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<H2><A NAME="SECTION03531000000000000000"></A><A NAME="subsecconventional"></A>
<BR>
Conventional Storage
</H2>

<P>
The default scheme for storing matrices<A NAME="19617"></A>
is the obvious one described in subsection&nbsp;<A HREF="node116.html#subsecarrayargs">5.1.6</A>:
a matrix <B><I>A</I></B> is stored in a two-dimensional array A, with
matrix element <B><I>a</I><SUB><I>ij</I></SUB></B> stored in array element A(<B><I>i</I>,<I>j</I></B>).

<P>
If a matrix is <B>triangular</B><A NAME="19621"></A>
(upper or lower, as specified by
the argument UPLO), only the elements of the relevant triangle
are accessed. The remaining elements of the array need not be set.
Such elements are indicated by <IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img914.png"
 ALT="$\ast$">
in the examples below.
For example, when <B><I>n</I> = 4</B>:

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER">UPLO</TD>
<TD ALIGN="CENTER">Triangular matrix <B><I>A</I></B></TD>
<TD ALIGN="CENTER">Storage in array A</TD>
</TR>
<TR><TD ALIGN="CENTER">`U'</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\left( \begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
       & a_{22} & a_{23} & a_{24} \\
       &        & a_{33} & a_{34} \\
       &        &        & a_{44}
\end{array} \right)$
 -->
<IMG
 WIDTH="190" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img918.png"
 ALT="$
\left( \begin{array}{cccc}
a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14} \\
&amp; a_{22} &amp; a_{23} &amp; a_{24} \\
&amp; &amp; a_{33} &amp; a_{34} \\
&amp; &amp; &amp; a_{44}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
 \ast  & a_{22} & a_{23} & a_{24} \\
 \ast  &  \ast  & a_{33} & a_{34} \\
 \ast  &  \ast  &  \ast  & a_{44}
\end{array}$
 -->
<IMG
 WIDTH="162" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img919.png"
 ALT="$
\begin{array}{cccc}
a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14} \\
\ast &amp; a_{22} &amp; a_{...
...\\
\ast &amp; \ast &amp; a_{33} &amp; a_{34} \\
\ast &amp; \ast &amp; \ast &amp; a_{44}
\end{array}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">`L'</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\left( \begin{array}{cccc}
a_{11} &        &        &       \\
a_{21} & a_{22} &        &       \\
a_{31} & a_{32} & a_{33} &       \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array} \right)$
 -->
<IMG
 WIDTH="190" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img920.png"
 ALT="$
\left( \begin{array}{cccc}
a_{11} &amp; &amp; &amp; \\
a_{21} &amp; a_{22} &amp; &amp; \\
a_{31} &amp; a_{32} &amp; a_{33} &amp; \\
a_{41} &amp; a_{42} &amp; a_{43} &amp; a_{44}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\begin{array}{cccc}
a_{11} &  \ast  &  \ast  &  \ast  \\
a_{21} & a_{22} &  \ast  &  \ast  \\
a_{31} & a_{32} & a_{33} &  \ast  \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}$
 -->
<IMG
 WIDTH="162" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img921.png"
 ALT="$
\begin{array}{cccc}
a_{11} &amp; \ast &amp; \ast &amp; \ast \\
a_{21} &amp; a_{22} &amp; \ast &amp; \...
...{31} &amp; a_{32} &amp; a_{33} &amp; \ast \\
a_{41} &amp; a_{42} &amp; a_{43} &amp; a_{44}
\end{array}$"></TD>
</TR>
</TABLE>
</DIV>

<P>
Similarly, if the matrix is upper Hessenberg, elements below the
first subdiagonal need not be set.

<P>
Routines that handle <B>symmetric</B><A NAME="19680"></A> 
or <B>Hermitian</B><A NAME="19682"></A> matrices
allow for either the upper or lower triangle of the matrix 
(as specified by UPLO) to
be stored in the corresponding elements of the array; the remaining
elements of the array need not be set.
For example, when <B><I>n</I> = 4</B>:

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER">UPLO</TD>
<TD ALIGN="CENTER">Hermitian matrix <B><I>A</I></B></TD>
<TD ALIGN="CENTER">Storage in array A</TD>
</TR>
<TR><TD ALIGN="CENTER">`U'</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\left( \begin{array}{cccc}
a_{11}       & a_{12}       & a_{13}       & a_{14} \\
\bar{a}_{12} & a_{22}       & a_{23}       & a_{24} \\
\bar{a}_{13} & \bar{a}_{23} & a_{33}       & a_{34} \\
\bar{a}_{14} & \bar{a}_{24} & \bar{a}_{34} & a_{44}
\end{array} \right)$
 -->
<IMG
 WIDTH="190" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img922.png"
 ALT="$
\left( \begin{array}{cccc}
a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14} \\
\bar{a}_{12} ...
...4} \\
\bar{a}_{14} &amp; \bar{a}_{24} &amp; \bar{a}_{34} &amp; a_{44}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
 \ast  & a_{22} & a_{23} & a_{24} \\
 \ast  &  \ast  & a_{33} & a_{34} \\
 \ast  &  \ast  &  \ast  & a_{44}
\end{array}$
 -->
<IMG
 WIDTH="162" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img919.png"
 ALT="$
\begin{array}{cccc}
a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14} \\
\ast &amp; a_{22} &amp; a_{...
...\\
\ast &amp; \ast &amp; a_{33} &amp; a_{34} \\
\ast &amp; \ast &amp; \ast &amp; a_{44}
\end{array}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">`L'</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\left( \begin{array}{cccc}
a_{11} & \bar{a}_{21} & \bar{a}_{31} & \bar{a}_{41} \\
a_{21} & a_{22}       & \bar{a}_{32} & \bar{a}_{42} \\
a_{31} & a_{32}       & a_{33}       & \bar{a}_{43} \\
a_{41} & a_{42}       & a_{43}       & a_{44}
\end{array} \right)$
 -->
<IMG
 WIDTH="190" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img923.png"
 ALT="$
\left( \begin{array}{cccc}
a_{11} &amp; \bar{a}_{21} &amp; \bar{a}_{31} &amp; \bar{a}_{41}...
..._{33} &amp; \bar{a}_{43} \\
a_{41} &amp; a_{42} &amp; a_{43} &amp; a_{44}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\begin{array}{cccc}
a_{11} &  \ast  &  \ast  &  \ast  \\
a_{21} & a_{22} &  \ast  &  \ast  \\
a_{31} & a_{32} & a_{33} &  \ast  \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}$
 -->
<IMG
 WIDTH="162" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
 SRC="img921.png"
 ALT="$
\begin{array}{cccc}
a_{11} &amp; \ast &amp; \ast &amp; \ast \\
a_{21} &amp; a_{22} &amp; \ast &amp; \...
...{31} &amp; a_{32} &amp; a_{33} &amp; \ast \\
a_{41} &amp; a_{42} &amp; a_{43} &amp; a_{44}
\end{array}$"></TD>
</TR>
</TABLE>
</DIV>

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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