!DECK DSLUBC SUBROUTINE DSLUBC (N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL, & ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) !***BEGIN PROLOGUE DSLUBC !***PURPOSE Incomplete LU BiConjugate Gradient Sparse Ax=b Solver. ! Routine to solve a linear system Ax = b using the ! BiConjugate Gradient method with Incomplete LU ! decomposition preconditioning. !***LIBRARY SLATEC (SLAP) !***CATEGORY D2A4, D2B4 !***TYPE DOUBLE PRECISION (SSLUBC-S, DSLUBC-D) !***KEYWORDS ITERATIVE INCOMPLETE LU PRECONDITION, ! NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE !***AUTHOR Greenbaum, Anne, (Courant Institute) ! Seager, Mark K., (LLNL) ! Lawrence Livermore National Laboratory ! PO BOX 808, L-60 ! Livermore, CA 94550 (510) 423-3141 ! seager@llnl.gov !***DESCRIPTION ! ! *Usage: ! INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX ! INTEGER ITER, IERR, IUNIT, LENW, IWORK(NL+NU+4*N+2), LENIW ! DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, RWORK(NL+NU+8*N) ! ! CALL DSLUBC(N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL, ! $ ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) ! ! *Arguments: ! N :IN Integer. ! Order of the Matrix. ! B :IN Double Precision B(N). ! Right-hand side vector. ! X :INOUT Double Precision X(N). ! On input X is your initial guess for solution vector. ! On output X is the final approximate solution. ! NELT :IN Integer. ! Number of Non-Zeros stored in A. ! IA :INOUT Integer IA(NELT). ! JA :INOUT Integer JA(NELT). ! A :INOUT Double Precision A(NELT). ! These arrays should hold the matrix A in either the SLAP ! Triad format or the SLAP Column format. See "Description", ! below. If the SLAP Triad format is chosen it is changed ! internally to the SLAP Column format. ! ISYM :IN Integer. ! Flag to indicate symmetric storage format. ! If ISYM=0, all non-zero entries of the matrix are stored. ! If ISYM=1, the matrix is symmetric, and only the upper ! or lower triangle of the matrix is stored. ! ITOL :IN Integer. ! Flag to indicate type of convergence criterion. ! If ITOL=1, iteration stops when the 2-norm of the residual ! divided by the 2-norm of the right-hand side is less than TOL. ! If ITOL=2, iteration stops when the 2-norm of M-inv times the ! residual divided by the 2-norm of M-inv times the right hand ! side is less than TOL, where M-inv is the inverse of the ! diagonal of A. ! ITOL=11 is often useful for checking and comparing different ! routines. For this case, the user must supply the "exact" ! solution or a very accurate approximation (one with an error ! much less than TOL) through a common block, ! COMMON /DSLBLK/ SOLN( ) ! If ITOL=11, iteration stops when the 2-norm of the difference ! between the iterative approximation and the user-supplied ! solution divided by the 2-norm of the user-supplied solution ! is less than TOL. Note that this requires the user to set up ! the "COMMON /DSLBLK/ SOLN(LENGTH)" in the calling routine. ! The routine with this declaration should be loaded before the ! stop test so that the correct length is used by the loader. ! This procedure is not standard Fortran and may not work ! correctly on your system (although it has worked on every ! system the authors have tried). If ITOL is not 11 then this ! common block is indeed standard Fortran. ! TOL :INOUT Double Precision. ! Convergence criterion, as described above. (Reset if IERR=4.) ! ITMAX :IN Integer. ! Maximum number of iterations. ! ITER :OUT Integer. ! Number of iterations required to reach convergence, or ! ITMAX+1 if convergence criterion could not be achieved in ! ITMAX iterations. ! ERR :OUT Double Precision. ! Error estimate of error in final approximate solution, as ! defined by ITOL. ! IERR :OUT Integer. ! Return error flag. ! IERR = 0 => All went well. ! IERR = 1 => Insufficient space allocated for WORK or IWORK. ! IERR = 2 => Method failed to converge in ITMAX steps. ! IERR = 3 => Error in user input. ! Check input values of N, ITOL. ! IERR = 4 => User error tolerance set too tight. ! Reset to 500*D1MACH(3). Iteration proceeded. ! IERR = 5 => Preconditioning matrix, M, is not positive ! definite. (r,z) < 0. ! IERR = 6 => Matrix A is not positive definite. (p,Ap) < 0. ! IERR = 7 => Incomplete factorization broke down and was ! fudged. Resulting preconditioning may be less ! than the best. ! IUNIT :IN Integer. ! Unit number on which to write the error at each iteration, ! if this is desired for monitoring convergence. If unit ! number is 0, no writing will occur. ! RWORK :WORK Double Precision RWORK(LENW). ! Double Precision array used for workspace. ! LENW :IN Integer. ! Length of the double precision workspace, RWORK. ! LENW >= NL+NU+8*N. ! NL is the number of non-zeros in the lower triangle of the ! matrix (including the diagonal). ! NU is the number of non-zeros in the upper triangle of the ! matrix (including the diagonal). ! IWORK :WORK Integer IWORK(LENIW). ! Integer array used for workspace. ! Upon return the following locations of IWORK hold information ! which may be of use to the user: ! IWORK(9) Amount of Integer workspace actually used. ! IWORK(10) Amount of Double Precision workspace actually used. ! LENIW :IN Integer. ! Length of the integer workspace, IWORK. ! LENIW >= NL+NU+4*N+12. ! NL is the number of non-zeros in the lower triangle of the ! matrix (including the diagonal). ! NU is the number of non-zeros in the upper triangle of the ! matrix (including the diagonal). ! ! *Description: ! This routine is simply a driver for the DBCGN routine. It ! calls the DSILUS routine to set up the preconditioning and ! then calls DBCGN with the appropriate MATVEC, MTTVEC and ! MSOLVE, MTSOLV routines. ! ! The Sparse Linear Algebra Package (SLAP) utilizes two matrix ! data structures: 1) the SLAP Triad format or 2) the SLAP ! Column format. The user can hand this routine either of the ! of these data structures and SLAP will figure out which on ! is being used and act accordingly. ! ! =================== S L A P Triad format =================== ! ! This routine requires that the matrix A be stored in the ! SLAP Triad format. In this format only the non-zeros are ! stored. They may appear in *ANY* order. The user supplies ! three arrays of length NELT, where NELT is the number of ! non-zeros in the matrix: (IA(NELT), JA(NELT), A(NELT)). For ! each non-zero the user puts the row and column index of that ! matrix element in the IA and JA arrays. The value of the ! non-zero matrix element is placed in the corresponding ! location of the A array. This is an extremely easy data ! structure to generate. On the other hand it is not too ! efficient on vector computers for the iterative solution of ! linear systems. Hence, SLAP changes this input data ! structure to the SLAP Column format for the iteration (but ! does not change it back). ! ! Here is an example of the SLAP Triad storage format for a ! 5x5 Matrix. Recall that the entries may appear in any order. ! ! 5x5 Matrix SLAP Triad format for 5x5 matrix on left. ! 1 2 3 4 5 6 7 8 9 10 11 ! |11 12 0 0 15| A: 51 12 11 33 15 53 55 22 35 44 21 ! |21 22 0 0 0| IA: 5 1 1 3 1 5 5 2 3 4 2 ! | 0 0 33 0 35| JA: 1 2 1 3 5 3 5 2 5 4 1 ! | 0 0 0 44 0| ! |51 0 53 0 55| ! ! =================== S L A P Column format ================== ! ! This routine requires that the matrix A be stored in the ! SLAP Column format. In this format the non-zeros are stored ! counting down columns (except for the diagonal entry, which ! must appear first in each "column") and are stored in the ! double precision array A. In other words, for each column ! in the matrix put the diagonal entry in A. Then put in the ! other non-zero elements going down the column (except the ! diagonal) in order. The IA array holds the row index for ! each non-zero. The JA array holds the offsets into the IA, ! A arrays for the beginning of each column. That is, ! IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the ! ICOL-th column in IA and A. IA(JA(ICOL+1)-1), ! A(JA(ICOL+1)-1) points to the end of the ICOL-th column. ! Note that we always have JA(N+1) = NELT+1, where N is the ! number of columns in the matrix and NELT is the number of ! non-zeros in the matrix. ! ! Here is an example of the SLAP Column storage format for a ! 5x5 Matrix (in the A and IA arrays '|' denotes the end of a ! column): ! ! 5x5 Matrix SLAP Column format for 5x5 matrix on left. ! 1 2 3 4 5 6 7 8 9 10 11 ! |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35 ! |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 ! | 0 0 33 0 35| JA: 1 4 6 8 9 12 ! | 0 0 0 44 0| ! |51 0 53 0 55| ! ! *Side Effects: ! The SLAP Triad format (IA, JA, A) is modified internally to ! be the SLAP Column format. See above. ! ! *Cautions: ! This routine will attempt to write to the Fortran logical output ! unit IUNIT, if IUNIT .ne. 0. Thus, the user must make sure that ! this logical unit is attached to a file or terminal before calling ! this routine with a non-zero value for IUNIT. This routine does ! not check for the validity of a non-zero IUNIT unit number. ! !***SEE ALSO DBCG, DSDBCG !***REFERENCES (NONE) !***ROUTINES CALLED DBCG, DCHKW, DS2Y, DSILUS, DSLUI, DSLUTI, DSMTV, ! DSMV !***REVISION HISTORY (YYMMDD) ! 890404 DATE WRITTEN ! 890404 Previous REVISION DATE ! 890915 Made changes requested at July 1989 CML Meeting. (MKS) ! 890921 Removed TeX from comments. (FNF) ! 890922 Numerous changes to prologue to make closer to SLATEC ! standard. (FNF) ! 890929 Numerous changes to reduce SP/DP differences. (FNF) ! 910411 Prologue converted to Version 4.0 format. (BAB) ! 920407 COMMON BLOCK renamed DSLBLK. (WRB) ! 920511 Added complete declaration section. (WRB) ! 921113 Corrected C***CATEGORY line. (FNF) !***END PROLOGUE DSLUBC ! .. Parameters .. INTEGER LOCRB, LOCIB PARAMETER (LOCRB=1, LOCIB=11) ! .. Scalar Arguments .. DOUBLE PRECISION ERR, TOL INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, LENIW, LENW, N, NELT ! .. Array Arguments .. DOUBLE PRECISION A(NELT), B(N), RWORK(LENW), X(N) INTEGER IA(NELT), IWORK(LENIW), JA(NELT) ! .. Local Scalars .. INTEGER ICOL, J, JBGN, JEND, LOCDIN, LOCDZ, LOCIL, LOCIU, LOCIW, & LOCJL, LOCJU, LOCL, LOCNC, LOCNR, LOCP, LOCPP, LOCR, & LOCRR, LOCU, LOCW, LOCZ, LOCZZ, NL, NU ! .. External Subroutines .. EXTERNAL DBCG, DCHKW, DS2Y, DSILUS, DSLUI, DSLUTI, DSMTV, DSMV !***FIRST EXECUTABLE STATEMENT DSLUBC ! IERR = 0 IF( N.LT.1 .OR. NELT.LT.1 ) THEN IERR = 3 RETURN ENDIF ! ! Change the SLAP input matrix IA, JA, A to SLAP-Column format. CALL DS2Y( N, NELT, IA, JA, A, ISYM ) ! ! Count number of Non-Zero elements preconditioner ILU matrix. ! Then set up the work arrays. NL = 0 NU = 0 DO 20 ICOL = 1, N ! Don't count diagonal. JBGN = JA(ICOL)+1 JEND = JA(ICOL+1)-1 IF( JBGN.LE.JEND ) THEN !VD$ NOVECTOR DO 10 J = JBGN, JEND IF( IA(J).GT.ICOL ) THEN NL = NL + 1 IF( ISYM.NE.0 ) NU = NU + 1 ELSE NU = NU + 1 ENDIF 10 CONTINUE ENDIF 20 CONTINUE ! LOCIL = LOCIB LOCJL = LOCIL + N+1 LOCIU = LOCJL + NL LOCJU = LOCIU + NU LOCNR = LOCJU + N+1 LOCNC = LOCNR + N LOCIW = LOCNC + N ! LOCL = LOCRB LOCDIN = LOCL + NL LOCU = LOCDIN + N LOCR = LOCU + NU LOCZ = LOCR + N LOCP = LOCZ + N LOCRR = LOCP + N LOCZZ = LOCRR + N LOCPP = LOCZZ + N LOCDZ = LOCPP + N LOCW = LOCDZ + N ! ! Check the workspace allocations. CALL DCHKW( 'DSLUBC', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR ) IF( IERR.NE.0 ) RETURN ! IWORK(1) = LOCIL IWORK(2) = LOCJL IWORK(3) = LOCIU IWORK(4) = LOCJU IWORK(5) = LOCL IWORK(6) = LOCDIN IWORK(7) = LOCU IWORK(9) = LOCIW IWORK(10) = LOCW ! ! Compute the Incomplete LU decomposition. CALL DSILUS( N, NELT, IA, JA, A, ISYM, NL, IWORK(LOCIL), & IWORK(LOCJL), RWORK(LOCL), RWORK(LOCDIN), NU, IWORK(LOCIU), & IWORK(LOCJU), RWORK(LOCU), IWORK(LOCNR), IWORK(LOCNC) ) ! ! Perform the incomplete LU preconditioned ! BiConjugate Gradient algorithm. CALL DBCG(N, B, X, NELT, IA, JA, A, ISYM, DSMV, DSMTV, & DSLUI, DSLUTI, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, & RWORK(LOCR), RWORK(LOCZ), RWORK(LOCP), & RWORK(LOCRR), RWORK(LOCZZ), RWORK(LOCPP), & RWORK(LOCDZ), RWORK, IWORK ) RETURN !------------- LAST LINE OF DSLUBC FOLLOWS ---------------------------- END