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<H3><A NAME="SECTION03242100000000000000">
<I>QR</I> Factorization</A>
</H3>
<P>
The most
common, and best known, of the factorizations
is the <B><I>QR</I></B> <B>factorization</B><A NAME="2569"></A>
given by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = Q\left( \begin{array}{c}R\\0\end{array}\right), \quad \mbox{if $m \ge n$,}
\end{displaymath}
-->
<IMG
WIDTH="204" HEIGHT="54" BORDER="0"
SRC="img102.png"
ALT="\begin{displaymath}
A = Q\left( \begin{array}{c}R\\ 0\end{array}\right), \quad \mbox{if $m \ge n$,}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>R</I></B> is an <B><I>n</I></B>-by-<B><I>n</I></B> upper triangular matrix and <B><I>Q</I></B> is an <B><I>m</I></B>-by-<B><I>m</I></B>
orthogonal (or unitary) matrix. If <B><I>A</I></B> is of full rank <B><I>n</I></B>, then <B><I>R</I></B> is
non-singular.
It is sometimes convenient to write the factorization as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{cc} Q_1 & Q_2\end{array} \right)
\left( \begin{array}{c}R\\0\end{array}\right)
\end{displaymath}
-->
<IMG
WIDTH="183" HEIGHT="54" BORDER="0"
SRC="img103.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{cc} Q_1 & Q_2\end{array} \right)
\left( \begin{array}{c}R\\ 0\end{array}\right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which reduces to
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = Q_1 R ,
\end{displaymath}
-->
<B>
<I>A</I> = <I>Q</I><SUB>1</SUB> <I>R</I> ,
</B>
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>Q</I><SUB>1</SUB></B> consists of the first <B><I>n</I></B> columns of <B><I>Q</I></B>, and <B><I>Q</I><SUB>2</SUB></B> the
remaining <B><I>m</I>-<I>n</I></B> columns.
<P>
If <B><I>m</I> < <I>n</I></B>, <B><I>R</I></B> is trapezoidal, and the factorization can be written
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = Q\left( \begin{array}{cc}R_1 & R_2\end{array}\right), \quad
\mbox{if $m < n$,}
\end{displaymath}
-->
<IMG
WIDTH="242" HEIGHT="37" BORDER="0"
SRC="img104.png"
ALT="\begin{displaymath}
A = Q\left( \begin{array}{cc}R_1 & R_2\end{array}\right), \quad
\mbox{if $m < n$,}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>R</I><SUB>1</SUB></B> is upper triangular and <B><I>R</I><SUB>2</SUB></B> is rectangular.
<P>
The routine xGEQRF<A NAME="2584"></A><A NAME="2585"></A><A NAME="2586"></A><A NAME="2587"></A>
computes the <B><I>QR</I></B> factorization. The matrix <B><I>Q</I></B> is not
formed explicitly, but is represented as a product of elementary reflectors,
<A NAME="2588"></A>
<A NAME="2589"></A>
as described in section <A HREF="node128.html#secorthog">5.4</A>.
Users need not be aware of the details of this representation,
because associated routines are provided to work with <B><I>Q</I></B>:
xORGQR<A NAME="2591"></A><A NAME="2592"></A> (or xUNGQR<A NAME="2593"></A><A NAME="2594"></A>
in the complex case) can generate all or part of <B><I>Q</I></B>,
while xORMQR<A NAME="2595"></A><A NAME="2596"></A> (or xUNMQR<A NAME="2597"></A><A NAME="2598"></A>) can pre- or post-multiply
a given matrix by <B><I>Q</I></B> or <B><I>Q</I><SUP><I>T</I></SUP></B>
(<B><I>Q</I><SUP><I>H</I></SUP></B> if complex).
<P>
The <B><I>QR</I></B> factorization can be used to solve the linear least squares
problem (<A HREF="node27.html#llsq">2.1</A>)<A NAME="2600"></A> when <IMG
WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img105.png"
ALT="$m \geq n$">
and
<B><I>A</I></B> is of full rank, since
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\|b - Ax\|_2 = \| Q^T b - Q^T A x\|_2 =
\left\|\begin{array}{c} c_1 - Rx \\c_2 \end{array}\right \|_2, \quad
\mbox{where } c \equiv
\left( \begin{array}{c} c_1 \\c_2 \end{array} \right) =
\left( \begin{array}{c} Q_1^T b \\Q_2^T b \end{array} \right) =
Q^T b;
\end{displaymath}
-->
<IMG
WIDTH="652" HEIGHT="55" BORDER="0"
SRC="img106.png"
ALT="\begin{displaymath}
\Vert b - Ax\Vert _2 = \Vert Q^T b - Q^T A x\Vert _2 =
\left...
...egin{array}{c} Q_1^T b \\ Q_2^T b \end{array} \right) =
Q^T b;
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<B><I>c</I></B> can be computed by xORMQR<A NAME="2611"></A><A NAME="2612"></A> (or xUNMQR
<A NAME="2613"></A><A NAME="2614"></A>), and <B><I>c</I><SUB>1</SUB></B> consists of its first
<B><I>n</I></B> elements. Then
<B><I>x</I></B> is the solution of the upper triangular system
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Rx = c_1
\end{displaymath}
-->
<B>
<I>Rx</I> = <I>c</I><SUB>1</SUB>
</B>
</DIV>
<BR CLEAR="ALL">
<P></P>
which can be computed by xTRTRS<A NAME="2615"></A><A NAME="2616"></A><A NAME="2617"></A><A NAME="2618"></A>.
The residual vector <B><I>r</I></B> is given by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
r = b - A x = Q \left( \begin{array}{c} 0 \\c_2 \end{array} \right) ,
\end{displaymath}
-->
<IMG
WIDTH="186" HEIGHT="54" BORDER="0"
SRC="img107.png"
ALT="\begin{displaymath}
r = b - A x = Q \left( \begin{array}{c} 0 \\ c_2 \end{array} \right) ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and may be computed using xORMQR<A NAME="2622"></A><A NAME="2623"></A> (or xUNMQR
<A NAME="2624"></A><A NAME="2625"></A>).
The residual sum of squares <B>|r|<SUB>2</SUB><SUP>2</SUP></B> may be computed without forming <B><I>r</I></B>
explicitly, since
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\|r\|_2 = \|b - Ax\|_2 = \|c_2\|_2.
\end{displaymath}
-->
<B>
|r|<SUB>2</SUB> = |b - <I>Ax</I>|<SUB>2</SUB> = |c<SUB>2</SUB>|<SUB>2</SUB>.
</B>
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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