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<H2><A NAME="SECTION0004169000000000000000">SLA_SVDSOL - Solution Vector from SVD</A>
<A NAME="xref_SLA_SVDSOL"> </A><A NAME="SLA_SVDSOL"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>From a given vector and the SVD of a matrix (as obtained from
the sla_SVD routine), obtain the solution vector.
This routine solves the equation:
<PRE><TT>
<B>A</B> <IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"> <B>x</B> = <B>b</B>
</TT></PRE>
where:
<PRE><TT>
<B>A</B> is a given <I>m</I> (rows) <IMG WIDTH="25" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img226.gif"
ALT="$\times n$"> (columns) matrix, where <IMG WIDTH="48" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img227.gif"
ALT="$m \geq n$">
<B>x</B> is the <I>n</I>-vector we wish to find, and
<B>b</B> is a given <I>m</I>-vector
</TT></PRE>
by means of the <I>Singular Value Decomposition</I> method (SVD).
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_SVDSOL (M, N, MP, NP, B, U, W, V, WORK, X)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>M,N</EM></TD>
<TH ALIGN="LEFT"><B>I</B></TH>
<TH ALIGN="LEFT" NOWRAP><I>m</I>, <I>n</I>, the numbers of rows and columns in matrix <B>A</B></TH>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>MP,NP</EM></TD>
<TD ALIGN="LEFT"><B>I</B></TD>
<TD ALIGN="LEFT" NOWRAP>physical dimensions of array containing matrix <B>A</B></TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>B</EM></TD>
<TD ALIGN="LEFT"><B>D(M)</B></TD>
<TD ALIGN="LEFT" NOWRAP>known vector <B>b</B></TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>U</EM></TD>
<TD ALIGN="LEFT"><B>D(MP,NP)</B></TD>
<TD ALIGN="LEFT" NOWRAP>array containing <IMG WIDTH="46" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img228.gif"
ALT="$m \times n$"> matrix <B>U</B></TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>W</EM></TD>
<TD ALIGN="LEFT"><B>D(N)</B></TD>
<TD ALIGN="LEFT" NOWRAP><IMG WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img73.gif"
ALT="$n \times n$"> diagonal matrix <B>W</B>
(diagonal elements only)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>V</EM></TD>
<TD ALIGN="LEFT"><B>D(NP,NP)</B></TD>
<TD ALIGN="LEFT" NOWRAP>array containing <IMG WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img73.gif"
ALT="$n \times n$"> orthogonal matrix <B>V</B></TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>WORK</EM></TD>
<TH ALIGN="LEFT"><B>D(N)</B></TH>
<TD ALIGN="LEFT" NOWRAP>workspace</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>X</EM></TD>
<TD ALIGN="LEFT"><B>D(N)</B></TD>
<TD ALIGN="LEFT" NOWRAP>unknown vector <B>x</B></TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>NOTES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>In the Singular Value Decomposition method (SVD),
the matrix <B>A</B> is first factorized (for example by
the routine sla_SVD) into the following components:
<PRE><TT>
<B>A</B> = <B>U</B> <IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"> <B>W</B> <IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"> <B>V</B><SUP><I>T</I></SUP>
</TT></PRE>
where:
<PRE><TT>
<B>A</B> is any <I>m</I> (rows) <IMG WIDTH="25" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img226.gif"
ALT="$\times n$"> (columns) matrix, where <I>m</I> > <I>n</I>
<B>U</B> is an <IMG WIDTH="46" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img228.gif"
ALT="$m \times n$"> column-orthogonal matrix
<B>W</B> is an <IMG WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img73.gif"
ALT="$n \times n$"> diagonal matrix with <IMG WIDTH="54" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img229.gif"
ALT="$w_{ii} \geq 0$">
<B>V</B><SUP><I>T</I></SUP> is the transpose of an <IMG WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img73.gif"
ALT="$n \times n$"> orthogonal matrix
</TT></PRE>
Note that <I>m</I> and <I>n</I> are the <I>logical</I> dimensions of the
matrices and vectors concerned, which can be located in
arrays of larger <I>physical</I> dimensions MP and NP.
The solution is then found from the expression:
<PRE><TT>
<B>x</B> = <B>V</B> <IMG WIDTH="71" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img230.gif"
ALT="$\cdot~[diag(1/$"><B>W</B><IMG WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img231.gif"
ALT="$_{j})]
\cdot ($"><B>U</B><IMG WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img70.gif"
ALT="$^{T}\cdot$"><B>b</B>)
</TT></PRE>
<DT>2.
<DD>If matrix <B>A</B> is square, and if the diagonal matrix <B>W</B> is not
altered, the method is equivalent to conventional solution
of simultaneous equations.
<DT>3.
<DD>If <I>m</I> > <I>n</I>, the result is a least-squares fit.
<DT>4.
<DD>If the solution is poorly determined, this shows up in the
SVD factorization as very small or zero <B>W</B><SUB><I>j</I></SUB> values. Where
a <B>W</B><SUB><I>j</I></SUB> value is small but non-zero it can be set to zero to
avoid ill effects. The present routine detects such zero
<B>W</B><SUB><I>j</I></SUB> values and produces a sensible solution, with highly
correlated terms kept under control rather than being allowed
to elope to infinity, and with meaningful values for the
other terms.
</DL></DL>
<P> <DL>
<DT><STRONG>REFERENCE:</STRONG>
<DD><I>Numerical Recipes</I>, section 2.9.
</DL>
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<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
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