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    <LI>  if NIVEAU = 3, the complete solution <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img55.gif"> 
   is performed,
   <LI>  if NIVEAU = 4, the backward substitution <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img56.gif"> is performed,
</UL>
<LI> NCLRL: the number of boundary conditions in terms of linear relations,
<LI> NFNDL1:  the number of the support file of I.D.S. NDL1,
<LI> NINDL1: its level number,
<LI> NFMUA: the number of the support file of I.D.S.  MUA,
<LI> NIMUA: its level number,
<LI> NFBDCL: the number of the support file of I.D.S.   BDCL,
<LI> NIBDCL:  its level number,
<LI> NFBE: the number of the support file of I.D.S. B (containing the RHS),
<LI> NIBE:  its level number,
<LI> NFBS:  the number of the support file of O.D.S. B (containing the solution),
<LI> NIBS:  its level number,
<DL COMPACT><DT>Note:
<DD> If NIBS = NIBE, the solution overwrites the RHS !
<P>
 </DL></UL> 
 </DL>
<P>
<H2><A NAME=SECTION041213000000000000000>1.2.13 DRGAPC: Gauss forward- and backsubstitution</A></H2>
<P>
<A NAME=1640>&#160;</A>
<P>
 <DL COMPACT><DT>Aim:
<DD> Performs  the solution of a linear system in main memory,
using Gauss factorization<A NAME=1642>&#160;</A>.
The lower triangular matrix  <b>L</b> and the upper triangular matrix <b>U</b> are already stored in 
structure MUA.
<P>
<DT>Utilization:
<DD>  
<UL><LI> dimension array M (in the blank common),
<LI> call INITI,
<LI> create the input data structures MUA and B (and eventually NDL1),
<LI> execute module GAUSPC, to perform the Gauss factorization of matrix <b>A</b>:
<IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img57.gif"></UL>
<P>
Call the module:
<P>
<PRE>  
      CALL DRGAPC(M,NIVEAU,NCLRL,NFNDL1,NINDL1,
     +            NFMUA,NIMUA,NFBE,NIBE,
     +            NFBDCL,NIBDCL,NFBS,NIBS)
</PRE>
<P>
where:
<P>
<UL><LI> M: the super array,
<LI> NIVEAU: the type of solution chosen:
<UL><LI>  if NIVEAU = 0, only the backward substitution <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img58.gif">  is performed,
    <LI>  if NIVEAU = 1, only the forward substitution <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img51.gif"> is performed,
    <LI>  if NIVEAU = 2, the complete solution <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img59.gif"> 
   is performed,
</UL>
<LI> NCLRL: the number of boundary conditions in terms of linear relations,
<LI> NFNDL1:  the number of the support file of I.D.S. NDL1,
<LI> NINDL1: its level number,
<LI> NFMUA: the number of the support file of I.D.S.  MUA,
<LI> NIMUA: its level number,
<LI> NFBDCL: the number of the support file of I.D.S.   BDCL,
<LI> NIBDCL:  its level number,
<LI> NFBE: the number of the support file of I.D.S. B (containing the RHS),
<LI> NIBE:  its level number,
<LI> NFBS:  the number of the support file of O.D.S. B (containing the solution),
<LI> NIBS:  its level number,
<DL COMPACT><DT>Note:
<DD> If NIBS = NIBE, the solution overwrites the RHS !
<P>
 </DL></UL> 
 </DL>
<P>
<H2><A NAME=SECTION041214000000000000000>1.2.14 FRONT: frontal method</A></H2>
<P>
<A NAME=1653>&#160;</A>
<P>
<DL COMPACT><DT>Aim:
<DD> Assembles the linear system, imposes the boundary conditions, and
performs the solution in main memory using the frontal method.
<P>
<DT>Utilization:
<DD>  
<UL><LI> dimension array M (in the blank common),
<LI> call INITI,
<LI> create file NFFRON using module PREPAF.
</UL>
<P>
Call the module:
<P>
<PRE>  
      CALL FRONT(M,NFFRON,NFTAEM,NITAEM,NFTAES,NITAES,
     +           NFGAUS,NFBDCL,NIBDCL,NFB,NIB,NTB,
     +           NFNDL1,NINDL1,NTYPFR,LF,
     +           NOT1,NOT2,NDSM,EPS,VTG)
</PRE>
<P>
where:
<P>
<UL><LI> M: the super array,
<LI> NFFRON: the number of the data support file for the frontal method
(created by module PREPAF, always <b>&gt;0</b>),
<LI> NFTAEM: the number of the support file of structure TAE containing the element matrices,
<LI> NITAEM: its level number,
<LI> NFTAES: the number of the support file of structure TAE containing the element RHSs
 (can be equal to  NFTAEM)
<LI> NITAES: its level number
 (can be equal to NITAEM),
<LI> NFGAUS: the number of the support file containing the results of the frontal method 
(created by  module FRONT)
<LI> NFBDCL: the number of the support file of I.D.S. BDCL,
<LI> NIBDCL: its level number,
<LI> NFB:  the number of the support file of I.D.S. B (containing the RHS),
<LI> NIB: its level number,
<LI> NTB: the number of associated arrays,
<LI> NFNDL1: the number of the support file of I.D.S. NDL1,
<LI> NINDL1: its level number, 
<LI> NTYPFR: the number of frontal types,
<LI> LF: integer array of length NTYPFR, (created by PREPAF)
<LI> NOT1: the number of the element matrix array in structure  TAE,
<LI> NOT2: the number of the element RHS array in structure TAE,
<LI> NDSM: the RHSs,
<LI> EPS: the minimum value allowed for an element in the frontal matrix
(in absolute value),
<LI> VTG:  multiplication value of the diagonal coefficients with prescribed degrees of freedom.
<DL COMPACT><DT>Note:
<DD> Parameters NTYPFR and LF (array of length NTYPFR) must be input by the user
    if modules     <b> PREPAF</b> and <b> FRONT</b> are not executed during the same job.
<P>
 </DL></UL> 
 </DL>
<P>
<H2><A NAME=SECTION041215000000000000000>1.2.15 GAUSPC:  Gauss factorization</A></H2>
<P>
<A NAME=1665>&#160;</A>
<P>
<DL COMPACT><DT>Aim:
<DD> Perform the Gauss factorization,  in main memory, of
a non-singular matrix <b>A</b>, already 
assembled in structure MUA.
<P>
<DT>Utilization:
<DD>  
<UL><LI> dimension array M (in the blank common),
<LI> call INITI,
<LI> create the input data structure MUA (and eventually NDL1).
</UL>
(eg  execute module PREPAC, then module ASSMUA to create the input data structure MUA)
<P>
Call the module:
<P>
<PRE>  
      CALL GAUSPC(M,EPS,NENTRE,NFNDL1,NINDL1,
     +            NFMUAE,NIMUAE,NFMUAS,NIMUAS,NRETOU)
</PRE>
<P>
where:
<P>
<UL><LI> M: the super array,
<LI> EPS: <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img46.gif">, the minimum value accepted for a pivot <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img35.gif"> according to the criterion
<IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img60.gif">
<LI> NENTRE: if NENTRE = 0 the module returns to the calling program as soon as a pivot is smaller that
 EPS, or continue the computations if  NENTRE = 1,
<LI> NFNDL1:  the number of the support file of I.D.S. NDL1,
<LI> NINDL1: its level number,
<LI> NFMUAE:  the number of the support file of I.D.S.  MUA,
<LI> NIMUAE: its level number,
<LI> NFMUAS:  the number of the support file of O.D.S. MUA,
<LI> NIMUAS: its level number (the O.D.S. can be  distinct from the I.D.S. (NIMUAS <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img3.gif"> NIMUAE), or overwrite it
(NIMUAS = NIMUAE)),
<LI> NRETOU: output parameter: NRETOU = 1 if at least one pivot was judged too small,
 NRETOU = 0 if not.
</UL> 
 </DL>
<P>
<H2><A NAME=SECTION041216000000000000000>1.2.16 PREPAC: computation of pointers for MUA</A></H2>
<P>
<A NAME=1676>&#160;</A>
<P>
<DL COMPACT><DT>Aim:
<DD> Determine the pointers of matrix <b>A</b>,  and compute the sizes of data structures MUA and B.
<P>
<DT>Utilization:
<DD>  
<UL><LI> dimension array M (in the blank common),
<LI> call INITI,
<LI> create the input data structure MAIL and, eventually,  NDL1 and BDCL.
<P>
</UL>
<P>
Call the module:
<P>
<PRE>  
      CALL PREPAC(M,NNOMAX,ND,NDSM,NCLRL,NCODSA,NTY,
     +            NFMAIL,NIMAIL,NFNDL1,NINDL1,
     +            NFBDCL,NIBDCL,NFMUA,NIMUA,NTMUA,
     +            LBDP,LBDPDL,MCNDL1,MCMUA,MCB)
</PRE>
<P>
where:
<P>
<UL><LI> M:  the super array,
<LI> NNOMAX: the maximum number of nodes among all the element types,
<LI> ND: the number of degrees of freedom at each node, if constant (0 if not)
<LI> NDSM: the number of RHSs,
<LI> NCLRL: the number of boundary conditions in terms of linear relations,
<LI> NCODSA: the matrix storage code,
<LI>NCODSA  <b>&gt;</b> 0: symmetric storage of the lower triangular part,
<LI>NCODSA  = 0:  diagonal storage,
<LI>NCODSA  <b>&lt;</b> 0: non-symmetric storage of the entire matrix,
<LI> NTY: the type of arrays MUA5 and B4 to assemble,
<LI> NFMAIL: the number of the support file of I.D.S. MAIL,
<LI> NIMAIL: its level number,
<LI> NFNDL1: the number of the support file of I.D.S. NDL1,
<LI> NINDL1: its level number,
<LI> NFBDCL: the number of the support file of I.D.S. BDCL,
<LI> NIBDCL: its level number,
<LI> NFMUA: the number of the support file of O.D.S.  MUA,
<LI> NIMUA: its level number,
<LI> NTMUA: the number of associated arrays,
<LI> LBDP: output parameter: the largest difference between the node numbers of any element,
<LI> LBDPL: output parameter: the largest difference between the
degree of freedom numbers of any element,
<LI> MCNDL1: the number of words in structure NDL1,
<LI> MCMUA: the number of words in structure MUA,
<LI> MCB: the number of words in structure B.
</UL> 
 </DL>
<P>
<H2><A NAME=SECTION041217000000000000000>1.2.17 PREPAF: computation of pointers for the frontal method</A></H2>
<P>
<A NAME=1684>&#160;</A>  <A NAME=1685>&#160;</A>
<P>
<DL COMPACT><DT>Aim:
<DD> Determine the pointers of matrix <b>A</b> and RHSs <b>B</b>,  and compute the memory 
space required for the solution of a linear system by the frontal method.
<P>
<DT>Utilization:
<DD>  
<UL><LI> dimension array M (in the blank common),
<LI> call INITI,
<LI> create the input data structure MAIL.
<P>
</UL>
<P>
Call the module:
<P>
<PRE>                                               
      CALL PREPAF(M,NFMAIL,NIMAIL,NFFRON,LFRON,NTYPFR,LF)
</PRE>
<P>
where:
<P>
<UL><LI> M: the super array,
<LI> NFMAIL: the number of the support file of I.D.S.  MAIL
<LI> NIMAIL: its level number,
<LI> NFFRON: the number of the support file containing the results (always <b>&gt;</b> 0)
<LI> LFRON: the  length of the front expected by the user,
<LI> NTYFR: the number of front types (i.e. the type of nodes in the mesh),
<LI> LF: integer array of length NTYFR,
<UL><LI> LF(I) contains the length of the  Ith front. If, during execution,
 one of these values exceed LFRON, the program stops and prints a message.
</UL></UL> 
 </DL>
<P>
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