c----------------------------------------------------------------------c c S P A R S K I T c c----------------------------------------------------------------------c c REORDERING ROUTINES -- STRONGLY CONNECTED COMPONENTS c c----------------------------------------------------------------------c c Contributed by: C Laura C. Dutto - email: dutto@cerca.umontreal.ca c July 1992 - Update: March 1994 C----------------------------------------------------------------------- c CONTENTS: c -------- c blccnx : Driver routine to reduce the structure of a matrix c to its strongly connected components. c cconex : Main routine to compute the strongly connected components c of a (block diagonal) matrix. c anccnx : We put in ICCNEX the vertices marked in the component MCCNEX. c newcnx : We put in ICCNEX the vertices marked in the component c MCCNEX. We modify also the vector KPW. c blccn1 : Parallel computation of the connected components of a c matrix. The parallel loop is performed only if the matrix c has a block diagonal structure. c ccnicopy:We copy an integer vector into anothoer. c compos : We calculate the composition between two permutation c vectors. c invlpw : We calculate the inverse of a permutation vector. c numini : We initialize a vector to the identity. c tbzero : We initialize to ZERO an integer vector. c iplusa : Given two integers IALPHA and IBETA, for an integer vector c IA we calculate IA(i) = ialpha + ibeta * ia(i) C c----------------------------------------------------------------------c subroutine BLCCNX(n, nbloc, nblcmx, nsbloc, job, lpw, amat, ja, * ia, iout, ier, izs, nw) C----------------------------------------------------------------------- c c This routine determines if the matrix given by the structure c IA et JA is irreductible. If not, it orders the unknowns such c that all the consecutive unknowns in KPW between NSBLOC(i-1)+1 c and NSBLOC(i) belong to the ith component of the matrix. c The numerical values of the matrix are in AMAT. They are modified c only if JOB = 1 and if we have more than one connected component. c c On entry: c -------- c n = row and column dimension of the matrix c nblcmx = maximum number of connected components allowed. The size c of NSBLOC is nblcmx + 1 (in fact, it starts at 0). c job = integer indicating the work to be done: c job = 1 if the permutation LPW is modified, we c permute not only the structure of the matrix c but also its numerical values. c job.ne.1 if the permutation LPW is modified, we permute c the structure of the matrix ignoring real values. c iout = impression parameter. If 0 < iout < 100, we print c comments and error messages on unit IOUT. c nw = length of the work vector IZS. c c Input / output: c -------------- c nbloc = number of connected components of the matrix. If the c matrix is not irreductible, nbloc > 1. We allow c nbloc > 1 on entry; in this case we calculate the c number of connected components in each previous one. c nsbloc = integer array of length NBLOC + 1 containing the pointers c to the first node of each component on the old (input) c and on the new (output) ordering. c lpw = integer array of length N corresponding to the c permutation of the unknowns. We allow LPW to be a vector c different from the identity on input. c amat = real*8 values of the matrix given by the structure IA, JA. c ja = integer array of length NNZERO (= IA(N+1)-IA(1)) corresponding c to the column indices of nonzero elements of the matrix, stored c rowwise. It is modified only if the matrix has more c than one connected component. c ia = integer array of length N+1 corresponding to the c pointer to the beginning of each row in JA (compressed c sparse row storage). It is modified only if c the matrix has more than one connected component. c c On return: c ---------- c ier = integer. Error message. Normal return ier = 0. c c Work space: c ---------- c izs = integer vector of length NW c C----------------------------------------------------------------------- C Laura C. Dutto - email: dutto@cerca.umontreal.ca c July 1992 - Update: March 1994 C----------------------------------------------------------------------- integer izs(nw), lpw(n), nsbloc(0:nblcmx), ia(n+1), ja(*) real*8 amat(*) logical impr character*6 chsubr C----------------------------------------------------------------------- ier = 0 impr = iout.gt.0.and.iout.le.99 ntb = ia(n+1) - 1 mxccex = max(nblcmx,20) c.....The matrix AMAT is a real*8 vector ireal = 2 c c.....MXPTBL: maximal number of vertices by block mxptbl = 0 do ibloc = 1, nbloc mxptbl = max( mxptbl, nsbloc(ibloc) - nsbloc(ibloc-1)) enddo c long1 = nbloc * mxptbl long2 = nbloc * (mxccex+1) c.....Dynamic allocation of memory iend = 1 iiend = iend ilpw = iiend ikpw = ilpw + n ilccnx = ikpw + long1 imark = ilccnx + long2 iend = imark + n if(iend .gt. nw) go to 220 c nbloc0 = nbloc chsubr = 'BLCCN1' c.....We determine if the matrix has more than NBLOC0 connected components. call BLCCN1(n, nbloc, nblcmx, nsbloc, izs(ilpw), izs(ikpw), ia, * ja, izs(imark), mxccex, izs(ilccnx), mxptbl, iout, * ier) if(ier.ne.0) go to 210 c if(nbloc .gt. nbloc0) then c..........The matrix has more than NBLOC0 conneted components. So, we c..........modify the vectors IA and JA to take account of the new permutation. nfree = iend - ikpw call tbzero(izs(ikpw), nfree) iiat = ikpw ijat = iiat + n + 1 iamat = ijat + ntb iend = iamat if(job .eq. 1) iend = iamat + ireal * ntb if(iend .gt. nw) go to 220 c c..........We copy IA and JA on IAT and JAT respectively call ccnicopy(n+1, ia, izs(iiat)) call ccnicopy(ntb, ja, izs(ijat)) if(job .eq. 1) call dcopy(ntb, amat, 1, izs(iamat), 1) call dperm(n, izs(iamat), izs(ijat), izs(iiat), amat, * ja, ia, izs(ilpw), izs(ilpw), job) ipos = 1 c..........We sort columns inside JA. call csrcsc(n, job, ipos, amat, ja, ia, izs(iamat), * izs(ijat), izs(iiat)) call csrcsc(n, job, ipos, izs(iamat), izs(ijat), izs(iiat), * amat, ja, ia) endif c.....We modify the ordering of unknowns in LPW call compos(n, lpw, izs(ilpw)) c 120 nfree = iend - iiend call tbzero(izs(iiend), nfree) iend = iiend return c 210 IF(IMPR) WRITE(IOUT,310) chsubr,ier go to 120 220 IF(IMPR) WRITE(IOUT,320) nw, iend if(ier.eq.0) ier = -1 go to 120 c 310 FORMAT(' ***BLCCNX*** ERROR IN ',a6,'. IER = ',i8) 320 FORMAT(' ***BLCCNX*** THERE IS NOT ENOUGH MEMORY IN THE WORK', 1 ' VECTOR.'/13X,' ALLOWED MEMORY = ',I10,' - NEEDED', 2 ' MEMORY = ',I10) end c ********************************************************************** subroutine CCONEX(n, icol0, mxccnx, lccnex, kpw, ia, ja, mark, * iout, ier) C----------------------------------------------------------------------- c c This routine determines if the matrix given by the structure c IA and JA is irreductible. If not, it orders the unknowns such c that all the consecutive unknowns in KPW between LCCNEX(i-1)+1 c and LCCNEX(i) belong to the ith component of the matrix. c The structure of the matrix could be nonsymmetric. c The diagonal vertices (if any) will belong to the last connected c component (convention). c c On entry: c -------- c n = row and column dimension of the matrix c icol0 = the columns of the matrix are between ICOL0+1 and ICOL0+N c iout = impression parameter. If 0 < IOUT < 100, we print c comments and error messages on unit IOUT. c ia = integer array of length N+1 corresponding to the c pointer to the beginning of each row in JA (compressed c sparse row storage). c ja = integer array of length NNZERO (= IA(N+1)-IA(1)) c corresponding to the column indices of nonzero elements c of the matrix, stored rowwise. c c Input/Output: c ------------ c mxccnx = maximum number of connected components allowed on input, c and number of connected components of the matrix, on output. c c On return: c ---------- c lccnex = integer array of length MXCCNX + 1 containing the pointers c to the first node of each component, in the vector KPW. c kpw = integer array of length N corresponding to the c inverse of permutation vector. c ier = integer. Error message. Normal return ier = 0. c c Work space: c ---------- c mark = integer vector of length N c C----------------------------------------------------------------------- C Laura C. Dutto - email: dutto@cerca.umontreal.ca c July 1992 - Update: March 1994 C----------------------------------------------------------------------- dimension ia(n+1), lccnex(0:mxccnx), kpw(n), ja(*), mark(n) logical impr C----------------------------------------------------------------------- ier = 0 ipos = ia(1) - 1 impr = iout.gt.0.and.iout.le.99 c nccnex = 0 c.....We initialize MARK to zero. At the end of the algorithm, it would c.....indicate the number of connected component associated with the vertex. c.....The number (-1) indicates that the row associated with this vertex c.....is a diagonal row. This value could be modified because we accept c.....a non symmetric matrix. All the diagonal vertices will be put in c.....the same connected component. call tbzero(mark, n) c 5 do i = 1,n if(mark(i) .eq. 0) then ideb = i go to 15 endif enddo go to 35 c 15 if( ia(ideb+1) - ia(ideb) .eq. 1) then c..........The row is a diagonal row. mark(ideb) = -1 go to 5 endif iccnex = nccnex + 1 if(iccnex .gt. mxccnx) go to 220 index = 0 newind = 0 jref = 0 mark(ideb) = iccnex index = index + 1 kpw(index) = ideb c 20 jref = jref + 1 ideb = kpw(jref) do 30 ir = ia(ideb)-ipos, ia(ideb+1)-ipos-1 j = ja(ir) - icol0 mccnex = mark(j) if(mccnex .le. 0) then index = index + 1 kpw(index) = j mark(j) = iccnex else if( mccnex .eq. iccnex) then go to 30 else if( mccnex .gt. iccnex) then c.............We realize that the connected component MCCNX is, c.............in fact, included in this one. We modify MARK and KPW. call NEWCNX(n, mccnex, iccnex, index, kpw, mark) if(mccnex .eq. nccnex) nccnex = nccnex - 1 else c.............We realize that the previously marked vertices belong, c.............in fact, to the connected component ICCNX. We modify MARK. call ANCCNX(n, iccnex, mccnex, mark, nwindx) iccnex = mccnex newind = newind + nwindx endif 30 continue if(jref .lt. index) go to 20 c c.....We have finished with this connected component. index = index + newind if(iccnex .eq. nccnex+1) nccnex = nccnex + 1 go to 5 c....................................................................... c c We have partitioned the graph in its connected components! c c....................................................................... 35 continue c c.....All the vertices have been already marked. Before modifying KPW c.....(if necessary), we put the diagonal vertex (if any) in the last c.....connected component. call tbzero(lccnex(1), nccnex) c idiag = 0 do i = 1, n iccnex = mark(i) if(iccnex .eq. -1) then idiag = idiag + 1 if(idiag .eq. 1) then nccnex = nccnex + 1 if(nccnex .gt. mxccnx) go to 220 if(impr) write(iout,340) endif mark(i) = nccnex else lccnex(iccnex) = lccnex(iccnex) + 1 endif enddo if(idiag .ge. 1) lccnex(nccnex) = idiag c if(nccnex .eq. 1) then lccnex(nccnex) = n go to 40 endif c iccnex = 1 8 if(iccnex .gt. nccnex) go to 12 if(lccnex(iccnex) .le. 0) then do i = 1, n if(mark(i) .ge. iccnex) mark(i) = mark(i) - 1 enddo nccnex = nccnex - 1 do mccnex = iccnex, nccnex lccnex(mccnex) = lccnex(mccnex + 1) enddo else iccnex = iccnex + 1 endif go to 8 c 12 index = 0 do iccnex = 1, nccnex noeicc = lccnex(iccnex) lccnex(iccnex) = index index = index + noeicc enddo if(index .ne. n) go to 210 c c.....We define correctly KPW do i = 1,n iccnex = mark(i) index = lccnex(iccnex) + 1 kpw(index) = i lccnex(iccnex) = index enddo c 40 mxccnx = nccnex lccnex(0) = nccnex if(nccnex .eq. 1) call numini(n, kpw) return c 210 if(impr) write(iout,310) index,n go to 235 220 if(impr) write(iout,320) nccnex, mxccnx go to 235 235 ier = -1 return c 310 format(' ***CCONEX*** ERROR TRYING TO DETERMINE THE NUMBER', * ' OF CONNECTED COMPONENTS.'/13X,' NUMBER OF MARKED', * ' VERTICES =',i7,3x,'TOTAL NUMBER OF VERTICES =',I7) 320 format(' ***CCONEX*** THE ALLOWED NUMBER OF CONNECTED COMPONENTS', * ' IS NOT ENOUGH.'/13X,' NECESSARY NUMBER = ',I4, * 5x,' ALLOWED NUMBER = ',I4) 323 format(' ***CCONEX*** ERROR IN ',A6,'. IER = ',I8) 340 format(/' ***CCONEX*** THE LAST CONNECTED COMPONENT WILL', * ' HAVE THE DIAGONAL VERTICES.') end c ********************************************************************** subroutine ANCCNX(n, mccnex, iccnex, mark, ncount) C----------------------------------------------------------------------- c c We put in ICCNEX the vertices marked in the component MCCNEX. C C----------------------------------------------------------------------- c include "NSIMPLIC" dimension mark(n) C----------------------------------------------------------------------- C Laura C. Dutto - email: dutto@cerca.umontreal.ca - December 1993 C----------------------------------------------------------------------- ncount = 0 do i = 1, n if( mark(i) .eq. mccnex) then mark(i) = iccnex ncount = ncount + 1 endif enddo c return end c ********************************************************************** subroutine NEWCNX(n, mccnex, iccnex, index, kpw, mark) C----------------------------------------------------------------------- c c We put in ICCNEX the vertices marked in the component MCCNEX. We c modify also the vector KPW. C C----------------------------------------------------------------------- c include "NSIMPLIC" dimension kpw(*), mark(n) C----------------------------------------------------------------------- C Laura C. Dutto - email: dutto@cerca.umontreal.ca - December 1993 C----------------------------------------------------------------------- do i = 1, n if( mark(i) .eq. mccnex) then mark(i) = iccnex index = index + 1 kpw(index) = i endif enddo c return end c ********************************************************************** subroutine BLCCN1(n, nbloc, nblcmx, nsbloc, lpw, kpw, ia, ja, * mark, mxccex, lccnex, mxptbl, iout, ier) C----------------------------------------------------------------------- c c This routine determines if the matrix given by the structure c IA et JA is irreductible. If not, it orders the unknowns such c that all the consecutive unknowns in KPW between NSBLOC(i-1)+1 c and NSBLOC(i) belong to the ith component of the matrix. c c On entry: c -------- c n = row and column dimension of the matrix c nblcmx = The size of NSBLOC is nblcmx + 1 (in fact, it starts at 0). c ia = integer array of length N+1 corresponding to the c pointer to the beginning of each row in JA (compressed c sparse row storage). c ja = integer array of length NNZERO (= IA(N+1)-IA(1)) corresponding c to the column indices of nonzero elements of the matrix, c stored rowwise. c mxccex = maximum number of connected components allowed by block. c mxptbl = maximum number of points (or unknowns) in each connected c component (mxptbl .le. n). c iout = impression parameter. If 0 < iout < 100, we print c comments and error messages on unit IOUT. c c Input/Output: c ------------ c nbloc = number of connected components of the matrix. If the c matrix is not irreductible, nbloc > 1. We allow c nbloc > 1 on entry; in this case we calculate the c number of connected components in each previous one. c nsbloc = integer array of length NBLOC + 1 containing the pointers c to the first node of each component on the new ordering. c Normally, on entry you put: NBLOC = 1, NSBLOC(0) = 0, c NSBLOC(NBLOC) = N. c c On return: c ---------- c lpw = integer array of length N corresponding to the c permutation vector (the row i goes to lpw(i)). c ier = integer. Error message. Normal return ier = 0. c c Work space: c ---------- c kpw = integer vector of length MXPTBL*NBLOC necessary for parallel c computation. c mark = integer vector of length N c lccnex = integer vector of length (MXCCEX+1)*NBLOC necessary for parallel c computation. c C----------------------------------------------------------------------- C Laura C. Dutto - e-mail: dutto@cerca.umontreal.ca c Juillet 1992. Update: March 1994 C----------------------------------------------------------------------- dimension lpw(n), kpw(mxptbl*nbloc), ia(n+1), ja(*), * lccnex((mxccex+1)*nbloc), nsbloc(0:nbloc), mark(n) logical impr character chsubr*6 C----------------------------------------------------------------------- ier = 0 impr = iout.gt.0.and.iout.le.99 isor = 0 c chsubr = 'CCONEX' newblc = 0 C$DOACROSS if(nbloc.gt.1), LOCAL(ibloc, ik0, ik1, ins0, ntb0, C$& nccnex, ilccnx, info, kpibl), REDUCTION(ier, newblc) do 100 ibloc = 1,nbloc ik0 = nsbloc(ibloc - 1) ik1 = nsbloc(ibloc) ntb0 = ia(ik0+1) if(ia(ik1+1) - ntb0 .le. 1) go to 100 ntb0 = ntb0 - 1 ins0 = ik1 - ik0 c........We need more memory place for KPW1 because of parallel computation kpibl = (ibloc-1) * mxptbl call numini( ins0, kpw(kpibl+1)) nccnex = mxccex ilccnx = (mxccex+1) * (ibloc-1) + 1 c....................................................................... c c Call to the main routine: CCONEX c c....................................................................... call cconex(ins0, ik0, nccnex, lccnex(ilccnx), kpw(kpibl+1), * ia(ik0+1), ja(ntb0+1), mark(ik0+1), isor, info) ier = ier + info if(info .ne. 0 .or. nccnex .lt. 1) go to 100 c c........We add the new connected components on NEWBLC newblc = newblc + nccnex c........We define LPW different from the identity only if there are more c........than one connected component in this block if(nccnex .eq. 1) then call numini(ins0, lpw(ik0+1)) else call invlpw(ins0, kpw(kpibl+1), lpw(ik0+1)) endif call iplusa(ins0, ik0, 1, lpw(ik0+1)) 100 continue c if(ier .ne. 0) go to 218 if(newblc .eq. nbloc) go to 120 if(newblc .gt. nblcmx) go to 230 c c.....We modify the number of blocks to indicate the number of connected c.....components in the matrix. newblc = 0 nsfin = 0 CDIR$ NEXT SCALAR do ibloc = 1, nbloc ilccnx = (mxccex+1) * (ibloc-1) + 1 nccnex = lccnex(ilccnx) if(nccnex .gt. 1 .and. impr) write(iout,420) ibloc,nccnex lcc0 = 0 CDIR$ NEXT SCALAR do icc = 1,nccnex newblc = newblc + 1 nsb = lccnex(ilccnx+icc) c...........Be careful! In LCCNEX we have the cumulated number of vertices nsbloc(newblc) = nsfin + nsb if(nccnex .gt. 1 .and. impr) write(iout,425) icc,nsb-lcc0 lcc0 = nsb enddo nsfin = nsfin + nsb enddo nbloc = newblc c 120 return c 218 if(impr) write(iout,318) chsubr,ier go to 120 230 if(impr) write(iout,330) newblc,nblcmx if(ier.eq.0) ier = -1 go to 120 c 318 format(' ***BLCCN1*** ERROR IN ',a6,'. IER = ',i8) 330 format(' ***BLCCN1*** THE MEMORY SPACE ALLOWED FOR NSBLOC IS', * ' NOT ENOUGH.'/13X,' NUMBER (NECESSARY) OF CONNECTED', * ' COMPONENTS = ',I5/13X,' MAXIMAL NUMBER OF BLOCKS',14x, * '= ',i5) 420 FORMAT(' *** The block ',i3,' has ',i3,' strongly connected', * ' components. The number of vertices by component is:') 425 format(5x,'Component No.',i3,' - Number of vertices = ',i6) end C*********************************************************************** SUBROUTINE CCNICOPY(N,IX,IY) C....................................................................... C We copy the vector IX on the vector IY C....................................................................... DIMENSION IX(n),IY(n) C....................................................................... IF(N.LE.0) RETURN C$DOACROSS if(n .gt. 250), local(i) DO 10 I = 1,N IY(I) = IX(I) 10 CONTINUE C RETURN END c*********************************************************************** SUBROUTINE COMPOS(n, lpw0, lpw1) C----------------------------------------------------------------------- c c We take account of the original order of unknowns. We put the c final result on LPW0. c C----------------------------------------------------------------------- DIMENSION lpw0(n), lpw1(n) C----------------------------------------------------------------------- c Laura C. Dutto - Mars 1994 C----------------------------------------------------------------------- C$DOACROSS if(n .gt. 250), local(i0) do i0 = 1, n lpw0(i0) = lpw1(lpw0(i0)) enddo c return end C ********************************************************************** SUBROUTINE INVLPW(n, lpw, kpw) c....................................................................... c c KPW is the inverse of LPW c c....................................................................... dimension lpw(n), kpw(n) c....................................................................... c Laura C. Dutto - Novembre 1993 c....................................................................... C$DOACROSS if(n .gt. 200), local(i0, i1) do i0 = 1, n i1 = lpw(i0) kpw(i1) = i0 enddo c return end C ********************************************************************** subroutine NUMINI(n, lpw) c....................................................................... dimension lpw(n) c....................................................................... c c The vector LPW is initialized as the identity. c c....................................................................... c Laura C. Dutto - Novembre 1993 c....................................................................... C$DOACROSS if(n .gt. 250), local(i) do i=1,n lpw(i) = i enddo c return end C*********************************************************************** SUBROUTINE TBZERO(M,NMOT) C....................................................................... C We initialize to ZERO an integer vector of length NMOT. C....................................................................... DIMENSION M(NMOT) C....................................................................... IF(NMOT.le.0) return C$DOACROSS if(nmot.gt.500), LOCAL(i) DO 1 I=1,NMOT M(I)=0 1 CONTINUE RETURN END C ********************************************************************** SUBROUTINE IPLUSA (n, nalpha, nbeta, ia) c....................................................................... C c We add NALPHA to each element of NBETA * IA: c c ia(i) = nalpha + nbeta * ia(i) c c....................................................................... integer ia(n) c....................................................................... c Laura C. Dutto - February 1994 c....................................................................... if(n .le. 0) return c nmax = 500 if(nalpha .eq. 0) then if(nbeta .eq. 1) return if(nbeta .eq. -1) then C$DOACROSS if(n .gt. nmax), local (i) do i = 1, n ia(i) = - ia(i) enddo else C$DOACROSS if(n .gt. nmax/2), local (i) do i = 1, n ia(i) = nbeta * ia(i) enddo endif return endif if(nbeta .eq. 0) then C$DOACROSS if(n .gt. nmax), local (i) do i = 1, n ia(i) = nalpha enddo return endif if(nbeta .eq. -1) then C$DOACROSS if(n .gt. nmax/2), local (i) do i = 1, n ia(i) = nalpha - ia(i) enddo else if(nbeta .eq. 1) then C$DOACROSS if(n .gt. nmax/2), local (i) do i = 1, n ia(i) = nalpha + ia(i) enddo else C$DOACROSS if(n .gt. nmax/3), local (i) do i = 1, n ia(i) = nalpha + nbeta * ia(i) enddo endif c return end