{PDORMLQ} & PZUNMLQ\index{PZUNMLQ} \\  
\hline
$QL$, general    
& fact., no pivoting     & PSGEQLF\index{PSGEQLF} & PCGEQLF\index{PCGEQLF} & PDGEQLF\index{PDGEQLF} & PZGEQLF\index{PZGEQLF} \\ 
& generate $Q$               & PSORGQL\index{PSORGQL} & PCUNGQL\index{PCUNGQL} & PDORGQL\index{PDORGQL} & PZUNGQL\index{PZUNGQL} \\ 
& multiply by $Q$     & PSORMQL\index{PSORMQL} & PCUNMQL\index{PCUNMQL} & PDORMQL\index{PDORMQL} & PZUNMQL\index{PZUNMQL} \\  
\hline
$RQ$, general    
& fact., no pivoting     & PSGERQF\index{PSGERQF} & PCGERQF\index{PCGERQF} & PDGERQF\index{PDGERQF} & PZGERQF\index{PZGERQF} \\ 
& generate $Q$               & PSORGRQ\index{PSORGRQ} & PCUNGRQ\index{PCUNGRQ} & PDORGRQ\index{PDORGRQ} & PZUNGRQ\index{PZUNGRQ} \\ 
& multiply by $Q$     & PSORMRQ\index{PSORMRQ} & PCUNMRQ\index{PCUNMRQ} & PDORMRQ\index{PDORMRQ} & PZUNMRQ\index{PZUNMRQ} \\  
\hline
$RZ$, trapezoidal
& fact., no pivoting     & PSTZRZF\index{PSTZRZF} & PCTZRZF\index{PCTZRZF} & PDTZRZF\index{PDTZRZF} & PZTZRZF\index{PZTZRZF} \\ 
& multiply by $Z$     & PSORMRZ\index{PSORMRZ} & PCUNMRZ\index{PCUNMRZ} & PDORMRZ\index{PDORMRZ} & PZUNMRZ\index{PZUNMRZ} \\  
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
\stepcounter{subsubsection}
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}T = \bordermatrix{    & p      &  n-p    \cr
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\begin{displaymath}A B^{-1} = ( R T^{-1} ) Z^T 
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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$BQ^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline13821: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{table}[tbhp]

\label{tabcompeig}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Type of Matrix & Operation & \multicolumn{2}{c|}{Single Precision} &
\multicolumn{2}{c||}{Double Precision}  \\  \cline{3-6}
		 & & Real & Complex & Real & Complex \\  \hline
dense symmetric & tridiagonal & PSSYTRD\index{PSSYTRD} & PCHETRD\index{PCHETRD} & PDSYTRD\index{PDSYTRD} & PZHETRD\index{PZHETRD} \\ 
(or Hermitian)  & reduction & & & &  \\  \hline
orthogonal/ & multiply matrix & PSORMTR\index{PSORMTR} & PCUNMTR\index{PCUNMTR} & PDORMTR\index{PDORMTR} & PZUNMTR\index{PZUNMTR} \\ 
unitary & after reduction & & & & \\ 
        & by PxSYTRD & & & & \\  \hline
symmetric             & eigenvalues/ & SSTEQR2\index{SSTEQR2} &  & DSTEQR2\index{DSTEQR2} &  \\ 
tridiagonal           & eigenvectors via & & & & \\ 
                      & look-ahead QR & & & & \\ 
symmetric              & eigenvalues only  & PSSTEBZ\index{PSSTEBZ} & & PDSTEBZ\index{PDSTEBZ} & \\ 
tridiagonal            & via bisection & & & & \\ 
                      & eigenvectors by & PSSTEIN\index{PSSTEIN} & PCSTEIN\index{PCSTEIN} & PDSTEIN\index{PDSTEIN} & PZSTEIN\index{PZSTEIN} \\ 
                      & inverse iteration & & & & \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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\stepcounter{subsubsection}
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{table}[htbp]

\label{tabcompeig2}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Type of & Operation & \multicolumn{2}{c|}{Single Precision} &
\multicolumn{2}{c||}{Double Precision}  \\  \cline{3-6}
Matrix & & Real & Complex & Real & Complex \\  \hline
general & Hessenberg reduction & PSGEHRD\index{PSGEHRD} & PCGEHRD\index{PCGEHRD} &PDGEHRD\index{PDGEHRD} & PZGEHRD\index{PZGEHRD} \\ 
orthogonal/ & multiply matrix after & PSORMHR\index{PSORMHR} & PCUNMHR\index{PCUNMHR} & PDORMHR\index{PDORMHR} & PZUNMHR\index{PZUNMHR} \\ 
unitary & Hessenberg reduction & & & & \\  \hline
Hessenberg & eigenvalues and & PSLAHQR\index{PSLAHQR} & & PDLAHQR\index{PDLAHQR} &  \\ 
           & Schur decomposition &  & &  &  \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$m \ll n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14233: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{\Sigma}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14247: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}x = \hat{V} \hat{\Sigma}^{-1} \hat{c}_1 ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$c = U^T b = 
U_{2}^T U_{1}^T b$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14265: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[htbp]

\label{tabcompsvd}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Type of & Operation & \multicolumn{2}{c|}{Single Precision} &
\multicolumn{2}{c||}{Double Precision}  \\  \cline{3-6}
Matrix & & Real & Complex & Real & Complex \\  \hline
general & bidiagonal reduction & PSGEBRD\index{PSGEBRD} & PCGEBRD\index{PCGEBRD} &PDGEBRD\index{PDGEBRD} & PZGEBRD\index{PZGEBRD} \\  \hline
orthogonal/ & multiply matrix after & PSORMBR\index{PSORMBR} & PCUNMBR\index{PCUNMBR} & PDORMBR\index{PDORMBR} & PZUNMBR\index{PZUNMBR} \\ 
unitary & bidiagonal reduction & & & & \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
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{\newpage\clearpage\samepage
\begin{displaymath}Az=\lambda Bz \quad \Rightarrow \quad (L ^{-1}AL ^{-T})(L^Tz)= \lambda(L^Tz).
 \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$Cy=\lambda y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14329: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$y =
 L^H z$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14341: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[thb]
 
 \label{tabgst}
 \begin{center}
 \begin{tabular}{||l|c|l|l|l||} \hline
 & Type of    & \multicolumn{1}{c|}{Factorization} & 
 \multicolumn{1}{c|}{Reduction} & \multicolumn{1}{c||}{Recovery of} \\ 
 & Problem    & \multicolumn{1}{c|}{of $B$}    &           & 
 \multicolumn{1}{c||}{Eigenvectors} \\  
 \hline
 1. & $Az = \lambda Bz$ & $B = LL^T$ & $C = L^{-1} A L^{-T}$ & $z = L^{-T} y$ \\ 
    &                   & $B = U^TU$ & $C = U^{-T} A U^{-1}$ & $z = U^{-1} y$ \\ 
 \hline
 2. & $ABz = \lambda z$ & $B = LL^T$ & $C = L^T A L$         & $z = L^{-T} y$ \\ 
    &                   & $B = U^TU$ & $C = U A U^T$         & $z = U^{-1} y$ \\ 
 \hline
 3. & $BAz = \lambda z$ & $B = LL^T$ & $C = L^T A L$         & $z = L y$ \\ 
    &                   & $B = U^TU$ & $C = U A U^T$         & $z = U^T y$ \\ 
 \hline
 \end{tabular}
 \end{center}
 \end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[tbh]

\label{tabcompgeig}
\begin{center}
\begin{tabular}{||l|l|l|l|l|l||} \hline
Type of Matrix & Operation & \multicolumn{2}{c|}{Single Precision} &
\multicolumn{2}{c||}{Double Precision}  \\  \cline{3-6}
 & & Real & Complex & Real & Complex  \\  \hline
symmetric/Hermitian & reduction  & PSSYGST\index{PSSYGST} & PCHEGST\index{PCHEGST} & PDSYGST\index{PDSYGST} & PZHEGST\index{PZHEGST} \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}Q = H_{1} H_{2} \ldots H_{k}. \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}H = I - \tau v v^{H}  , \label{Hdef}
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
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\begin{figure}% latex2html id marker 2256
[htp]
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\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htbp]
 \centering
 
 \begin{tabular}{||l|l||}\hline
  Scope & Meaning\\ \hline
  Row    & All processes in a process row participate.\\ \hline
  Column & All processes in a process column participate.\\ \hline
  All    & All processes in the process grid participate.\\ \hline
 \end{tabular}
\end{table}}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \begin{center}
  \begin{tabular}{||l | l||} \hline
DESC\_( DTYPE\_ ) & Designation \\  \hline
1             & dense matrices \\ 
501           & narrow band and tridiagonal coefficient matrices \\ 
502           & narrow band and tridiagonal right-hand-side matrices \\ 
601           & out-of-core dense matrices \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{figure}[thb]
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           \label{fig:gauss}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[h]
 \centerline{\hbox{\psfig{figure=FIG/bcol.ps,height=1.4in,width=1.4in}
 \hspace*{1.0in} \psfig{figure=FIG/cycliccol.ps,height=1.4in,width=1.4in}}}
  
                                                          \label{fig:bcol}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lfloor k / 
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{\newpage\clearpage\samepage
\begin{figure}[h]
 \centerline{\hbox{\psfig{figure=FIG/bcycliccol.ps,height=1.4in,width=1.4in}
  \hspace*{1.0in} \psfig{figure=FIG/bcyclicroc.ps,height=1.4in,width=1.4in}}}
    \label{fig:bcycliccol}
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{\newpage\clearpage\samepage
\begin{displaymath}I = k~NB + x = ( l~P + p ) * NB + x,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}\label{eqn:indexmap}
 \displaystyle p = \lfloor (I-1) / NB \rfloor~{\rm mod}~P,~
               l = \lfloor (I-1) / ( P \ast NB ) \rfloor,~{\rm and}~
               x = {\rm mod}( I-1, NB ) + 1.
\end{equation}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$l \ast NB + x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14721: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]
 
                                                          \label{tab:bmap}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$I$    & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$    & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 
$l$    & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
$x$    & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\  \hline
$l*NB+x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 
       \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]
  \label{tab:cmap}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$I$   & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$   & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 
$l$   & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ 
$x$   & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  \hline
$l*NB+x$ & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 & 8 & 8 \\ 
         \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]
                            \label{tab:bcmap}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$I$    & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$    & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 
$l$    & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 \\ 
$x$    & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 \\  \hline
$l*NB+x$ & 1 & 2 & 3 & 1 & 2 & 3 & 4 & 5 & 6 & 4 & 5 & 6 & 7 & 8 & 9 & 7 \\ 
        \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}\label{eqn:indexmap1}
 \left\{
  \begin{array}{l}
   \displaystyle p = ( SRC + \lfloor (I-1) / NB \rfloor )~{\rm mod}~P, \\ 
   \displaystyle l = \lfloor (I-1) / ( P \ast NB ) \rfloor,  \\ 
   \displaystyle x = {\rm mod}( I-1, NB ) + 1.
  \end{array}
 \right.
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$P~=~2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14791: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[h]
                \label{tab:bcmap0}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$I$    & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$    & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 
$l$    & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 \\ 
$x$    & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 \\  \hline
$l*NB+x$ & 1 & 2 & 3 & 1 & 2 & 3 & 4 & 5 & 6 & 4 & 5 & 6 & 7 & 8 & 9 & 7 \\ 
        \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$MB \times NB$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14817: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\left\{
  \begin{array}{lcl}
   ( l, m )     & = & \displaystyle
          ( \lfloor (I-1) / ( P_r \ast MB ) \rfloor,
            \lfloor (J-1) / ( P_c \ast NB ) \rfloor ), \\ 
   ( p_r, p_c ) & = & \displaystyle
          ( ( RSRC + \lfloor (I-1) / MB \rfloor )~{\rm mod}~P_{r},
            ( CSRC + \lfloor (J-1) / NB \rfloor )~{\rm mod}~P_{c} ), \\ 
   ( x, y )     & = & \displaystyle
          ( {\rm mod}( I-1, MB ) + 1, {\rm mod}( J-1, NB ) + 1 ).
  \end{array}
 \right.
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$M\_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14831: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$N\_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14833: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$MB\_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14837: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$NB\_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14839: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$~\times~$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14845: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$M\_A=N\_A=5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14851: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$MB\_A=NB\_A=2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14855: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$5 \times 5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14857: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$2 \times 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14859: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[htb]
 \centerline{\psfig{figure=mat5.ps,height=1.8in,width=4.0in}}
 
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$a_{25}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14869: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}LOC_{r}() \approx \displaystyle \frac{\displaystyle
           \frac{M\_A + MB\_A - 1}{MB\_A}+P_{r}-1}{P_{r}} \ast MB\_A
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}LOC_{r}() \approx \lceil \lceil  M\_A / MB\_A \rceil / P_{r} \rceil
           \ast MB\_A.
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}LOC_{c}() \approx \displaystyle \frac{\displaystyle 
            \frac{N\_A + NB\_A - 1}{NB\_A}+ P_{c} -1}{P_{c}} \ast NB\_A
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}LOC_{c}() \approx \lceil \lceil  N\_A / NB\_A \rceil / P_{c} \rceil
           \ast NB\_A.
\end{displaymath}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \begin{center}
  \begin{tabular}{||l | l | l | l||} \hline
DESC\_() & Symbolic & Scope    & Definition \\ 
   & Name     &          &              \\  \hline
1  & DTYPE\_A & (global) & Descriptor type DTYPE\_A=1 for \\ 
   &          &          & dense matrices. \\ 
2  & CTXT\_A  & (global) & BLACS context handle, indicating the \\ 
   &          &          & BLACS process grid over which the global \\ 
   &          &          & matrix A is distributed. The context itself\\ 
   &          &          & is global, but the handle (the integer value)\\ 
   &          &          & may vary. \\ 
3  & M\_A     & (global) & Number of rows in the global array A. \\ 
4  & N\_A     & (global) & Number of columns in the global array A.\\ 
5  & MB\_A    & (global) &  Blocking factor used to distribute \\ 
   &          &          & the rows of the array. \\ 
6  & NB\_A    & (global) & Blocking factor used to distribute \\ 
   &          &          & the columns of the array. \\ 
7  & RSRC\_A  & (global) & Process row over which the first row \\ 
   &          &          & of the array A is distributed. \\ 
8  & CSRC\_A  & (global) & Process column over which the first \\ 
   &          &          & column of the array A is distributed. \\ 
9  & LLD\_A   & (local)  & Leading dimension of the local \\ 
   &          &          & array.  LLD\_A $\geq$ MAX(1,LOC$_{r}$(M\_A)).\\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$2 \times 3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline14922: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[htb]
 \centerline{\hbox{\psfig{figure=mat9.ps,height=1.6in,width=1.8in}
 \hspace*{1.0in} \psfig{figure=grid9.ps,height=1.6in,width=1.8in}}}
 
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                 \label{tableofldas}
 \begin{center}
  \begin{tabular}{||c|c|c|c||} \hline
   Process Coordinates & LLD\_ & LOC$_{r}$(M\_) & LOC$_{c}$(N\_) \\  \hline
   (0,0)                & 5     & 5    & 4    \\ 
   (0,1)                & 5     & 5    & 3    \\ 
   (0,2)                & 5     & 5    & 2    \\ 
   (1,0)                & 4     & 4    & 4    \\ 
   (1,1)                & 4     & 4    & 3    \\ 
   (1,2)                & 4     & 4    & 2    \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{figure}[htb]
 \centerline{\psfig{figure=FIG/view.ps,height=1.0in,width=2.0in}}
                 \label{fig:view}
\end{figure}}\hbox{}\vfil


\setlength{\labelwidth}{0.5in}%


\setlength{\labelsep}{0.1in}%


\setlength{\leftmargin}{0.6in}%


\setlength{\parsep}{0in}%


\setlength{\topsep}{0in}%

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\stepcounter{subsection}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{figure}[h]
 \centerline{\hbox{\psfig{figure=FIG/blockcol.ps,height=1.4in,width=1.4in}
 \hspace*{1.0in} \psfig{figure=FIG/blockrow.ps,height=1.4in,width=1.4in}}}
 
                                                  \label{fig:block}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$1 \times P$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15088: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$NB \ge \lceil N / P \rceil$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15090: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lfloor k / NB \rfloor$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15094: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lceil N / P \rceil$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15096: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$M \times N$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15127: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$p^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15153: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}J = p~NB + x,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}\label{eqn:indexblockmap}
 \displaystyle p = \lfloor (J-1) / NB \rfloor,~
               x = {\rm mod}( J-1, NB ) + 1.
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]
 
                                                         \label{tab:blockmap}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$J$  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 
$x$  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$N \times NRHS$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15189: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}\label{eqn:indexblockmap1}
 \left\{
  \begin{array}{l}
   \displaystyle p = ( SRC + \lfloor (J-1) / NB \rfloor ), \\ 
   \displaystyle x = {\rm mod}( J-1, NB ) + 1.
  \end{array}
 \right.
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]
                           \label{tab:bcblockmap0}
 \begin{center}
  \begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$J$  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12 &13 &14 &15 &16 \\  \hline
$p$  & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
$x$  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$1 \times P_{c}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15233: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\frac{N\_A}{NB\_A} = P_{c}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15237: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$mod(N\_A,NB\_A) \neq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15245: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$int(\frac{N\_A}{NB\_A})=K$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15251: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$(0,0),\ldots,(0,K-1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15253: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$mod(M\_B,MB\_B) \neq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15287: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$int(\frac{M\_B}{MB\_B})=K$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15293: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$(0,0),\ldots,(K-1,0)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15295: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$K < P_{r}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15307: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$(K+1,0),\ldots,(P_{r}-1,0)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15309: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$7 \times 7$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15332: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$N\_A=7$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccccccc}
      a_{11} & a_{12} & a_{13} & 0 & 0 & 0 & 0 \\ 
      a_{21} & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 \\ 
      a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & 0 & 0 \\ 
         0   & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} & 0 \\ 
         0   &    0   & a_{53} & a_{54} & a_{55} & a_{56} & a_{57} \\ 
         0   &    0   &   0    & a_{64} & a_{65} & a_{66} & a_{67} \\ 
         0   &    0   &   0    &   0    & a_{75} & a_{76} & a_{77} \\ 
       \end{array}\right)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$NB\_A~=~3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15352: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{ccc|ccc|c}
   \multicolumn{7}{c}{Processes} \\ 
   \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
   $\ast$   &$\ast$   &$a_{13}$ & $a_{24}$ &$a_{35}$ &$a_{46}$ & $a_{57}$ \\ 
   $\ast$   &$a_{12}$ &$a_{23}$ & $a_{34}$ &$a_{45}$ &$a_{56}$ & $a_{67}$ \\ 
   $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
   $a_{21}$ &$a_{32}$ &$a_{43}$ & $a_{54}$ &$a_{65}$ &$a_{76}$ &   $\ast$ \\ 
   $a_{31}$ &$a_{42}$ &$a_{53}$ & $a_{64}$ &$a_{75}$ & $\ast$  &   $\ast$ \\ 
  \end{tabular}
 \end{center}
                            \label{fig:nonsymmbandnp7}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$NB\_A=3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline15430: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{ccc|ccc|c}
   \multicolumn{7}{c}{Processes} \\ 
   \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
      F  &  F      &  F      &  F       &  F      &  F      &  F \\ 
      F  &  F      &  F      &  F       &  F      &  F      &  F \\ 
      F  &  F      &  F      &  F       &  F      &  F      &  F \\ 
      F  &  F      &  F      &  F       &  F      &  F      &  F \\ 
 $\ast$  &$\ast$   &$a_{13}$ & $a_{24}$ &$a_{35}$ &$a_{46}$ & $a_{57}$ \\ 
 $\ast$  &$a_{12}$ &$a_{23}$ & $a_{34}$ &$a_{45}$ &$a_{56}$ & $a_{67}$ \\ 
$a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
$a_{21}$ &$a_{32}$ &$a_{43}$ & $a_{54}$ &$a_{65}$ &$a_{76}$ &   $\ast$ \\ 
$a_{31}$ &$a_{42}$ &$a_{53}$ & $a_{64}$ &$a_{75}$ & $\ast$  &   $\ast$ \\ 
  \end{tabular}
 \end{center}
                  \label{fig:nonsymmbandpp7}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{ccc|ccc|c}
   \multicolumn{7}{c}{Processes} \\ 
   \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
   $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
   $a_{21}$ &$a_{32}$ &$a_{43}$ & $a_{54}$ &$a_{65}$ &$a_{76}$ &   $\ast$ \\ 
   $a_{31}$ &$a_{42}$ &$a_{53}$ & $a_{64}$ &$a_{75}$ & $\ast$  &   $\ast$ \\ 
  \end{tabular}
 \end{center}
                    \label{fig:Lsymmband7}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{ccc|ccc|c}
   \multicolumn{7}{c}{Processes} \\ 
   \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
    $\ast$  & $\ast$  &$a_{31}$ & $a_{42}$ &$a_{53}$ &$a_{64}$ & $a_{75}$ \\ 
    $\ast$  &$a_{21}$ &$a_{32}$ & $a_{43}$ &$a_{54}$ &$a_{65}$ & $a_{76}$ \\ 
   $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
  \end{tabular}
 \end{center}
                     \label{fig:Usymmband7}
\end{figure}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccccccc}
      a_{11} & a_{12} &    0   &    0   &    0   &    0   &    0 \\ 
      a_{21} & a_{22} & a_{23} &    0   &    0   &    0   &    0 \\ 
         0   & a_{32} & a_{33} & a_{34} &    0   &    0   &    0 \\ 
         0   &    0   & a_{43} & a_{44} & a_{45} &    0   &    0 \\ 
         0   &    0   &    0   & a_{54} & a_{55} & a_{56} &    0 \\ 
         0   &    0   &    0   &   0    & a_{65} & a_{66} & a_{67} \\ 
         0   &    0   &    0   &   0    &   0    & a_{76} & a_{77} \\ 
       \end{array}\right)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{c|ccc|ccc|c}
   & \multicolumn{7}{c}{Processes} \\ 
   & \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
DL & $\ast$   &$a_{21}$ &$a_{32}$ & $a_{43}$ &$a_{54}$ &$a_{65}$ & $a_{76}$ \\ 
D  & $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
DU & $a_{12}$ &$a_{23}$ &$a_{34}$ & $a_{45}$ &$a_{56}$ &$a_{67}$ & \\ 
  \end{tabular}
 \end{center}
                                    \label{fig:tridiagnp7}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{c|ccc|ccc|c}
   & \multicolumn{7}{c}{Processes} \\ 
   & \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
 D & $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
 E & $a_{21}$ &$a_{32}$ &$a_{43}$ & $a_{54}$ &$a_{65}$ &$a_{76}$ &  \\ 
  \end{tabular}
 \end{center}
                 \label{fig:Lbidiag7}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{figure}[thb]
 \begin{center}
  \begin{tabular}{c|ccc|ccc|c}
   & \multicolumn{7}{c}{Processes} \\ 
   & \multicolumn{3}{c}{0} & \multicolumn{3}{c}{1} & 2 \\  \hline
 D & $a_{11}$ &$a_{22}$ &$a_{33}$ & $a_{44}$ &$a_{55}$ &$a_{66}$ & $a_{77}$ \\ 
 E & $a_{12}$ &$a_{23}$ &$a_{34}$ & $a_{45}$ &$a_{56}$ &$a_{67}$ &  \\ 
  \end{tabular}
 \end{center}
                 \label{fig:Ubidiag7}
\end{figure}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \begin{center}
  \begin{tabular}{||l | l | l | l||} \hline
DESC\_() & Symbolic & Scope    & Definition \\ 
         & Name     &          &              \\  \hline
1        & DTYPE\_A & (global) & The descriptor type (DTYPE\_A=501) \\ 
         &          &          & for $1 \times P_{c}$ process grid for \\ 
         &          &          & band and tridiagonal matrices \\ 
         &          &          & block-column distributed.\\ 
2        & CTXT\_A  & (global) & The BLACS context handle, indicating the \\ 
         &          &          & BLACS process grid over which the global \\ 
         &          &          & matrix A is distributed. The context itself\\ 
         &          &          & is global, but the handle (the integer value)\\ 
         &          &          & may vary. \\ 
3        & N\_A     & (global) & The number of columns in the global matrix A.\\ 
4        & NB\_A    & (global) & The column block size. \\ 
5        & CSRC\_A  & (global) & The process column over which the first \\ 
         &          &          & column of the global matrix A is distributed.\\ 
6        & LLD\_A   & (local)  & The leading dimension of the local \\ 
         &          &          & array.  For the tridiagonal \\ 
         &          &          & subroutines, this entry is ignored. \\ 
7        &          &          & Unused, reserved  \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \begin{center}
 \begin{tabular}{||l | l | l | l||} \hline
DESC\_() & Symbolic & Scope    & Definition \\ 
         & Name     &          &              \\  \hline
1        & DTYPE\_B & (global) & The descriptor type (DTYPE\_B=502) \\ 
         &          &          & for $P_{r} \times 1$ process grid for \\ 
         &          &          & block-row distributed matrices.\\ 
2        & CTXT\_B  & (global) & The BLACS context handle, indicating the \\ 
         &          &          & BLACS process grid over which the global \\ 
         &          &          & matrix B is distributed. The context itself\\ 
         &          &          & is global, but the handle (the integer value)\\ 
         &          &          & may vary. \\ 
3        & M\_B     & (global) & The number of rows in the global matrix B.\\ 
4        & MB\_B    & (global) & The row block size. \\ 
5        & RSRC\_B  & (global) & The process row over which the first \\ 
         &          &          & row of the global matrix B is distributed. \\ 
6        & LLD\_B   & (local)  & The leading dimension of the local \\ 
         &          &          & array.  LLD\_B $\geq$ MAX(1,LOCr(M\_B)). \\ 
         &          &          & For the tridiagonal subroutines, this \\ 
         &          &          & entry is ignored. \\ 
7        &          &          & Reserved \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}

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{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \begin{center}
  \begin{tabular}{||l | l | l | l||} \hline
DESC\_() & Symbolic & Scope    & Definition \\ 
      & Name     &          &              \\  \hline
1     & DTYPE\_A & (global) & Descriptor type DTYPE\_A=601 \\ 
      &          &          &  for an out-of-core matrix. \\ 
2     & CTXT\_A  & (global) & BLACS context handle, indicating the \\ 
      &          &          & $MP \times NQ$ BLACS process grid over  \\ 
      &          &          & which the global matrix A is distributed.\\ 
      &          &          & Context itself is global,\\ 
      &          &          & but the handle (the integer value) may vary. \\ 
3     & M\_A     & (global) & Number of rows in the global array A. \\ 
4     & N\_A     & (global) & Number of columns in the global array A.\\ 
5     & MB\_A    & (global) & Blocking factor used to distribute \\ 
      &          &          & the rows of the $MMB \times NNB$ submatrix. \\ 
6     & NB\_A    & (global) & Blocking factor used to distribute \\ 
      &          &          & the columns of the $MMB \times NNB$ submatrix. \\ 
7     & RSRC\_A  & (global) & Process row over which the first row \\ 
      &          &          & of the array A is distributed. \\ 
8     & CSRC\_A  & (global) & Process column over which the first \\ 
      &          &          & column of the array A is distributed.  \\ 
9     & LLD\_A   & (global) & The conceptual leading dimension of the global \\ 
      &          &          & array.\\ 
10    & IODEV\_A & {global} & I/O unit device number associated with \\ 
      &          &          & the out-of-core matrix A. \\ 
11    & SIZE\_A  & {local}  & Amount of local in-core memory available for \\ 
      &          &          & the factorization of A. \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{tabular}{ll}
 Value & Meaning \\  \hline
 {\tt `L'} & Multiply general distributed matrix by symmetric or triangular \\ 
           & distributed matrix on the left. \\ 
 {\tt `R'} & Multiply general distributed matrix by symmetric or triangular \\ 
           & distributed matrix on the right. \\ 
 \end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{ll}
 Value & Meaning \\  \hline
 {\tt `N'} & Operate with the distributed matrix. \\ 
 {\tt `T'} & Operate with the transpose of the distributed matrix. \\ 
 {\tt `C'} & Operate with the conjugate transpose of the distributed matrix. \\ 
 \end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{ll}
 Value & Meaning \\  \hline
 {\tt `U'} & Upper triangle \\ 
 {\tt `L'} & Lower triangle \\ 
 \end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{ll}
 Value & Meaning \\  \hline
 {\tt `U'} & Unit triangular \\ 
 {\tt `R'} & Nonunit triangular \\ 
 \end{tabular}}\hbox{}\vfil


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\stepcounter{subsection}
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\stepcounter{chapter}
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\setbox\sizebox=\hbox{$P=M \times N / 1000000$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16164: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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\stepcounter{subsection}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$C_f$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16191: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$C \leftarrow \alpha A B + \beta C$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16223: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$C_v\,N^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16256: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\stepcounter{subsubsection}
{\newpage\clearpage\samepage
\begin{displaymath}E(N,P)= {1\over{P}}{{T_{\rm seq}(N)}\over{T(N,P)}},
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$T_{\rm
seq}(N)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$_{\rm seq}()$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16302: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{figure}[h]
   \centerline{\psfig{file=iso_lu.ps,height=2.5in,width=4.0in}}
    \label{fig:iso_pgon}
\end{figure}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}% latex2html id marker 3694
[h]
  \label{tab:vars}
 \begin{center}
  \begin{tabular}{||l|l|l||} \hline
	Variable & Description & Details \\  \hline
	$C_f\,N^3$\index{C$_f$} & Total number of floating-point operations
	                        & Table~\ref{standardflopcount}\\  \hline
	$C_v\,N^2 / \sqrt{P}$\index{C$_v$} & Total number of data items
        communicated & Table~\ref{standardflopcount}\\  \hline
	$C_m\,N / NB$\index{C$_m$}& Total number of messages 
	                   & Table~\ref{standardflopcount}\\  \hline
	$t_f$\index{t$_f$} & Time per floating-point operation 
	(typically $F_{mm}$) & Table~\ref{tab:blacs} \\  \hline
	$t_v$\index{t$_v$} & Time per data item communicated
	 & Table~\ref{tab:blacs} \\  \hline
	$t_m$\index{t$_m$} & Time per message
	 & Table~\ref{tab:blacs} \\  \hline
	$N$ 	& Matrix size 
	& -- \\  \hline
	$P$	& Number of processors
	& -- \\  \hline
	$NB$	& Data distribution block size
	& -- \\  \hline
	$T()$\index{T()} & Parallel execution time (estimated)
	 & Equation~\ref{eqn:tim} \\  \hline
	$T_{\rm seq}()$\index{T$_{\rm seq}()$} & Serial execution time (estimated)
	 &  Equation~\ref{eqn:tim} \\  \hline
	$E()$\index{E()} & Efficiency (estimated) 
	 &  Equation~\ref{eqn:eff} \\  \hline
	$F_{MM}$\index{F$_{MM}$} & Time per floating-point operation in matrix-matrix
	multiply & Table~\ref{tab:blacs} \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}T( N, P ) = \frac{C_f\,N^3}{P}\,t_f + \frac{C_v\,N^2}{\sqrt{P}}\,t_v +
 \frac{C_m\,N}{NB}\,t_m, \qquad T_{seq}( N, P ) = C_f\,N^3\,t_f.
                                                           \label{eqn:tim}
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{equation}E( N, P ) = \left( 1 + \frac{1}{NB}\,\frac{C_m\,t_m}{C_f\,t_f}\,\frac{P}{N^2} +
                   \frac{C_v\,t_v}{C_f\,t_f}\,\frac{\sqrt{P}}{N} \right) ^{-1}.
                                                           \label{eqn:eff}
\end{equation}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$(t_m / t_f)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16374: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
 \label{tab:node}
 \begin{center}
  \begin{tabular}{||l|c|c||} \hline
               & Cray T3E & IBM SP2 \\  \hline
Processor	& Dec Alpha EV5 & POWER2 590 \\  \hline
Clock speed (MHz) & 300		& 66	     \\  \hline
Processors per node & 1		& 1	     \\  \hline
Memory per node (MB) & 256	& 128	      \\  \hline
Operating system & UNICOS/mk 1.4.1 & AIX 4.1.4     \\  \hline
BLAS		& LIBSCI 3.0    & ESSL 2.2.2.2  \\  \hline
BLACS		& MPI BLACS $1.1\alpha$	& MPI BLACS $1.1\alpha$	\\  \hline
Communication Software & Cray MPI & POE (2.1.0.12) \\  \hline
C compiler	& cc    	& mpcc (3.1.4.3)  \\  \hline
C flags		& -O3           & -qarch=pwr2 -qtune=pwr2s -O2 \\  \hline
Fortran compiler& f90           & mpxlf (3.2.4.2) \\  \hline
Fortran flags	& -O3           & -qarch=pwr2 -qtune=pwr2s -O2 \\  \hline
Precision	& single (64-bit)& double (64-bit) \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
  \label{tab:node2}
 \begin{center}
  \begin{tabular}{||l|c|c||} \hline
  		& Intel XP/S & Intel ASCI Red \\ 
		& MP Paragon & Supercomputer \\  \hline 
Processor	& i860 XP    &  Intel Pentium Pro \\  \hline
Clock speed (MHz) &  50	     &  200 \\  \hline
Processors per node &  2     &  2 \\  \hline
Memory per node (MB) &  64   &  128 \\  \hline
Operating system & R1.4-2    & Alpha Release \\  \hline
BLAS		&  Basic Math Library & Alpha Release \\ 
    		&  Software (Release 5.0)&  \\  \hline
BLACS		&  NX BLACS 1.0	& NX BLACS 1.0 \\  \hline
Communication Software & NX     & NX on Portals \\  \hline
C compiler	& icc (R5.0.2)  & pgcc Rel 1.3-4a \\  \hline
C flags		&  -O4  	& -O \\  \hline
Fortran compiler& if77 (R5.0.2) & pgf77 Rel 1.3-4a \\  \hline
Fortran flags	& -O4       	& -O \\  \hline
Precision	& double (64-bit) &double (64-bit)\\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
 
  \label{tab:node3}
 \begin{center}
  \begin{tabular}{||l|c|c||} \hline
  		& Cluster of & Berkeley \\ 
		& Sun Ultra 2's & NOW \\  \hline 
Processor	& UltraSPARC-1 &  UltraSPARC-1 \\  \hline
Clock speed (MHz) &  167    &  167 \\  \hline
Processors per node &   2      &  1 \\  \hline
Memory per node (MB) &  256    &  128 \\  \hline
Operating system &  SunOS 2.5.1 & SunOS 2.5.1 \\  \hline
BLAS		&       & Sun Performance Library v1.2\\  \hline
BLACS		&   $1.1\alpha$ & $1.1\alpha$ \\  \hline
Communication Software &  MPICH 1.0.13 & MPI on AM-II \\  \hline
C compiler	&   gcc (v2.7.2.1) & cc (Sun Workshop 4.2)\\  \hline
C flags		&   -O3          & -dalign -xarch=v8plusa\\ 
       		&                & -xchip=ultra -fast\\ 
       		&                & -xtarget=native -xO5\\  \hline
Fortran compiler&  f77 (SC4.0)  & f77 (SC4.0) \\  \hline
Fortran flags	&  -O4 -f       & -dalign -xarch=v8plusa\\ 
       		&               & -xchip=ultra -fast\\ 
       		&               & -xtarget=native -xO5\\  \hline
Precision	& double (64-bit)& double (64-bit) \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
 
\label{tab:blacs}
 \begin{center}
  \begin{tabular}{||l|r|r|r|r|r|r||} \hline
             & \multicolumn{2}{c|}{Mflop/s}
             & \multicolumn{2}{c|}{$t_m~(\mu s)$}
             & \multicolumn{2}{c||}{$1/t_v$ (MB/s)} \\  \cline{2-7}
   & \multicolumn{1}{c|}{$F_{MM}$} & \multicolumn{1}{c|}{Peak}
   & \multicolumn{1}{c|}{BLACS} & \multicolumn{1}{c|}{Native}
   & \multicolumn{1}{c|}{BLACS} & \multicolumn{1}{c||}{Native} \\  \hline
   Cray T3E (MPI)
   &    360   &  600  &  17.8 & 15.8   & 117.5& 117.5  \\  \hline
   IBM SP2 (MPL)
   &    200   &  266  &  63.7 &  49.7  &  21.8& 30.8\\  \hline
   Intel XP/S MP Paragon (NX)
   &     90   &  100  &  37.7 &  31.5  & 133.6 & 143.0 \\  \hline
   Intel ASCI Option Red
   &          &       &       &        &       &      \\ 
   Supercomputer (NX)
   &    264   &  400  &  29.8 &  28.7  & 391.2 &  391.2 \\  \hline
   Cluster of Sun Ultra 2s (MPI)
   &    125   &  334    & 369.0 &  313.0  &  9.8 & 10.9 \\   \hline
   Berkeley NOW (MPI on AM-II)
   &    129   &  334    & 48.0  &  43.8 &  27.9  &  28.1 \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
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\stepcounter{subsubsection}
{\newpage\clearpage\samepage
\begin{table}[thb]
 
	\label{tab:pblas2}
 \begin{minipage}{\linewidth}
  \begin{center}
   \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
      & Process & Block & \multicolumn{5}{c||}{Values of $N$} \\  \cline{4-8}
      &  grid   & size  & 2000 & 4000 & 6000 &  8000 & 10000 \\  \hline
   Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
       & $2 \times 2$ & 32 & 516 & 475 & 477 & 459 &       \\ 
PSGEMV      & $4 \times 4$ & 32 & 1715 & 1422 & 2039 & 1892 & 1896 \\ 
      & $8 \times 8$ & 32 & 2986 & 6129 & 7157 & 5253 & 7887 \\  \hline
   IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
      & $2 \times 2$ & 50 &  552 &  643 &  663 &       &       \\ 
PDGEMV      & $4 \times 4$ & 50 & 1497 & 2081 & 2244 &  2493 &  2380 \\ 
      & $8 \times 8$ & 50 & 1379 & 2178 & 2355 &  2708 &  2575 \\  \hline
   Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
      & $2 \times 2$ & 32 &  162 &      &      &       &       \\ 
PDGEMV      & $4 \times 4$ & 32 &  543 &  620 &  666 &       &       \\ 
      & $8 \times 8$ & 32 & 1597 & 2117 & 2356 &  2461 &  2563 \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
      & $2 \times 2$ & 64 &  107 &  121 &  122 &   112 &       \\ 
PDGEMV      & $2 \times 4$ & 64 &  198 &  233 &  237 &   219 &   204 \\ 
      & $3 \times 4$ & 64 &  275 &  332 &  345 &   261 &   307 \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
      & $2 \times 2$ & 32 &  118 &  123 &  121 &    &       \\ 
PDGEMV      & $2 \times 4$ & 32 &  232 &  221 &  240 &  220 &   \\ 
      & $4 \times 4$ & 32 &  226 &  406 & 401 &  439 &  474  \\ 
      & $4 \times 8$ & 32 &  711 &  645 & 612 &  723 &  836  \\ 
      & $8 \times 8$ & 32 &  328 &  816 & 1033 & 1274 & 1532  \\  \hline
   \end{tabular}
  \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$y \leftarrow y + A\,x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16483: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
 
	              \label{tab:pblas3}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$} \\  \cline{4-8}
     &  grid   & size  & 2000 & 4000 & 6000 &  8000 & 10000 \\  \hline
  Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $2 \times 2$ & 32 & 1055 & 1070 & 1075 &       &       \\ 
PSGEMM     & $4 \times 4$ & 32 & 3630 & 4005 & 4258 & 4171 & 4292 \\ 
     & $8 \times 8$ & 32 & 13456 & 14287 & 15419 & 15858 & 16755 \\  \hline
  IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $2 \times 2$ & 50 &  755 &      &      &       &       \\ 
PDGEMM     & $4 \times 4$ & 50 & 2514 & 2850 & 3040 &       &       \\ 
     & $8 \times 8$ & 50 & 6205 & 8709 & 9862 & 10468 & 10774 \\  \hline
  Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
     & $2 \times 2$ & 32 &  330 &      &      &       &       \\ 
PDGEMM     & $4 \times 4$ & 32 & 1233 & 1281 & 1334 &       &       \\ 
     & $8 \times 8$ & 32 & 4496 & 4864 & 5030 &  5103 &  5257 \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$ & 64 &  406 &  442 &      &       &       \\ 
PDGEMM     & $2 \times 4$ & 64 &  529 &  770 &  851 &       &       \\ 
     & $3 \times 4$ & 64 &  500 &  940 & 1119 &  1220 &       \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
      & $2 \times 2$ & 32 & 463 & 470 &    &    &       \\ 
PDGEMM      & $2 \times 4$ & 32 &  496 & 524 &  &  &   \\ 
      & $4 \times 4$ & 32 &  926 & 1031 & 632 & 1754 &   \\ 
      & $4 \times 8$ & 32 & 2490 & 2822 & 3316 & 3306 & 3450 \\ 
      & $8 \times 8$ & 32 & 4130 & 5457 & 6041 & 6360 & 6647  \\  \hline
  \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

\stepcounter{subsubsection}
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{\newpage\clearpage\samepage
\begin{table}[htb]
 
                                                    \label{standardflopcount}
 \vspace{0.1in}
 \begin{center}
  \begin{tabular}{||l|l|r|c|c||}               \hline
   Driver  & Options & \multicolumn{1}{c|}{$C_f$} &
             \multicolumn{1}{c|}{$C_v$} & \multicolumn{1}{c||}{$C_m$} \\  \hline
   PxGESV  & 1 right hand side                     & $2 / 3$ &
                $3 + 1/4\,\log_2P$ & $NB\,(6 + \log_2P)$ \\ 
   PxPOSV  & 1 right hand side                     & $1 / 3$ &
                $2 + 1/2\,\log_2P$ & $4 + \log_2P$          \\ 
   PxGELS  & 1 right hand side                     & $4 / 3$ &
                $3 + \log_2P$      & $2\,(NB\,\log_2P + 1)$ \\ 
   PxSYEVX & eigenvalues only                      & $  4/3 $ &
                $5/2 \log_2P$             & $17/2\,NB + 2$             \\ 
   PxSYEVX & eigenvalues and eigenvectors          & $  10/3 $ &
                $5 \log_2P$             & $17/2\,NB + 2$             \\ 
   PxSYEV & eigenvalues only                      & $  4/3 $ &
                $5/2 \log_2P$             & $17/2\,NB + 2$             \\ 
   PxSYEV & eigenvalues and eigenvectors          & $  22/3 $ &
                $5 \log_2P$             & $17/2\,NB + 2$             \\ 
  PxGESVD & singular values only             &   26/3      &
            $10 \log_2P$                     &   $17 NB$   \\ 
  PxGESVD & singular values and left and     &             &
                                             &             \\ 
          & right singular vectors           &   38/3      &
            $14 \log_2P$                     &   $17 NB$   \\ 
   PxLAHQR & eigenvalues only                      & $ 5 $ &
               $9/2 \, (\sqrt{P}) \ast \log_2P$ 
             & $9\,(2 + \log_2P)\,N$ \\ 
           &                                       &       &
               $+ 8\,N/NB$ 
             &                        \\ 
   PxLAHQR & full Schur form    & $  18 $ &
               $9/2 \, (\sqrt{P}) \ast \log_2P$ 
             & $9\,(2 + \log_2P)\,N$ \\ 
           &                    &         &
               $+ 8\,N/NB$ 
             &                        \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
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\stepcounter{subsection}
{\newpage\clearpage\samepage
\begin{table}[htb]
                           \label{tab:luperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$}  \\  \cline{4-8}
     &  Grid   & Size   & 2000 & 5000 & 7500 & 10000 & 15000 \\  \hline
  Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 32 & 702 & 884 & 932 &       &       \\ 
PSGESV     & $2 \times 8$  & 32 & 1608 & 2680 & 3218 & 3356 & 3602 \\ 
     & $4 \times 16$ & 32 & 2419 & 6912 & 9028 & 10299 & 12547 \\  \hline
  IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 50 &  421 &  603 &      &       &       \\ 
PDGESV     & $2 \times 8$  & 50 &  722 & 1543 & 1903 &  2149 &       \\ 
     & $4 \times 16$ & 50 &  924 & 3017 & 4295 &  5596 &  7057 \\  \hline
  Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
     & $1 \times 4$  & 32 &  212 &  282 &      &       &       \\ 
PDGESV     & $2 \times 8$  & 32 &  460 &  865 & 1010 &       &       \\ 
     & $4 \times 16$ & 32 &  721 & 2084 & 2837 &  3344 &  3963 \\  \hline
  Intel ASCI Red Supercomputer\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel Pentium Pro
processors, which are used by the multi-threaded BLAS.}
     & $1 \times 4$  & 32 & 566 &    &      &       &       \\ 
PDGESV     & $2 \times 8$  & 32 & 1379 & 2497 &    &       &       \\ 
     & $4 \times 8$ & 32 & 1912 & 4030 &    &      &  \\ 
     & $4 \times 16$ & 32 & 2389 & 6329 &    & 9731 &  \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$  & 64 &  203 &  340 &  380 &       &       \\ 
PDGESV     & $2 \times 4$  & 64 &  196 &  499 &  618 &   689 &       \\ 
     & $2 \times 6$  & 64 &  212 &  608 &  816 &   934 &       \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
     & $1 \times 4$  & 32 &  350 &  &   &       &       \\ 
PDGESV     & $1 \times 8$  & 32 &  560 &  744 &  &   &       \\ 
     & $2 \times 8$  & 32 & 811 & 1310 & 1472 & 1547 &       \\ 
     & $4 \times 8$  & 32 & 948 & 1913 & 2149 & 2324 & 2436   \\ 
     & $4 \times 16$ & 32 & 1171 & 3091 & 3937 & 4263 & 4560   \\  \hline
  \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                      \label{tab:lltperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$} \\  \cline{4-8}
     &  Grid   & Size  & 2000 & 5000 & 7500 & 10000 & 15000 \\  \hline
  Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $2 \times 2$ & 32 & 822 & 988 & 1018 & 1039 &       \\ 
PSPOSV     & $4 \times 4$ & 32 & 2203 & 3322 & 3586 & 3828 & 4050 \\ 
     & $8 \times 8$ & 32 & 4664 & 9520 & 11534 & 12809 & 14048 \\  \hline
  IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $2 \times 2$ & 50 &  462 &  615 &      &       &       \\ 
PDPOSV     & $4 \times 4$ & 50 & 1081 & 1811 & 2118 &  2312 &       \\ 
     & $8 \times 8$ & 50 & 1807 & 4431 & 5727 &  6826 &  8084 \\  \hline
  Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
     & $2 \times 2$ & 32 &  193 &  255 &      &       &       \\ 
PDPOSV     & $4 \times 4$ & 32 &  499 &  822 &  942 &   964 &       \\ 
     & $8 \times 8$ & 32 & 1092 & 2258 & 2829 &  3220 &  3725 \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$ & 64 &  177 &  347 &  397 &       &       \\ 
PDPOSV     & $2 \times 4$ & 64 &  141 &  415 &  569 &   669 &       \\ 
     & $3 \times 4$ & 64 &  100 &  410 &  610 &   755 &       \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
     & $2 \times 2$ & 32 &  341 &    &    &       &       \\ 
PDPOSV     & $2 \times 4$ & 32 & 491 & 703 & 769 &   &       \\ 
     & $4 \times 4$ & 32 &  828 & 1301 & 1446 & 1542 &       \\ 
     & $4 \times 8$ & 32 &  890 & 1914 & 2332 & 2602 & 2889  \\ 
     & $8 \times 8$ & 32 & 1487 & 3263 & 4047 & 4667 & 5369  \\  \hline
  \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

\stepcounter{subsubsection}
{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}H = I - \tau v v^T ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}H_1 H_2 \ldots H_K = I - V T V^T ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$v_1, v_2, \ldots , v_K$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16800: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$H_1, H_2, \ldots , H_K$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16802: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                                   \label{tab:lsperf}
 \begin{minipage}{\linewidth}
  \begin{center}
   \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
      & Process & Block & \multicolumn{5}{c||}{Values of $N$} \\  \cline{4-8}
      &  Grid   & Size  & 2000 & 5000 & 7500 & 10000 & 15000 \\  \hline
   Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
      & $1 \times 4$  & 32 & 568 & 650 & 659 & 692 &       \\ 
PSGELS      & $2 \times 8$  & 32 & 1652 & 2290 & 2434 & 2594 & 2674 \\ 
      & $4 \times 16$ & 32 & 3433 & 7020 & 8072 & 9012 & 9746 \\  \hline
   IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
      & $1 \times 4$  & 50 &  387 &  564 &      &       &       \\ 
PDGELS      & $2 \times 8$  & 50 &  725 & 1506 & 1829 &  2041 &       \\ 
      & $4 \times 16$ & 50 & 1157 & 3367 & 4715 &  5836 &  7182 \\  \hline
   Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
      & $1 \times 4$  & 32 &  201 &  255 &      &       &       \\ 
PDGELS      & $2 \times 8$  & 32 &  528 &  825 &  898 &   930 &       \\ 
      & $4 \times 16$ & 32 & 1004 & 2354 & 2937 &  3263 &  3598 \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
      & $2 \times 2$  & 64 &  206 &  281 &  318 &       &       \\ 
PDGELS      & $2 \times 4$  & 64 &  220 &  519 &  585 &   612 &       \\ 
      & $2 \times 6$  & 64 &  241 &  616 &  794 &   905 &       \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
      & $1 \times 4$  & 32 & 238 & 363 &    &       &       \\ 
PDGELS      & $1 \times 8$  & 32 & 482 & 635 & 682 &    &       \\ 
      & $2 \times 8$  & 32 & 836 & 1246 & 1351 & 1415 &       \\ 
      & $4 \times 8$  & 32 & 1213 & 2026 & 2575 & 2433 &   \\ 
      & $8 \times 8$  & 32 & 1513 & 3446 & 4296 & 4754 & 5242 \\  \hline
  \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

\stepcounter{subsubsection}
{\newpage\clearpage\samepage
\begin{displaymath}A \leftarrow A - U X^T - X U^T , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A \leftarrow A - U X^T - Y V^T , \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A \leftarrow (I - V T^T V^T)(A - X V^T).\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                             \label{tab:evxperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$}  \\  \cline{4-8}
     &  Grid   & Size  & 1000 & 2000 & 3000 & 4000 & 5000 \\  \hline
 Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $1 \times 4$        & 32 & 15& 76 &    &     &      \\ 
PSSYEVX     & $2 \times 8$ & 32 & 7 & 29 & 76 & 164 &      \\ 
     & $4 \times 16$       & 32 & 5 & 17 & 34 &  64 & 123 \\  \hline
 IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 50 &   36 &      &      &      &      \\ 
PDSYEVX     & $2 \times 8$  & 50 &  18 &  76  &      &      &      \\ 
     & $4 \times 16$ & 50 & 16 & 52 &  112 &  213 &  397 \\  \hline
 Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
     & $1 \times 4$  & 32 &   96 &      &      &      &      \\ 
PDSYEVX     & $2 \times 8$  & 32 &   35 &  142 &      &      &      \\ 
     & $4 \times 16$ & 32 &   19 &   60 &  132 &  260 &      \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$  & 64 &   45 &  263 &      &      &      \\ 
PDSYEVX     & $2 \times 4$  & 64 &   44 &  389 &      &      &      \\ 
     & $2 \times 6$  & 64 &   79 &  849 & 1486 &      &      \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
     & $1 \times 4$  & 32 &   28 &      &      &      &      \\ 
PDSYEVX     & $1 \times 8$  & 32 &   23 &      &      &      &      \\ 
     & $2 \times 8$  & 32 & 15 &  64 & 172 &      &      \\ 
     & $4 \times 8$  & 32 & 14 &  47 & 128 & 608 & 663 \\  \hline
 \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                              \label{tab:evperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$}  \\  \cline{4-8}
     &  Grid   & Size  & 1000 & 2000 & 3000 & 4000 & 5000 \\  \hline
 Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $1 \times 4$        & 32 & 38 & 263 &     &     &      \\ 
PSSYEV     & $2 \times 8$  & 32 & 15 &  81 & 239 & 552 &      \\ 
     & $4 \times 16$       & 32 &  9 &  34 & 102 & 183 & 366 \\  \hline
 IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 50 &  49 &      &      &      &      \\ 
PDSYEV     & $2 \times 8$  & 50 &  27 &  121  &      &      &      \\ 
     & $4 \times 16$ & 50 & 23  &  75 &  162 &  344 &  \\  \hline
 Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Each node consists of two computational Intel i860 XP RISC
processors, which are used by the multi-threaded BLAS.}
     & $1 \times 4$  & 32 &  189 &      &      &      &      \\ 
PDSYEV     & $2 \times 8$  & 32 &   84 &  645 &      &      &      \\ 
     & $4 \times 16$ & 32 &   60 &  344 &  980 &      &      \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$  & 64 &   81 &      &      &      &      \\ 
PDSYEV     & $2 \times 4$  & 64 &   66 &      &      &      &      \\ 
     & $2 \times 6$  & 64 &  110 &  933 & 2308 &      &      \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
     & $1 \times 4$  & 32 &  42 &      &      &      &      \\ 
PDSYEV     & $1 \times 8$  & 32 & 30 &      &      &      &      \\ 
     & $2 \times 8$  & 32 & 19 & 99 & 285 &      &      \\ 
     & $4 \times 8$  & 32 & 15 & 67 & 271 & 829 & 650  \\  \hline
 \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
                             \label{tab:svdperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$}  \\  \cline{4-8}
     &  Grid   & Size  & 1000 & 2000 & 3000 & 4000 & 5000 \\  \hline
 Cray T3E\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational DEC Alpha EV5 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 32 &  71  & 508  &  1810 &      &      \\ 
PSGESVD     & $2 \times 8$  & 32 &  24  &  145 & 454  & 1020 & 1970  \\ 
     & $4 \times 16$ & 32 &  13  &  55  &  141 & 314  & 542  \\  \hline
 IBM SP2\footnote{A ScaLAPACK process is running on each node.  Each
node consists of one computational IBM POWER2 590 processor, which is
used by the BLAS.}
     & $1 \times 4$  & 50 & 86   & 1031 &      &      &      \\ 
PDGESVD     & $2 \times 8$  & 50 & 37   & 190  & 547  &      &      \\ 
     & $4 \times 16$ & 50 &  28  & 104  & 242  &      &      \\  \hline
   Cluster of Sun Ultra 2s\footnote{A ScaLAPACK process is running on
each node.  Each node consists of two computational UltraSPARC-1 processors,
but only one processor is used by the BLAS.}
     & $2 \times 2$  & 64 & 473 & 4417 &      &      &      \\ 
PDGESVD     & $2 \times 4$  & 64 & 283  & 2375 &      &      &      \\ 
     & $2 \times 6$  & 64 &  225 & 1819  &      &      &      \\  \hline
   Berkeley NOW\footnote{A ScaLAPACK process is running on each node.
Each node consists of one computational UltraSPARC-1 processor, which
is used by the BLAS.}
     & $1 \times 4$  & 32 & 153 & 1250 &      &      &      \\ 
PDGESVD    & $1 \times 8$  & 32 & 88 & 671 & 1999 &      &      \\ 
     & $2 \times 8$  & 32 & 46 & 324 & 1004 &      &      \\ 
     & $4 \times 8$  & 32 & 25 & 189 & 584 &     &      \\  \hline
 \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}T = Q^T H Q \end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$18N^3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline16994: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[htb]
 
\label{tab:pdlahqr}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|c|r|r|r|r|r||} \hline
     & Process & Block & \multicolumn{5}{c||}{Values of $N$}  \\  \cline{4-8}
     &  Grid   & Size  & 100 & 300 & 500 & 1000 & 1500 \\  \hline
 Intel XP/S MP Paragon\footnote{A ScaLAPACK process is running on each
node.  Only one computational Intel i860 XP RISC processor cpu was used
per node for these timings.}
     & $1 \times 1$  & 50 &  3.2 & 53.7  &      &       &       \\ 
     & $2 \times 2$  & 50 &  2.9 & 25.9  & 89.8 &       &       \\ 
     & $3 \times 3$  & 50 &  3.4 & 17.9  & 60.2 & 300.5 &       \\ 
     & $4 \times 4$  & 50 &  4.5 & 17.7  & 47.8 & 200.2 & 514.5 \\ 
\hline
 Intel ASCI Option Red Supercomputer\footnote{A ScaLAPACK process is running
on each node.  Only one computational Intel Pentium Pro processor cpu
was used per node for these timings.}
     & $1 \times 1$  & 50  &  0.5 & 11.2  & 54.0 &       &       \\ 
     & $2 \times 2$  & 50  &  0.4 & 4.7   & 19.5 & 131.2 & 442.5 \\ 
     & $3 \times 3$  & 50  &  0.5 & 3.2   & 12.3 & 74.2  & 220.4 \\ 
     & $4 \times 4$  & 50  &  0.7 & 3.0   & 8.7  & 43.8  & 137.1 \\ 
     & $5 \times 5$  & 50  &  0.8 & 3.0   & 6.5  & 31.0  & 95.6  \\ 
\hline
 \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[htb]
   \label{tab:estperf}
 \begin{minipage}{\linewidth}
 \begin{center}
  \begin{tabular}{||l|c|r|r|r|r|r|r|r|r|r|r||} \hline
  IBM SP2\footnote{One process spawned per node and one computational IBM POWER2 590 processor per node.}
     & $P$ & \multicolumn{10}{c||}{Values of $N$} \\  \cline{3-12}
     &     & \multicolumn{2}{c|}{2000} & \multicolumn{2}{c|}{5000}  &
             \multicolumn{2}{c|}{7500} & \multicolumn{2}{c|}{10000} &
             \multicolumn{2}{c||}{15000} \\  \cline{3-12}
     &     & Est &  Obt &  Est &  Obt &  Est &  Obt &
             Est &  Obt &  Est &  Obt \\  \hline
  PDGESV
     &  4 &  357 &  421 &  632 &  603 &      &      &
                 &      &      &      \\ 
     & 16 &  497 &  722 & 1581 & 1543 & 2116 & 1903 &
            2424 & 2149 &      &      \\ 
     & 64 &  502 &  924 & 2432 & 3017 & 4235 & 4295 &
            5793 & 5596 & 7992 & 7057 \\  \hline
  PDPOSV
     &  4 &  530 &  462 &  669 &  615 &      &      &
                 &      &      &      \\ 
     & 16 & 1315 & 1081 & 2083 & 1811 & 2366 & 2118 &
            2535 & 2312 &      &      \\ 
     & 64 & 2577 & 1807 & 5327 & 4431 & 6709 & 5727 &  
            7661 & 6826 & 8887 & 8084 \\  \hline
  \end{tabular}
 \end{center}
 \end{minipage}
\end{table}}\hbox{}\vfil

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\stepcounter{subsection}
\stepcounter{subsection}
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{\newpage\clearpage\samepage
\begin{table}[htb]
 
                                                       \label{tab:sugg}
 \begin{center}
  \begin{tabular}{||l|c|l||} \hline
   Driver       &    Process Grid        & Comments \\  \hline
   PxGESV       &    $P_{r} \ll P_{c}$           & 1D distribution for small $P$ ($\leq 8$) \\ 
   PxPOSV       &    $P_{r} = P_{c}$             &          \\ 
   PxGELS       & $P_{r} \ll P_{c}$ if $M \ge N$ & 1D distribution for small $P$ ($\leq 8$) \\ 
                & $P_{r} \gg P_{c}$ otherwise    &                                    \\ 
   PxSYEVX      &    $P_{r} \leq P_{c}$             & $lcm(P_{r},P_{c}) < 10 \times max(P_{r},P_{c})$         \\ 
   PxSYEV      &    $P_{r} \leq P_{c}$             & $lcm(P_{r},P_{c}) < 10 \times max(P_{r},P_{c})$         \\ 
   PxGESVD      &    $P_{r} = P_{c}$             &  $lcm(P_{r},P_{c}) < 10 \times max(P_{r},P_{c})$         \\ 
   PxLAHQR      &    $P_{r} = P_{c}$             &          \\  \hline
  \end{tabular}
 \end{center}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\max(\epsilon, 10^{-5})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17220: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| fl(a \oplus b ) - (a \oplus b ) | \leq \epsilon \cdot | a \oplus b | \; \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\oplus$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17228: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}| fl( \sqrt{a} ) - (\sqrt{a} ) | \leq \epsilon \cdot | \sqrt{a} | \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| fl(a \pm b ) - (a \pm b ) | \leq \epsilon \cdot ( |a| + |b| ).
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[h]

\label{tabIEEEvalues}
\begin{center}
\begin{tabular} { || l | l | l || }
\hline
Machine Parameter & Single Precision (32 bits) & Double Precision (64 bits) \\  
\hline
Relative machine precision $\epsilon$ 
& $2^{-24}   \approx 5.96 \cdot 10^{-8}$
& $2^{-53}   \approx 1.11 \cdot 10^{-16}$ \\ 
\hspace*{.25in} = PxLAMCH(ICTXT, 'E') & & \\  
\hline
Underflow threshold 
& $2^{-126}  \approx 1.18 \cdot 10^{-38}$
& $2^{-1022} \approx 2.23 \cdot 10^{-308}$ \\ 
\hspace*{.25in} = PxLAMCH(ICTXT, 'U') & & \\ 
\hline
Overflow  threshold 
& $2^{128} (1-\epsilon) \approx 3.40 \cdot 10^{38}$
& $2^{1024}(1-\epsilon) \approx 1.79 \cdot 10^{308}$ \\ 
\hspace*{.25in} = PxLAMCH(ICTXT, 'O') & & \\ 
\hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[h]

\index{norm!vector}
\index{norm!matrix}
\index{norm!one norm}
\index{norm!two norm}
\index{norm!infinity norm}
\index{norm!Frobenius norm}
\label{tabnorms}
\begin{center}
\begin{tabular}{||l|l|l||} \hline
   Norm &  Vector  & Matrix \\  \hline
One-norm      & $\|x\|_{1} = \sum_i |x_i|$ &
                $\|A\|_{1} = \max_j \sum_i |a_{ij}|$ \\ 
Two-norm      & $\|x\|_2 = ( \sum_i |x_i|^2 )^{1/2}$ &
                $\|A\|_2 = \max_{x \neq 0} \|Ax\|_2 / \|x\|_2$ \\ 
Frobenius norm& $\|x\|_F = \|x\|_2$   &
                $\|A\|_F = ( \sum_{ij} |a_{ij}|^2 )^{1/2}$ \\  
Infinity-norm & $\|x\|_{\infty} = \max_i |x_i|$ &
                $\|A\|_{\infty} = \max_i \sum_j |a_{ij}|$ \\ 
\hline
\end{tabular} 
\end{center}
\end{table}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}x = \left( \begin{array}{c} 1 \\  100 \\  9 \end{array} \right) \; \; , \; \; 
\hat{x} = \left( \begin{array}{c} 1.1 \\  99 \\  11 \end{array} \right)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{eqnarray*}\| \hat{x} - x \|_{\infty} = 2 \; , \;  & \displaystyle{
\frac{\| \hat{x} - x \|_{\infty}}{\| x \|_{\infty}} = .02 }  \; , \; &
\frac{\| \hat{x} - x \|_{\infty}}{\| \hat{x} \|_{\infty}} = .0202  \\ 
\| \hat{x} - x \|_{2} = 2.238  \; , \;  & \displaystyle{
\frac{\| \hat{x} - x \|_{2}}{\| x \|_{2}} = .0223 }  \; , \; &
\frac{\| \hat{x} - x \|_{2}}{\| \hat{x} \|_{2}} = .0225  \\ 
\| \hat{x} - x \|_{1} = 3.1  \; , \; & \displaystyle{
\frac{\| \hat{x} - x \|_{1}}{\| x \|_{1}} = .0282 }  \; , \; & 
\frac{\| \hat{x} - x \|_{1}}{\| \hat{x} \|_{1}} = .0279  
\end{eqnarray*}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 1 & 2 & 3 \\  4 & 5 & 6 \\  7 & 8 & 10 \end{array} \right) \; \; , \; \;
\hat{A} = \left( \begin{array}{ccc} .44 & 2.36 & 3.04 \\  3.09 & 5.87 & 6.66 \\  7.36 & 7.77 & 9.07 \end{array} \right)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{eqnarray*}\| \hat{A} - A \|_{\infty} = 2.44 \; , \;  & \displaystyle{
\frac{\| \hat{A} - A \|_{\infty}}{\| A \|_{\infty}} = .0976 }  \; , \;  &
\frac{\| \hat{A} - A \|_{\infty}}{\| \hat{A} \|_{\infty}} = .1008 \\ 
\| \hat{A} - A \|_{2} = 1.75 \; , \;  & \displaystyle{
\frac{\| \hat{A} - A \|_{2}}{\| A \|_{2}} = .1007 }  \; , \;  &
\frac{\| \hat{A} - A \|_{2}}{\| \hat{A} \|_{2}} = .1020 \\ 
\| \hat{A} - A \|_{1} = 1.83 \; , \;  & \displaystyle{
\frac{\| \hat{A} - A \|_{1}}{\| A \|_{1}} = .0963 }  \; , \;  &
\frac{\| \hat{A} - A \|_{1}}{\| \hat{A} \|_{1}} = .0975  \\ 
\| \hat{A} - A \|_{F} = 1.87 \; , \;  & \displaystyle{
\frac{\| \hat{A} - A \|_{F}}{\| A \|_{F}} = .1075 }  \; , \;  &
\frac{\| \hat{A} - A \|_{F}}{\| \hat{A} \|_{F}} = .1082  
\end{eqnarray*}}\hbox{}\vfil

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\begin{displaymath}A^{-1} \approx \left( \begin{array}{ccc} -.667 & -1.333 & 1 \\  -.667 & 3.667 & -2 \\  1 & -2 & 1 \end{array} \right)
\; \; {\rm and} \; \; \kappa_{\infty} (A) = 158.33 \; \; .
\end{displaymath}}\hbox{}\vfil

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{\newpage\clearpage\samepage
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\setbox\sizebox=\hbox{$\{ \beta x \; , \; \beta {\rm ~a~scalar} \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17604: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\begin{displaymath}\theta ( \hat{\cal S}, {\cal S} ) =
\theta ( \hat{x} , x ) \equiv \arccos
\frac{| \hat{x}^T x |}{ \| \hat{x} \|_2 \cdot \| x \|_2 } \; \; .
\end{displaymath}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\theta ( \hat{x} , x )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17626: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}x = \left( \begin{array}{c} 1 \\  100 \\  9 \end{array} \right) \; \; {\rm and} \; \;
\hat{x} = \left( \begin{array}{c} 1.1 \\  99 \\  11 \end{array} \right)
\end{displaymath}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\theta ( \gamma \hat{x} , \beta x ) = .0209$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17632: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}\| \hat{x} - \beta x \|_2 = \frac{\sqrt{2} \sin \theta}{\sqrt{1+ \cos \theta}} \approx \theta
\; .
\end{displaymath}}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}\hat{x} = \left( \begin{array}{c} 1.1 \\  99 \\  11 \end{array} \right) \cdot (1.1^2 + 99^2 + 11^2)^{-1/2}
\; {\rm so} \; \| \hat{x} \|_2 = 1 
\; \; {\rm and} \; \;
x = \left( \begin{array}{c} 1 \\  100 \\  9 \end{array} \right) 
\end{displaymath}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\| \hat{x} - \beta x \|_2 \approx .0209$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17664: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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{\newpage\clearpage\samepage
\begin{displaymath}.9 \frac{| \hat{\alpha} - \alpha |}{| \hat{\alpha} |} \leq
\frac{| \hat{\alpha} - \alpha |}{| {\alpha} |} \leq
1.1 \frac{| \hat{\alpha} - \alpha |}{| \hat{\alpha} |} \; \; ,
\end{displaymath}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\|Ax\|_p \leq \|A\|_p \cdot \|x\|_p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17702: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{table}[h]

\label{tableVectorNormFpq}

\begin{center}
Values of $f_{pq}(n)$ such that $\| x \|_p \leq f_{pq}(n) \|x \|_q$, where $x$ is an $n$-vector

\begin{tabular}{||cc|c|c|c||} 
\hline
    &          & \multicolumn{3}{|c||}{$q$} \\ 
    &          & \multicolumn{1}{|c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c||}{$\infty$}   
\\  \hline
    &     1    & 1 & $\sqrt{n}$ & $n$        \\  \cline{3-5}
$p$ &     2    & 1 &     1      & $\sqrt{n}$ \\  \cline{3-5}
    & $\infty$ & 1 &     1      &     1      \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$f_{pq}(m,n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17746: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\| A \|_p \leq f_{pq}(m,n) \| A \|_q$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17748: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{table}[t]

\label{tableMatrixNormFpq}

\begin{center}
Values of $f_{pq}(m,n)$ such that $\| A \|_p \leq f_{pq}(m,n) \| A \|_q$, where $A$ is $m$ by $n$

\begin{tabular}{||cc|c|c|c|c||}
\hline
    &          & \multicolumn{4}{|c||}{$q$} \\ 
    &          & \multicolumn{1}{|c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{$F$} 
    & \multicolumn{1}{c||}{$\infty$} \\  \hline
    &     1    & 1          &     $\sqrt{m}$      & $\sqrt{m}$ &    $m$      \\  \cline{3-6}
$p$ &     2    & $\sqrt{n}$ &         1           &       1    & $\sqrt{m}$  \\  \cline{3-6}
    &   $F$    & $\sqrt{n}$ & $\sqrt{\min (m,n)}$ &       1    & $\sqrt{m}$  \\  \cline{3-6}
    & $\infty$ & $n$        &     $\sqrt{n}$      & $\sqrt{n}$ &     1       \\  \hline
\end{tabular}
\end{center}
\end{table}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\|A\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17802: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sigma_{\max}(A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17806: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sigma_{\min}(A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17810: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\min_{x \neq 0} \|Ax\|_2 / \|x\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17814: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\min_{x \neq 0} \|A^Tx\|_2 / \|x\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17818: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\{ x_1 , ... , x_k \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17826: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\{ \sum_{i=1}^k \beta_i x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17830: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\{ \hat{x}_1 , ... , \hat{x}_k \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17846: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\hat{x}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17850: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
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\setbox\sizebox=\hbox{$\theta (\hat{\cal S}, {\cal S})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17870: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{eqnarray*}\theta (\hat{\cal S}, {\cal S}) & \equiv  &
\max_{\hat{x} \in {\hat{\cal S}} \atop \hat{x} \neq 0}
\min_{x \in {\cal S} \atop x \neq 0}
\theta ( \hat{x},x ) \; \; .
\end{eqnarray*}}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\hat{X}=[\hat{x}_1 , ... , \hat{x}_k]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17904: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\hat{X}^H \hat{X} = I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17906: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$X = [ x_1 , ... , x_k ]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17912: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$X^HX = I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17914: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \hat{\cal S}, {\cal S} ) = \arccos \sigma_{\min} ( \hat{X}^H X ) \; \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\hat{X} = \left( \begin{array}{cc} -.79996 & .60005 \\  -.59997 & -.79990 \\  -.01 & -.01 \end{array} \right) 
\; \; {\rm and} \; \;
X = \left( \begin{array}{cc} 1 & 0 \\  0 & 1 \\  0 & 0 \end{array} \right) 
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\theta ( \hat{\cal S}, {\cal S} ) = .01414$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline17920: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\stepcounter{section}
\stepcounter{subsection}
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\setbox\sizebox=\hbox{${\rm alg} (z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18005: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$f(z+\delta) \approx f(z) + f'(z) \cdot \delta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18033: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}{\rm alg} (z)-f(z) = f(z+\delta) - f(z) \approx f'(z) \cdot \delta .
\end{displaymath}}\hbox{}\vfil

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\setbox\sizebox=\hbox{${\rm alg} (z)-f(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18039: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\left| \frac{{\rm alg} (z)-f(z)}{f(z)} \right|
\approx
\left| \frac{f'(z) \cdot z}{f(z)} \right| \cdot 
\left| \frac{\delta}{z} \right| \equiv \kappa (f,z) 
\cdot \left| \frac{\delta}{z} \right|
.\end{displaymath}}\hbox{}\vfil

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\setcounter{footnoteapproxerrorbound}{\value{footnote}}%

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\| \hat{x} - x \|_{\infty}}{\|x\|_{\infty}} \leq {\tt ERRBD}
\end{displaymath}}\hbox{}\vfil


\setcounter{footnoteDEC5000}{\value{footnote}}%

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\setbox\sizebox=\hbox{${\tt PSLAMCH(ICTXT, 'E')} = 2^{-24} = 5.961 \cdot 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18242: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 4 & 16000 & 17000 \\  2 & 5 & 8 \\  3 & 6 & 10 \end{array} \right)
\; \; {\rm and} \; \;
b = \left( \begin{array}{c} 100.1 \\  .1 \\  .01 \end{array} \right) \; . \; \;
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}x = \left( \begin{array}{c} -.3974 \\  -.3349 \\  .3211 \end{array} \right) \; , \; \; 
\hat{x} = \left( \begin{array}{c} -.3968 \\  -.3344 \\  .3207 \end{array} \right) \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt ANORM} = 3.300 \cdot 10^4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18244: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt RCOND} = 3.907 \cdot 10^{-6}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18246: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$= 3.902 \cdot 10^{-6}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18248: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt ERRBD} =  1.5 \cdot 10^{-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18250: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$= 1.5 \cdot 10^{-3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18252: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\| \hat{x} - x \|_{\infty}}{\|x\|_{\infty}} \leq {\tt FERR}
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{x} = \left( \begin{array}{c} -.3974 \\  -.3349 \\  .3211 \end{array} \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18258: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt FERR} = 3.0 \cdot 10^{-5}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18260: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$4.3 \cdot 10^{-7}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18262: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$r = b - A \hat{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18270: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\max \left( \frac{\| E \|_{\infty}}{\| A \|_{\infty}} ,
            \frac{\| f \|_{\infty}}{\| b \|_{\infty}} \right) 
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\max \left( \frac{\| E \|_{\infty}}{\| A \|_{\infty}} , 
\frac{\| f \|_{\infty}}{\| b \|_{\infty}} \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18284: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\omega_{\infty} =
\frac{\| r \|_{\infty} }{\| A \|_{\infty} \cdot \| \hat{x} \|_{\infty} + \| b \|_{\infty} } \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\omega_{\infty} \leq p(n) \cdot \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18288: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_{\infty} (A) \equiv \|A\|_{\infty} \cdot \|A^{-1}\|_{\infty}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18294: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$x-\hat{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18296: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\|x- \hat{x} \|_{\infty}}{\| x \|_{\infty}}
\mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} 2 \cdot \omega_{\infty} \cdot \kappa_{\infty} (A) = {\tt ERRBD} \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$2 \cdot \omega_{\infty}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$4.504 \cdot 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18302: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\epsilon  = 2^{-24} = 5.960 \cdot 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18304: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\epsilon =$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt R} = (5.882 \cdot 10^{-5}, .125, .1 )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18314: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt RCOND} = 3.454 \cdot 10^{-3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18316: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\max_{i,j,k} \left( \frac{| e_{ij} |}{|a_{ij}|} ,
            \frac{| f_{k} |}{|b_{k}|} \right) 
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\max_{i,j,k} \left( \frac{| e_{ij} |}{|a_{ij}|} ,
\frac{| f_{k} |}{|b_{k}|} \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18330: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\omega_{c} = \max_i \frac{|r_i|}{ (|A| \cdot |\hat{x} | + |b|)_i} \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\omega_{c} \leq p(n) \cdot \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_{c} (A,b,\hat{x}) \equiv {\| \, |A^{-1}| ( |A| \cdot | \hat{x} | + |b| ) 
\, \|_{\infty}}/{\| \hat{x} \|_{\infty}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18352: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\| x- \hat{x} \|_{\infty}}{\| \hat{x} \|_{\infty}} 
\leq \omega_{c} \cdot \kappa_{c} (A,b,\hat{x})  .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\omega_{c}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18356: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|x - \hat{x}\|_{\infty} / \| \hat{x} \|_{\infty}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18358: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\omega_{c} \cdot \kappa_{c} (A,b,\hat{x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18360: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\| x- \hat{x} \|_{\infty}}{\| \hat{x} \|_{\infty}} \leq {\tt FERR} = 
\frac{\| \, |A^{-1}| ( |\hat{r}| + n \epsilon (|A| \cdot |\hat{x}| + |b|) ) 
\|_{\infty} } {\| \hat{x} \|_{\infty}}  \; \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{r}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18362: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$b-A \hat{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18364: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$4.6 \cdot 10^{-8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18366: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\| Ax-b \|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18475: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\| \hat{x} - x \|_2}{\| x \|_2} \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} {\tt ERRBD}
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 4 & 3 & 5 \\  2 & 5 & 8 \\  3 & 6 & 10 \\  4 & 5 & 11 \end{array} \right)
\; {\rm and} \;
b = \left( \begin{array}{c} 100.1 \\  .1 \\  .01 \\  .01 \end{array} \right) \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}x = \hat{x} = \left( \begin{array}{c} 38.49 \\  21.59 \\  -23.88 \end{array} \right) \; \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt BNORM} = 100.1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18495: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt RNORM} = 8.843$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18497: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt RCOND} = 4.712 \cdot 10^{-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18499: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt ERRBD} = 4.9 \cdot 10^{-6}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18501: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$4.6 \cdot 10^{-7}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18503: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A \ast X-B$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18507: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|(A+E) \hat{x} - (b+f)\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18519: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\max \left( \frac{\| E \|_2}{\| A \|_2} ,
            \frac{\| f \|_2}{\| b \|_2} \right) \leq p(n) \epsilon
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18531: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\rho = \|A \hat{x} -b\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18533: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sin ( \theta ) = \rho / \|b\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18535: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A \hat{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18539: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$p(n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18543: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{x} - x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18545: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\frac{\|x-\hat{x}\|_2}{\|x\|_2} \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} p(n) \epsilon 
\left\{ \frac{2 \kappa_2 (A)}{\cos ( \theta )} + \tan ( \theta ) \kappa_2^2 (A)
\right\},
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\cos ( \theta ) $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18549: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\tan ( \theta )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18551: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_{\infty}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18559: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18561: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18563: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A= Z \Lambda Z^H$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18618: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\Lambda = {\mbox {\rm diag}} ( \lambda_1 , \ldots , \lambda_n )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18622: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18624: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lambda_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18626: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18630: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A z_i = \lambda_i z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18634: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{\lambda}_1 \leq \cdots \leq \hat{\lambda}_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18636: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD} \; \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{z}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18638: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \hat{z}_i , z_i ) \leq {\tt ZERRBD} (i) \; .
\end{displaymath}}\hbox{}\vfil


\addtocounter{footnote}{2}%

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 1 & 2 & 3 \\  2 & 4 & 5 \\  3 & 5 & 6 \end{array} \right) \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{||c|c||c|c||c|c||}
\hline
i & $\hat{\lambda}_i$ & {\tt EERRBD} & true $| \hat{\lambda}_i - \lambda_i |$ &
{\tt ZERRBD}(i) & true $\theta ( \hat{z}_i , z_i )$ \\  \hline
1 & $-.5157$ & $6.7 \cdot 10^{-7}$ & $1.6 \cdot 10^{-7}$ &
$9.8 \cdot 10^{-7}$ & $1.2 \cdot 10^{-7}$ \\ 
2 & $.1709$  & $6.7 \cdot 10^{-7}$ & $3.2 \cdot 10^{-7}$ &
$9.8 \cdot 10^{-7}$ & $7.0 \cdot 10^{-8}$ \\ 
3 & $11.34$  & $6.7 \cdot 10^{-7}$ & $2.8 \cdot 10^{-6}$ &
$6.1 \cdot 10^{-8}$ & $9.7 \cdot 10^{-8}$ \\  \hline
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{Z} \hat{\Lambda} \hat{Z}^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18680: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A+E = (\hat{Z}+ \delta \hat{Z}) \hat{\Lambda} (\hat{Z}+ \delta \hat{Z})^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18684: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{Z}+ \delta \hat{Z}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18686: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|E\|_2 / \|A\|_2 \leq p(n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18688: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\| \delta \hat{Z} \|_2 \leq p(n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18690: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lambdahat_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18698: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| \lambdahat_i - \lambda_i |  \leq p(n) \cdot \epsilon \cdot \|A\|_2
= {\tt EERRBD} \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\max_i | \lambda_i | = \|A\|_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18702: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\zhat_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18704: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \zhat_i , z_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} \frac{p(n) \epsilon \|A\|_2}{\gap_i}
= {\tt ZERRBD}(i)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\gap_i = \min_{j \neq i} | \lambda_i - \lambda_j |$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18712: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\{\hat{z}_i \, , \, i \in {\cal I}\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18722: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\cal I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18724: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( {\hat{\cal S}} , {\cal S} ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} \frac{p(n) \epsilon \|A\|_2}
{\gap_{\cal I}} ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\gap_{\cal I} = \min_{i \in {\cal I} \atop j \not\in {\cal I}} 
| \lambda_i - \lambda_j |
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18736: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\theta ( \hat{z}_i , z_i ) \leq {\tt ZERRBD}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18738: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$| \hat{\lambda}_i - \hat{\lambda}_j| \leq {\tt ORTOL} 
\cdot \max_k | \hat{\lambda}_k |$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18742: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt ORTOL} = 10^{-3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18744: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$O(nm^2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18748: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$O( \epsilon )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18764: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$D = {\mbox {\rm diag}}( t_{11}^{1/2} , \ldots , t_{nn}^{1/2})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18774: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$h_{ii}= 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18776: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| \lambdahat_i - \lambda_i | \leq p(n) \cdot \epsilon \cdot \kappa_2(H)
\cdot \lambda_i ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_2 (H)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18786: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{section}
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\setbox\sizebox=\hbox{$\Sigma = {\mbox {\rm diag}} ( \sigma_1 , \ldots , \sigma_{r} )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18852: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{r} \geq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18854: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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{\newpage\clearpage\samepage
\begin{displaymath}| \hat{\sigma}_i - \sigma_i | \leq {\tt SERRBD} \; \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{v}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18872: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{u}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18874: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{eqnarray*}\theta ( \hat{v}_i , v_i ) & \leq & {\tt VERRBD}(i) \\ 
\theta ( \hat{u}_i , u_i ) & \leq & {\tt UERRBD}(i) \; \; .
\end{eqnarray*}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 4 & 3 & 5 \\  2 & 5 & 8 \\  3 & 6 & 10 \\  4 & 5 & 11 \end{array} \right) \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{||c|c||c|c||c|c||c|c||}
\hline 
$i$ & $\hat{\sigma}_i$ & {\tt SERRBD} & true $| \hat{\sigma}_i - \sigma_i |$ &
{\tt VERRBD}($i$) & true $\theta ( \hat{v}_i , v_i )$ &
{\tt UERRBD}($i$) & true $\theta ( \hat{u}_i , u_i )$  \\  \hline
1 & 21.05 & $1.3 \cdot 10^{-6}$ & $1.7 \cdot 10^{-6}$ & 
$6.7 \cdot 10^{-8}$ & $8.1 \cdot 10^{-8}$ &
$6.7 \cdot 10^{-8}$ & $1.5 \cdot 10^{-7}$ \\ 
2 & 2.370 & $1.3 \cdot 10^{-6}$ & $5.8 \cdot 10^{-7}$ &
$1.0 \cdot 10^{-6}$ & $2.9 \cdot 10^{-7}$ &
$1.0 \cdot 10^{-6}$ & $2.4 \cdot 10^{-7}$  \\ 
3 & 1.143 & $1.3 \cdot 10^{-6}$ & $3.2 \cdot 10^{-7}$ &
$1.0 \cdot 10^{-6}$ & $3.0 \cdot 10^{-7}$ &
$1.1 \cdot 10^{-6}$ & $2.4 \cdot 10^{-7}$ \\  \hline 
\end{tabular}}\hbox{}\vfil

\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{U} \hat{\Sigma} \hat{V}^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18949: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|E\|_2 / \|A\|_2 \leq p(m,n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18953: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$A+E = (\hat{U} + \delta \hat{U}) \hat{\Sigma} (\hat{V}+ \delta \hat{V})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18961: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{U}+ \delta \hat{U}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18963: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{V}+ \delta
\hat{V}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18965: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\| \delta \hat{U} \| \leq p(m,n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18967: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\| \delta \hat{V} \| \leq p(m,n) \epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18969: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sigmahat_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18971: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| \sigmahat_i - \sigma_i | \leq p(m,n) \cdot \epsilon \cdot \sigma_1
= {\tt SERRBD} \; ,
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\sigma_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18977: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\uhat_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18979: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \uhat_i , u_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} \frac{p(m,n) \epsilon \|A\|_2}{\gap_i}
= {\tt UERRBD}(i)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\gap_i = \min_{j \neq i} | \sigma_i - \sigma_j |$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18985: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\gap_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18997: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

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\setbox\sizebox=\hbox{$\min ( \min_{j \neq n} ( | \sigma_n - \sigma_j | , \sigma_n ) )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline18999: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\vhat_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19001: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\{\hat{u}_i \, , \, i \in {\cal I}\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19007: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( {\hat{\cal S}} , {\cal S} ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} \frac{p(m,n) \epsilon \|A\|_2}
{\gap_{\cal I}}  .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\gap_{\cal I} = \min_{i \in {\cal I} \atop j \not\in {\cal I}} 
| \sigma_i - \sigma_j |
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\{\hat{v}_i \, , \, i \in {\cal I}\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19021: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\left( \begin{array}{cc} 0 & A^H \\  A & 0 \end{array} \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19025: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lambda_1 \leq \cdots \leq \lambda_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19079: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$Az_i = \lambda_i B z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19103: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$Z^{-1} A Z^{-T} = \Lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19107: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$Z^{-1} A Z^{-H} = \Lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19111: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$ABz_i = \lambda_i z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19119: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$BAz_i = \lambda_i z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19135: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD}(i) \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}A = \left( \begin{array}{ccc} 100000 & 10099 & 2109 \\  10099 & 100020 & 10012 \\  
2109 & 10112 & -48461 \end{array} \right)
\; \; {\rm and} \; \;
B = \left( \begin{array}{ccc} 99 & 10 & 2 \\  10 & 100 & 10 \\  2 & 10 & 100 \end{array} \right)
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt ANORM} = 120231$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19145: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt BNORM} = 120$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19147: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{${\tt RCONDB} = .8326$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19149: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{||c|c||c|c||c|c||}
\hline
$i$ & $\lambda_i$ & 
{\tt EERRBD}$(i)$ & true $| \hat{\lambda}_i - \lambda_i |$ &
{\tt ZERRBD}$(i)$ & true $\theta ( \hat{v}_i , v_i )$ \\  \hline
1 & $-500.0$ & $1.3 \cdot 10^{-4}$ & $9.0 \cdot 10^{-6}$ &
$9.8 \cdot 10^{-8}$ & $1.0 \cdot 10^{-9}$ \\ 
2 & $1000.$ & $1.7 \cdot 10^{-4}$ & $4.6 \cdot 10^{-5}$ &
$1.9 \cdot 10^{-5}$ & $1.2 \cdot 10^{-7}$ \\ 
3 & $1010.$ & $1.7 \cdot 10^{-4}$ & $1.0 \cdot 10^{-4}$ &
$1.9 \cdot 10^{-5}$ & $1.1 \cdot 10^{-7}$ \\  \hline
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{||c|c||c|c||c|c||}
\hline
$i$ & $\lambda_i$ &
{\tt EERRBD}$(i)$ & true $| \hat{\lambda}_i - \lambda_i |$ &
{\tt ZERRBD}$(i)$ & true $\theta ( \hat{v}_i , v_i )$ \\  \hline
1 & $-4.817 \cdot 10^6$ & $1.3$ & $6.0 \cdot 10^{-3}$ &
$1.7 \cdot 10^{-7}$ & $7.0 \cdot 10^{-9}$ \\ 
2 & $8.094 \cdot 10^6$ & $1.6$ & $1.5$ &
$3.4 \cdot 10^{-7}$ & $3.3 \cdot 10^{-8}$ \\ 
3 & $1.219 \cdot 10^7$ & $1.9$ & $4.5$ &
$3.4 \cdot 10^{-7}$ & $4.7 \cdot 10^{-8}$ \\  \hline
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{\lambda}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19245: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}|\hat{\lambda}_i - \lambda_i | \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} p(n) \epsilon 
\cdot ( \|B^{-1}\|_2 \|A\|_2  + \kappa_2 (B) \cdot | \hat{\lambda}_i | ) 
 = {\tt EERRBD}(i) \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \hat{z_i} , z_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}
p(n) \epsilon 
\frac{\|B^{-1} \|_2 \|A\|_2 ( \kappa_2 (B) )^{1/2} 
+ \kappa_2 (B) | \hat{\lambda}_i |}
{ \gap_i } = {\tt ZERRBD}(i) \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}|\hat{\lambda}_i - \lambda_i | \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}} p(n) \epsilon \cdot
( \|B\|_2 \|A\|_2 + \kappa_2 (B) \cdot | \hat{\lambda}_i | )
= {\tt EERRBD}(i) \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\theta ( \hat{z_i} ,  z_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}
p(n) \epsilon \left(
\frac{\|B\|_2 \|A\|_2 ( \kappa_2 (B) )^{1/2}}
{ \gap_i }
+ \kappa_2 (B) \right) = {\tt ZERRBD}(i) \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\kappa_2 (B)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19277: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$D = {\mbox {\rm diag}} (b_{11}^{-1/2} , \ldots , b_{nn}^{-1/2})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19283: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$D^{-1}AD^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19305: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\chi ( \hat{\lambda}_i , \lambda_i ) = 
\frac{| \hat{\lambda}_i - \lambda_i |}
{\sqrt{1 + \hat{\lambda}_i^2} \cdot \sqrt{1 +  \lambda_i^2} }
.\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\lambda_i = \tan \theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19313: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\hat{\lambda}_i = \tan \hat{\theta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19315: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\chi ( \hat{\lambda}_i , \lambda_i ) = | \sin ( \hat{\theta} - \theta ) |$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19317: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\chi ( \hat{\lambda}_i , \lambda_i )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19329: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\chi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19333: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$(A+E) - \lambda (B+F)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19345: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|x_i\|_2=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19349: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|E\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19353: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|A\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19355: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|F\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19357: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$\|B\|$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19359: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{displaymath}\chi ( \hat{\lambda}_i , \lambda_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}
\frac{\|E\| + \|F\|}{\sqrt{(x_i^H Ax_i )^2 + (x_i^H Bx_i )^2}} \; \; .
\end{displaymath}}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$1/{\sqrt{(x_i^H Ax_i )^2 + (x_i^H Bx_i )^2}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19363: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$j^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19432: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$i^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19434: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{subsection}
\stepcounter{section}
\stepcounter{section}
{\newpage\clearpage\samepage
\setbox\sizebox=\hbox{$100 \ast i+j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline19471: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox}\hbox{}\vfil

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{section}
\stepcounter{section}
\appendix
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage\samepage
\begin{tabular}% latex2html id marker 6406
{| l   l | p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex& \\ 
\hline \hline
{\bf PSDBSV\index{PSDBSV}}&{\bf PCDBSV\index{PCDBSV}}&
Solves a general banded system of linear equations
$AX=B$ without pivoting.\\ 
PSDBTRF\index{PSDBTRF}&PCDBTRF\index{PCDBTRF}& 
Computes an $LU$ factorization of a general band matrix without pivoting.\\ 
PSDBTRS\index{PSDBTRS}&PCDBTRS\index{PCDBTRS}& 
Solves a general banded system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
using the $LU$ factorization computed by PSDBTRF\index{PSDBTRF}/PCDBTRF\index{PCDBTRF}.\\ 
{\bf PSDTSV\index{PSDTSV}}&{\bf PCDTSV\index{PCDTSV}}&
Solves a general tridiagonal system of linear equations $AX=B$ without pivoting.\\ 
PSDTTRF\index{PSDTTRF}&PCDTTRF\index{PCDTTRF}& 
Computes an $LU$ factorization of a general tridiagonal matrix
without pivoting. \\ 
PSDTTRS\index{PSDTTRS}&PCDTTRS\index{PCDTTRS}& 
Solves a general tridiagonal system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
using the $LU$ factorization computed by PSDTTRF\index{PSDTTRF}/PCDTTRF\index{PCDTTRF}.\\ 
{\bf PSGBSV\index{PSGBSV}}&{\bf PCGBSV\index{PCGBSV}}&
Solves a general banded system of linear equations
$AX=B$ using partial pivoting with local row interchanges.\\ 
PSGBTRF\index{PSGBTRF}&PCGBTRF\index{PCGBTRF}& 
Computes an $LU$ factorization of a general band matrix,
using partial pivoting with local row interchanges.\\ 
PSGBTRS\index{PSGBTRS}&PCGBTRS\index{PCGBTRS}& 
Solves a general banded system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
using the $LU$ factorization computed by PSGBTRF\index{PSGBTRF}/PCGBTRF\index{PCGBTRF}.\\ 
PSGEBRD\index{PSGEBRD}&PCGEBRD\index{PCGEBRD}& 
Reduces a general rectangular matrix to real bidiagonal form 
by orthogonal/unitary transformations.\\ 
PSGECON\index{PSGECON}&PCGECON\index{PCGECON}& 
Estimates the reciprocal of the condition number
of a general matrix, 
in either the 1-norm or the infinity-norm,
using the $LU$ factorization computed by PSGETRF\index{PSGETRF}/PCGETRF\index{PCGETRF}.\\ 
PSGEEQU\index{PSGEEQU}&PCGEEQU\index{PCGEEQU}&  
Computes row and column scalings to equilibrate a general rectangular matrix 
and reduce its condition number.\\ 
PSGEHRD\index{PSGEHRD}&PCGEHRD\index{PCGEHRD}& 
Reduces a general matrix to upper Hessenberg form 
by an orthogonal/unitary similarity transformation.\\ 
PSGELQF\index{PSGELQF}&PCGELQF\index{PCGELQF}& 
Computes an $LQ$ factorization of a general rectangular matrix.\\ 
{\bf PSGELS\index{PSGELS} }&{\bf PCGELS\index{PCGELS}}& 
Computes the least squares solution to an overdetermined system of linear
equations, $A X=B$ or $A^H X=B$, 
or the minimum norm solution of an underdetermined system,
where $A$ is a general rectangular matrix of full rank, 
using a $QR$ or $LQ$ factorization of $A$.\\ 
PSGEQLF\index{PSGEQLF}&PCGEQLF\index{PCGEQLF}& 
Computes a $QL$ factorization of a general rectangular matrix.\\ 
PSGEQPF\index{PSGEQPF}&PCGEQPF\index{PCGEQPF}& 
Computes a $QR$ factorization with column pivoting of a general rectangular 
matrix.\\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}% latex2html id marker 6460
{| l   l |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex& \\ 
\hline \hline
PSGEQRF\index{PSGEQRF}&PCGEQRF\index{PCGEQRF}& 
Computes a $QR$ factorization of a general rectangular matrix.\\ 
PSGERFS\index{PSGERFS}&PCGERFS\index{PCGERFS}& 
Improves the computed solution to a general system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
and provides forward and backward error bounds for the solution.\\ 
PSGERQF\index{PSGERQF}&PCGERQF\index{PCGERQF}& 
Computes an $RQ$ factorization of a general rectangular matrix.\\ 
{\bf PSGESV\index{PSGESV}}&{\bf PCGESV\index{PCGESV}}&
Solves a general system of linear equations $AX=B$.\\ 
{\bf PSGESVD\index{PSGESVD}}& & 
Computes the singular value decomposition (SVD) of a general rectangular 
matrix.\\ 
{\bf PSGESVX\index{PSGESVX}}&{\bf PCGESVX\index{PCGESVX}}& 
Solves a general system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
and provides an estimate of the condition number 
and error bounds on the solution.\\ 
PSGETRF\index{PSGETRF}&PCGETRF\index{PCGETRF}& 
Computes an $LU$ factorization of a general matrix,
using partial pivoting with row interchanges.\\ 
PSGETRI\index{PSGETRI}&PCGETRI\index{PCGETRI}& 
Computes the inverse of a general matrix, 
using the $LU$ factorization computed by PSGETRF\index{PSGETRF}/PCGETRF\index{PCGETRF}.\\ 
PSGETRS\index{PSGETRS}&PCGETRS\index{PCGETRS}& 
Solves a general system of linear equations
$AX=B$, $A^T X=B$ or $A^H X=B$,
using the $LU$ factorization computed by PSGETRF\index{PSGETRF}/PCGETRF\index{PCGETRF}.\\ 
PSGGQRF\index{PSGGQRF}&PCGGQRF\index{PCGGQRF}&
Computes a generalized QR factorization of a pair of matrices. \\ 
PSGGRQF\index{PSGGRQF}&PCGGRQF\index{PCGGRQF}&
Computes a generalized RQ factorization of a pair of matrices. \\ 
PSORGLQ\index{PSORGLQ}&PCUNGLQ\index{PCUNGLQ}& 
Generates all or part of the orthogonal/unitary matrix $Q$ 
from an $LQ$ factorization determined by PSGELQF\index{PSGELQF}/PCGELQF\index{PCGELQF}.\\ 
PSORGQL\index{PSORGQL}&PCUNGQL\index{PCUNGQL}& 
Generates all or part of the orthogonal/unitary matrix $Q$ 
from a $QL$ factorization determined by PSGEQLF\index{PSGEQLF}/PCGEQLF\index{PCGEQLF}.\\ 
PSORGQR\index{PSORGQR}&PCUNGQR\index{PCUNGQR}& 
Generates all or part of the orthogonal/unitary matrix $Q$ 
from a $QR$ factorization determined by PSGEQRF\index{PSGEQRF}/PCGEQRF\index{PCGEQRF}.\\ 
PSORGRQ\index{PSORGRQ}&PCUNGRQ\index{PCUNGRQ}& 
Generates all or part of the orthogonal/unitary matrix $Q$ 
from an $RQ$ factorization determined by PSGERQF\index{PSGERQF}/PCGERQF\index{PCGERQF}.\\ 
PSORMBR\index{PSORMBR}&PCUNMBR\index{PCUNMBR}& 
Multiplies a general matrix by one of the orthogonal/unitary transformation 
matrices 
from a reduction to bidiagonal form determined by PSGEBRD\index{PSGEBRD}/PCGEBRD\index{PCGEBRD}.\\ 
PSORMHR\index{PSORMHR}&PCUNMHR\index{PCUNMHR}& 
Multiplies a general matrix by the orthogonal/unitary transformation matrix
from a reduction to Hessenberg form determined by PSGEHRD\index{PSGEHRD}/PCGEHRD\index{PCGEHRD}.\\ 
PSORMLQ\index{PSORMLQ}&PCUNMLQ\index{PCUNMLQ}& 
Multiplies a general matrix by the orthogonal/unitary matrix 
from an $LQ$ factorization determined by PSGELQF\index{PSGELQF}/PCGELQF\index{PCGELQF}.\\ 
PSORMQL\index{PSORMQL}&PCUNMQL\index{PCUNMQL}& 
Multiplies a general matrix by the orthogonal/unitary matrix 
from a $QL$ factorization determined by PSGEQLF\index{PSGEQLF}/PCGEQLF\index{PCGEQLF}.\\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}% latex2html id marker 6529
{| l   l |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex& \\ 
\hline \hline
PSORMQR\index{PSORMQR}&PCUNMQR\index{PCUNMQR}& 
Multiplies a general matrix by the orthogonal/unitary matrix 
from a $QR$ factorization determined by PSGEQRF\index{PSGEQRF}/PCGEQRF\index{PCGEQRF}.\\ 
PSORMRQ\index{PSORMRQ}&PCUNMRQ\index{PCUNMRQ}& 
Multiplies a general matrix by the orthogonal/unitary matrix 
from an $RQ$ factorization determined by PSGERQF\index{PSGERQF}/PCGERQF\index{PCGERQF}.\\ 
PSORMRZ\index{PSORMRZ}&PCUNMRZ\index{PCUNMRZ}& 
Multiplies a general matrix by the orthogonal/unitary matrix 
from an $RZ$ factorization determined by PSTZRZF\index{PSTZRZF}/PCTZRZF\index{PCTZRZF}.\\ 
PSORMTR\index{PSORMTR}&PCUNMTR\index{PCUNMTR}& 
Multiplies a general matrix by the orthogonal/unitary transformation matrix
from a reduction to tridiagonal form determined by PSSYTRD\index{PSSYTRD}/PCHETRD\index{PCHETRD}.\\ 
{\bf PSPBSV\index{PSPBSV}}&{\bf PCPBSV\index{PCPBSV}}&
Solves a symmetric/Hermitian positive definite banded system of linear 
equations $A X=B$.\\ 
PSPBTRF\index{PSPBTRF}&PCPBTRF\index{PCPBTRF}&
Computes the Cholesky factorization of 
a symmetric/Hermitian positive definite band matrix.\\ 
PSPBTRS\index{PSPBTRS}&PCPBTRS\index{PCPBTRS}&
Solves a symmetric/Hermitian positive definite banded system of linear 
equations $A X=B$, 
using the Cholesky factorization computed by PSPBTRF\index{PSPBTRF}/PCPBTRF\index{PCPBTRF}.\\ 
PSPOCON\index{PSPOCON}&PCPOCON\index{PCPOCON}&
Estimates the reciprocal of the condition number of 
a symmetric/Hermitian positive definite matrix,
using the Cholesky factorization computed by PSPOTRF\index{PSPOTRF}/PCPOTRF\index{PCPOTRF}.\\ 
PSPOEQU\index{PSPOEQU}&PCPOEQU\index{PCPOEQU}&
Computes row and column scalings to equilibrate 
a symmetric/Hermitian positive definite matrix 
and reduce its condition number.\\ 
PSPORFS\index{PSPORFS}&PCPORFS\index{PCPORFS}&
Improves the computed solution to 
a symmetric/Hermitian positive definite system of linear equations $A X=B$,
and provides forward and backward error bounds for the solution.\\ 
{\bf PSPOSV\index{PSPOSV}}&{\bf PCPOSV\index{PCPOSV}}&
Solves a symmetric/Hermitian positive definite system of linear equations
$A X=B$.\\ 
{\bf PSPOSVX\index{PSPOSVX}}&{\bf PCPOSVX\index{PCPOSVX}}& 
Solves a symmetric/Hermitian positive definite system of linear equations
$A X=B$,
and provides an estimate of the condition number 
and error bounds on the solution.\\ 
PSPOTRF\index{PSPOTRF}&PCPOTRF\index{PCPOTRF}&
Computes the Cholesky factorization of 
a symmetric/Hermitian positive definite matrix.\\ 
PSPOTRI\index{PSPOTRI}&PCPOTRI\index{PCPOTRI}&
Computes the inverse of 
a symmetric/Hermitian positive definite matrix,
using the Cholesky factorization computed by PSPOTRF\index{PSPOTRF}/PCPOTRF\index{PCPOTRF}.\\ 
PSPOTRS\index{PSPOTRS}&PCPOTRS\index{PCPOTRS}&
Solves a symmetric/Hermitian positive definite system of linear equations
$A X=B$,
using the Cholesky factorization computed by PSPOTRF\index{PSPOTRF}/PCPOTRF\index{PCPOTRF}.\\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}% latex2html id marker 6587
{| l  p{0.85in} |p{4.40in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex& \\ 
\hline \hline
{\bf PSPTSV\index{PSPTSV}}&{\bf PCPTSV\index{PCPTSV}}&
Solves a symmetric/Hermitian positive definite tridiagonal system of linear 
equations
$A X=B$.\\ 
PSPTTRF\index{PSPTTRF}&PCPTTRF\index{PCPTTRF}& 
Computes the $LDL^H$ factorization of 
a symmetric/Hermitian positive definite tridiagonal matrix.\\ 
PSPTTRS\index{PSPTTRS}&PCPTTRS\index{PCPTTRS}& 
Solves a symmetric/Hermitian positive definite tridiagonal system of linear 
equations, 
using the $LDL^H$ factorization computed by PSPTTRF\index{PSPTTRF}/PCPTTRF\index{PCPTTRF}.\\ 
PSSTEBZ\index{PSSTEBZ}& &
Computes selected eigenvalues of a real symmetric tridiagonal matrix
by bisection. \\ 
PSSTEIN\index{PSSTEIN}&PCSTEIN\index{PCSTEIN}&
Computes selected eigenvectors of a real symmetric tridiagonal matrix 
by inverse iteration.\\ 
{\bf PSSYEV\index{PSSYEV} }& &
Computes all eigenvalues and, optionally, eigenvectors of 
a symmetric/Hermitian matrix.\\ 
{\bf PSSYEVX\index{PSSYEVX}}&{\bf PCHEEVX\index{PCHEEVX}}& 
Computes selected eigenvalues and eigenvectors of 
a symmetric/Hermitian matrix.\\ 
PSSYGST\index{PSSYGST}&PCHEGST\index{PCHEGST}& 
Reduces a symmetric/Hermitian definite generalized eigenproblem 
$Ax=\lambda Bx$,  $ABx=\lambda x$,  or $BAx=\lambda x$,
to standard form, where $B$ has been factorized by PSPOTRF\index{PSPOTRF}/PCPOTRF\index{PCPOTRF}.\\ 
{\bf PSSYGVX\index{PSSYGVX}}&{\bf PCHEGVX\index{PCHEGVX}}& 
Computes all or selected eigenvalues and the eigenvectors of 
a generalized symmetric/Hermitian definite generalized eigenproblem, 
$Ax=\lambda Bx$,  $ABx=\lambda x$,  or $BAx=\lambda x$.\\ 
PSSYTRD\index{PSSYTRD}&PCHETRD\index{PCHETRD}& 
Reduces a symmetric/Hermitian matrix to 
real symmetric tridiagonal form 
by an orthogonal/unitary similarity transformation.\\ 
PSTRCON\index{PSTRCON}&PCTRCON\index{PCTRCON}& 
Estimates the reciprocal of the condition number of a triangular matrix,
in either the 1-norm or the infinity-norm.\\ 
PSTRRFS\index{PSTRRFS}&PCTRRFS\index{PCTRRFS}& 
Provides forward and backward error bounds for the solution of
a triangular system of linear equations
$A X=B$, $A^T X=B$ or $A^H X=B$.\\ 
PSTRTRI\index{PSTRTRI}&PCTRTRI\index{PCTRTRI}& 
Computes the inverse of a triangular matrix.\\ 
PSTRTRS\index{PSTRTRS}&PCTRTRS\index{PCTRTRS}& 
Solves a triangular system of linear equations
$A X=B$, $A^T X=B$ or $A^H X=B$.\\ 
PSTZRZF\index{PSTZRZF}&PCTZRZF\index{PCTZRZF}& 
Reduces an upper trapezoidal matrix to upper triangular form using
orthogonal/unitary transformations.\\ 
\hline
\end{tabular}}\hbox{}\vfil

\stepcounter{section}
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\begin{tabular}{| l   l | p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex&\\ 
\hline \hline
&PCLACGV&
Conjugates a complex vector.\\ 
&{\it PCMAX1}&
Finds the index of the element whose real part has maximum absolute value 
(similar to the Level 1 PBLAS PCAMAX,
but using the absolute value of the real part).\\ 
&{\it PSCSUM1}&
Forms the 1-norm of a complex vector 
(similar to the Level 1 PBLAS PSCASUM,
but using the true absolute value).\\ 
PSDBTRSV & PCDBTRSV &
Is called by PSDBTRS/PCDBTRS. \\ 
PSDTTRSV & PCDTTRSV &
Is called by PSDTTRS/PCDTTRS. \\ 
PSGEBD2&PCGEBD2&
Reduces a general rectangular matrix to  real bidiagonal form 
by an orthogonal/unitary transformation
(unblocked algorithm).\\ 
PSGEHD2&PCGEHD2& 
Reduces a general matrix to upper Hessenberg form 
by an orthogonal/unitary similarity transformation
(unblocked algorithm).\\ 
PSGELQ2&PCGELQ2&
Computes an $LQ$ factorization of a general rectangular matrix
(unblocked algorithm).\\ 
PSGEQL2&PCGEQL2&
Computes a $QL$ factorization of a general rectangular matrix
(unblocked algorithm).\\ 
PSGEQR2&PCGEQR2&
Computes a $QR$ factorization of a general rectangular matrix
(unblocked algorithm).\\ 
PSGERQ2&PCGERQ2&
Computes an $RQ$ factorization of a general rectangular matrix
(unblocked algorithm).\\ 
PSGETF2&PCGETF2&
Computes an $LU$ factorization of a general matrix,
using partial pivoting with row interchanges
(local unblocked algorithm).\\ 
PSLABAD&& 
Returns the square root of the underflow and overflow thresholds
if the exponent-range is very large.\\ 
PSLABRD&PCLABRD&
Reduces the first $nb$ rows and columns of a general rectangular matrix $A$
to real bidiagonal form by an orthogonal/unitary transformation,
and returns auxiliary matrices 
that are needed to apply the transformation to the unreduced part of $A$.\\ 
PSLACHKIEE& &
(real) performs a simple check for the features of the IEEE standard \\ 
PSLACON&PCLACON&
Estimates the 1-norm of a square matrix,
using reverse communication for evaluating matrix-vector products.\\ 
PSLACONSB & & (real) looks for two consecutive small subdiagonal elements \\ 
PSLACP2 & PCLACP2 & copies all or part of a distributed matrix to another distributed matrix \\ 
PSLACP3 & & (real) copies from a global parallel array into a local \\ 
        & & replicated array or vice versa. \\ 
PSLACPY&PCLACPY& 
Copies all or part of one two-dimensional array to another.\\ 
PSLAEVSWP& PCLAEVSWP &
Moves the eigenvectors from where they are computed, to a ScaLAPACK
standard block cyclic array.\\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}{| l p{0.75in} |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex&\\ 
\hline \hline
PSLAHQR&  &
Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, 
using the double-shift/single-shift $QR$ algorithm.\\ 
PSLAHRD&PCLAHRD&
Reduces the first $nb$ columns of a general rectangular matrix $A$
so that elements below the $k^{th}$ subdiagonal are zero,
by an orthogonal/unitary transformation,
and returns auxiliary matrices 
which are needed to apply the transformation to the unreduced part of $A$.\\ 
PSLAIECT& &
Exploits IEEE arithmetic to accelerate the computations of
eigenvalues.\\ 
PSLAMCH\index{PSLAMCH}\index{PDLAMCH}&& 
Determines machine parameters for floating-point arithmetic.\\ 
PSLANGE&PCLANGE& 
Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a general rectangular matrix.\\ 
PSLANHS&PCLANHS&  
Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of an upper Hessenberg matrix.\\  
PSLANSY&PCLANSY PCLANHE&
Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a real symmetric/complex symmetric/complex Hermitian matrix.\\ 
PSLANTR&PCLANTR&  
Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a triangular matrix.\\ 
PSLAPIV&PCLAPIV&
Applies a permutation matrix to a general distributed matrix, resulting
in row or column pivoting. \\ 
PSLAQGE&PCLAQGE& 
Scales a general rectangular matrix, 
using row and column scaling factors computed by PSGEEQU\index{PSGEEQU}/PCGEEQU\index{PCGEEQU}.\\ 
PSLAQSY&PCLAQSY& 
Scales a symmetric/Hermitian matrix,
using scaling factors computed by PSPOEQU\index{PSPOEQU}/PCPOEQU\index{PCPOEQU}.\\ 
PSLARED1D& &
Redistributes an array assuming that the input array, BYCOL, is
distributed across
rows and that all process columns contain the same copy of
BYCOL. \\ 
PSLARED2D& &
Redistributes an array assuming that the input array, BYROW, is
distributed across
columns and that all process rows contain the same copy of
BYROW.  The output array, BYALL, will be identical on all
processes.\\ 
PSLARF&PCLARF&
Applies an elementary reflector to a general rectangular matrix.\\ 
PSLARFB&PCLARFB& 
Applies a block reflector or its transpose/conjugate-transpose 
to a general rectangular matrix.\\ 
       & PCLARFC & (complex) Applies (multiplies by) the conjugate
transpose of an elementary reflector to a general matrix. \\ 
PSLARFG&PCLARFG& 
Generates an elementary reflector (Householder matrix).\\ 
PSLARFT&PCLARFT&
Forms the triangular factor $T$ of a block reflector $H = I - V T V^H$.\\ 
\hline
\end{tabular}}\hbox{}\vfil

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{| l p{0.75in} |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex&\\ 
\hline \hline
PSLARZ&PCLARZ&
Applies an elementary reflector as returned by PSTZRZF/PCTZRZF to a general matrix.\\ 
PSLARZB&PCLARZB& 
Applies a block reflector or its transpose/conjugate-transpose as
returned by PSTZRZF/PCTZRZF to a general matrix.\\ 
    & PCLARZC & (complex) Applies (multiplies by) the conjugate
transpose of an elementary reflector as returned by PSTZRZF/PCTZRZF to a
general matrix.\\ 
PSLARZT&PCLARZT&
Forms the triangular factor $T$ of a block reflector $H = I - V T V^H$
as returned by PSTZRZF/PCTZRZF.\\ 
PSLASCL&PCLASCL& 
Multiplies a general rectangular matrix by a real scalar defined as
$c_{to}/c_{from}$.\\ 
PSLASET&PCLASET& 
Initializes the off-diagonal elements of a matrix to $\alpha$
and the diagonal elements to $\beta$.\\ 
PSLASMSUB & & (real) Looks for a small subdiagonal element from the bottom \\ 
        & & of the matrix that it can safely set to zero. \\ 
PSLASNBT& &
Computes the position of the sign bit of a floating-point number. \\ 
PSLASSQ&PCLASSQ& 
Updates a sum of squares represented in scaled form.\\ 
PSLASWP&PCLASWP& 
Performs a sequence of row interchanges on a general rectangular matrix.\\ 
PSLATRA & PCLATRA & Computes the trace of a general square distributed matrix.\\ 
PSLATRD&PCLATRD& 
Reduces the first $nb$ rows and columns of a symmetric/Hermitian matrix $A$
to real tridiagonal form by an orthogonal/unitary similarity transformation,
and returns auxiliary matrices
that are needed to apply the transformation to the unreduced part of $A$.\\ 
PSLATRS&PCLATRS& 
Solves a triangular system of equations
$A  x = \sigma b$,  $A^T  x = \sigma b$,  or  $A^H  x = \sigma b$,
where $\sigma$ is a scale factor set to prevent overflow.\\ 
PSLATRZ&PCLATRZ&
Reduces the $M$-by-$N$ real/complex upper trapezoidal matrix to
upper triangular form by orthogonal/unitary transformations.\\ 
PSLAUU2&PCLAUU2& 
Computes the product $U  U^H$ or $L^H L$, 
where $U$ and $L$ are upper or lower triangular matrices
(local unblocked algorithm).\\ 
PSLAUUM&PCLAUUM& 
Computes the product $U  U^H$ or $L^H L$, 
where $U$ and $L$ are upper or lower triangular matrices.\\ 
PSLAWIL & & (real) Forms the Wilkinson transform.\\ 
PSORG2L&PCUNG2L&
Generates all or part of the orthogonal/unitary matrix $Q$ 
from a $QL$ factorization determined by
PSGEQLF\index{PSGEQLF}/PCGEQLF\index{PCGEQLF}
(unblocked algorithm).\\ 
PSORG2R&PCUNG2R&
Generates all or part of the orthogonal/unitary matrix $Q$ 
from a $QR$ factorization determined by
PSGEQRF\index{PSGEQRF}/PCGEQRF\index{PCGEQRF}
(unblocked algorithm).\\ 
PSORGL2&PCUNGL2&
Generates all or part of the orthogonal/unitary matrix $Q$ 
from an $LQ$ factorization determined by
PSGELQF\index{PSGELQF}/PCGELQF\index{PCGELQF}
(unblocked algorithm).\\ 
\hline
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{| l   p{0.75in} |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex&\\ 
\hline \hline
PSORGR2&PCUNGR2&
Generates all or part of the orthogonal/unitary matrix $Q$ 
from an $RQ$ factorization determined by
PSGERQF\index{PSGERQF}/PCGERQF\index{PCGERQF}
(unblocked algorithm).\\ 
PSORM2L&PCUNM2L&
Multiplies a general matrix by the orthogonal/unitary matrix 
from a $QL$ factorization determined by
PSGEQLF\index{PSGEQLF}/PCGEQLF\index{PCGEQLF}
(unblocked algorithm).\\ 
PSORM2R&PCUNM2R&
Multiplies a general matrix by the orthogonal/unitary matrix 
from a $QR$ factorization determined by
PSGEQRF\index{PSGEQRF}/PCGEQRF\index{PCGEQRF}
(unblocked algorithm).\\ 
PSORML2&PCUNML2&
Multiplies a general matrix by the orthogonal/unitary matrix 
from an $LQ$ factorization determined by
PSGELQF\index{PSGELQF}/PCGELQF\index{PCGELQF}
(unblocked algorithm).\\ 
PSORMR2&PCUNMR2&
Multiplies a general matrix by the orthogonal/unitary matrix 
from an $RQ$ factorization determined by
PSGERQF\index{PSGERQF}/PCGERQF\index{PCGERQF}
(unblocked algorithm).\\ 
PSPBTRSV & PCPBTRSV &
Solves a single triangular linear system via frontsolve
or backsolve where the triangular matrix is a factor 
of a banded matrix computed by PSPBTRF/PCPBTRF.\\ 
PSPTTRSV & PCPTTRSV &
Solves a single triangular linear system via frontsolve
or backsolve where the triangular matrix is a factor 
of a tridiagonal matrix computed by PSPTTRF/PCPTTRF.\\ 
PSPOTF2&PCPOTF2&
Computes the Cholesky factorization of 
a symmetric/Hermitian positive definite matrix
(local unblocked algorithm).\\ 
PSRSCL&PCSRSCL&
Multiplies a vector by the reciprocal of a real scalar.\\ 
PSSYGS2&PCHEGS2&
Reduces a symmetric/Hermitian definite generalized eigenproblem 
$Ax=\lambda Bx$,  $ABx=\lambda x$,  or $BAx=\lambda x$,
to standard form, where $B$ has been factorized by
PSPOTRF\index{PSPOTRF}/PCPOTRF\index{PCPOTRF}
(local unblocked algorithm).\\ 
PSSYTD2&PCHETD2&
Reduces a symmetric/Hermitian matrix to 
real symmetric tridiagonal form 
by an orthogonal/unitary similarity transformation
(local unblocked algorithm).\\ 
PSTRTI2&PCTRTI2&
Computes the inverse of a triangular matrix
(local unblocked algorithm).\\ 
PXERBLA\index{PXERBLA} &&
Error-handling routine called by ScaLAPACK routines 
if an input parameter has an invalid value.\\ 
SDBTF2 & CDBTF2 &
Computes an $LU$ factorization of a general band matrix with
no pivoting (local unblocked algorithm). \\ 
SDBTRF & CDBTRF &
Computes an $LU$ factorization of a general band matrix with
no pivoting (local blocked algorithm). \\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}% latex2html id marker 6720
{| l p{0.75in} |p{4.5in}    |}
\hline \multicolumn{2}{|c|}{Routine}&\multicolumn{1}{c|}{Description} \\  
\cline{1-2} Real&Complex&\\ 
\hline \hline
SDTTRF & CDTTRF &
Computes an $LU$ factorization of a general tridiagonal matrix with
no pivoting (local blocked algorithm). \\ 
SDTTRSV & CDTTRSV &
Solves a general tridiagonal system of linear equations $AX=B$,
$A^{T}X=B$ or $A^{H}X=B$, using the $LU$ factorization computed
by SDTTRF/CDTTRF. \\ 
SLAMSH & &
Sends multiple shifts through a small (single node) matrix to
see how consecutive small subdiagonal elements are modified by
subsequent shifts in an effort to maximize the number of bulges
that can be sent through.\\ 
SLAREF & &
Applies one or several Householder reflectors of size 3
to one or two matrices (if column is specified) on either their
rows or columns. \\ 
SLASORTE & &
Sorts eigenpairs so that real eigenpairs are together and
complex are together.  This way one can employ 2-by-2 shifts easily
since every 2nd subdiagonal is guaranteed to be zero. \\ 
SLASRT2 & &
Modified LAPACK routine SLASRT, which sorts numbers
in increasing or decreasing order using Quick Sort,
reverting to Insertion sort on arrays of size $\leq$ 20.  \\ 
SPTTRSV & CPTTRSV &
Solves a symmetric/Hermitian positive definite tridiagonal system of
linear equations, using the $LDL^{H}$ factorization computed
by SPTTRF/CPTTRF. \\ 
SSTEIN2 & &
Modified LAPACK routine SSTEIN, which computes the
eigenvectors of a real symmetric tridiagonal matrix T corresponding
to specified eigenvalues, using inverse iteration. \\ 
SSTEQR2 & &
Modified LAPACK routine SSTEQR, which computes all
eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the implicit QL or QR method. \\ 
\hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}{| l | l | l |}\hline
\_    &  Data and Precision       & Type \\  \hline
s     &  single-precision real    & float \\ 
d     &  double-precision real    & double \\ 
c     &  single-precision complex & float \\ 
z     &  double-precision complex & double \\ 
i     &  integer                  & int \\  \hline
\end{tabular}}\hbox{}\vfil

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\begin{tabular}{ l l } 
\multicolumn{2}{l}{\Large\bf Simple Drivers}\\ 
\multicolumn{2}{c}{\bf Simple Driver Routines for Linear Equations} \\ 
\\ [0.05in]
Matrix Type                        & Routine\\  \hline
General                               
& \verb*+PSGESV( N, NRHS, A, IA, JA, IPIV, B, IB, JB, INFO )+ \\ 
& \verb*+PCGESV( N, NRHS, A, IA, JA, IPIV, B, IB, JB, INFO )+ \\  
& \\ 
General Band
& \verb*+PSDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK,LWORK, INFO ) +\\ 
(no pivoting)
& \verb*+PCDBSV( N, BWL, BWU, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) +\\ 
& \\ 
General Band
& \verb*+PSGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) +\\ 
(partial pivoting)
& \verb*+PCGBSV( N, BWL, BWU, NRHS, A, JA, DESCA, IPIV, B, IB, DESCB, WORK, LWORK, INFO ) +\\ 
& \\ 
General Tridiagonal 
& \verb*+PSDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO )+\\ 
(no pivoting)
& \verb*+PCDTSV( N, NRHS, DL, D, DU, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO )+\\ 
& \\ 
Symmetric/Hermitian 
& \verb*+PSPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO )+\\  
Positive Definite
& \verb*+PCPOSV( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, INFO )+\\  
& \\ 
Symmetric/Hermitian 
& \verb*+PSPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO )+ \\ 
Positive Definite Band
& \verb*+PCPBSV( UPLO, N, BW, NRHS, A, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO )+ \\ 
& \\ 
Symmetric/Hermitian 
& \verb*+PSPTSV( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) +\\  
Positive Definite Tridiagonal
& \verb*+PCPTSV( N, NRHS, D, E, JA, DESCA, B, IB, DESCB, WORK, LWORK, INFO ) +\\  
& \\ 
\multicolumn{2}{c}{\bf Simple Driver Routines for Standard and Generalized Linear Least Squares Problems} \\ 
\\ [0.05in]
Problem Type                        & Routine\\  \hline
Solve Using Orthogonal Factor, 
& \verb*+PSGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO )+ \\ 
Assuming Full Rank
& \verb*+PCGELS( TRANS, M, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB, WORK, LWORK, INFO )+ \\ 
& \\ 

\multicolumn{2}{c}{\bf Simple Driver Routines for Standard Eigenvalue and Singular Value Problems} \\ 
\\ [0.025in]
Matrix/Problem Type                   & Routine\\  \hline
Symmetric/Hermitian
& \verb*+PSSYEV( JOBZ, UPLO, N, A, IA, JA, DESCA, W, Z, IZ, JZ, DESCZ, WORK, LWORK, INFO )+\\ 
Eigenvalues/vectors
& \\  \\ 


General 
& \verb*+PSGESVD( JOBU, JOBVT, M, N, A, IA, JA, DESCA, S, U, IU, JU, DESCU, VT, IVT, JVT, +\\ 
Singular Values/Vectors 
& \verb*+         DESCVT, WORK, LWORK, INFO ) +\\ 
& \\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{l l} 
\multicolumn{2}{c}{\bf Expert Driver Routines for Linear Equations} \\  
\\ [0.1in]
Matrix Type                        & Routine\\  \hline
General                               
&\verb*+PSGESVX( FACT,  TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF,+\\ 
&\verb*+         IPIV,  EQUED, R, C,    B, IB, JB, DESCB, X,  IX,  JX,  DESCX,+\\ 
&\verb*+         RCOND, FERR,  BERR, WORK, LWORK,  IWORK, LIWORK,  INFO )+\\ 
&\verb*+PCGESVX( FACT,  TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF,+\\ 
&\verb*+         IPIV,  EQUED, R, C,    B, IB, JB, DESCB, X,  IX,  JX, DESCX, +\\ 
&\verb*+         RCOND, FERR,  BERR, WORK, LWORK,  RWORK, LRWORK,  INFO )+\\  \\ 

Symmetric/Hermitian 
&\verb*+PSPOSVX( FACT,  UPLO, N, NRHS,  A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, +\\ 
Positive Definite 
&\verb*+         EQUED,             S,  B, IB, JB, DESCB, X,  IX,  JX,  DESCX, +\\ 
&\verb*+         RCOND, FERR, BERR, WORK,  LWORK,  IWORK, LIWORK,  INFO )+\\ 
&\verb*+PCPOSVX( FACT, UPLO,  N, NRHS,  A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, +\\ 
&\verb*+         EQUED,             S,  B, IB, JB, DESCB, X,  IX,  JX, DESCX, +\\ 
&\verb*+         RCOND, FERR, BERR, WORK,  LWORK,  RWORK, LRWORK,  INFO )+\\ 
& \\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{ l l }
\multicolumn{2}{c}{\bf Expert Driver Routines for Standard and Generalized Symmetric Eigenvalue Problems} \\  
\\ [0.1in]
Matrix/Problem Type                   & Routine\\  \hline

Symmetric
& \verb*+PSSYEVX( JOBZ,   RANGE,    UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU,+\\ 
Eigenvalues/vectors 
& \verb*+         ABSTOL, M, NZ, W, ORFAC,   Z, IZ, JZ, DESCZ, WORK, LWORK,+\\ 
& \verb*+         IWORK,  LIWORK,                       IFAIL, ICLUSTR, GAP, INFO )+\\ 
Hermitian
& \verb*+PCHEEVX( JOBZ,  RANGE,     UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, +\\ 
Eigenvalues/vectors
& \verb*+         ABSTOL, M, NZ, W, ORFAC,   Z, IZ, JZ, DESCZ, WORK, LWORK,+\\ 
& \verb*+         RWORK,  LRWORK,   IWORK,   LIWORK,    IFAIL, ICLUSTR, GAP, INFO )+\\  \\ 

Symmetric
& \verb*+PSSYGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, +\\ 
Eigenvalues/vectors 
& \verb*+         VL, VU,  IL, IU,   ABSTOL, M, NZ, W,     ORFAC, Z, IZ, JZ, DESCZ, +\\  
& \verb*+         WORK, LWORK, IWORK, LIWORK,    IFAIL,   ICLUSTR, GAP, INFO )+\\ 
Hermitian
& \verb*+PCHEGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, +\\ 
Eigenvalues/vectors
& \verb*+         VL, VU, IL, IU,    ABSTOL, M, NZ, W,     ORFAC, Z, IZ, JZ, DESCZ,+\\ 
& \verb*+         WORK,  LWORK,  RWORK,  LRWORK, IWORK, LIWORK, +\\ 
& \verb*+         IFAIL,         ICLUSTR,   GAP, INFO )+\\ 
& \\ 
\end{tabular}}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\begin{tabular}{l l|l}
Name   &Operation                     & Prefixes \\ 
\hline \\  \\ 
P\_SWAP & $x \leftrightarrow y$ & S, D, C, Z \\ 
P\_SCAL & $x \leftarrow \alpha x$ & S, D, C, Z, CS, ZD \\ 
P\_COPY & $y \leftarrow x$ & S, D, C, Z \\ 
P\_AXPY & $y \leftarrow \alpha x + y$ & S, D, C, Z \\ 
P\_DOT  & $dot \leftarrow x^{T} y$ & S, D \\ 
P\_DOTU & $dotu \leftarrow x^{T} y$ & C, Z \\ 
P\_DOTC & $dotc \leftarrow x^{H} y$ & C, Z \\ 
P\_NRM2 & $norm2 \leftarrow \|x\|_{2}$ & S, D, SC, DZ \\ 
P\_ASUM & $asum \leftarrow \|re(x)\|_1 + \|im(x)\|_1$ & S, D, SC, DZ \\ 
P\_AMAX & $indx \leftarrow 1^{st}~k \ni |Re(x_{k})| + |Im(x_{k})|$ & S, D, C, Z \\ 
        & \verb*|            | $ = max(|Re(x_{i})| + |Im(x_{i})|) = amax $ & \\ 
\\  \\  \\ 
P\_GEMV & $y \leftarrow \alpha op(A)x + \beta y, op(A)=A,A^{T},A^{H}, A - m \times n$ & S, D, C, Z \\ 
P\_HEMV & $y \leftarrow \alpha Ax + \beta y$ & C, Z \\ 
P\_SYMV & $y \leftarrow \alpha Ax + \beta y$ & S, D \\ 
P\_TRMV & $x \leftarrow Ax, x \leftarrow A^{T}x,
x \leftarrow A^{H}x,$ & S, D, C, Z \\ 
P\_TRSV & $x \leftarrow \alpha A^{-1}x, x \leftarrow \alpha A^{-T}x,
x \leftarrow \alpha A^{-H}x,$ & S, D, C, Z \\ 
\\  \\ 
P\_GER & $A \leftarrow \alpha xy^{T} + A, A - m \times n$ & S, D \\ 
P\_GERU & $A \leftarrow \alpha xy^{T} + A, A - m \times n$ & C, Z \\ 
P\_GERC & $A \leftarrow \alpha xy^{H} + A, A - m \times n$ & C, Z \\ 
P\_HER & $A \leftarrow \alpha xx^{H} + A$ & C, Z \\ 
P\_HER2 & $A \leftarrow \alpha xy^{H} + y(\alpha x)^{H} + A$ & C, Z \\ 
P\_SYR & $A \leftarrow \alpha xx^{T} + A$ & S, D \\ 
P\_SYR2 & $A \leftarrow \alpha xy^{T} + \alpha yx^{T} + A$ & S, D \\ 
\\  \\  \\  \\  \\ 
P\_GEMM & $C \leftarrow \alpha op(A)op(B) + \beta C, op(X)=X,X^{T},X^{H},
C - m \times n$ & S, D, C, Z \\ 
P\_SYMM & $C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C,
C - m \times n, A = A^{T}$ & S, D, C, Z \\ 
P\_HEMM & $C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C,
C - m \times n, A = A^{H}$ & C, Z \\ 
P\_SYRK & $C \leftarrow \alpha AA^{T} + \beta C, C \leftarrow \alpha A^{T}A + \beta C,
C - n \times n$ & S, D, C, Z \\ 
P\_HERK & $C \leftarrow \alpha AA^{H} + \beta C, C \leftarrow \alpha A^{H}A + \beta C,
C - n \times n$ & C, Z \\ 
P\_SYR2K & $C \leftarrow \alpha AB^{T} + \alpha BA^{T} + \beta C,
C \leftarrow \alpha A^{T}B + \alpha B^{T}A + \beta C, C - n \times n$ &
S, D, C, Z \\ 
P\_HER2K & $C \leftarrow \alpha AB^{H} + \bar{\alpha} BA^{H} + \beta C,
C \leftarrow \alpha A^{H}B + \bar{\alpha} B^{H}A + \beta C,
C - n \times n$ & C, Z \\ 
P\_TRAN & $C \leftarrow \beta C + \alpha A^T, A - n \times m, C - m \times n$ & S, D \\ 
P\_TRANU & $C \leftarrow \beta C + \alpha A^T, A - n \times m, C - m \times n$ & C, Z \\ 
P\_TRANC & $C \leftarrow \beta C + \alpha A^H, A - n \times m, C - m \times n$ & C, Z \\ 
P\_TRMM & $B \leftarrow \alpha op(A)B, B \leftarrow \alpha Bop(A), op(A)=A,A^{T},A^{H},
B - m \times n$ & S, D, C, Z \\ 
P\_TRSM & $B \leftarrow \alpha op(A^{-1})B, B \leftarrow \alpha Bop(A^{-1}),
op(A)=A,A^{T},A^{H}, B - m \times n$ & S, D, C, Z
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.5in}}
{\large \bf Meaning of prefixes} \\  \\ 
\begin{tabular}{ll}
S - \verb*#REAL#             & C - \verb*#COMPLEX# \\ 
D - \verb*#DOUBLE PRECISION# & Z - \verb*#COMPLEX*16# \\ 
                     & (may not be supported by all machines)
\end{tabular}

\\ 
\\ 

{\large \bf Level 2 and Level 3 PBLAS} \\ 
{\large \bf Matrix Types} \\  \\ 

GE - GEneral \\ 
SY - SYmmetric \\ 
HE - HErmitian \\ 
TR - TRiangular

\\ 
\\ 
\\ 


{\large \bf Level 2 and Level 3 PBLAS} \\ 
{\large \bf Options} \\  \\ 

Dummy options arguments are declared as CHARACTER$\ast$1 and may be passed as
character strings.

TRANS\_ = 'No transpose', 'Transpose', 'Conjugate transpose',
($X,~X^{T},~X^{H}$) \\ 
UPLO = 'Upper triangular', 'Lower Triangular' \\ 
DIAG = 'Non-unit triangular', 'Unit triangular' \\ 
SIDE = 'Left' or 'Right' (A or op(A) on the left, or A or op(A) on the right) \\ 
\\ 

For real matrices, TRANS\_ = 'T' and TRANS\_ = 'C' have the same 
meaning. \\ 
\\ 

For Hermitian matrices, TRANS\_ ='T' is not allowed. \\ 

\\ 
For complex symmetric matrices, TRANS\_ ='C' is not allowed. \\ 

\\ 
\\ 
\\ 

{\large \bf Array Descriptor, Increment} \\   \\ 

The array descriptor $DESCA$ is an integer array of dimension 9.
It describes the two-dimensional block-cyclic mapping of the matrix A.  \\ 
\\ 

The first two entries are the descriptor type and the BLACS context.
The third and fourth entries are the dimensions of the matrix (row, column).
The fifth and sixth entries are the row- and column block sizes used
to distribute the matrix.  The seventh and eighth are the coordinates
of the process containing the first entry of the matrix.
The last entry contains the leading dimension of the local array containing
the matrix elements.  \\ 
\\ 

The increment specified for vectors is always global. So far only 1 and
DESCA( M\_ ) are supported.
\end{tabular}}\hbox{}\vfil

\stepcounter{section}
{\newpage\clearpage\samepage
\begin{tabular}[t]{l}
\multicolumn{1}{l}{\Large\bf Fortran Interface }\\  \\ [0.28in]
\underline{{\large Initialization}}                        \\  
\verb*+BLACS_PINFO   (+ \underline{MYPNUM}, \underline{NPROCS} )\\  
\verb*+BLACS_SETUP   (+ \underline{MYPNUM}, \underline{NPROCS} )\\  
\verb*+BLACS_GET     (+ ICTXT, WHAT, \underline{VAL} )\\ 
\verb*+BLACS_SET     (+ ICTXT, WHAT, VAL )\\ 
\verb*+BLACS_GRIDINIT(+ \underline{ICTXT},\verb*+ ORDER,           +NPROW, NPCOL ) \\ 
\verb*+BLACS_GRIDMAP (+ \underline{ICTXT}, USERMAP, LDUMAP, NPROW, NPCOL )\\  \\ 
\underline{{\large Destruction}}\\  
\verb*+BLACS_FREEBUFF( ICTXT, WAIT )+\\ 
\verb*+BLACS_GRIDEXIT( ICTXT )+\\ 
\verb*+BLACS_ABORT   ( ICTXT, ERRORNUM )+\\ 
\verb*+BLACS_EXIT    ( DONEFLAG )+\\  \\ 

\underline{{\large Sending}}                        \\  
\tt
$\Box$GESD2D( ICTXT,\verb*+                         +M, N, A, LDA, RDEST, CDEST ) \\ 
 \\ 
\tt
$\Box$GEBS2D( ICTXT,\verb*+ SCOPE, TOP,             +M, N, A, LDA\verb*+               +) \\ 
 \\ 
\tt
$\Box$TRSD2D( ICTXT,\verb*+             +UPLO, DIAG, M, N, A, LDA, RDEST, CDEST ) \\ 
 \\ 
\tt
$\Box$TRBS2D(  ICTXT, SCOPE, TOP, UPLO, DIAG, M, N, A, LDA\verb*+               +) \\ 
 \\ 
\underline{{\large Receiving}}                      \\  
\tt
$\Box$GERV2D( ICTXT, \verb*+                        +M, N, \underline{A}, LDA, RSRC, CSRC ) \\ 
 \\ 
\tt
$\Box$GEBR2D( ICTXT, \verb*+SCOPE, TOP,             +M, N, \underline{A}, LDA, RSRC, CSRC ) \\ 
 \\ 
\tt
$\Box$TRRV2D( ICTXT, \verb*+            UPLO, DIAG, +M, N, \underline{A}, LDA, RSRC, CSRC ) \\ 
 \\ 
\tt
$\Box$TRBR2D( ICTXT, \verb*+SCOPE, TOP, UPLO, DIAG, +M, N, \underline{A}, LDA, RSRC, CSRC ) \\  \\ 
\underline{{\large Combine Operations}}                        \\  
\tt
$\Box$GAMX2D( ICTXT, SCOPE, TOP, M, N, \underline{A}, LDA, \underline{RA}, \underline{CA}, RCFLAG, RDEST, CDEST ) \\ 
 \\ 
\tt
$\Box$GAMN2D( ICTXT, SCOPE, TOP, M, N, \underline{A}, LDA, \underline{RA}, \underline{CA}, RCFLAG, RDEST, CDEST ) \\ 
 \\ 
\tt
$\Box$GSUM2D( ICTXT, SCOPE, TOP, M, N, \underline{A}, LDA, \verb*+                +RDEST, CDEST ) \\ 
\\ 
\it All routines preceded by a $\Box$ have the following
{\it prefixes:}  {\tt S, D, C, Z, I.} \\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{ l }
\underline{{\large Informational and Miscellaneous}}\\  
BLACS\_GRIDINFO( ICTXT, \underline{NPROW}, \underline{NPCOL}, \underline{MYROW}, \underline{MYCOL} ) \\ 
\underline{BLACS\_PNUM}\verb*+    ( ICTXT,               PROW,   PCOL )+\\ 
BLACS\_PCOORD\verb*+  +( ICTXT, PNUM,\verb*+         +\underline{PROW},\verb*+  + \underline{PCOL} ) \\ 
\verb*+BLACS_BARRIER ( ICTXT, SCOPE )+\\  \\ 
\underline{{\rm \large Non-standard}}\\ 
\verb*+SETPVMTIDS ( NTASKS, TIDS )+\\ 
\underline{DCPUTIME00} ( )\\ 
\underline{DWALLTIME00}( )\\ 
\underline{KSENDID}\verb*+    ( ICTXT,        RDEST, CDEST )+ \\ 
\underline{KRECVID}\verb*+    ( ICTXT,        RSRC,  CSRC  )+ \\ 
\underline{KBSID}\verb*+      ( ICTXT, SCOPE               )+ \\ 
\underline{KBRID}\verb*+      ( ICTXT, SCOPE, RSRC,  CSRC  )+ \\  \\ 
 \\  \\ 
\underline{{\large Declarations}}\\  
\verb*+CHARACTER          DIAG, ORDER, SCOPE, TOP, UPLO+\\ 
\verb*+INTEGER            BLACS_PNUM, CDEST, ICTXT, CSRC, DONEFLAG+\\ 
\verb*+INTEGER            ERRORNUM, LDA, RCFLAG, M, MAXID, MINID, N+\\  \verb*+INTEGER            NBRANCHES, NPCOL, NPROW +\\ 
\verb*+INTEGER            PCOL, PNUM, PROW, RDEST, RSRC, WAIT +\\ 
\verb*+INTEGER            CA( * ), RA( * )+\\ 
\verb*+DOUBLE PRECISION   DCPUTIME00, DWALLTIME00+\\ 
\verb*+REAL/DOUBLE        A( LDA, * )+\\ 
\verb*+COMPLEX/COMPLEX*16 A( LDA, * )+\\ 
\verb*+ or +\\ 
\verb*+INTEGER            A( LDA, * )+\\  \\ 
\underline{{\large Options}}\\ 
\verb*+UPLO  = 'Upper triangular', 'Lower triangular';  +\\ 
\verb*+DIAG  = 'Non-unit triangular', 'Unit triangular';  +\\ 
\verb*+SCOPE = 'All', 'row', 'column'; +\\ 
\verb*+TOP   = (SEE DESCRIPTION BELOW). +\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\large Broadcast Topologies}} \\  \\ 
\verb*+TOP   = ' ' : System dependent default topology;+\\ 
\verb*+      = 'I' : increasing ring; +\\ 
\verb*+      = 'D' : decreasing ring; +\\ 
\verb*+      = 'H' : hypercube (minimum spanning tree);  +\\ 
\verb*+      = 'S' : split-ring;  +\\ 
\verb*+      = 'F' : fully connected;  +\\ 
\verb*+      = 'M' : nodes divided into I increasing +\\ 
\verb*+              rings, where I is set with call +\\ 
\verb*+              to BLACS_SET;+\\ 
\verb*+      = 'T' : tree broadcast with NBRANCHES = I, +\\ 
\verb*+              where I is set with call to +\\ 
\verb*+              BLACS_SET;+\\ 
\verb*+      = '1' : tree broadcast with NBRANCHES = 1;+\\ 
\verb*+      = '2' : tree broadcast with NBRANCHES = 2;+\\ 
\verb*+         +\vdots\\ 
\verb*+      = '9' : tree broadcast with NBRANCHES = 9.+\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\large Global Topologies}} \\  \\ 
\verb*+TOP   = ' ' : System dependent default topology;+\\ 
\verb*+      = '1' : tree gather with NBRANCHES = 1;+\\ 
\verb*+      = '2' : tree gather with NBRANCHES = 2;+\\ 
\verb*+         +\vdots\\ 
\verb*+      = '9' : tree gather with NBRANCHES = 9;+\\ 
\verb*+      = 'T' : tree gather with NBRANCHES = I, +\\ 
\verb*+              where I is set with call to +\\ 
\verb*+              BLACS_SET;+\\ 
\verb*+      = 'F' : Fully connected;+\\ 
\verb*+      = 'H' : if RDEST = -1, a specialized +\\ 
\verb*+              "leave on all" hypercube topology+\\ 
\verb*+              called bidirectional exchange is used. +\\ 
\verb*+              Otherwise, TOP = '1' is substituted. +\\ 
\\  \\ 
\underline{{\large Notation}}                    \\   \\ 
\rm
Underlined parameters are output arguments.
If a routine is underlined, it is
a function that returns a value.
The prefix {\bf P} usually stands for process.
Other standard notations are\\ 
\verb*+GE - GENERAL     TR - TRAPEZOIDAL  +\\ 
\verb*+SD - SEND        BS - BROADCAST/SEND+\\ 
\verb*+RV - RECEIVE     BR - BROADCAST/RECEIVE+\\ 
\verb*+GAMX - General element-wise Absolute value MAXIMUM+\\ 
\verb*+GAMN - General element-wise Absolute value MINIMUM+\\ 
\verb*+GSUM - General element-wise SUMMATION+\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\rm \large Key Ideas:}}\\  \\ 
\rm
A BLACS context is created via a call to either {\tt BLACS\_GRIDINIT} or 
{\tt BLACS\_GRIDMAP}.
No routine requiring a context may be used until one of these routines
has been called.  Multiple calls to {\tt BLACS\_GRIDINIT} or {\tt BLACS\_GRIDMAP} result
in the creation of new contexts.  To preserve resources, the user should free
unused contexts by calling {\tt BLACS\_GRIDEXIT}.  When all BLACS operations
are done, a call to {\tt BLACS\_EXIT} frees any remaining contexts and shuts
down all BLACS operations.\\ 

\rm
Please note that {\tt BLACS\_GRIDINIT} and {\tt BLACS\_GRIDMAP} accept
system contexts as input.  A default system context encompassing all available
processes may be obtained by a call to {\tt BLACS\_GET}.\\ 

\rm
{\tt BLACS\_SET} can only be used to change the message ID range before
the creation of the first context.  Subsequent calls will be ignored.
\\  \\  
\underline{{\rm \large Topology Hints}}  \\   \\ 
\rm
Topologies allow the user to optimize communication patterns for a
particular operation.  If the user does not have a communication
pipe to maintain, the default {\tt TOP = ' '} is recommended.
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{l}
\multicolumn{1}{l}{\Large\bf C Interface }\\ 
\\ [0.28in]
\underline{{\rm \large Initialization}}                        \\  
\verb*+void Cblacs_pinfo   (+ int *\underline{mypnum}, int *\underline{nprocs} )\\  
\verb*+void Cblacs_setup   (+ int *\underline{mypnum}, int *\underline{nprocs} )\\  
\verb*+void Cblacs_get     (+ int ictxt, int what, int *\underline{val} ) \\ 
\verb*+void Cblacs_set     (+ int ictxt, int what, int *val ) \\ 
\verb*+void Cblacs_gridinit(+ int *\underline{ictxt},\verb*+ char *order,           +int nprow, int npcol ) \\ 
\verb*+void Cblacs_gridmap (+ int *\underline{ictxt}, int *pmap, int ldpmap, int nprow, int npcol )\\  \\ 
\underline{{\rm \large Destruction}}\\  
\verb*+void Cblacs_freebuff(+ int ictxt, int wait )\\ 
\verb*+void Cblacs_gridexit(+ int ictxt )\\ 
\verb*+void Cblacs_abort   (+ int ictxt, int errornum )\\ 
\verb*+void Cblacs_exit    (+ int doneflag )\\  \\ 
\underline{{\rm \large Sending}}                        \\  
void C$\Box$gesd2d( int ictxt,\verb*+                                                 +int m, int n, TYPE *A, int lda, \\ 
\verb*+                +int rdest, int cdest ) \\  \\ 
void C$\Box$gebs2d( int ictxt, char *scope, char *top,\verb*+                         +int m, int n, TYPE *A, int lda ) \\  \\ 
void C$\Box$trsd2d( int ictxt, char *uplo, char *diag,\verb*+                         +int m, int n, TYPE *A, int lda, \\ 
\verb*+                +int rdest, int cdest ) \\  \\ 
void C$\Box$trbs2d( int ictxt, char *scope, char *top, char *uplo, char *diag, int m, int n, TYPE *A, int lda ) \\  \\ 
\underline{{\rm \large Receiving}}                      \\  
void C$\Box$gerv2d( int ictxt,\verb*+                                                 +int m, int n, TYPE *\underline{A}, int lda, \\ 
\verb*+                +int rsrc, int csrc ) \\  \\ 
void C$\Box$gebr2d( int ictxt, char *scope, char *top,\verb*+                         +int m, int n, TYPE *\underline{A}, int lda, \\ 
\verb*+                +int rsrc, int csrc ) \\  \\ 
void C$\Box$trrv2d( int ictxt, char *uplo, char *diag,\verb*+                         +int m, int n, TYPE *\underline{A}, int lda, \\ 
\verb*+                +int rsrc, int csrc ) \\  \\ 
void C$\Box$trbr2d( int ictxt, char *scope, char *top, char *uplo, char *diag, int m, int n, TYPE *\underline{A}, int lda, \\ 
\verb*+                +int rsrc, int csrc ) \\  \\ 
\underline{{\large Combine Operations}}                        \\  
void C$\Box$gamx2d( int ictxt, char *scope, char *top, int m, int n, TYPE *\underline{A}, int lda, int *\underline{RA}, int *\underline{CA}, \\ 
\verb*+                +int RCflag, int rdest, int cdest ) \\ 
 \\ 
void C$\Box$gamn2d( int ictxt, char *scope, char *top, int m, int n, TYPE *\underline{A}, int lda, int *\underline{RA}, int *\underline{CA}, \\ 
\verb*+                +int RCflag, int rdest, int cdest ) \\ 
 \\ 
void C$\Box$gsum2d( int ictxt, char *scope, char *top, int m, int n, TYPE *\underline{A}, int lda,\\ 
\verb*+                            +int rdest, int cdest ) \\  \\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}{ l@{\hspace{2.55in}} l }
\begin{tabular}[t]{l}
{\rm \large Definition of $\Box$}\\  
\begin{tabular}{||l|l|l||}\hline\hline
\tt
$\Box${\rm is} & {\rm Data operated on is}      & TYPE {\rm is}\\ \hline\hline
  s            & {\rm single precision real}    & float\\ \hline
  d            & {\rm double precision real}    & double\\ \hline
  c            & {\rm single precision complex} & float\\ \hline
  z            & {\rm double precision complex} & double\\ \hline
  i            & {\rm integer}                  & int   \\ \hline\hline
\end{tabular}
\end{tabular}
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{l}
\underline{{\rm \large Informational and Miscellaneous}}\\  
void Cblacs\_gridinfo( int ictxt, int *\underline{nprow}, int *\underline{npcol}, int *\underline{myprow}, int *\underline{mypcol} ) \\ 
\verb*+int  +\underline{Cblacs\_pnum}\verb*+    +( int ictxt,\verb*+           +int\verb*+  +prow, int\verb*+  +pcol ) \\ 
void Cblacs\_pcoord\verb*+  +( int ictxt, int pnum, int *\underline{prow}, int *\underline{pcol} )\\ 
void Cblacs\_barrier( int ictxt, char *scope )\\  \\ 
\underline{{\rm \large Non-standard}}\\  
\verb*+void   Csetpvmtids (+ int ntasks, int *tids )\\ 
\verb*+double Cdcputime00 ( )+\\ 
\verb*+double Cdwalltime00( )+\\ 
\verb*+int    +\underline{Cksendid}\verb*+    ( int ictxt,              int rdest, int cdest )+\\ 
\verb*+int    +\underline{Ckrecvid}\verb*+    ( int ictxt,              int rsrc,  int csrc  )+\\ 
\verb*+int    +\underline{Ckbsid}\verb*+      ( int ictxt, char *scope                       )+\\ 
\verb*+int    +\underline{Ckbrid}\verb*+      ( int ictxt, char *scope, int rsrc,  int csrc  )+\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{ l }
\underline{{\large Options}}                    \\  \\ 
\verb*+UPLO  = "Upper triangular", "Lower triangular";  +\\ 
\verb*+DIAG  = "Non-unit triangular", "Unit triangular";  +\\ 
\verb*+SCOPE = "All", "row", "column"; +\\ 
\verb*+TOP   = (SEE DESCRIPTION BELOW). +\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\rm \large Broadcast Topologies}} \\  \\ 
\verb*+TOP   = " " : System dependent default topology;+\\ 
\verb*+      = "I" : increasing ring;+\\ 
\verb*+      = "D" : decreasing ring;+\\ 
\verb*+      = "H" : hypercube (minimum spanning tree);+\\ 
\verb*+      = "S" : split-ring;+\\ 
\verb*+      = "F" : fully connected;+\\ 
\verb*+      = "M" : nodes divided into I increasing +\\ 
\verb*+              rings, where I is set with call +\\ 
\verb*+              to Cblacs_set;+\\ 
\verb*+      = "T" : tree broadcast with NBRANCHES = I, +\\ 
\verb*+              where I is set with call to +\\ 
\verb*+              Cblacs_set;+\\ 
\verb*+      = "1" : tree broadcast with NBRANCHES = 1;+\\ 
\verb*+      = "2" : tree broadcast with NBRANCHES = 2;+\\ 
\verb*+         +\vdots\\ 
\verb*+      = "9" : tree broadcast with NBRANCHES = 9.+\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\rm \large Global Topologies}} \\  \\ 
\verb*+TOP   = " " : System dependent default topology;+\\ 
\verb*+      = "1" : tree gather with NBRANCHES = 1;+\\ 
\verb*+      = "2" : tree gather with NBRANCHES = 2;+\\ 
\verb*+         +\vdots\\ 
\verb*+      = "9" : tree gather with NBRANCHES = 9;+\\ 
\verb*+      = "T" : tree gather with NBRANCHES = I, +\\ 
\verb*+              where I is set with call to +\\ 
\verb*+              Cblacs_set;+\\ 
\verb*+      = "F" : Fully connected;+\\ 
\verb*+      = "H" : if rdest = -1, a specialized +\\ 
\verb*+              "leave on all" hypercube topology+\\ 
\verb*+              called bidirectional exchange is used. +\\ 
\verb*+              Otherwise, TOP = "1" is substituted. +\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\rm\large Notation}}                    \\   \\ 
\rm
Underlined parameters are output arguments.
If a routine is underlined it is
a function that returns a value.
The prefix {\bf p} usually stands for process.  
Other standard notations are:\\ 
\verb*+GE - GENERAL     TR - TRAPEZOIDAL  +\\ 
\verb*+SD - SEND        BS - BROADCAST/SEND+\\ 
\verb*+RV - RECEIVE     BR - BROADCAST/RECEIVE+\\ 
\verb*+GAMX - General element-wise Absolute value MAXIMUM+\\ 
\verb*+GAMN - General element-wise Absolute value MINIMUM+\\ 
\verb*+GSUM - General element-wise SUMMATION+\\ 
\end{tabular}}\hbox{}\vfil

{\newpage\clearpage\samepage
\begin{tabular}[t]{p{4.0in}}
\underline{{\rm \large Key Ideas:}}               \\   \\ 
\rm
A BLACS context is created via a call to either {\tt Cblacs\_gridinit} or 
{\tt Cblacs\_gridmap}.  No routine requiring a context may be used until 
one of these routines has been called.  Multiple calls to 
{\tt Cblacs\_gridinit} or {\tt Cblacs\_gridmap} result in the
creation of new contexts.  To preserve resources, the user should free
unused contexts by calling {\tt Cblacs\_gridexit}.  When all BLACS operations
are done, a call to {\tt Cblacs\_exit} frees any remaining contexts, and shuts
down all BLACS operations.

Please note that {\tt Cblacs\_gridinit} and {\tt Cblacs\_gridmap} accept
system contexts as input.  A default system context encompassing all available
processes may be obtained by a call to {\tt Cblacs\_get}.
\\  \\ 
\rm
{\tt Cblacs\_set} can only be used to set the BLACS' message ID range before the 
creation of the first context.  Subsequent calls will be ignored.
\\  \\  
\underline{{\rm \large Topology Hints}}  \\   \\ 
\rm
Topologies allow the user to optimize communication patterns for a
particular operation.  If the user does not have a communication pipe
to maintain, the default TOP = " " is recommended.
\end{tabular}}\hbox{}\vfil

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