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<H2><A NAME="SECTION04332000000000000000">Orthogonal Factorizations and Linear Least Squares Problems</A></H2>
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<A NAME="subseccomporthog"> </A>
<P>
ScaLAPACK provides a number of routines for factorizing a general
rectangular <I>m</I>-by-<I>n</I> matrix <I>A</I>,
as the product of an <B>orthogonal</B> matrix (<B>unitary</B> if complex)
and a <B>triangular</B> (or possibly trapezoidal) matrix.
<A NAME="1483"> </A>
<P>
A real matrix <I>Q</I> is <B>orthogonal</B> if <IMG WIDTH=69 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13210" SRC="img112.gif"><A NAME="1486"> </A>;
a complex matrix <I>Q</I> is <B>unitary</B> if <IMG WIDTH=71 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13214" SRC="img113.gif"><A NAME="1489"> </A>.
Orthogonal or unitary matrices<A NAME="1490"> </A><A NAME="1491"> </A> have the important property that they leave the
two-norm of a vector invariant:
<BR><IMG WIDTH=425 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath13200" SRC="img114.gif"><BR>
As a result, they help to maintain numerical stability because they do not
<A NAME="1493"> </A>
amplify rounding errors.
<P>
Orthogonal factorizations<A NAME="1494"> </A><A NAME="1495"> </A><A NAME="1496"> </A> are used in
the solution of linear least squares problems<A NAME="1497"> </A>.
They may also be used to perform preliminary
steps in the solution of eigenvalue or
singular value problems.
<P>
Table <A HREF="node57.html#tabcompof">3.7</A> lists all routines provided by ScaLAPACK to
perform orthogonal factorizations and the generation or pre- or
post-multiplication of the matrix <I>Q</I> for each matrix type and storage
scheme.
<P>
<BR> <HR>
<UL><A NAME="CHILD_LINKS"> </A>
<LI> <A NAME="tex2html2810" HREF="node53.html#SECTION04332100000000000000"><I>QR</I> Factorization</A>
<LI> <A NAME="tex2html2811" HREF="node54.html#SECTION04332200000000000000"><I>LQ</I> Factorization</A>
<LI> <A NAME="tex2html2812" HREF="node55.html#SECTION04332300000000000000"><I>QR</I> Factorization with Column Pivoting</A>
<LI> <A NAME="tex2html2813" HREF="node56.html#SECTION04332400000000000000">Complete Orthogonal Factorization</A>
<LI> <A NAME="tex2html2814" HREF="node57.html#SECTION04332500000000000000">Other Factorizations</A>
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<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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