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| author | Joseph Hunkeler <jhunkeler@gmail.com> | 2015-03-04 21:21:30 -0500 |
|---|---|---|
| committer | Joseph Hunkeler <jhunkeler@gmail.com> | 2015-03-04 21:21:30 -0500 |
| commit | d54fe7c1f704a63824c5bfa0ece65245572e9b27 (patch) | |
| tree | afc52015ffc2c74e0266653eecef1c8ef8ba5d91 /src/slalib/sun67.htx/node180.html | |
| download | calfuse-d54fe7c1f704a63824c5bfa0ece65245572e9b27.tar.gz | |
Initial commit
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| -rw-r--r-- | src/slalib/sun67.htx/node180.html | 184 |
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diff --git a/src/slalib/sun67.htx/node180.html b/src/slalib/sun67.htx/node180.html new file mode 100644 index 0000000..2372b60 --- /dev/null +++ b/src/slalib/sun67.htx/node180.html @@ -0,0 +1,184 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN"> +<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997) + by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds +* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan +* with significant contributions from: + Jens Lippman, Marek Rouchal, Martin Wilck and others --> +<HTML> +<HEAD> +<TITLE>SLA_SVD - Singular Value Decomposition</TITLE> +<META NAME="description" CONTENT="SLA_SVD - Singular Value Decomposition"> +<META NAME="keywords" CONTENT="sun67"> +<META NAME="resource-type" CONTENT="document"> +<META NAME="distribution" CONTENT="global"> +<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso_8859_1"> +<LINK REL="STYLESHEET" HREF="sun67.css"> +<LINK REL="next" HREF="node181.html"> +<LINK REL="previous" HREF="node179.html"> +<LINK REL="up" HREF="node13.html"> +<LINK REL="next" HREF="node181.html"> +</HEAD> +<BODY > +<BR> <HR> +<A NAME="tex2html2226" HREF="node181.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2224" HREF="node13.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2218" HREF="node179.html"> +<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0" + SRC="contents_motif.gif"></A> +<BR> +<B> Next:</B> <A NAME="tex2html2227" HREF="node181.html">SLA_SVDCOV - Covariance Matrix from SVD</A> +<BR> +<B>Up:</B> <A NAME="tex2html2225" HREF="node13.html">SUBPROGRAM SPECIFICATIONS</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2219" HREF="node179.html">SLA_SUPGAL - Supergalactic to Galactic</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<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> +<BR> <HR> +<A NAME="tex2html2226" HREF="node181.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2224" HREF="node13.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2218" HREF="node179.html"> +<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0" + SRC="contents_motif.gif"></A> +<BR> +<B> Next:</B> <A NAME="tex2html2227" HREF="node181.html">SLA_SVDCOV - Covariance Matrix from SVD</A> +<BR> +<B>Up:</B> <A NAME="tex2html2225" HREF="node13.html">SUBPROGRAM SPECIFICATIONS</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2219" HREF="node179.html">SLA_SUPGAL - Supergalactic to Galactic</A> +<BR> <HR> <P> +<P><!--End of Navigation Panel--> +<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> +</ADDRESS> +</BODY> +</HTML> |