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<H2><A NAME="SECTION03246000000000000000"></A><A NAME="subseccompsvd"></A>
<BR>
Singular Value Decomposition
</H2>

<P>
Let <B><I>A</I></B> be a general real <B><I>m</I></B>-by-<B><I>n</I></B> matrix. The <B>singular value
decomposition (SVD)</B> of <B><I>A</I></B> is the factorization<A NAME="3410"></A>  
<!-- MATH
 $A=U \Sigma V^T$
 -->
<IMG
 WIDTH="92" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img157.png"
 ALT="$A=U \Sigma V^T$">,
where
<B><I>U</I></B> and <B><I>V</I></B> are orthogonal, and

<!-- MATH
 $\Sigma = {\mbox {\rm diag}}( \sigma_1 , \ldots , \sigma_r )$
 -->
<IMG
 WIDTH="159" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img158.png"
 ALT="$\Sigma = {\mbox {\rm diag}}( \sigma_1 , \ldots , \sigma_r )$">,

<!-- MATH
 $r = \min (m,n)$
 -->
<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img159.png"
 ALT="$r = \min (m,n)$">,
with 
<!-- MATH
 $\sigma_1 \geq \cdots \geq \sigma_r \geq 0$
 -->
<IMG
 WIDTH="138" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img160.png"
 ALT="$\sigma_1 \geq \cdots \geq \sigma_r \geq 0$">.
If <B><I>A</I></B> is complex, then
its SVD is 
<!-- MATH
 $A=U \Sigma V^H$
 -->
<IMG
 WIDTH="94" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img161.png"
 ALT="$A=U \Sigma V^H$">
where <B><I>U</I></B> and <B><I>V</I></B> are unitary,
and <IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.png"
 ALT="$\Sigma$">
is as before with real
diagonal elements.
The <IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img36.png"
 ALT="$\sigma _ i $">
are called the <B>singular values</B><A NAME="3412"></A>,
the first <B><I>r</I></B> columns of <B><I>V</I></B>
the <B>right singular vectors</B><A NAME="3414"></A> and
the first <B><I>r</I></B> columns of <B><I>U</I></B>
the <B>left singular vectors</B><A NAME="3416"></A>.

<P>
The routines described in this section, and listed in Table <A HREF="node53.html#tabcompsvd">2.12</A>,
are used to compute this decomposition.
The computation proceeds in the following stages:

<P>
<DL COMPACT>
<DT>1.
<DD>The matrix <B><I>A</I></B> is reduced to bidiagonal<A NAME="3419"></A>
form: <B><I>A</I>=<I>U</I><SUB>1</SUB> <I>B V</I><SUB>1</SUB><SUP><I>T</I></SUP></B> if
<B><I>A</I></B> is real (<B><I>A</I>=<I>U</I><SUB>1</SUB> <I>B V</I><SUB>1</SUB><SUP><I>H</I></SUP></B> if <B><I>A</I></B> is complex), where <B><I>U</I><SUB>1</SUB></B> and <B><I>V</I><SUB>1</SUB></B>
are orthogonal (unitary if <B><I>A</I></B> is complex), and <B><I>B</I></B> is real and
upper-bidiagonal when <IMG
 WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img105.png"
 ALT="$m \geq n$">
and lower bidiagonal when <B><I>m</I> &lt; <I>n</I></B>, so
that <B><I>B</I></B> is nonzero only on the main diagonal and either on the first
superdiagonal (if <IMG
 WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img105.png"
 ALT="$m \geq n$">)
or the first subdiagonal (if <B><I>m</I>&lt;<I>n</I></B>).
<P>
<DT>2.
<DD>The SVD of the bidiagonal matrix <B><I>B</I></B> is computed: 
<!-- MATH
 $B=U_2 \Sigma V_2^T$
 -->
<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img162.png"
 ALT="$B=U_2 \Sigma V_2^T$">,
where <B><I>U</I><SUB>2</SUB></B> and <B><I>V</I><SUB>2</SUB></B> are orthogonal and <IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.png"
 ALT="$\Sigma$">
is diagonal as described
above. The singular vectors of <B><I>A</I></B> are then <B><I>U</I> = <I>U</I><SUB>1</SUB> <I>U</I><SUB>2</SUB></B> and
<B><I>V</I> = <I>V</I><SUB>1</SUB> <I>V</I><SUB>2</SUB></B>.
</DL>

<P>
The reduction to bidiagonal form is performed by the subroutine xGEBRD,
<A NAME="3421"></A>
<A NAME="3422"></A><A NAME="3423"></A><A NAME="3424"></A><A NAME="3425"></A>
 or by xGBBRD
<A NAME="3426"></A><A NAME="3427"></A><A NAME="3428"></A><A NAME="3429"></A>
for a band matrix.

<P>
The routine xGEBRD represents
<B><I>U</I><SUB>1</SUB></B> and <B><I>V</I><SUB>1</SUB></B> in factored form as products of elementary reflectors,
<A NAME="3430"></A>
<A NAME="3431"></A>
as described in section&nbsp;<A HREF="node128.html#secorthog">5.4</A>.
If <B><I>A</I></B> is real,
the matrices <B><I>U</I><SUB>1</SUB></B> and <B><I>V</I><SUB>1</SUB></B> may be computed explicitly using routine xORGBR,
<A NAME="3433"></A><A NAME="3434"></A>
or multiplied by other matrices without forming <B><I>U</I><SUB>1</SUB></B> and <B><I>V</I><SUB>1</SUB></B> using routine xORMBR<A NAME="3435"></A><A NAME="3436"></A>.
If <B><I>A</I></B> is complex, one instead uses xUNGBR<A NAME="3437"></A><A NAME="3438"></A>
and xUNMBR<A NAME="3439"></A><A NAME="3440"></A>, respectively.

<P>
If <B><I>A</I></B> is banded and xGBBRD is used to reduce it to bidiagonal form,
<B><I>U</I><SUB>1</SUB></B> and <B><I>V</I><SUB>1</SUB></B> are determined as products of Givens rotations
<A NAME="3441"></A>, rather than
as products of elementary reflectors. If <B><I>U</I><SUB>1</SUB></B> or <B><I>V</I><SUB>1</SUB></B> is required, it
must be formed explicitly by xGBBRD. xGBBRD uses a vectorizable
algorithm, similar to that used by xSBTRD (see Kaufman&nbsp;[<A
 HREF="node151.html#vbandr">77</A>]).
xGBBRD may be much faster than xGEBRD when the bandwidth is narrow.

<P>
The SVD of the bidiagonal matrix is computed either by subroutine
xBDSQR<A NAME="3443"></A><A NAME="3444"></A><A NAME="3445"></A><A NAME="3446"></A>
or by subroutine
xBDSDC<A NAME="3447"></A><A NAME="3448"></A>.
xBDSQR uses QR iteration when singular vectors are desired
[<A
 HREF="node151.html#demmelkahan">32</A>,<A
 HREF="node151.html#deiftdemmellitomei">23</A>],
and otherwise uses the dqds algorithm [<A
 HREF="node151.html#fernandoparlett">51</A>].
xBDSQR is more accurate than its counterparts in LINPACK and EISPACK:
barring underflow and overflow, it computes all the singular values of
<B><I>B</I></B> to nearly full relative precision, independent of their magnitudes.
It also computes the singular vectors much more accurately. See
section&nbsp;<A HREF="node96.html#secsvd">4.9</A> and
[<A
 HREF="node151.html#demmelkahan">32</A>,<A
 HREF="node151.html#deiftdemmellitomei">23</A>,<A
 HREF="node151.html#fernandoparlett">51</A>] for details.

<P>
xBDSDC uses a variation of Cuppen's divide and conquer algorithm to find
singular values and singular vectors [<A
 HREF="node151.html#jessupsorensen">69</A>,<A
 HREF="node151.html#gueisenstat3">58</A>].
It is much faster than xBDSQR if singular vectors of large matrices are desired.
When singular values only are desired, it uses the same dqds algorithm as xBDSQR
[<A
 HREF="node151.html#fernandoparlett">51</A>].
Divide-and-conquer is not guaranteed to compute singular values to nearly
full relative precision, but in practice xBDSDC is often at least as
accurate as xBDSQR.
xBDSDC represents the singular vector matrices <B><I>U</I><SUB>2</SUB></B> and <B><I>V</I><SUB>2</SUB></B>
in a compressed format requiring only <IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img163.png"
 ALT="$O(n \log n)$">
space instead
of <B><I>n</I><SUP>2</SUP></B>. <B><I>U</I><SUB>2</SUB></B> and <B><I>V</I><SUB>2</SUB></B> may subsequently be generated explicitly
using routine xLASDQ, or multiplied by vectors without forming
them explicitly using routine xLASD0.

<P>
If <IMG
 WIDTH="57" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img164.png"
 ALT="$m \gg n$">,
it may be more efficient to first perform a <B><I>QR</I></B> factorization
of <B><I>A</I></B>, using the routine xGEQRF<A NAME="3455"></A><A NAME="3456"></A><A NAME="3457"></A>
<A NAME="3458"></A>,
and then to compute the SVD of the <B><I>n</I></B>-by-<B><I>n</I></B> matrix <B><I>R</I></B>, since
if <B><I>A</I> = <I>QR</I></B> and 
<!-- MATH
 $R = U \Sigma V^T$
 -->
<IMG
 WIDTH="92" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img165.png"
 ALT="$R = U \Sigma V^T$">,
then the SVD of <B><I>A</I></B> is given by

<!-- MATH
 $A = (QU) \Sigma V^T$
 -->
<IMG
 WIDTH="119" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img166.png"
 ALT="$A = (QU) \Sigma V^T$">.
Similarly, if <IMG
 WIDTH="57" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img167.png"
 ALT="$m \ll n$">,
it may be more efficient to first perform an
<B><I>LQ</I></B> factorization of <B><I>A</I></B>, using xGELQF. These preliminary <B><I>QR</I></B> and <B><I>LQ</I></B>
<A NAME="3459"></A><A NAME="3460"></A><A NAME="3461"></A><A NAME="3462"></A>
factorizations are performed by the drivers
xGESVD <A NAME="3463"></A><A NAME="3464"></A><A NAME="3465"></A><A NAME="3466"></A>
and
xGESDD.<A NAME="3467"></A><A NAME="3468"></A><A NAME="3469"></A><A NAME="3470"></A>

<P>
The SVD may be used to find a minimum norm solution<A NAME="3471"></A> to a (possibly)
rank-deficient linear least squares
<A NAME="3472"></A>
problem (<A HREF="node27.html#llsq">2.1</A>). The effective rank, <B><I>k</I></B>, of <B><I>A</I></B> can be determined as the
number of singular values which exceed a suitable threshold.
Let <IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img168.png"
 ALT="$\hat{\Sigma}$">
be the leading <B><I>k</I></B>-by-<B><I>k</I></B> submatrix of <IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.png"
 ALT="$\Sigma$">,
and
<IMG
 WIDTH="18" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img169.png"
 ALT="$\hat{V}$">
be the matrix consisting of the first <B><I>k</I></B> columns of <B><I>V</I></B>.
Then the solution is given by:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
x = \hat{V} \hat{\Sigma}^{-1} \hat{c}_1
\end{displaymath}
 -->


<IMG
 WIDTH="91" HEIGHT="29" BORDER="0"
 SRC="img170.png"
 ALT="\begin{displaymath}
x = \hat{V} \hat{\Sigma}^{-1} \hat{c}_1
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img117.png"
 ALT="$\hat{c}_1$">
consists of the first <B><I>k</I></B> elements of 
<!-- MATH
 $c = U^T b =
U_{2}^T U_{1}^T b$
 -->
<B><I>c</I> = <I>U</I><SUP><I>T</I></SUP> <I>b</I> =
<I>U</I><SUB>2</SUB><SUP><I>T</I></SUP> <I>U</I><SUB>1</SUB><SUP><I>T</I></SUP> <I>b</I></B>. <B><I>U</I><SUB>1</SUB><SUP><I>T</I></SUP> <I>b</I></B> can be computed using xORMBR, and
<A NAME="3484"></A><A NAME="3485"></A>
xBDSQR has an option to multiply a vector by <B><I>U</I><SUB>2</SUB><SUP><I>T</I></SUP></B>.
<A NAME="3487"></A><A NAME="3488"></A><A NAME="3489"></A><A NAME="3490"></A>

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="tabcompsvd"></A>
<DIV ALIGN="CENTER">
<A NAME="3492"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.12:</STRONG>
Computational routines for the singular value decomposition</CAPTION>
<TR><TD ALIGN="LEFT">Type of matrix</TD>
<TD ALIGN="LEFT">Operation</TD>
<TD ALIGN="CENTER" COLSPAN=2>Single precision</TD>
<TD ALIGN="CENTER" COLSPAN=2>Double precision</TD>
</TR>
<TR><TD ALIGN="LEFT">and storage scheme</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
</TR>
<TR><TD ALIGN="LEFT">general</TD>
<TD ALIGN="LEFT">bidiagonal reduction</TD>
<TD ALIGN="LEFT">SGEBRD<A NAME="3504"></A></TD>
<TD ALIGN="LEFT">CGEBRD<A NAME="3505"></A></TD>
<TD ALIGN="LEFT">DGEBRD<A NAME="3506"></A></TD>
<TD ALIGN="LEFT">ZGEBRD<A NAME="3507"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">general band</TD>
<TD ALIGN="LEFT">bidiagonal reduction</TD>
<TD ALIGN="LEFT">SGBBRD<A NAME="3508"></A></TD>
<TD ALIGN="LEFT">CGBBRD<A NAME="3509"></A></TD>
<TD ALIGN="LEFT">DGBBRD<A NAME="3510"></A></TD>
<TD ALIGN="LEFT">ZGBBRD<A NAME="3511"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">orthogonal/unitary</TD>
<TD ALIGN="LEFT">generate matrix after</TD>
<TD ALIGN="LEFT">SORGBR<A NAME="3512"></A></TD>
<TD ALIGN="LEFT">CUNGBR<A NAME="3513"></A></TD>
<TD ALIGN="LEFT">DORGBR<A NAME="3514"></A></TD>
<TD ALIGN="LEFT">ZUNGBR<A NAME="3515"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">bidiagonal reduction</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix after</TD>
<TD ALIGN="LEFT">SORMBR<A NAME="3516"></A></TD>
<TD ALIGN="LEFT">CUNMBR<A NAME="3517"></A></TD>
<TD ALIGN="LEFT">DORMBR<A NAME="3518"></A></TD>
<TD ALIGN="LEFT">ZUNMBR<A NAME="3519"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">bidiagonal reduction</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">bidiagonal</TD>
<TD ALIGN="LEFT">SVD using</TD>
<TD ALIGN="LEFT">SBDSQR<A NAME="3520"></A></TD>
<TD ALIGN="LEFT">CBDSQR<A NAME="3521"></A></TD>
<TD ALIGN="LEFT">DBDSQR<A NAME="3522"></A></TD>
<TD ALIGN="LEFT">ZBDSQR<A NAME="3523"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">QR or dqds</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">SVD using</TD>
<TD ALIGN="LEFT">SBDSDC<A NAME="3524"></A></TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">DBDSDC<A NAME="3525"></A></TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">divide-and-conquer</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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