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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<H1><A NAME="SECTION034120000000000000000"></A>
<A NAME="secGSVDbound"></A>
<BR>
Error Bounds for the Generalized Singular Value Decomposition
</H1>
<P>
The generalized (or quotient) singular value decomposition
of an <B><I>m</I></B>-by-<B><I>n</I></B> matrix
<B><I>A</I></B> and a <B><I>p</I></B>-by-<B><I>n</I></B> matrix <B><I>B</I></B> is the pair of factorizations
<A NAME="12895"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = U \Sigma_1 [0,R] Q^T \; \; {\rm and} \; \;
B = V \Sigma_2 [0,R] Q^T
\end{displaymath}
-->
<IMG
WIDTH="316" HEIGHT="31" BORDER="0"
SRC="img865.png"
ALT="\begin{displaymath}
A = U \Sigma_1 [0,R] Q^T \; \; {\rm and} \; \;
B = V \Sigma_2 [0,R] Q^T
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>U</I></B>, <B><I>V</I></B>, <B><I>Q</I></B>, <B><I>R</I></B>, <IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$\Sigma_1$">
and <IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img68.png"
ALT="$\Sigma_2$">
are defined
as follows.
<UL><LI><B><I>U</I></B> is <B><I>m</I></B>-by-<B><I>m</I></B>, <B><I>V</I></B> is <B><I>p</I></B>-by-<B><I>p</I></B>, <B><I>Q</I></B> is <B><I>n</I></B>-by-<B><I>n</I></B>,
and all three matrices are orthogonal. If <B><I>A</I></B> and
<B><I>B</I></B> are complex, these matrices are unitary instead
of orthogonal, and <B><I>Q</I><SUP><I>T</I></SUP></B> should be
replaced by <B><I>Q</I><SUP><I>H</I></SUP></B> in the pair of factorizations.
<LI><B><I>R</I></B> is <B><I>r</I></B>-by-<B><I>r</I></B>, upper triangular and nonsingular.
<B>[0,<I>R</I>]</B> is <B><I>r</I></B>-by-<B><I>n</I></B>. The integer <B><I>r</I></B> is the rank of
<!-- MATH
$\left( \begin{array}{c} A \\B \end{array} \right)$
-->
<IMG
WIDTH="60" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$ \left( \begin{array}{c}
A \\
B
\end{array} \right) $">,
and satisfies <IMG
WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$r \leq n$">.
<LI><IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$\Sigma_1$">
is <B><I>m</I></B>-by-<B><I>r</I></B>,
<IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img68.png"
ALT="$\Sigma_2$">
is <B><I>p</I></B>-by-<B><I>r</I></B>, both are real, nonnegative and diagonal,
and
<!-- MATH
$\Sigma_1^T \Sigma_1 + \Sigma_2^T \Sigma_2 = I$
-->
<IMG
WIDTH="145" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$\Sigma_1^T \Sigma_1 + \Sigma_2^T \Sigma_2 = I$">.
Write
<!-- MATH
$\Sigma_1^T \Sigma_1 = {\rm diag} ( \alpha_1^2 , \ldots , \alpha_r^2 )$
-->
<IMG
WIDTH="193" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$\Sigma_1^T \Sigma_1 = {\rm diag} ( \alpha_1^2 , \ldots , \alpha_r^2 )$">
and
<!-- MATH
$\Sigma_2^T \Sigma_2 = {\rm diag} ( \beta_1^2 , \ldots , \beta_r^2 )$
-->
<IMG
WIDTH="192" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$\Sigma_2^T \Sigma_2 = {\rm diag} ( \beta_1^2 , \ldots , \beta_r^2 )$">,
where <IMG
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img72.png"
ALT="$\alpha_i$">
and <IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img73.png"
ALT="$\beta_i$">
lie in the interval from 0 to 1.
The ratios
<!-- MATH
$\alpha_1 / \beta_1 , \ldots,, \alpha_r / \beta_r$
-->
<IMG
WIDTH="140" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img866.png"
ALT="$\alpha_1 / \beta_1 , \ldots,, \alpha_r / \beta_r$">
are called the <B>generalized singular values</B> of the pair <B><I>A</I></B>, <B><I>B</I></B>.
If <IMG
WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$\beta_i = 0$">,
then the generalized singular value
<!-- MATH
$\alpha_i / \beta_i$
-->
<IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$\alpha_i / \beta_i$">
is <B>infinite</B>.
For details on the structure of <IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.png"
ALT="$\Sigma_1$">,
<IMG
WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img68.png"
ALT="$\Sigma_2$">
and <B><I>R</I></B>, see
section <A HREF="node36.html#sectionGSVDdriver">2.3.5.3</A>.
</UL>
<P>
The generalized singular value decomposition is
computed by driver routine xGGSVD (see section <A HREF="node36.html#sectionGSVDdriver">2.3.5.3</A>).
<A NAME="12906"></A><A NAME="12907"></A><A NAME="12908"></A><A NAME="12909"></A>
We will give error bounds for the generalized
<A NAME="12910"></A>
singular values in the
common case where
<!-- MATH
$\left( \begin{array}{c} A \\B \end{array} \right)$
-->
<IMG
WIDTH="60" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$ \left( \begin{array}{c}
A \\
B
\end{array} \right) $">
has full
rank <B><I>r</I>=<I>n</I></B>.
Let
<!-- MATH
$\hat{\alpha}_i$
-->
<IMG
WIDTH="21" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img867.png"
ALT="$\hat{\alpha}_i$">
and <IMG
WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img868.png"
ALT="$\hat{\beta}_i$">
be the values of <IMG
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img72.png"
ALT="$\alpha_i$">
and <IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img73.png"
ALT="$\beta_i$">,
respectively,
computed by xGGSVD.
The approximate error
bound<A NAME="footfnm 0"><SUP>4.10</SUP></A>for these values is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
| \hat{\alpha}_i - \alpha_i | +
| \hat{\beta}_i - \beta_i | \leq {\tt SERRBD} \; \; .
\end{displaymath}
-->
<IMG
WIDTH="237" HEIGHT="31" BORDER="0"
SRC="img869.png"
ALT="\begin{displaymath}
\vert \hat{\alpha}_i - \alpha_i \vert +
\vert \hat{\beta}_i - \beta_i \vert \leq {\tt SERRBD} \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Note that if <IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img73.png"
ALT="$\beta_i$">
is close to zero, then a true
generalized singular value
<!-- MATH
$\alpha_i / \beta_i$
-->
<IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$\alpha_i / \beta_i$">
can differ greatly in magnitude from
the computed generalized singular value
<!-- MATH
$\hat{\alpha}_i / \hat{\beta}_i$
-->
<IMG
WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img870.png"
ALT="$\hat{\alpha}_i / \hat{\beta}_i$">,
even if <TT>SERRBD</TT> is
close to its minimum <IMG
WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$\epsilon$">.
<A NAME="12921"></A>
<P>
Here is another way to interpret <TT>SERRBD</TT>:
if we think of <IMG
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img72.png"
ALT="$\alpha_i$">
and <IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img73.png"
ALT="$\beta_i$">
as representing the <EM>subspace</EM> <IMG
WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img318.png"
ALT="$\cal S$">
consisting of the straight line through the origin with slope
<!-- MATH
$\alpha_i / \beta_i$
-->
<IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$\alpha_i / \beta_i$">,
and similarly
<!-- MATH
$\hat{\alpha}_i$
-->
<IMG
WIDTH="21" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img867.png"
ALT="$\hat{\alpha}_i$">
and <IMG
WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img868.png"
ALT="$\hat{\beta}_i$">
representing the subspace <IMG
WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img320.png"
ALT="$\hat{\cal S}$">,
then <IMG
WIDTH="59" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img871.png"
ALT="${\tt SERRBD}$">
bounds the acute angle between
<IMG
WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img318.png"
ALT="$\cal S$">
and <IMG
WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img320.png"
ALT="$\hat{\cal S}$">.
<A NAME="12929"></A>
<A NAME="12930"></A>
Note that any two
lines through the origin with nearly vertical slopes
(very large
<!-- MATH
$\alpha / \beta$
-->
<IMG
WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img872.png"
ALT="$\alpha / \beta$">)
are close together in angle.
(This is related to the <EM>chordal distance</EM> in
section <A HREF="node99.html#secGSEPFurtherDetails">4.10.1</A>.)
<P>
<TT>SERRBD</TT> can be computed by the following code fragment,
which for simplicity assumes <IMG
WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img105.png"
ALT="$m \geq n$">.
(The assumption <B><I>r</I>=<I>n</I></B> implies only that <IMG
WIDTH="84" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img873.png"
ALT="$p+m \geq n$">.
Error bounds can also be computed when
<!-- MATH
$p+m \geq n > m$
-->
<IMG
WIDTH="122" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img874.png"
ALT="$p+m \geq n > m$">,
with slightly more complicated code.)
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Compute generalized singular values of A and B
CALL SGGSVD( 'N', 'N', 'N', M, N, P, K, L, A, LDA, B,
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
$ WORK, IWORK, INFO )
* Compute rank of [A',B']'
RANK = K+L
IF( INFO.GT.0 ) THEN
PRINT *,'SGGSVD did not converge'
ELSE IF( RANK.LT.N ) THEN
PRINT *,'[A**T,B**T]**T not full rank'
ELSE IF ( M .GE. N .AND. N .GT. 0 ) THEN
* Compute reciprocal condition number RCOND of R
CALL STRCON( 'I', 'U', 'N', N, A, LDA, RCOND, WORK, IWORK,
$ INFO )
RCOND = MAX( RCOND, EPSMCH )
SERRBD = EPSMCH / RCOND
END IF
</PRE>
<A NAME="12936"></A>
<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>, if
<!-- MATH
${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
-->
<IMG
WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img397.png"
ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{ccc} 1 & 2 & 3 \\3 & 2 & 1 \\4 & 5 & 6 \\7 & 8 & 8 \end{array} \right)
\; \; {\rm and} \; \;
B = \left( \begin{array}{ccc} -2 & -3 & 3 \\4 & 6 & 5 \end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="349" HEIGHT="93" BORDER="0"
SRC="img875.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 4 & ...
...egin{array}{ccc} -2 & -3 & 3 \\ 4 & 6 & 5 \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then, to 4 decimal places,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\left( \begin{array}{c} \alpha_1 \\\alpha_2 \\\alpha_3 \end{array} \right) =
\left( \begin{array}{c} \hat{\alpha}_1 \\\hat{\alpha}_2 \\\hat{\alpha}_3 \end{array} \right) =
\left( \begin{array}{c} 1.000 \\.7960 \\7.993 \cdot 10^{-2} \end{array} \right)
\; \; {\rm and} \; \;
\left( \begin{array}{c} \beta_1 \\\beta_2 \\\beta_3 \end{array} \right) =
\left( \begin{array}{c} \hat{\beta}_1 \\\hat{\beta}_2 \\\hat{\beta}_3 \end{array} \right) =
\left( \begin{array}{c} 0 \\.6053 \\.9968 \end{array} \right) ,
\end{displaymath}
-->
<IMG
WIDTH="611" HEIGHT="76" BORDER="0"
SRC="img876.png"
ALT="\begin{displaymath}
\left( \begin{array}{c} \alpha_1 \\ \alpha_2 \\ \alpha_3 \en...
...ft( \begin{array}{c} 0 \\ .6053 \\ .9968 \end{array} \right) ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<!-- MATH
${\tt SERRBD} = 1.4 \cdot 10^{-6}$
-->
<IMG
WIDTH="153" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img877.png"
ALT="${\tt SERRBD} = 1.4 \cdot 10^{-6}$">,
and the true errors
are <B>0</B>,
<!-- MATH
$4.3 \cdot 10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img408.png"
ALT="$4.3 \cdot 10^{-7}$">
and
<!-- MATH
$1.5 \cdot 10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img642.png"
ALT="$1.5 \cdot 10^{-7}$">.
<P>
<BR><HR>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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