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diff --git a/src/slalib/sun67.htx/node182.html b/src/slalib/sun67.htx/node182.html new file mode 100644 index 0000000..f99ea3f --- /dev/null +++ b/src/slalib/sun67.htx/node182.html @@ -0,0 +1,201 @@ +<!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_SVDSOL - Solution Vector from SVD</TITLE> +<META NAME="description" CONTENT="SLA_SVDSOL - Solution Vector from SVD"> +<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="node183.html"> +<LINK 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Navigation Panel--> +<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> +<BR> <HR> +<A NAME="tex2html2246" HREF="node183.html"> +<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A> +<A NAME="tex2html2244" HREF="node13.html"> +<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A> +<A NAME="tex2html2238" HREF="node181.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="tex2html2247" HREF="node183.html">SLA_TP2S - Tangent Plane to Spherical</A> +<BR> +<B>Up:</B> <A NAME="tex2html2245" HREF="node13.html">SUBPROGRAM SPECIFICATIONS</A> +<BR> +<B> Previous:</B> <A NAME="tex2html2239" HREF="node181.html">SLA_SVDCOV - Covariance Matrix from SVD</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> |