psldu(3S)

NAME
     PSLDU_Destroy, PSLDU_ExtractPerm, PSLDU_Factor, PSLDU_FactorOOC,
     PSLDU_OOCLimit, PSLDU_OOCPath, PSLDU_Ordering, PSLDU_Preprocess,
     PSLDU_PreprocessZ, PSLDU_Solve, PSLDU_SolveM, PSLDU_Storage - Parallel
     sparse unsymmetric linear system solver

SYNOPSIS
     Fortran synopsis:

          SUBROUTINE PSLDU_DESTROY (token)
          INTEGER token

          SUBROUTINE PSLDU_EXTRACTPERM (token, perm)
          INTEGER token, perm(*)

          SUBROUTINE PSLDU_FACTOR (token, n, pointers, indices, values)
          INTEGER token, n, pointers(*), indices(*)
          DOUBLE PRECISION values(*)

          SUBROUTINE PSLDU_FACTOROOC (token, n, pointers, indices, values)
          INTEGER token, n, pointers(*), indices(*)
          DOUBLE PRECISION values(*)

          SUBROUTINE PSLDU_OOCLIMIT (token, ooclimit)
          INTEGER token
          DOUBLE PRECISION ooclimit

          SUBROUTINE PSLDU_OOCPATH (token, oocpath)
          INTEGER token
          CHARACTER oocpath(*)

          SUBROUTINE PSLDU_ORDERING (token, method)
          INTEGER token, method

          SUBROUTINE PSLDU_PREPROCESS (token, n, pointers, indices, non_zeros,
          ops)
          INTEGER token, n, pointers(*), indices(*)
          INTEGER*8 non_zeros
          DOUBLE PRECISION ops

          SUBROUTINE PSLDU_PREPROCESSZ (token, n, pointers, indices, mask,
          non_zeros, ops)
          INTEGER token, n, pointers(*), indices(*), mask(*)
          INTEGER*8 non_zeros
          DOUBLE PRECISION ops

          SUBROUTINE PSLDU_SOLVE (token, x, b)
          INTEGER token
          DOUBLE PRECISION x(*), b(*)

          SUBROUTINE PSLDU_SOLVEM (token, X, B, nrhs)
          INTEGER token, nrhs
          DOUBLE PRECISION X(*), B(*)

          DOUBLE PRECISION FUNCTION PSLDU_STORAGE(token)
          INTEGER token

     C/C++ synopsis:

          #include <scsl_sparse.h>

          void PSLDU_Destroy (int token );

          void PSLDU_ExtractPerm (int token, int perm[] );

          void PSLDU_Factor (int token, int n, int pointers[], int indices[],
          double values[] );

          void PSLDU_FactorOOC (int token, int n, int pointers[], int
          indices[], double values[] );

          void PSLDU_OOCLimit (int token, double ooclimit );

          void PSLDU_OOCPath (int token, char oocpath[] );

          void PSLDU_Ordering (int token, int method );

          void PSLDU_Preprocess (int token, int n, int pointers[], int
          indices[], long long *non_zeros, double *ops );

          void PSLDU_PreprocessZ (int token, int n, int pointers[], int
          indices[], int mask[], long long *non_zeros, double *ops );

          void PSLDU_Solve (int token, double x[], double b[] );

          void PSLDU_SolveM (int token, double X[], double B[], int nrhs);

          double PSLDU_Storage (int token);

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits). This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option.  A program may
     use only one of the two versions; 4-byte integer and 8-byte integer
     library calls cannot be mixed.

     The C and C++ prototypes shown above are appropriate for the 4-byte
     integer version of SCSL. When using the 8-byte integer version, the
     variables of type int become long long and the <scsl_sparse_i8.h> header
     file should be included.

DESCRIPTION
     NOTE:  these interfaces are obsolete and will no longer be supported in
     future versions of SCSL.  Please use the routines described in the DPSLDU
     man page instead.  For complex data types, see the ZPSLDU man page.

     PSLDU solves sparse unsymmetric linear systems of the form Ax = b where A
     is an n-by-n input matrix having symmetric non-zero pattern but
     unsymmetric non-zero values, b is an input vector of length n, and x is
     an unknown vector of length n.

     PSLDU uses a direct method. A is factored into the following form:

          A = L D U

     where L is a lower triangular matrix with unit diagonal, D is a diagonal
     matrix, and U is an upper triangular matrix with unit diagonal.

     Note that NO PIVOTING FOR STABILITY is performed during factorization.

     The PSLDU library contains five main routines.

     *   PSLDU_Ordering() allows the user to select one of five possible
         reordering methods to be used in the matrix preprocessing phase.

     *   PSLDU_Preprocess() performs preprocessing operations on the structure
         of A (heuristic reordering to reduce fill in L and U, symbolic
         factorization, etc.).

     *   PSLDU_Factor() factors the matrix A into L and U, using the
         previously computed preprocessing data.

     *   PSLDU_Solve() solves for a vector x, given an input vector b.

     *   PSLDU_Destroy() frees all storage associated with the matrix A
         (including L, D, U, and various data structures computed during
         preprocessing).

     The user can call PSLDU_Factor() several times after a single call to
     PSLDU_Preprocess() to factor multiple matrices with identical non-zero
     structures but different values.  Similarly, the user can call
     PSLDU_Solve() several times after a single call to PSLDU_Factor() to
     solve for multiple right-hand-sides.  Also, the user can call
     PSLDU_SolveM() to solve for multiple right-hand-sides all stored in a
     single array.

   Sparse Matrix Format
     Sparse matrix A must be input to PSLDU in Harwell-Boeing format (also
     known as Compressed Column Storage format).

     The matrix is held in three arrays: pointers, indices, and values.  The
     indices array contains the row indices of the non-zeros in A. The values
     array holds the corresponding non-zero values. The pointers array
     contains the index in indices for the first non-zero in each column of A.
     Thus, the row indices for the non-zeros in column i can be found in
     locations indices[pointers[i]] through indices[pointers[i+1]-1]. The
     corresponding values can be found in location values[pointers[i]] through
     values[pointers[i+1]-1].

     PSLDU imposes one constraint on the representation of the A matrix. The
     non-zeros within each column must appear in order of increasing row
     number.

     In the following example, the unsymmetric matrix

     1.0 0.0 5.0 0.0
     0.0 3.0 0.0 8.0
     2.0 0.0 7.0 0.0
     0.0 4.0 0.0 9.0


     would be represented in FORTRAN as follows:

      INTEGER pointers(5), indices(8), i
      DOUBLE PRECISION values(8)
      DATA (pointers(i), i = 1, 5) / 1, 3, 5, 7, 9 /
      DATA (indices(i),  i = 1, 8) / 1, 3, 2, 4, 1, 3, 2, 4 /
      DATA (values(i),   i = 1, 8) / 1.0, 2.0, 3.0, 4.0, 5.0,
     &                               7.0, 8.0, 9.0 /


     Zero-based indexing is used in C, so the pointers, indices, and values
     arrays would contain the following:

     int pointers[]  = {0, 2, 4, 6, 8};
     int indices[]   = {0, 2, 1, 3, 0, 2, 1, 3};
     double values[] = {1.0, 2.0, 3.0, 4.0, 5.0, 7.0, 8.0, 9.0};


   Ordering Methods
     The PSLDU_Ordering(token, method) routine allows the user to change the
     ordering method used to pre-order the matrix before factorization.  This
     routine must be called before calling PSLDU_Preprocess. Five options are
     currently available for the method parameter:

     *   Method 0 performs no pre-ordering

     *   Method 1 performs Approximate Minimum Fill ordering

     *   Method 2 performs a single nested dissection ordering (default).
         This method is often called "Extreme matrix ordering".

     *   Method 3 performs multiple nested dissection orderings (in parallel)

     *   Method 4 performs multiple nested dissection (the same as in Method
         3), but it uses a feedback file to "learn" from the previous solves
         of the same matrix structure and it performs more orderings. The
         multiple nested dissection technique of Methods 3 and 4 is also
         referred to as "Extreme2 matrix ordering".

     Method 2 is significantly more expensive than Method 1, but it usually
     produces significantly better orderings.  Method 3 is especially
     effective on multi-processor systems.  It computes OMP_NUM_THREADS (where
     OMP_NUM_THREADS is an environment variable indicating the number of
     processors to be used for parallel computation) matrix orderings using
     different starting points for the algorithm and uses the ordering that
     will lead to the fewest floating-point operations to factorize the
     matrix.

     Method 4 is useful only when the same non-zero structure is used for
     multiple solves.  Method 4 keeps a record in a "feedback" file of a
     signature for non-zero structures for a maximum of 200 matrices and of
     the starting point that was saved from a previous solve for that
     structure.  In the next Method 4 ordering for that non-zero structure,
     that best starting point and 2 * OMP_NUM_THREADS - 1 new ones generate
     orderings.  The best ordering is used.  In this way, the quality of
     orderings stay the same or improve over time.

     Methods 3 and 4 typically take more time for the matrix preprocessing
     than the default.  However, on large systems or on repeated
     factorizations, significant overall speedups (1.1X to 2X) can be obtained
     compared to Method 2.

   Extracting the permutation vector
     Unless ordering Method 0 is used, PSLDU applies a symmetric permutation
     to matrix A before the factorization step; the resulting permuted matrix
     generally has significantly less fill-in than the original matrix.  The
     user can obtain the permutation matrix associated with a given token by
     calling PSLDU_ExtractPerm(token, perm). The permutation is returned as an
     integer array of length n, with 1 <= perm(i) <= n (0 <= perm[i] < n for C
     code).

     A value of k for perm(i) implies that node k in the original ordering is
     node i in the new ordering.

   Matrices with zeros on the diagonal
     As noted above, no pivoting is done for stability during factorization;
     when zero or near-zero pivots are encountered, PSLDU usually fails. In
     these cases, it may be possible to use PSLDU_PreprocessZ() to obtain a
     slightly different, but stable, ordering.  The user provides an
     additional integer array, mask, as an argument to PSLDU_PreprocessZ().
     If mask(i)=0, then PSLDU will attempt to maximize the diagonal element
     |Aii|.

   Memory usage
     The returned value of PSLDU_Storage() is an estimate of the amount of
     storage required (in millions of bytes) by the solver's data structures
     for a given matrix system.

   Out-of-core Factorization
     The storage associated with the factor can be managed in two ways.  The
     PSLDU_Factor() routine allocates memory for the factor and manages it
     internally, releasing it only when PSLDU_Destroy() is called.  The
     alternative is to do out-of-core factorization by calling
     PSLDU_FactorOOC(). This routine uses a small amount of in-core memory,
     placing the remainder of the factor matrix on disk as it is computed.
     The user can call PSLDU_OOCPath() to indicate the directory in which the
     factor file should be written, and PSLDU_OOCLimit() to indicate how much
     memory to use to hold portions of the factor matrix in-core.  More in-
     core memory generally leads to less disk I/O and higher performance
     during the factorization.  The only required change is to move from in-
     core factorization to out-of-core factorization is the change from
     PSLDU_Factor() to PSLDU_FactorOOC().  The other routines (PSLDU_Solve(),
     PSLDU_Destroy(), etc.) handle out-of-core factors transparently.  Note
     that PSLDU_FactorOOC and subsequent calls to PSLDU_Solve are not
     parallelized.

   Multiple Right-Hand-Sides
     PSLDU can solve for large numbers of right-hand-sides with one call to
     PSLDU_SolveM().  It solves these right hand sides in parallel, with each
     processor solving up to four at a time.

   Arguments
     These routines have the following arguments:

     token     (input) PSLDU can handle multiple matrices simultaneously. The
               token distinguishes between active matrices.  The token passed
               to PSLDU_Factor() must match the token used in some previous
               call to PSLDU_Preprocess().  Similarly, the token passed to
               PSLDU_Solve() must match the token used in some previous call
               to PSLDU_Factor().  0 <= token <= 19.

     method    (input) An integer specifying the ordering method used during
               preprocessing.  0 <= method <= 4.

     n         (input) The number of rows and columns in the matrix A.  n >=
               0.

     pointers, indices, values
               (input) The pointers and indices arrays store the non-zero
               structure of sparse input matrix A in Harwell-Boeing or
               Compressed Sparse Column (CSC) format.

               The pointers array stores n+1 integers, where pointers[i] gives
               the index in indices of the first non-zero in column i of A.
               The indices array stores the row indices of the non-zeros in A.
               The values array stores the non-zero values in the matrix A.

     non_zeros (output) The number of non-zero values in L and U.

     ops       (output) The number of floating-point operations required to
               factor A.

     mask      (input) An integer array of length n used in
               PSLDU_PreprocessZ().  If mask(i)=0, then node i of matrix A is
               ordered after all of its neighbors in an attempt to avoid a
               zero pivot.

     b         (input) The right-hand-side vector in a PSLDU_Solve call.

     x         (output) The solution vector in a PSLDU_Solve call.

     nrhs      (input) The number of right-hand side vectors present in a
               PSLDU_SolveM() call.

     B         (input) The right-hand-side matrix in a PSLDU_SolveM() call.
               Must be stored in column-major order, and each of the nrhs
               columns must have length n.

     X         (output) The solution matrix in a PSLDU_SolveM() call. Must be
               stored in column-major order, and each of the nrhs columns must
               have length n.

     oocpath   (input) A character array/string with a path to the directory
               where the temporary out-of-core factor files should be stored.
               If this path is on a striped (or raid-0) file system, the
               performance of the out-of-core solves can be considerably
               improved.  The default path is /usr/tmp.

     ooclimit  (input) A double precision number indicating the number of
               Mbytes of random access memory that should be used for factor
               storage during a call to PSLDU_FactorOOC. Note that there are
               many other arrays used besides those directly used to store the
               factorization, so total RAM usage by the solve will exceed this
               number.  The default is 64 MB.

     perm      (output) An integer array of length n containing the
               permutation used to reorder matrix A.

ENVIRONMENT VARIABLES
     Two environment variables can affect the operation of ordering methods 3
     and 4.  SPARSE_NUM_ORDERS can be used to change the number of orderings
     performed from the default of OMP_NUM_THREADS for Method 3 and
     (2*OMP_NUM_THREADS) for Method 4.  SPARSE_FEEDBACK_FILE can be set to the
     path and file name where the feedback information will be kept;
     otherwise, the default feedback file is $HOME/.sparseFeedback.  This file
     will be less than 5K bytes.

     The environment variable OMP_NUM_THREADS determines the number of
     processors that are used for the numerical factorization. The out-of-core
     solve is limited to one processor. Setting the environment variable
     PSLDU_VERBOSE causes PSLDU to output information about the factorization.

NOTES
     These routines are optimized and parallelized for the SGI R8000 and
     R1x000 platforms.

SEE ALSO
     INTRO_SCSL(3S), INTRO_SOLVERS(3S), DPSLDLT(3S), ZPSLDLT(3S), DPSLDU(3S),
     ZPSLDU(3S)