*DECK SLPDOC
SUBROUTINE SLPDOC
C***BEGIN PROLOGUE SLPDOC
C***PURPOSE Sparse Linear Algebra Package Version 2.0.2 Documentation.
C Routines to solve large sparse symmetric and nonsymmetric
C positive definite linear systems, Ax = b, using precondi-
C tioned iterative methods.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2A4, D2B4, Z
C***TYPE SINGLE PRECISION (SLPDOC-S, DLPDOC-D)
C***KEYWORDS BICONJUGATE GRADIENT SQUARED, DOCUMENTATION,
C GENERALIZED MINIMUM RESIDUAL, ITERATIVE IMPROVEMENT,
C NORMAL EQUATIONS, ORTHOMIN,
C PRECONDITIONED CONJUGATE GRADIENT, SLAP,
C SPARSE ITERATIVE METHODS
C***AUTHOR Seager, Mark. K., (LLNL)
C User Systems Division
C Lawrence Livermore National Laboratory
C PO BOX 808, L-60
C Livermore, CA 94550
C (FTS) 543-3141, (510) 423-3141
C seager@llnl.gov
C***DESCRIPTION
C The
C Sparse Linear Algebra Package
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C
C =================================================================
C ========================== Introduction =========================
C =================================================================
C This package was originally derived from a set of iterative
C routines written by Anne Greenbaum, as announced in "Routines
C for Solving Large Sparse Linear Systems", Tentacle, Lawrence
C Livermore National Laboratory, Livermore Computing Center
C (January 1986), pp 15-21.
C
C This document contains the specifications for the SLAP Version
C 2.0 package, a Fortran 77 package for the solution of large
C sparse linear systems, Ax = b, via preconditioned iterative
C methods. Included in this package are "core" routines to do
C Iterative Refinement (Jacobi's method), Conjugate Gradient,
C Conjugate Gradient on the normal equations, AA'y = b, (where x =
C A'y and A' denotes the transpose of A), BiConjugate Gradient,
C BiConjugate Gradient Squared, Orthomin and Generalized Minimum
C Residual Iteration. These "core" routines do not require a
C "fixed" data structure for storing the matrix A and the
C preconditioning matrix M. The user is free to choose any
C structure that facilitates efficient solution of the problem at
C hand. The drawback to this approach is that the user must also
C supply at least two routines (MATVEC and MSOLVE, say). MATVEC
C must calculate, y = Ax, given x and the user's data structure for
C A. MSOLVE must solve, r = Mz, for z (*NOT* r) given r and the
C user's data structure for M (or its inverse). The user should
C choose M so that inv(M)*A is approximately the identity and the
C solution step r = Mz is "easy" to solve. For some of the "core"
C routines (Orthomin, BiConjugate Gradient and Conjugate Gradient
C on the normal equations) the user must also supply a matrix
C transpose times vector routine (MTTVEC, say) and (possibly,
C depending on the "core" method) a routine that solves the
C transpose of the preconditioning step (MTSOLV, say).
C Specifically, MTTVEC is a routine which calculates y = A'x, given
C x and the user's data structure for A (A' is the transpose of A).
C MTSOLV is a routine which solves the system r = M'z for z given r
C and the user's data structure for M.
C
C This process of writing the matrix vector operations can be time
C consuming and error prone. To alleviate these problems we have
C written drivers for the "core" methods that assume the user
C supplies one of two specific data structures (SLAP Triad and SLAP
C Column format), see below. Utilizing these data structures we
C have augmented each "core" method with two preconditioners:
C Diagonal Scaling and Incomplete Factorization. Diagonal scaling
C is easy to implement, vectorizes very well and for problems that
C are not too ill-conditioned reduces the number of iterations
C enough to warrant its use. On the other hand, an Incomplete
C factorization (Incomplete Cholesky for symmetric systems and
C Incomplete LU for nonsymmetric systems) may take much longer to
C calculate, but it reduces the iteration count (for most problems)
C significantly. Our implementations of IC and ILU vectorize for
C machines with hardware gather scatter, but the vector lengths can
C be quite short if the number of non-zeros in a column is not
C large.
C
C =================================================================
C ==================== Supplied Data Structures ===================
C =================================================================
C The following describes the data structures supplied with the
C package: SLAP Triad and Column formats.
C
C ====================== S L A P Triad format =====================
C
C In the SLAP Triad format only the non-zeros are stored. They may
C appear in *ANY* order. The user supplies three arrays of length
C NELT, where NELT is the number of non-zeros in the matrix:
C (IA(NELT), JA(NELT), A(NELT)). If the matrix is symmetric then
C one need only store the lower triangle (including the diagonal)
C and NELT would be the corresponding number of non-zeros stored.
C For each non-zero the user puts the row and column index of that
C matrix element in the IA and JA arrays. The value of the
C non-zero matrix element is placed in the corresponding location
C of the A array. This is an extremely easy data structure to
C generate. On the other hand, it is not very efficient on vector
C computers for the iterative solution of linear systems. Hence,
C SLAP changes this input data structure to the SLAP Column format
C for the iteration (but does not change it back).
C
C Here is an example of the SLAP Triad storage format for a
C nonsymmetric 5x5 Matrix. NELT=11. Recall that the entries may
C 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 the SLAP Column format the non-zeros are stored counting down
C columns (except for the diagonal entry, which must appear first
C in each "column") and are stored in the real array A. In other
C words, for each column in the matrix first put the diagonal entry
C in A. Then put in the other non-zero elements going down the
C column (except the diagonal) in order. The IA array holds the
C row index for each non-zero. The JA array holds the offsets into
C the IA, A arrays for the beginning of each column. That is,
C IA(JA(ICOL)), A(JA(ICOL)) are the first elements of the ICOL-th
C column in IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) are the
C last elements of the ICOL-th column. Note that we always have
C JA(N+1) = NELT+1, where N is the number of columns in the matrix
C and NELT is the number of non-zeros in the matrix. If the matrix
C is symmetric one need only store the lower triangle (including
C the diagonal) and NELT would be the corresponding number of
C non-zeros stored.
C
C Here is an example of the SLAP Column storage format for a
C nonsymmetric 5x5 Matrix (in the A and IA arrays '|' denotes the
C end of a 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 =================================================================
C ====================== Which Method To Use ======================
C =================================================================
C
C BACKGROUND
C In solving a large sparse linear system Ax = b using an iterative
C method, it is not necessary to actually store the matrix A.
C Rather, what is needed is a procedure for multiplying the matrix
C A times a given vector y to obtain the matrix-vector product, Ay.
C SLAP has been written to take advantage of this fact. The higher
C level routines in the package require storage only of the non-zero
C elements of A (and their positions), and even this can be
C avoided, if the user writes his own subroutine for multiplying
C the matrix times a vector and calls the lower-level iterative
C routines in the package.
C
C If the matrix A is ill-conditioned, then most iterative methods
C will be slow to converge (if they converge at all!). To improve
C the convergence rate, one may use a "matrix splitting," or,
C "preconditioning matrix," say, M. It is then necessary to solve,
C at each iteration, a linear system with coefficient matrix M. A
C good preconditioner M should have two properties: (1) M should
C "approximate" A, in the sense that the matrix inv(M)*A (or some
C variant thereof) is better conditioned than the original matrix
C A; and (2) linear systems with coefficient matrix M should be
C much easier to solve than the original system with coefficient
C matrix A. Preconditioning routines in the SLAP package are
C separate from the iterative routines, so that any of the
C preconditioners provided in the package, or one that the user
C codes himself, can be used with any of the iterative routines.
C
C CHOICE OF PRECONDITIONER
C If you willing to live with either the SLAP Triad or Column
C matrix data structure you can then choose one of two types of
C preconditioners to use: diagonal scaling or incomplete
C factorization. To choose between these two methods requires
C knowing something about the computer you're going to run these
C codes on and how well incomplete factorization approximates the
C inverse of your matrix.
C
C Let us suppose you have a scalar machine. Then, unless the
C incomplete factorization is very, very poor this is *GENERALLY*
C the method to choose. It will reduce the number of iterations
C significantly and is not all that expensive to compute. So if
C you have just one linear system to solve and "just want to get
C the job done" then try incomplete factorization first. If you
C are thinking of integrating some SLAP iterative method into your
C favorite "production code" then try incomplete factorization
C first, but also check to see that diagonal scaling is indeed
C slower for a large sample of test problems.
C
C Let us now suppose you have a vector computer with hardware
C gather/scatter support (Cray X-MP, Y-MP, SCS-40 or Cyber 205, ETA
C 10, ETA Piper, Convex C-1, etc.). Then it is much harder to
C choose between the two methods. The versions of incomplete
C factorization in SLAP do in fact vectorize, but have short vector
C lengths and the factorization step is relatively more expensive.
C Hence, for most problems (i.e., unless your problem is ill
C conditioned, sic!) diagonal scaling is faster, with its very
C fast set up time and vectorized (with long vectors)
C preconditioning step (even though it may take more iterations).
C If you have several systems (or right hand sides) to solve that
C can utilize the same preconditioner then the cost of the
C incomplete factorization can be amortized over these several
C solutions. This situation gives more advantage to the incomplete
C factorization methods. If you have a vector machine without
C hardware gather/scatter (Cray 1, Cray 2 & Cray 3) then the
C advantages for incomplete factorization are even less.
C
C If you're trying to shoehorn SLAP into your favorite "production
C code" and can not easily generate either the SLAP Triad or Column
C format then you are left to your own devices in terms of
C preconditioning. Also, you may find that the preconditioners
C supplied with SLAP are not sufficient for your problem. In this
C situation we would recommend that you talk with a numerical
C analyst versed in iterative methods about writing other
C preconditioning subroutines (e.g., polynomial preconditioning,
C shifted incomplete factorization, SOR or SSOR iteration). You
C can always "roll your own" by using the "core" iterative methods
C and supplying your own MSOLVE and MATVEC (and possibly MTSOLV and
C MTTVEC) routines.
C
C SYMMETRIC SYSTEMS
C If your matrix is symmetric then you would want to use one of the
C symmetric system solvers. If your system is also positive
C definite, (Ax,x) (Ax dot product with x) is positive for all
C non-zero vectors x, then use Conjugate Gradient (SCG, SSDCG,
C SSICSG). If you're not sure it's SPD (symmetric and Positive
C Definite) then try SCG anyway and if it works, fine. If you're
C sure your matrix is not positive definite then you may want to
C try the iterative refinement methods (SIR) or the GMRES code
C (SGMRES) if SIR converges too slowly.
C
C NONSYMMETRIC SYSTEMS
C This is currently an area of active research in numerical
C analysis and there are new strategies being developed.
C Consequently take the following advice with a grain of salt. If
C you matrix is positive definite, (Ax,x) (Ax dot product with x
C is positive for all non-zero vectors x), then you can use any of
C the methods for nonsymmetric systems (Orthomin, GMRES,
C BiConjugate Gradient, BiConjugate Gradient Squared and Conjugate
C Gradient applied to the normal equations). If your system is not
C too ill conditioned then try BiConjugate Gradient Squared (BCGS)
C or GMRES (SGMRES). Both of these methods converge very quickly
C and do not require A' or M' (' denotes transpose) information.
C SGMRES does require some additional storage, though. If the
C system is very ill conditioned or nearly positive indefinite
C ((Ax,x) is positive, but may be very small), then GMRES should
C be the first choice, but try the other methods if you have to
C fine tune the solution process for a "production code". If you
C have a great preconditioner for the normal equations (i.e., M is
C an approximation to the inverse of AA' rather than just A) then
C this is not a bad route to travel. Old wisdom would say that the
C normal equations are a disaster (since it squares the condition
C number of the system and SCG convergence is linked to this number
C of infamy), but some preconditioners (like incomplete
C factorization) can reduce the condition number back below that of
C the original system.
C
C =================================================================
C ======================= Naming Conventions ======================
C =================================================================
C SLAP iterative methods, matrix vector and preconditioner
C calculation routines follow a naming convention which, when
C understood, allows one to determine the iterative method and data
C structure(s) used. The subroutine naming convention takes the
C following form:
C P[S][M]DESC
C where
C P stands for the precision (or data type) of the routine and
C is required in all names,
C S denotes whether or not the routine requires the SLAP Triad
C or Column format (it does if the second letter of the name
C is S and does not otherwise),
C M stands for the type of preconditioner used (only appears
C in drivers for "core" routines), and
C DESC is some number of letters describing the method or purpose
C of the routine. The following is a list of the "DESC"
C fields for iterative methods and their meaning:
C BCG,BC: BiConjugate Gradient
C CG: Conjugate Gradient
C CGN,CN: Conjugate Gradient on the Normal equations
C CGS,CS: biConjugate Gradient Squared
C GMRES,GMR,GM: Generalized Minimum RESidual
C IR,R: Iterative Refinement
C JAC: JACobi's method
C GS: Gauss-Seidel
C OMN,OM: OrthoMiN
C
C In the single precision version of SLAP, all routine names start
C with an S. The brackets around the S and M designate that these
C fields are optional.
C
C Here are some examples of the routines:
C 1) SBCG: Single precision BiConjugate Gradient "core" routine.
C One can deduce that this is a "core" routine, because the S and
C M fields are missing and BiConjugate Gradient is an iterative
C method.
C 2) SSDBCG: Single precision, SLAP data structure BCG with Diagonal
C scaling.
C 3) SSLUBC: Single precision, SLAP data structure BCG with incom-
C plete LU factorization as the preconditioning.
C 4) SCG: Single precision Conjugate Gradient "core" routine.
C 5) SSDCG: Single precision, SLAP data structure Conjugate Gradient
C with Diagonal scaling.
C 6) SSICCG: Single precision, SLAP data structure Conjugate Gra-
C dient with Incomplete Cholesky factorization preconditioning.
C
C
C =================================================================
C ===================== USER CALLABLE ROUTINES ====================
C =================================================================
C The following is a list of the "user callable" SLAP routines and
C their one line descriptions. The headers denote the file names
C where the routines can be found, as distributed for UNIX systems.
C
C Note: Each core routine, SXXX, has a corresponding stop routine,
C ISSXXX. If the stop routine does not have the specific stop
C test the user requires (e.g., weighted infinity norm), then
C the user should modify the source for ISSXXX accordingly.
C
C ============================= sir.f =============================
C SIR: Preconditioned Iterative Refinement Sparse Ax = b Solver.
C SSJAC: Jacobi's Method Iterative Sparse Ax = b Solver.
C SSGS: Gauss-Seidel Method Iterative Sparse Ax = b Solver.
C SSILUR: Incomplete LU Iterative Refinement Sparse Ax = b Solver.
C
C ============================= scg.f =============================
C SCG: Preconditioned Conjugate Gradient Sparse Ax=b Solver.
C SSDCG: Diagonally Scaled Conjugate Gradient Sparse Ax=b Solver.
C SSICCG: Incomplete Cholesky Conjugate Gradient Sparse Ax=b Solver.
C
C ============================= scgn.f ============================
C SCGN: Preconditioned CG Sparse Ax=b Solver for Normal Equations.
C SSDCGN: Diagonally Scaled CG Sparse Ax=b Solver for Normal Eqn's.
C SSLUCN: Incomplete LU CG Sparse Ax=b Solver for Normal Equations.
C
C ============================= sbcg.f ============================
C SBCG: Preconditioned BiConjugate Gradient Sparse Ax = b Solver.
C SSDBCG: Diagonally Scaled BiConjugate Gradient Sparse Ax=b Solver.
C SSLUBC: Incomplete LU BiConjugate Gradient Sparse Ax=b Solver.
C
C ============================= scgs.f ============================
C SCGS: Preconditioned BiConjugate Gradient Squared Ax=b Solver.
C SSDCGS: Diagonally Scaled CGS Sparse Ax=b Solver.
C SSLUCS: Incomplete LU BiConjugate Gradient Squared Ax=b Solver.
C
C ============================= somn.f ============================
C SOMN: Preconditioned Orthomin Sparse Iterative Ax=b Solver.
C SSDOMN: Diagonally Scaled Orthomin Sparse Iterative Ax=b Solver.
C SSLUOM: Incomplete LU Orthomin Sparse Iterative Ax=b Solver.
C
C ============================ sgmres.f ===========================
C SGMRES: Preconditioned GMRES Iterative Sparse Ax=b Solver.
C SSDGMR: Diagonally Scaled GMRES Iterative Sparse Ax=b Solver.
C SSLUGM: Incomplete LU GMRES Iterative Sparse Ax=b Solver.
C
C ============================ smset.f ============================
C The following routines are used to set up preconditioners.
C
C SSDS: Diagonal Scaling Preconditioner SLAP Set Up.
C SSDSCL: Diagonally Scales/Unscales a SLAP Column Matrix.
C SSD2S: Diagonal Scaling Preconditioner SLAP Normal Eqns Set Up.
C SS2LT: Lower Triangle Preconditioner SLAP Set Up.
C SSICS: Incomplete Cholesky Decomp. Preconditioner SLAP Set Up.
C SSILUS: Incomplete LU Decomposition Preconditioner SLAP Set Up.
C
C ============================ smvops.f ===========================
C Most of the incomplete factorization (LL' and LDU) solvers
C in this file require an intermediate routine to translate
C from the SLAP MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK,
C IWORK) calling convention to the calling sequence required
C by the solve routine. This generally is accomplished by
C fishing out pointers to the preconditioner (stored in RWORK)
C from the IWORK array and then making a call to the routine
C that actually does the backsolve.
C
C SSMV: SLAP Column Format Sparse Matrix Vector Product.
C SSMTV: SLAP Column Format Sparse Matrix (transpose) Vector Prod.
C SSDI: Diagonal Matrix Vector Multiply.
C SSLI: SLAP MSOLVE for Lower Triangle Matrix (set up for SSLI2).
C SSLI2: Lower Triangle Matrix Backsolve.
C SSLLTI: SLAP MSOLVE for LDL' (IC) Fact. (set up for SLLTI2).
C SLLTI2: Backsolve routine for LDL' Factorization.
C SSLUI: SLAP MSOLVE for LDU Factorization (set up for SSLUI2).
C SSLUI2: SLAP Backsolve for LDU Factorization.
C SSLUTI: SLAP MTSOLV for LDU Factorization (set up for SSLUI4).
C SSLUI4: SLAP Backsolve for LDU Factorization.
C SSMMTI: SLAP MSOLVE for LDU Fact of Normal Eq (set up for SSMMI2).
C SSMMI2: SLAP Backsolve for LDU Factorization of Normal Equations.
C
C =========================== slaputil.f ==========================
C The following utility routines are useful additions to SLAP.
C
C SBHIN: Read Sparse Linear System in the Boeing/Harwell Format.
C SCHKW: SLAP WORK/IWORK Array Bounds Checker.
C SCPPLT: Printer Plot of SLAP Column Format Matrix.
C SS2Y: SLAP Triad to SLAP Column Format Converter.
C QS2I1R: Quick Sort Integer array, moving integer and real arrays.
C (Used by SS2Y.)
C STIN: Read in SLAP Triad Format Linear System.
C STOUT: Write out SLAP Triad Format Linear System.
C
C
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 (NONE)
C***REVISION HISTORY (YYMMDD)
C 880715 DATE WRITTEN
C 890404 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 -----( This produced Version 2.0.1. )-----
C 891003 Rearranged list of user callable routines to agree with
C order in source deck. (FNF)
C 891004 Updated reference.
C 910411 Prologue converted to Version 4.0 format. (BAB)
C -----( This produced Version 2.0.2. )-----
C 910506 Minor improvements to prologue. (FNF)
C 920511 Added complete declaration section. (WRB)
C 920929 Corrected format of reference. (FNF)
C 921019 Improved one-line descriptions, reordering some. (FNF)
C***END PROLOGUE SLPDOC
C***FIRST EXECUTABLE STATEMENT SLPDOC
C
C This is a *DUMMY* subroutine and should never be called.
C
RETURN
C------------- LAST LINE OF SLPDOC FOLLOWS -----------------------------
END