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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 3.2 Final//FR"><!-- Converted with LaTeX2HTML 95.1 (Fri Jan 20 1995) --><!-- by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><!-- Modified Simulog 03/97 --><HTML><HEAD><TITLE>1.2 Solution by a direct method</TITLE><LINK REL=STYLESHEET TYPE="text/css" HREF="./Modulef.css" TITLE="Modulef CSS"><meta name="description" value="1.2 Solution by a direct method"><meta name="keywords" value="Guide5"><meta name="resource-type" value="document"><meta name="distribution" value="global"></HEAD><BODY BGCOLOR="#FFFFFF"><P> <IMG SRC="../icons/smallmod.gif" WIDTH=211 HEIGHT=50 ALIGN=BOTTOM ALT="Modulef"><A NAME=tex2html389 HREF="node6.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/previous_motif.gif" ALT="previous"></A><A NAME=tex2html395 HREF="node5.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/up_motif.gif" ALT="up"></A><A NAME=tex2html397 HREF="node8.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/next_motif.gif" ALT="next"></A><A NAME=tex2html399 HREF="node2.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/contents_motif.gif" ALT="contents"></A><A HREF="../Guide5-18/node7.html"><IMG BORDER=0 SRC="../icons/zoom18.gif" ALIGN=BOTTOM ALT="[BIG]"></A><A HREF="../Guide5-14/node7.html"><IMG BORDER=0 SRC="../icons/zoom14.gif" ALIGN=BOTTOM ALT="[Normal]"></A><A HREF="../Guide5-10/node7.html"><IMG BORDER=0 SRC="../icons/zoom10.gif" ALIGN=BOTTOM ALT="[small]"></A><BR><B> Next: </B> <A NAME=tex2html398 HREF="node8.html">1.3 Solution by an iterative method</A><B>Up: </B> <A NAME=tex2html396 HREF="node5.html">1 Solution of linear systems</A><B> Prev: </B> <A NAME=tex2html390 HREF="node6.html">1.1 Introduction</A><B><A HREF="node2.html" >Contents</A></B><HR SIZE=3 WIDTH="75%"><H1><A NAME=SECTION03120000000000000000>1.2 Solution by a direct method</A></H1><P><P><P><H2><A NAME=SECTION03121000000000000000>1.2.1 Introduction</A></H2><P><P><P>The methods presented in this chapter are the three classical solution methodsfor a linear system by matrix factorization<A NAME=397> </A>.We can therefore distinguish between:<P><UL><LI> the <b> Cholesky</b><A NAME=400> </A> method, is used when matrix <b>A</b> of the linear systemto be solved is <b> symmetric positive definite<A NAME=401> </A> <A NAME=402> </A></b> : <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img1.gif">, where<b>L</b> is the lower triangular matrix.<LI> the <b> Crout</b> method<A NAME=404> </A>, is used when the matrix is <b> symmetric<A NAME=405> </A></b> : <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img2.gif">, where <b>L</b> is the lower triangular matrix with unit diagonal, and <b>D</b> is the diagonal matrix.<LI> the <b> Gauss</b><A NAME=407> </A> method, is used for all <b> non-singular<A NAME=408> </A></b> matrices : <b>A=L U</b>, where <b>L</b> is the lower triangular matrix with unit diagonal, and <b>U</b> is the upper triangular matrix.</UL><P>Several versions of these three methods are implemented in the MODULEF code, according to the memory spacenecessary to store matrix <b>A</b>.<P>To start with, we assume that matrix <b>A</b> can be stored entirely in main memory, so that the storage of thematrix and the right-hand-side vector can be described exactly.<P><P><P><H2><A NAME=SECTION03122000000000000000>1.2.2 The MUA data structure</A></H2><P><P><P>For more details consult [<A HREF="node44.html#guide2"><A NAME=tex2html208 HREF="../Guide2/welcome.html">MODULEF User Guide - 2</A></A>].<P><H3><A NAME=SECTION03122100000000000000> Contents</A></H3><P><P><P>This DS stores, for each line, the coefficients lying between the first column with a <em> a priori</em> non-zerocoefficient and the diagonal of a sparse finite element matrix. This type of storage is called <em> profile</em>or <em> skyline</em> storage.<P>DS <b> MUA</b> consists of 6 arrays of predefined order.<P><DL COMPACT><DT>Array MUA0:<DD> General information. <BR><P>This integer array contains 32 variables, consisting of a general descriptionof the job (title, date, name), of DS <b> MUA</b> (type, level, ...), andindicates the presence or absence of array <b> MUA1</b>.<P><DL COMPACT><DT> 1:20 <tt> TITRE</tt><DD><P>the job title in 20 words of 4 characters,<P> </DL><P><DL COMPACT><DT> 21:22 <tt> DATE</tt><DD><P>the date of creation in 2 words of 4 characters,<P> </DL><P><DL COMPACT><DT> 23:28 <tt> NOMCRE</tt><DD><P>the creator's name in 6 words of 4 characters,<P> </DL><P><DL COMPACT><DT> 29 <tt> 'MUA '</tt><DD><P>the DS type,<P> </DL><P><DL COMPACT><DT> 30 <tt> NIVEAU</tt><DD><P>the DS level,<P> </DL><P><DL COMPACT><DT> 31 <tt> ETAT</tt><DD><P>a reserved parameter,<P> </DL><P><DL COMPACT><DT> 32 <tt> NTACM</tt><DD><P>the number of supplementary arrays associated with the DS(they are described in array <b> MUA1</b>).<P> </DL><P><DT>Array MUA1:<DD> Description of any supplementary arrays. <BR><P>This array is analogous to array <b> B1</b> of DS <b> B</b> (see this DS).<P><DT>Array MUA2:<DD> General description of the matrix. <BR><P>This integer array contains 12 values.<P><DL COMPACT><DT> 1 <tt> NTYP</tt><DD><P>the type of matrix coefficients,<P> </DL><P><DL COMPACT><DT> 2 <tt> NBLOC</tt><DD><P>the number of blocks or pages in the matrix,<P> </DL><P><DL COMPACT><DT> 3 <tt> NTCOL</tt><DD><P>the number of columns of the largest block,<P> </DL><P><DL COMPACT><DT> 4 <tt> NMOPB</tt><DD><P>the number of words required in main memory to store the largest block,<P> </DL><P><DL COMPACT><DT> 5 <tt> NMATA</tt><DD><P>the number of matrices on file,<P> </DL><P><DL COMPACT><DT> 6 <tt> NCODSA</tt><DD><P>the type of matrix storage:<UL><LI> 1: symmetric matrix, only the lower triangle is stored, line by line,<LI> 0: diagonal matrix, or<LI> -1: non-symmetric matrix, ordered line after line, with the diagonal coefficient placed at the end. <BR> </UL> </DL><P><DL COMPACT><DT> 7 <tt> LBDP</tt><DD><P>the largest difference + 1 between the 2 node numbers in the same element ifNCODSA is non-zero, 1 if NCODSA=0 i.e. the half band-width in terms of nodes,<P> </DL><P><DL COMPACT><DT> 8 <tt> LBDPDL</tt><DD><P>the largest difference + 1 between the degree of freedom numbers of 2 nodes belonging to the same element ifNCODSA is non-zero, 1 if NCODSA=0 i.e. the half band-width in terms of degrees of freedom,<P> </DL><P><DL COMPACT><DT> 9 <tt> ND</tt><DD><P>the number of degrees of freedom per node if it is constant, 0 otherwise,<P> </DL><P><DL COMPACT><DT> 10 <tt> NOE</tt><DD><P>the number of nodes,<P> </DL><P><DL COMPACT><DT> 11 <tt> NTDL</tt><DD><P>the matrix order,<P> </DL><P><DL COMPACT><DT> 12 <tt> MUA5AR</tt><DD><P>the assembly is already performed if 1,<BR> the assembly is not yet performed if 0.<P> </DL><P><DT>Array MUA3:<DD> Pointer to the last degree of freedom of each block. <BR><P>This integer array , of length NBLOC+1, contains:<P><UL><LI> MUA3(1) = 0,<LI> MUA3(i+1) = the number of the last degree of freedom of block i in the matrix.</UL><P><DT>Array MUA4:<DD> Pointer to the diagonal coefficient of each line. <BR><P>This integer array, of length NTDL+1 or 2 depending on NCODSA, contains:<P><UL><LI> NCODSA = 0<UL><LI> MUA4(1) = 0<LI> MUA4(2) = NTDL,</UL><LI> NCODSA <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img3.gif"> 0<UL><LI> MUA4(1) = 0<LI> MUA4(i+1) = the address of diagonal coefficient i.</UL></UL><P><DT>Array MUA5:<DD> The matrix coefficients. <BR><P>This NTYP type array contains the matrix coefficients:<P><UL><LI> MUA5(k) = value of the k-th matrix coefficient (if it is not paginated),<LI> or MUA5(k-MUA3(i)) the k-th coefficient of block i (if it is paginated).</UL><P> </DL><P><P><H3><A NAME=SECTION03122200000000000000> Example 1: Symmetric matrix</A></H3><P><P><P>Consider the 12-th order matrix given below (the values correspond to the the rows in the storage and not to the matrix coefficients) constituting 3 pages:<P><DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img4.gif"></DIV> <P><P><P><P>Array <b> MUA5</b>, of length MUA4(13), is subdivided into 3 pages, line 1 to 5, 6 to 10 and11 to 12. Each page contains a maximum of 32 words. Thus, we have:<P><UL><LI> NCODSA = 1 <LI> array <b> MUA3</b>:<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img5.gif"></DIV> <P> <LI> array <b> MUA4</b>:<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img6.gif"></DIV> <P> </UL><P><P><P>⌨️ 快捷键说明
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