*DECK SCGN
SUBROUTINE SCGN (N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MTTVEC,
+ MSOLVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R, Z, P, ATP,
+ ATZ, DZ, ATDZ, RWORK, IWORK)
C***BEGIN PROLOGUE SCGN
C***PURPOSE Preconditioned CG Sparse Ax=b Solver for Normal Equations.
C Routine to solve a general linear system Ax = b using the
C Preconditioned Conjugate Gradient method applied to the
C normal equations AA'y = b, x=A'y.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2A4, D2B4
C***TYPE SINGLE PRECISION (SCGN-S, DCGN-D)
C***KEYWORDS ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM SOLVE,
C NORMAL EQUATIONS., SLAP, SPARSE
C***AUTHOR Greenbaum, Anne, (Courant Institute)
C Seager, Mark K., (LLNL)
C Lawrence Livermore National Laboratory
C PO BOX 808, L-60
C Livermore, CA 94550 (510) 423-3141
C seager@llnl.gov
C***DESCRIPTION
C
C *Usage:
C INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX
C INTEGER ITER, IERR, IUNIT, IWORK(USER DEFINED)
C REAL B(N), X(N), A(NELT), TOL, ERR, R(N), Z(N)
C REAL P(N), ATP(N), ATZ(N), DZ(N), ATDZ(N)
C REAL RWORK(USER DEFINED)
C EXTERNAL MATVEC, MTTVEC, MSOLVE
C
C CALL SCGN(N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MTTVEC,
C $ MSOLVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R,
C $ Z, P, ATP, ATZ, DZ, ATDZ, RWORK, IWORK)
C
C *Arguments:
C N :IN Integer
C Order of the Matrix.
C B :IN Real B(N).
C Right-hand side vector.
C X :INOUT Real X(N).
C On input X is your initial guess for solution vector.
C On output X is the final approximate solution.
C NELT :IN Integer.
C Number of Non-Zeros stored in A.
C IA :IN Integer IA(NELT).
C JA :IN Integer JA(NELT).
C A :IN Real A(NELT).
C These arrays contain the matrix data structure for A.
C It could take any form. See "Description", below,
C for more details.
C ISYM :IN Integer.
C Flag to indicate symmetric storage format.
C If ISYM=0, all non-zero entries of the matrix are stored.
C If ISYM=1, the matrix is symmetric, and only the upper
C or lower triangle of the matrix is stored.
C MATVEC :EXT External.
C Name of a routine which performs the matrix vector multiply
C y = A*X given A and X. The name of the MATVEC routine must
C be declared external in the calling program. The calling
C sequence to MATVEC is:
C CALL MATVEC( N, X, Y, NELT, IA, JA, A, ISYM )
C Where N is the number of unknowns, Y is the product A*X
C upon return X is an input vector, NELT is the number of
C non-zeros in the SLAP-Column IA, JA, A storage for the matrix
C A. ISYM is a flag which, if non-zero, denotes that A is
C symmetric and only the lower or upper triangle is stored.
C MTTVEC :EXT External.
C Name of a routine which performs the matrix transpose vector
C multiply y = A'*X given A and X (where ' denotes transpose).
C The name of the MTTVEC routine must be declared external in
C the calling program. The calling sequence to MTTVEC is the
C same as that for MATVEC, viz.:
C CALL MTTVEC( N, X, Y, NELT, IA, JA, A, ISYM )
C Where N is the number of unknowns, Y is the product A'*X
C upon return X is an input vector, NELT is the number of
C non-zeros in the SLAP-Column IA, JA, A storage for the matrix
C A. ISYM is a flag which, if non-zero, denotes that A is
C symmetric and only the lower or upper triangle is stored.
C MSOLVE :EXT External.
C Name of a routine which solves a linear system MZ = R for
C Z given R with the preconditioning matrix M (M is supplied via
C RWORK and IWORK arrays). The name of the MSOLVE routine must
C be declared external in the calling program. The calling
C sequence to MSOLVE is:
C CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK)
C Where N is the number of unknowns, R is the right-hand side
C vector and Z is the solution upon return. NELT, IA, JA, A and
C ISYM are defined as above. RWORK is a real array that can
C be used to pass necessary preconditioning information and/or
C workspace to MSOLVE. IWORK is an integer work array for
C the same purpose as RWORK.
C ITOL :IN Integer.
C Flag to indicate type of convergence criterion.
C If ITOL=1, iteration stops when the 2-norm of the residual
C divided by the 2-norm of the right-hand side is less than TOL.
C If ITOL=2, iteration stops when the 2-norm of M-inv times the
C residual divided by the 2-norm of M-inv times the right hand
C side is less than TOL, where M-inv is the inverse of the
C diagonal of A.
C ITOL=11 is often useful for checking and comparing different
C routines. For this case, the user must supply the "exact"
C solution or a very accurate approximation (one with an error
C much less than TOL) through a common block,
C COMMON /SSLBLK/ SOLN( )
C If ITOL=11, iteration stops when the 2-norm of the difference
C between the iterative approximation and the user-supplied
C solution divided by the 2-norm of the user-supplied solution
C is less than TOL. Note that this requires the user to set up
C the "COMMON /SSLBLK/ SOLN(LENGTH)" in the calling routine.
C The routine with this declaration should be loaded before the
C stop test so that the correct length is used by the loader.
C This procedure is not standard Fortran and may not work
C correctly on your system (although it has worked on every
C system the authors have tried). If ITOL is not 11 then this
C common block is indeed standard Fortran.
C TOL :INOUT Real.
C Convergence criterion, as described above. (Reset if IERR=4.)
C ITMAX :IN Integer.
C Maximum number of iterations.
C ITER :OUT Integer.
C Number of iterations required to reach convergence, or
C ITMAX+1 if convergence criterion could not be achieved in
C ITMAX iterations.
C ERR :OUT Real.
C Error estimate of error in final approximate solution, as
C defined by ITOL.
C IERR :OUT Integer.
C Return error flag.
C IERR = 0 => All went well.
C IERR = 1 => Insufficient space allocated for WORK or IWORK.
C IERR = 2 => Method failed to converge in ITMAX steps.
C IERR = 3 => Error in user input.
C Check input values of N, ITOL.
C IERR = 4 => User error tolerance set too tight.
C Reset to 500*R1MACH(3). Iteration proceeded.
C IERR = 5 => Preconditioning matrix, M, is not positive
C definite. (r,z) < 0.
C IERR = 6 => Matrix A is not positive definite. (p,Ap) < 0.
C IUNIT :IN Integer.
C Unit number on which to write the error at each iteration,
C if this is desired for monitoring convergence. If unit
C number is 0, no writing will occur.
C R :WORK Real R(N).
C Z :WORK Real Z(N).
C P :WORK Real P(N).
C ATP :WORK Real ATP(N).
C ATZ :WORK Real ATZ(N).
C DZ :WORK Real DZ(N).
C ATDZ :WORK Real ATDZ(N).
C Real arrays used for workspace.
C RWORK :WORK Real RWORK(USER DEFINED).
C Real array that can be used by MSOLVE.
C IWORK :WORK Integer IWORK(USER DEFINED).
C Integer array that can be used by MSOLVE.
C
C *Description:
C This routine applies the preconditioned conjugate gradient
C (PCG) method to a non-symmetric system of equations Ax=b. To
C do this the normal equations are solved:
C AA' y = b, where x = A'y.
C In PCG method the iteration count is determined by condition
C -1
C number of the matrix (M A). In the situation where the
C normal equations are used to solve a non-symmetric system
C the condition number depends on AA' and should therefore be
C much worse than that of A. This is the conventional wisdom.
C When one has a good preconditioner for AA' this may not hold.
C The latter is the situation when SCGN should be tried.
C
C If one is trying to solve a symmetric system, SCG should be
C used instead.
C
C This routine does not care what matrix data structure is
C used for A and M. It simply calls MATVEC, MTTVEC and MSOLVE
C routines, with arguments as described above. The user could
C write any type of structure, and appropriate MATVEC, MTTVEC
C and MSOLVE routines. It is assumed that A is stored in the
C IA, JA, A arrays in some fashion and that M (or INV(M)) is
C stored in IWORK and RWORK) in some fashion. The SLAP
C routines SSDCGN and SSLUCN are examples of this procedure.
C
C Two examples of matrix data structures are the: 1) SLAP
C Triad format and 2) SLAP Column format.
C
C =================== S L A P Triad format ===================
C
C In this format only the non-zeros are stored. They may
C appear in *ANY* order. The user supplies three arrays of
C length NELT, where NELT is the number of non-zeros in the
C matrix: (IA(NELT), JA(NELT), A(NELT)). For each non-zero
C the user puts the row and column index of that matrix
C element in the IA and JA arrays. The value of the non-zero
C matrix element is placed in the corresponding location of
C the A array. This is an extremely easy data structure to
C generate. On the other hand it is not too efficient on
C vector computers for the iterative solution of linear
C systems. Hence, SLAP changes this input data structure to
C the SLAP Column format for the iteration (but does not
C change it back).
C
C Here is an example of the SLAP Triad storage format for a
C 5x5 Matrix. Recall that the entries may appear in any order.
C
C 5x5 Matrix SLAP Triad format for 5x5 matrix on left.
C 1 2 3 4 5 6 7 8 9 10 11
C |11 12 0 0 15| A: 51 12 11 33 15 53 55 22 35 44 21
C |21 22 0 0 0| IA: 5 1 1 3 1 5 5 2 3 4 2
C | 0 0 33 0 35| JA: 1 2 1 3 5 3 5 2 5 4 1
C | 0 0 0 44 0|
C |51 0 53 0 55|
C
C =================== S L A P Column format ==================
C
C In this format the non-zeros are stored counting down
C columns (except for the diagonal entry, which must appear
C first in each "column") and are stored in the real array A.
C In other words, for each column in the matrix put the
C diagonal entry in A. Then put in the other non-zero
C elements going down the column (except the diagonal) in
C order. The IA array holds the row index for each non-zero.
C The JA array holds the offsets into the IA, A arrays for the
C beginning of each column. That is, IA(JA(ICOL)),
C A(JA(ICOL)) points to the beginning of the ICOL-th column in
C IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) points to the
C end of the ICOL-th column. Note that we always have JA(N+1)
C = NELT+1, where N is the number of columns in the matrix and
C NELT is the number of non-zeros in the matrix.
C
C Here is an example of the SLAP Column storage format for a
C 5x5 Matrix (in the A and IA arrays '|' denotes the end of a
C column):
C
C 5x5 Matrix SLAP Column format for 5x5 matrix on left.
C 1 2 3 4 5 6 7 8 9 10 11
C |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35
C |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3
C | 0 0 33 0 35| JA: 1 4 6 8 9 12
C | 0 0 0 44 0|
C |51 0 53 0 55|
C
C *Cautions:
C This routine will attempt to write to the Fortran logical output
C unit IUNIT, if IUNIT .ne. 0. Thus, the user must make sure that
C this logical unit is attached to a file or terminal before calling
C this routine with a non-zero value for IUNIT. This routine does
C not check for the validity of a non-zero IUNIT unit number.
C
C***SEE ALSO SSDCGN, SSLUCN, ISSCGN
C***REFERENCES 1. Mark K. Seager, A SLAP for the Masses, in
C G. F. Carey, Ed., Parallel Supercomputing: Methods,
C Algorithms and Applications, Wiley, 1989, pp.135-155.
C***ROUTINES CALLED ISSCGN, R1MACH, SAXPY, SCOPY, SDOT
C***REVISION HISTORY (YYMMDD)
C 871119 DATE WRITTEN
C 881213 Previous REVISION DATE
C 890915 Made changes requested at July 1989 CML Meeting. (MKS)
C 890921 Removed TeX from comments. (FNF)
C 890922 Numerous changes to prologue to make closer to SLATEC
C standard. (FNF)
C 890929 Numerous changes to reduce SP/DP differences. (FNF)
C 891004 Added new reference.
C 910411 Prologue converted to Version 4.0 format. (BAB)
C 910502 Removed MATVEC, MTTVEC and MSOLVE from ROUTINES CALLED
C list. (FNF)
C 920407 COMMON BLOCK renamed SSLBLK. (WRB)
C 920511 Added complete declaration section. (WRB)
C 920929 Corrected format of reference. (FNF)
C 921019 Changed 500.0 to 500 to reduce SP/DP differences. (FNF)
C 921113 Corrected C***CATEGORY line. (FNF)
C***END PROLOGUE SCGN
C .. Scalar Arguments ..
REAL ERR, TOL
INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, N, NELT
C .. Array Arguments ..
REAL A(N), ATDZ(N), ATP(N), ATZ(N), B(N), DZ(N), P(N), R(N),
+ RWORK(*), X(N), Z(N)
INTEGER IA(NELT), IWORK(*), JA(NELT)
C .. Subroutine Arguments ..
EXTERNAL MATVEC, MSOLVE, MTTVEC
C .. Local Scalars ..
REAL AK, AKDEN, BK, BKDEN, BKNUM, BNRM, SOLNRM, TOLMIN
INTEGER I, K
C .. External Functions ..
REAL R1MACH, SDOT
INTEGER ISSCGN
EXTERNAL R1MACH, SDOT, ISSCGN
C .. External Subroutines ..
EXTERNAL SAXPY, SCOPY
C***FIRST EXECUTABLE STATEMENT SCGN
C
C Check user input.
C
ITER = 0
IERR = 0
IF( N.LT.1 ) THEN
IERR = 3
RETURN
ENDIF
TOLMIN = 500*R1MACH(3)
IF( TOL.LT.TOLMIN ) THEN
TOL = TOLMIN
IERR = 4
ENDIF
C Calculate initial residual and pseudo-residual, and check
C stopping criterion.
CALL MATVEC(N, X, R, NELT, IA, JA, A, ISYM)
DO 10 I = 1, N
R(I) = B(I) - R(I)
10 CONTINUE
CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK)
CALL MTTVEC(N, Z, ATZ, NELT, IA, JA, A, ISYM)
C
IF( ISSCGN(N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MTTVEC, MSOLVE,
$ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R, Z, P, ATP, ATZ,
$ DZ, ATDZ, RWORK, IWORK, AK, BK, BNRM, SOLNRM) .NE. 0 )
$ GO TO 200
IF( IERR.NE.0 ) RETURN
C
C ***** iteration loop *****
C
DO 100 K=1,ITMAX
ITER = K
C
C Calculate coefficient BK and direction vector P.
BKNUM = SDOT(N, Z, 1, R, 1)
IF( BKNUM.LE.0.0E0 ) THEN
IERR = 6
RETURN
ENDIF
IF(ITER .EQ. 1) THEN
CALL SCOPY(N, Z, 1, P, 1)
ELSE
BK = BKNUM/BKDEN
DO 20 I = 1, N
P(I) = Z(I) + BK*P(I)
20 CONTINUE
ENDIF
BKDEN = BKNUM
C
C Calculate coefficient AK, new iterate X, new residual R,
C and new pseudo-residual ATZ.
IF(ITER .NE. 1) CALL SAXPY(N, BK, ATP, 1, ATZ, 1)
CALL SCOPY(N, ATZ, 1, ATP, 1)
AKDEN = SDOT(N, ATP, 1, ATP, 1)
IF( AKDEN.LE.0.0E0 ) THEN
IERR = 6
RETURN
ENDIF
AK = BKNUM/AKDEN
CALL SAXPY(N, AK, ATP, 1, X, 1)
CALL MATVEC(N, ATP, Z, NELT, IA, JA, A, ISYM)
CALL SAXPY(N, -AK, Z, 1, R, 1)
CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK)
CALL MTTVEC(N, Z, ATZ, NELT, IA, JA, A, ISYM)
C
C check stopping criterion.
IF( ISSCGN(N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MTTVEC,
$ MSOLVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R,
$ Z, P, ATP, ATZ, DZ, ATDZ, RWORK, IWORK, AK, BK, BNRM,
$ SOLNRM) .NE. 0) GOTO 200
C
100 CONTINUE
C
C ***** end of loop *****
C
C stopping criterion not satisfied.
ITER = ITMAX + 1
C
200 RETURN
C------------- LAST LINE OF SCGN FOLLOWS ----------------------------
END