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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<TITLE>General Linear Model Problem</TITLE>
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<B> Next:</B> <A NAME="tex2html5411"
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<H2><A NAME="SECTION03462000000000000000">
General Linear Model Problem</A>
</H2>
<P>
The general linear model (GLM) problem is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\min_{x,y} \|y\| \quad \mbox{subject to} \quad d = Ax + By
\end{displaymath}
-->
<IMG
WIDTH="269" HEIGHT="40" BORDER="0"
SRC="img491.png"
ALT="\begin{displaymath}\min_{x,y} \Vert y\Vert \quad \mbox{subject to} \quad d = Ax + By \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>A</I></B> is an <B><I>n</I></B>-by-<B><I>m</I></B> matrix, <B><I>B</I></B> is an <B><I>n</I></B>-by-<B><I>p</I></B> matrix, and <B><I>d</I></B>
is a given <B><I>n</I></B>-vector, with
<!-- MATH
$m \leq n \leq m+p$
-->
<IMG
WIDTH="122" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img21.png"
ALT="$m \leq n \leq m+p$">.
<BR>
<P>
The GLM problem is solved by the driver routine xGGGLM
(see section <A HREF="node84.html#sec_lseglm_drivers">4.6</A>).
Let <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$\widehat {x}$">
and <IMG
WIDTH="14" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img492.png"
ALT="$\widehat {y}$">
be the computed values of <B><I>x</I></B> and <B><I>y</I></B>, respectively.
The approximate error bounds
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="441" HEIGHT="111" BORDER="0"
SRC="img493.png"
ALT="\begin{eqnarray*}
% latex2html id marker 11027\frac{ \Vert x - \widehat {x} \...
...hfil ...">
</DIV><P></P>
<BR CLEAR="ALL">
can be computed by the following code fragment
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Compute the 2-norm of the left hand side D
DNORM = SNRM2( N, D, 1 )
* Solve the generalized linear model problem
CALL SGGGLM( N, M, P, A, LDA, B, LDB, D, Xc, Yc, WORK,
$ LWORK, IWORK, INFO )
* Compute the F-norm of A and B
ANORM = SLANTR( 'F', 'U', 'N', M, M, A, LDA, WORK( M+NP+1 ) )
BNORM = SLANTR( 'F', 'U', 'N', N, P, B( 1, MAX( 1, P-N+1 ) ),
$ LDB, WORK( M+NP+1 ) )
* Compute the 2-norm of Xc
XNORM = SNRM2( M, Xc, 1 )
* Condition estimation
IF( N.EQ.M ) THEN
PBPSNM = ZERO
TNORM = SLANTR( '1', 'U', 'N', N, N, A, LDA, WORK( M+NP+M+1 ) )
CALL STRCON( '1', 'U', 'N', N, A, LDA, RCOND, WORK( M+NP+M+1 ),
$ IWORK, INFO )
ABPSNM = ONE / (RCOND * TNORM )
ELSE
* Compute norm of (PB)^+
TNORM = SLANTR( '1', 'U', 'N', N-M, N-M, B( M+1, P-N+M+1 ), LDB,
$ WORK( M+NP+1 ) )
CALL STRCON( '1', 'U', 'N', N-M, B( M+1, P-N+M+1 ), LDB, RCOND,
$ WORK( M+NP+1 ), IWORK, INFO )
PBPSNM = ONE / (RCOND * TNORM )
* Estimate norm of A^+_B
KASE = 0
CALL SLACON( N, WORK( M+NP+1 ), WORK( M+NP+N+1 ), IWORK, EST, KASE )
30 CONTINUE
CALL STRSV( 'Upper', 'No transpose', 'Non unit', N-M,
$ B( M+1, P-N+M+1 ), LDB, WORK( M+NP+N+M+1 ), 1 )
CALL SGEMV( 'No transpose', M, N-M, -ONE, B( 1, P-N+M+1 ),
$ LDB, WORK( M+NP+N+M+1 ), 1, ONE,
$ WORK( M+NP+N+1 ), 1 )
CALL STRSV( 'Upper', 'No transpose', 'Non unit', M, A, LDA,
$ WORK( M+NP+N+1 ), 1 )
DO I = 1, P
WORK( M+NP+I ) = WORK( M+NP+N+I )
END DO
CALL SLACON( M, WORK( M+NP+N+1 ), WORK( M+NP+1 ), IWORK, EST, KASE )
IF( KASE.EQ.0 ) GOTO 40
CALL STRSV( 'Upper', 'Transpose', 'Non unit', M, A, LDA,
$ WORK( M+NP+1 ), 1 )
CALL SGEMV( 'Transpose', M, N-M, -ONE, B( 1, P-N+M+1 ), LDB,
$ WORK( M+NP+1 ), 1, ZERO, WORK( M+NP+M+1 ), 1 )
CALL STRSV( 'Upper', 'Transpose', 'Non unit', N-M,
$ B( M+1, P-N+M+1 ), LDB, WORK( M+NP+M+1 ), 1 )
DO I = 1, N
WORK( M+NP+N+I ) = WORK( M+NP+I )
END DO
CALL SLACON( N, WORK( M+NP+1 ), WORK( M+NP+N+1 ), IWORK, EST, KASE )
IF( KASE.EQ.0 ) GOTO 40
GOTO 30
40 CONTINUE
ABPSNM = EST
END IF
* Estimate norm of (A^+_B)*B
IF( P+M.EQ.N ) THEN
EST = ZERO
ELSE
KASE = 0
CALL SLACON( P-N+M, WORK( M+NP+1 ), WORK( M+NP+M+1 ), IWORK, EST, KASE )
50 CONTINUE
*
IF( P.GE.N ) THEN
CALL STRMV( 'Upper', 'No trans', 'Non Unit', M,
$ B( 1, P-N+1 ), LDB, WORK( M+NP+M+P-N+1 ), 1 )
DO I = 1, M
WORK( M+NP+I ) = WORK( M+NP+M+P-N+I )
END DO
ELSE
CALL SGEMV( 'No transpose', N-P, P-N+M, ONE, B, LDB,
$ WORK( M+NP+M+1 ), 1, ZERO, WORK( M+NP+1 ), 1 )
CALL STRMV( 'Upper', 'No trans', 'Non Unit', P-N+M,
$ B( N-P+1, 1 ), LDB, WORK( M+NP+M+1 ), 1 )
DO I = N-P+1, M
WORK( M+NP+I ) = WORK( M+NP+M-N+P+I )
END DO
END IF
CALL STRSV( 'Upper', 'No transpose', 'Non unit', M, A, LDA,
$ WORK( M+NP+1 ), 1 )
CALL SLACON( M, WORK( M+NP+M+1 ), WORK( M+NP+1 ), IWORK, EST, KASE )
*
IF( KASE.EQ.0 ) GOTO 60
*
CALL STRSV( 'Upper', 'Transpose', 'Non unit', M, A, LDA,
$ WORK( M+NP+1 ), 1 )
IF( P.GE.N ) THEN
CALL STRMV( 'Upper', 'Trans', 'Non Unit', M,
$ B( 1, P-N+1 ), LDB, WORK( M+NP+1 ), 1 )
DO I = 1, M
WORK( M+NP+M+P-N+I ) = WORK( M+NP+I )
END DO
DO I = 1, P-N
WORK( M+NP+M+I ) = ZERO
END DO
ELSE
CALL STRMV( 'Upper', 'Trans', 'Non Unit', P-N+M,
$ B( N-P+1, 1 ), LDB, WORK( M+NP+N-P+1 ), 1 )
DO I = 1, P-N+M
WORK( M+NP+M+I ) = WORK( M+NP+N-P+I )
END DO
CALL SGEMV( 'Transpose', N-P, P-N+M, ONE, B, LDB,
$ WORK( M+NP+1 ), 1, ONE, WORK( M+NP+M+1 ), 1 )
END IF
CALL SLACON( P-N+M, WORK( M+NP+1 ), WORK( M+NP+M+1 ), IWORK, EST,
$ KASE )
*
IF( KASE.EQ.0 ) GOTO 60
GOTO 50
60 CONTINUE
END IF
ABPSBN = EST
* Get condition numbers and approximate error bounds
CNDAB = ANORM*ABPSNM
CNDBA = BNORM*PBPSNM
IF( PBPSNM.EQ.0.0E+0 ) THEN
* Then A is square and nonsingular
XERRBD = EPSMCH*( CNDAB*( ONE+DNORM/(ANORM*XNORM) ) )
YERRBD = 0.0E+0
ELSE
XERRBD = EPSMCH*( CNDAB*( ONE+DNORM/(ANORM*XNORM) ) +
$ 2.0E0*CNDAB*CNDBA*CNDBA*DNORM/(ANORM*XNORM) +
$ ABPSBN*ABPSBN*PBPSNM*PBPSNM*ANORM*DNORM/XNORM )
YERRBD = EPSMCH*( ABPSBN*ANORM*PBPSNM*PBPSNM +
$ PBPSNM*(ANORM*XNORM/DNORM + 2.0E0*CNDBA*CNDBA +
$ ONE) + CNDBA*PBPSNM )
END IF
</PRE>
<P>
For example, 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{tabular}{rrrr}
1 & 2 & 1 & 4 \\
1 & 3 & 2 & 1 \\
-1 & -2 & -1 & 1 \\
-1 & 2 & -1 & 5 \\
1 & 0 & 0 & 1
\end{tabular} \right), \quad
B = \left( \begin{tabular}{rrr}
1 & 2 & 2 \\
-1 & 1 & -2\\
3 & 1 & 6 \\
1 & -1 & 2 \\
2 & -2 & 4
\end{tabular} \right)
\quad \mbox{and} \quad
d = \left( \begin{array}{r}
1 \\
1 \\
1 \\
1 \\
1
\end{array} \right).
\end{displaymath}
-->
<IMG
WIDTH="544" HEIGHT="115" BORDER="0"
SRC="img494.png"
ALT="\begin{displaymath}A = \left( \begin{tabular}{rrrr}
1 & 2 & 1 & 4 \\
1 & 3 & ...
...array}{r}
1 \\
1 \\
1 \\
1 \\
1
\end{array} \right). \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Then (to 7 decimal places)
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\widehat {x} = \left( \begin{array}{r}
-0.5466667 \\
0.3200001 \\
0.7200000 \\
-0.0533334
\end{array} \right)
\quad \mbox{and} \quad
\widehat {y} = \left( \begin{array}{r}
0.1333334 \\
-0.1333334 \\
0.2666667
\end{array} \right).
\end{displaymath}
-->
<IMG
WIDTH="402" HEIGHT="93" BORDER="0"
SRC="img495.png"
ALT="\begin{displaymath}\widehat {x} = \left( \begin{array}{r}
-0.5466667 \\
0.320...
... 0.1333334 \\
-0.1333334 \\
0.2666667
\end{array} \right).\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The computed error bounds
<!-- MATH
${\tt XERRBD} = 1.2\cdot10^{-5}$
-->
<IMG
WIDTH="153" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img496.png"
ALT="${\tt XERRBD} = 1.2\cdot10^{-5}$">
and
<!-- MATH
${\tt YERRBD} = 6.4\cdot10^{-7}$
-->
<IMG
WIDTH="153" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img497.png"
ALT="${\tt YERRBD} = 6.4\cdot10^{-7}$">,
where
<!-- MATH
${\tt CNDAB} = 26.97$
-->
<IMG
WIDTH="113" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img498.png"
ALT="${\tt CNDAB} = 26.97$">,
<!-- MATH
${\tt CNDBA} = 1.78$
-->
<IMG
WIDTH="104" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img499.png"
ALT="${\tt CNDBA} = 1.78$">,
The true errors in <B><I>x</I></B> and <B><I>y</I></B> are
<!-- MATH
$1.2\cdot10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img500.png"
ALT="$1.2\cdot10^{-7}$">
and
<!-- MATH
$1.3\cdot10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img501.png"
ALT="$1.3\cdot10^{-7}$">,
respectively. Note that the exact solutions are
<!-- MATH
$x = \frac{1}{75}[-41,24,54,-4]^T$
-->
<IMG
WIDTH="187" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img502.png"
ALT="$x = \frac{1}{75}[-41,24,54,-4]^T$">
and
<!-- MATH
$y = \frac{1}{15}[2,-2,4]^T$
-->
<IMG
WIDTH="130" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img503.png"
ALT="$y = \frac{1}{15}[2,-2,4]^T$">.
<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>
<UL>
<LI><A NAME="tex2html5412"
HREF="node88.html">Further Details: Error Bounds for General Linear Model Problems</A>
</UL>
<!--End of Table of Child-Links-->
<BR><HR>
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
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