SCGN(3)

SYNOPSIS
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)

NAME
scgn - Preconditioned CG Sparse Ax=b Solver for Normal Equations. Routine to solve a general linear system Ax = b using the Preconditioned Conjugate Gradient method applied to the normal equations AA'y = b, x=A'y.

LIBRARY
SLATEC (SLAP)

CATEGORY
D2A4, D2B4

TYPE
SINGLE PRECISION (SCGN-S, DCGN-D)

KEYWORDS
ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM SOLVE, NORMAL EQUATIONS., 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, IWORK(USER DEFINED) REAL B(N), X(N), A(NELT), TOL, ERR, R(N), Z(N) REAL P(N), ATP(N), ATZ(N), DZ(N), ATDZ(N) REAL RWORK(USER DEFINED) EXTERNAL MATVEC, MTTVEC, MSOLVE CALL 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) *Arguments: N :IN Integer Order of the Matrix. B :IN Real B(N). Right-hand side vector. X :INOUT Real 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 :IN Integer IA(NELT). JA :IN Integer JA(NELT). A :IN Real A(NELT). These arrays contain the matrix data structure for A. It could take any form. See "Description", below, for more details. 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. MATVEC :EXT External. Name of a routine which performs the matrix vector multiply y = A*X given A and X. The name of the MATVEC routine must be declared external in the calling program. The calling sequence to MATVEC is: CALL MATVEC( N, X, Y, NELT, IA, JA, A, ISYM ) Where N is the number of unknowns, Y is the product A*X upon return X is an input vector, NELT is the number of non-zeros in the SLAP-Column IA, JA, A storage for the matrix A. ISYM is a flag which, if non-zero, denotes that A is symmetric and only the lower or upper triangle is stored. MTTVEC :EXT External. Name of a routine which performs the matrix transpose vector multiply y = A'*X given A and X (where ' denotes transpose). The name of the MTTVEC routine must be declared external in the calling program. The calling sequence to MTTVEC is the same as that for MATVEC, viz.: CALL MTTVEC( N, X, Y, NELT, IA, JA, A, ISYM ) Where N is the number of unknowns, Y is the product A'*X upon return X is an input vector, NELT is the number of non-zeros in the SLAP-Column IA, JA, A storage for the matrix A. ISYM is a flag which, if non-zero, denotes that A is symmetric and only the lower or upper triangle is stored. MSOLVE :EXT External. Name of a routine which solves a linear system MZ = R for Z given R with the preconditioning matrix M (M is supplied via RWORK and IWORK arrays). The name of the MSOLVE routine must be declared external in the calling program. The calling sequence to MSOLVE is: CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK) Where N is the number of unknowns, R is the right-hand side vector and Z is the solution upon return. NELT, IA, JA, A and ISYM are defined as above. RWORK is a real array that can be used to pass necessary preconditioning information and/or workspace to MSOLVE. IWORK is an integer work array for the same purpose as RWORK. 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 /SSLBLK/ 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 /SSLBLK/ 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 Real. 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 Real. 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*R1MACH(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. 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. R :WORK Real R(N). Z :WORK Real Z(N). P :WORK Real P(N). ATP :WORK Real ATP(N). ATZ :WORK Real ATZ(N). DZ :WORK Real DZ(N). ATDZ :WORK Real ATDZ(N). Real arrays used for workspace. RWORK :WORK Real RWORK(USER DEFINED). Real array that can be used by MSOLVE. IWORK :WORK Integer IWORK(USER DEFINED). Integer array that can be used by MSOLVE. *Description: This routine applies the preconditioned conjugate gradient (PCG) method to a non-symmetric system of equations Ax=b. To do this the normal equations are solved: AA' y = b, where x = A'y. In PCG method the iteration count is determined by condition -1 number of the matrix (M A). In the situation where the normal equations are used to solve a non-symmetric system the condition number depends on AA' and should therefore be much worse than that of A. This is the conventional wisdom. When one has a good preconditioner for AA' this may not hold. The latter is the situation when SCGN should be tried. If one is trying to solve a symmetric system, SCG should be used instead. This routine does not care what matrix data structure is used for A and M. It simply calls MATVEC, MTTVEC and MSOLVE routines, with arguments as described above. The user could write any type of structure, and appropriate MATVEC, MTTVEC and MSOLVE routines. It is assumed that A is stored in the IA, JA, A arrays in some fashion and that M (or INV(M)) is stored in IWORK and RWORK) in some fashion. The SLAP routines SSDCGN and SSLUCN are examples of this procedure. Two examples of matrix data structures are the: 1) SLAP Triad format and 2) SLAP Column format. =================== S L A P 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 ================== 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 real 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| *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
SSDCGN, SSLUCN, ISSCGN

REFERENCES
1. Mark K. Seager, A SLAP for the Masses, in G. F. Carey, Ed., Parallel Supercomputing: Methods, Algorithms and Applications, Wiley, 1989, pp.135-155.

ROUTINES CALLED
ISSCGN, R1MACH, SAXPY, SCOPY, SDOT

REVISION HISTORY (YYMMDD)
871119 DATE WRITTEN
881213 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)
891004 Added new reference.
910411 Prologue converted to Version 4.0 format. (BAB)
910502 Removed MATVEC, MTTVEC and MSOLVE from ROUTINES CALLED list. (FNF)
920407 COMMON BLOCK renamed SSLBLK. (WRB)
920511 Added complete declaration section. (WRB)
920929 Corrected format of reference. (FNF)
921019 Changed 500.0 to 500 to reduce SP/DP differences. (FNF)
921113 Corrected C***CATEGORY line. (FNF)