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<H2><A NAME="SECTION03533000000000000000"></A><A NAME="subsecband"></A>
<BR>
Band Storage
</H2>
<P>
An <B><I>m</I></B>-by-<B><I>n</I></B> band matrix<A NAME="19825"></A> with <B><I>kl</I></B> subdiagonals
and <B><I>ku</I></B> superdiagonals may be
stored compactly in a two-dimensional array with <B><I>kl</I>+<I>ku</I>+1</B> rows and <B><I>n</I></B> columns.
Columns of the matrix are stored in corresponding columns of the
array, and diagonals of the matrix are stored in rows of the array.
This storage scheme should be used in practice only if
<!-- MATH
$kl, ku \ll \min(m,n)$
-->
<IMG
WIDTH="150" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img928.png"
ALT="$kl, ku \ll \min(m,n)$">,
although LAPACK routines work correctly for all values of <B><I>kl</I></B> and <B><I>ku</I></B>.
In LAPACK, arrays that hold matrices in band storage have names
ending in `B'.
<P>
To be precise, <B><I>a</I><SUB><I>ij</I></SUB></B> is stored in AB(<B><I>ku</I>+1+<I>i</I>-<I>j</I>,<I>j</I></B>) for
<!-- MATH
$\max(1,j-ku) \leq i \leq \min(m,j+kl)$
-->
<IMG
WIDTH="279" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img929.png"
ALT="$\max(1,j-ku) \leq i \leq \min(m,j+kl)$">.
For example, when <B><I>m</I> = <I>n</I> = 5</B>, <B><I>kl</I> = 2</B> and <B><I>ku</I> = 1</B>:
<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER">Band matrix <B><I>A</I></B></TD>
<TD ALIGN="CENTER">Band storage in array AB</TD>
</TR>
<TR><TD ALIGN="CENTER">
<!-- MATH
$\left( \begin{array}{ccccc}
a_{11} & a_{12} & & & \\
a_{21} & a_{22} & a_{23} & & \\
a_{31} & a_{32} & a_{33} & a_{34} & \\
& a_{42} & a_{43} & a_{44} & a_{45} \\
& & a_{53} & a_{54} & a_{55}
\end{array} \right)$
-->
<IMG
WIDTH="229" HEIGHT="125" ALIGN="MIDDLE" BORDER="0"
SRC="img930.png"
ALT="$
\left( \begin{array}{ccccc}
a_{11} & a_{12} & & & \\
a_{21} & a_{22} & a_{23}...
..._{43} & a_{44} & a_{45} \\
& & a_{53} & a_{54} & a_{55}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$\begin{array}{ccccc}
\ast & a_{12} & a_{23} & a_{34} & a_{45} \\
a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\
a_{21} & a_{32} & a_{43} & a_{54} & \ast \\
a_{31} & a_{42} & a_{53} & \ast & \ast
\end{array}$
-->
<IMG
WIDTH="202" HEIGHT="103" ALIGN="MIDDLE" BORDER="0"
SRC="img931.png"
ALT="$
\begin{array}{ccccc}
\ast & a_{12} & a_{23} & a_{34} & a_{45} \\
a_{11} & a_...
... a_{43} & a_{54} & \ast \\
a_{31} & a_{42} & a_{53} & \ast & \ast
\end{array}$"></TD>
</TR>
</TABLE>
</DIV>
<P>
The elements marked <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img914.png"
ALT="$\ast$">
in the upper left and lower right
corners of the array AB need not be set, and are not referenced by
LAPACK routines.
<P>
<B>Note:</B> when a band matrix is supplied for <B><I>LU</I></B> factorization,
space<A NAME="19871"></A> must be allowed to store an
additional <B><I>kl</I></B> superdiagonals,
generated by fill-in as a result of row interchanges.
This means that the matrix is stored according to the above scheme,
but with <B><I>kl</I> + <I>ku</I></B> superdiagonals.
<P>
Triangular band matrices are stored in the same format, with either
<B><I>kl</I> = 0</B> if upper triangular, or <B><I>ku</I> = 0</B> if
lower triangular.
<P>
For symmetric or Hermitian band matrices with <B><I>kd</I></B> subdiagonals or
superdiagonals, only the upper or lower triangle (as specified by
UPLO) need be stored:
<P>
<UL><LI>if UPLO = `U', <B><I>a</I><SUB><I>ij</I></SUB></B> is stored in AB(<B><I>kd</I>+1+<I>i</I>-<I>j</I>,<I>j</I></B>) for
<!-- MATH
$\max(1,j-kd) \leq i \leq j$
-->
<IMG
WIDTH="176" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img932.png"
ALT="$\max(1,j-kd) \leq i \leq j$">;
<P>
<LI>if UPLO = `L', <B><I>a</I><SUB><I>ij</I></SUB></B> is stored in AB(<B>1+<I>i</I>-<I>j</I>,<I>j</I></B>) for
<!-- MATH
$j \leq i \leq \min(n,j+kd)$
-->
<IMG
WIDTH="174" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img933.png"
ALT="$j \leq i \leq \min(n,j+kd)$">.
<P>
</UL>
<P>
For example, when <B><I>n</I> = 5</B> and <B><I>kd</I> = 2</B>:
<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER">UPLO</TD>
<TD ALIGN="CENTER">Hermitian band matrix <B><I>A</I></B></TD>
<TD ALIGN="CENTER">Band storage in array AB</TD>
</TR>
<TR><TD ALIGN="CENTER">`U'</TD>
<TD ALIGN="CENTER">
<!-- MATH
$\left( \begin{array}{ccccc}
a_{11} & a_{12} & a_{13} & & \\
\bar{a}_{12} & a_{22} & a_{23} & a_{24} & \\
\bar{a}_{13} & \bar{a}_{23} & a_{33} & a_{34} & a_{35} \\
& \bar{a}_{24} & \bar{a}_{34} & a_{44} & a_{45} \\
& & \bar{a}_{35} & \bar{a}_{45} & a_{55}
\end{array} \right)$
-->
<IMG
WIDTH="229" HEIGHT="125" ALIGN="MIDDLE" BORDER="0"
SRC="img934.png"
ALT="$
\left( \begin{array}{ccccc}
a_{11} & a_{12} & a_{13} & & \\
\bar{a}_{12} & a_...
...44} & a_{45} \\
& & \bar{a}_{35} & \bar{a}_{45} & a_{55}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$\begin{array}{ccccc}
\ast & \ast & a_{13} & a_{24} & a_{35} \\
\ast & a_{12} & a_{23} & a_{34} & a_{45} \\
a_{11} & a_{22} & a_{33} & a_{44} & a_{55}
\end{array}$
-->
<IMG
WIDTH="202" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
SRC="img935.png"
ALT="$
\begin{array}{ccccc}
\ast & \ast & a_{13} & a_{24} & a_{35} \\
\ast & a_{12...
...3} & a_{34} & a_{45} \\
a_{11} & a_{22} & a_{33} & a_{44} & a_{55}
\end{array}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">`L'</TD>
<TD ALIGN="CENTER">
<!-- MATH
$\left( \begin{array}{ccccc}
a_{11} & \bar{a}_{21} & \bar{a}_{31} & & \\
a_{21} & a_{22} & \bar{a}_{32} & \bar{a}_{42} & \\
a_{31} & a_{32} & a_{33} & \bar{a}_{43} & \bar{a}_{53} \\
& a_{42} & a_{43} & a_{44} & \bar{a}_{54} \\
& & a_{53} & a_{54} & a_{55}
\end{array} \right)$
-->
<IMG
WIDTH="229" HEIGHT="125" ALIGN="MIDDLE" BORDER="0"
SRC="img936.png"
ALT="$
\left( \begin{array}{ccccc}
a_{11} & \bar{a}_{21} & \bar{a}_{31} & & \\
a_{21...
... & a_{44} & \bar{a}_{54} \\
& & a_{53} & a_{54} & a_{55}
\end{array} \right)
$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$\begin{array}{ccccc}
a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\
a_{21} & a_{32} & a_{43} & a_{54} & \ast \\
a_{31} & a_{42} & a_{53} & \ast & \ast
\end{array}$
-->
<IMG
WIDTH="202" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
SRC="img937.png"
ALT="$
\begin{array}{ccccc}
a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\
a_{21} & a...
... a_{43} & a_{54} & \ast \\
a_{31} & a_{42} & a_{53} & \ast & \ast
\end{array}$"></TD>
</TR>
</TABLE>
</DIV>
<P>
EISPACK<A NAME="19969"></A> routines use a different storage scheme for band matrices,
in which rows of the matrix are stored in corresponding rows of the
array, and diagonals of the matrix are stored in columns of the array
(see Appendix <A HREF="node146.html#chaplineis">D</A>).
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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