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<H2><A NAME="SECTION0004167000000000000000">SLA_SVD - Singular Value Decomposition</A>
<A NAME="xref_SLA_SVD"> </A><A NAME="SLA_SVD"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Singular value decomposition.
This routine expresses a given matrix <B>A</B> as the product of
three matrices <B>U</B>, <B>W</B>, <B>V</B><SUP><I>T</I></SUP>:
<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 <IMG WIDTH="48" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img227.gif"
ALT="$m \geq n$">
<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>
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_SVD (M, N, MP, NP, A, W, V, WORK, JSTAT)</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>A</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>A</B></TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>A</EM></TD>
<TH ALIGN="LEFT"><B>D(MP,NP)</B></TH>
<TH ALIGN="LEFT" NOWRAP>array containing <IMG WIDTH="46" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img228.gif"
ALT="$m \times n$"> column-orthogonal
matrix <B>U</B></TH>
</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> (<I>n.b.</I> not <B>V</B><SUP><I>T</I></SUP>)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>WORK</EM></TD>
<TD ALIGN="LEFT"><B>D(N)</B></TD>
<TD ALIGN="LEFT" NOWRAP>workspace</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>JSTAT</EM></TD>
<TD ALIGN="LEFT"><B>I</B></TD>
<TD ALIGN="LEFT" NOWRAP>0 = OK, -1 = array A wrong shape, >0 = index of W
for which convergence failed (see note 3, below)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>NOTES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>M and N are the <I>logical</I> dimensions of the
matrices and vectors concerned, which can be located in
arrays of larger <I>physical</I> dimensions, given by MP and NP.
<DT>2.
<DD>V contains matrix V, not the transpose of matrix V.
<DT>3.
<DD>If the status JSTAT is greater than zero, this need not
necessarily be treated as a failure. It means that, due to
chance properties of the matrix A, the QR transformation
phase of the routine did not fully converge in a predefined
number of iterations, something that very seldom occurs.
When this condition does arise, it is possible that the
elements of the diagonal matrix W have not been correctly
found. However, in practice the results are likely to
be trustworthy. Applications should report the condition
as a warning, but then proceed normally.
</DL></DL>
<P> <DL>
<DT><STRONG>REFERENCES:</STRONG>
<DD>The algorithm is an adaptation of the routine SVD in the <I>EISPACK</I>
library (Garbow <I>et al.</I> 1977, <I>EISPACK Guide Extension</I>,
Springer Verlag), which is a FORTRAN 66 implementation of the Algol
routine SVD of Wilkinson & Reinsch 1971 (<I>Handbook for Automatic
Computation</I>, vol 2, ed Bauer <I>et al.</I>, Springer Verlag). These
references give full details of the algorithm used here.
A good account of the use of SVD in least squares problems is given
in <I>Numerical Recipes</I> (Press <I>et al.</I> 1987, Cambridge
University Press), which includes another variant of the EISPACK code.
</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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