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<H2><A NAME="SECTION03532000000000000000"></A><A NAME="subsecpacked"></A>
<BR>
Packed Storage
</H2>

<P>
Symmetric, Hermitian or triangular matrices may be stored more
compactly<A NAME="19766"></A>, if the relevant 
triangle (again as specified by UPLO) is packed
<B>by columns</B> in a one-dimensional array. In LAPACK, arrays that hold
matrices in packed storage, have names ending in `P'. So:

<P>

<UL><LI>if UPLO = `U', <B><I>a</I><SUB><I>ij</I></SUB></B> is stored in AP(<B><I>i</I>+<I>j</I>(<I>j</I>-1)/2</B>) for <IMG
 WIDTH="42" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img924.png"
 ALT="$i \leq j$">;

<P>

<LI>if UPLO = `L', <B><I>a</I><SUB><I>ij</I></SUB></B> is stored in AP(
<!-- MATH
 $i+(2n-j)(j-1)/2$
 -->
<B><I>i</I>+(2<I>n</I>-<I>j</I>)(<I>j</I>-1)/2</B>) for 
<IMG
 WIDTH="42" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img925.png"
 ALT="$j \leq i$">.

<P>

</UL>

<P>
For example:

<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">Packed storage in array AP</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
 $a_{11} \; \underbrace{a_{12} \: a_{22}} \;
\underbrace{a_{13} \: a_{23} \: a_{33}} \;
\underbrace{a_{14} \: a_{24} \: a_{34} \: a_{44}}$
 -->
<IMG
 WIDTH="286" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="img926.png"
 ALT="$
a_{11} \; \underbrace{a_{12} \: a_{22}} \;
\underbrace{a_{13} \: a_{23} \: a_{33}} \;
\underbrace{a_{14} \: a_{24} \: a_{34} \: a_{44}}
$"></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
 $\underbrace{a_{11} \: a_{21} \: a_{31} \: a_{41}} \;
\underbrace{a_{22} \: a_{32} \: a_{42}} \;
\underbrace{a_{33} \: a_{43}} \; a_{44}$
 -->
<IMG
 WIDTH="286" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="img927.png"
 ALT="$
\underbrace{a_{11} \: a_{21} \: a_{31} \: a_{41}} \;
\underbrace{a_{22} \: a_{32} \: a_{42}} \;
\underbrace{a_{33} \: a_{43}} \; a_{44}
$"></TD>
</TR>
</TABLE>
</DIV>

<P>
Note that for real or complex symmetric matrices, 
packing the upper triangle
by columns is equivalent to packing the lower triangle by rows;
packing the lower triangle by columns is equivalent to packing
the upper triangle by rows. 
For complex Hermitian matrices,
packing the upper triangle
by columns is equivalent to packing the conjugate of the lower triangle by rows;
packing the lower triangle by columns is equivalent to packing
the conjugate of the upper triangle by rows. 

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