Contents


NAME

     zcscsm - compressed sparse column format triangular solve

SYNOPSIS

       SUBROUTINE ZCSCSM( TRANSA, M, N, UNITD, DV, ALPHA, DESCRA,
      *           VAL, INDX, PNTRB, PNTRE,
      *           B, LDB, BETA, C, LDC, WORK, LWORK)
       INTEGER    TRANSA, M, N, UNITD, DESCRA(5),
      *           LDB, LDC, LWORK
       INTEGER    INDX(NNZ), PNTRB(M), PNTRE(M)
       DOUBLE COMPLEX ALPHA, BETA
       DOUBLE COMPLEX DV(M), VAL(NNZ), B(LDB,*), C(LDC,*), WORK(LWORK)

       SUBROUTINE ZCSCSM_64( TRANSA, M, N, UNITD, DV, ALPHA, DESCRA,
      *           VAL, INDX, PNTRB, PNTRE,
      *           B, LDB, BETA, C, LDC, WORK, LWORK)
       INTEGER*8  TRANSA, M, N, UNITD, DESCRA(5),
      *           LDB, LDC, LWORK
       INTEGER*8  INDX(NNZ), PNTRB(M), PNTRE(M)
       DOUBLE COMPLEX ALPHA, BETA
       DOUBLE COMPLEX DV(M), VAL(NNZ), B(LDB,*), C(LDC,*), WORK(LWORK)

       where NNZ = PNTRE(M)-PNTRB(1)

     F95 INTERFACE

       SUBROUTINE CSCSM( TRANSA, M, [N], UNITD, DV, ALPHA, DESCRA, VAL, INDX,
      *   PNTRB, PNTRE, B, [LDB], BETA, C, [LDC], [WORK], [LWORK] )
       INTEGER TRANSA, M, UNITD
       INTEGER, DIMENSION(:) ::   DESCRA, INDX, PNTRB, PNTRE
       DOUBLE COMPLEX    ALPHA, BETA
       DOUBLE COMPLEX, DIMENSION(:) :: VAL, DV
       DOUBLE COMPLEX, DIMENSION(:, :) ::  B, C

       SUBROUTINE CSCSM_64(TRANSA, M, [N], UNITD, DV, ALPHA, DESCRA, VAL, INDX,
      *   PNTRB, PNTRE, B, [LDB], BETA, C, [LDC], [WORK], [LWORK] )
       INTEGER*8 TRANSA, M, UNITD
       INTEGER*8, DIMENSION(:) ::   DESCRA, INDX, PNTRB, PNTRE
       DOUBLE COMPLEX    ALPHA, BETA
       DOUBLE COMPLEX, DIMENSION(:) :: VAL, DV
       DOUBLE COMPLEX, DIMENSION(:, :) ::  B, C

     C INTERFACE

     #include <sunperf.h>

     void zcscsm(int transa, int m, int n, int unitd,
      doublecomplex *dv, doublecomplex *alpha, int *descra,
     doublecomplex *val,
      int *indx, int *pntrb, int *pntre, doublecomplex *b,
      int ldb, doublecomplex *beta, doublecomplex* c, int ldc);

     void  zcscsm_64(long transa, long m, long n, long unitd,
     doublecomplex *dv, doublecomplex *alpha, long *descra,
     doublecomplex *val, long *indx, long *pntrb, long *pntre,
     doublecomplex *b, long ldb, doublecomplex *beta,
     doublecomplex *c, long ldc);

DESCRIPTION

      zcscsm performs one of the matrix-matrix operations

        C <- alpha  op(A) B + beta C,     C <-alpha D op(A) B + beta C,
        C <- alpha  op(A) D B + beta C,

      where alpha and beta are scalars, C and B are m by n dense matrices,
      D is a diagonal scaling matrix,  A is a sparse m by m unit, or non-unit,
      upper or lower triangular matrix represented in the compressed sparse
      column format and op( A )  is one  of

       op( A ) = inv(A) or  op( A ) = inv(A')  or  op( A ) =inv(conjg( A' ))
       (inv denotes matrix inverse,  ' indicates matrix transpose).

ARGUMENTS

      TRANSA(input)   On entry, integer TRANSA indicates how to operate
                      with the sparse matrix:
                        0 : operate with matrix
                        1 : operate with transpose matrix
                        2 : operate with the conjugate transpose of matrix.
                          2 is equivalent to 1 if matrix is real.
                      Unchanged on exit.

      M(input)        On entry,  M  specifies the number of rows in
                      the matrix A. Unchanged on exit.

      N(input)        On entry,  N specifies the number of columns in
                      the matrix C. Unchanged on exit.

      UNITD(input)    On entry,  UNITD specifies the type of scaling:
                        1 : Identity matrix (argument DV[] is ignored)
                        2 : Scale on left (row scaling)
                        3 : Scale on right (column scaling)
                        4 : Automatic column scaling (see section NOTES for
                             further details)
                      Unchanged on exit.

      DV(input)       On entry, DV is an array of length M consisting of the
                      diagonal entries of the diagonal scaling matrix D.
                      If UNITD is 4, DV contains diagonal matrix by which
                      the columns of A have been scaled (see section NOTES for
                      further details). Otherwise, unchanged on exit.
      ALPHA(input)    On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

      DESCRA (input)  Descriptor argument.  Five element integer array:
                      DESCRA(1) matrix structure
                        0 : general
                        1 : symmetric (A=A')
                        2 : Hermitian (A= CONJG(A'))
                        3 : Triangular
                        4 : Skew(Anti)-Symmetric (A=-A')
                        5 : Diagonal
                        6 : Skew-Hermitian (A= -CONJG(A'))

                      Note: For the routine, DESCRA(1)=3 is only supported.

                      DESCRA(2) upper/lower triangular indicator
                        1 : lower
                        2 : upper
                      DESCRA(3) main diagonal type
                        0 : non-unit
                        1 : unit
                      DESCRA(4) Array base (NOT IMPLEMENTED)
                        0 : C/C++ compatible
                        1 : Fortran compatible
                      DESCRA(5) repeated indices? (NOT IMPLEMENTED)
                        0 : unknown
                        1 : no repeated indices

      VAL(input)      On entry, VAL is a scalar array of length
                      NNZ = PNTRE(M)-PNTRB(1) consisting of nonzero
                      entries of A. If UNITD is 4, VAL contains
                      the scaled matrix  A*D  (see section NOTES for
                      further details). Otherwise, unchanged on exit.

      INDX(input)     On entry, INDX is an integer array of length
                      NNZ = PNTRE(M)-PNTRB(1) consisting of the row
                      indices of nonzero entries of A.
                      Row indices MUST be sorted in increasing order
                      for each column. Unchanged on exit.

      PNTRB(input)    On entry, PNTRB is an integer array of length M
                      such that PNTRB(J)-PNTRB(1)+1 points to location
                      in VAL of the first nonzero element in column J.
                      Unchanged on exit.

      PNTRE(input)    On entry, PNTRE is an integer array of length M
                      such that PNTRE(J)-PNTRB(1) points to location
                      in VAL of the last nonzero element in column J.
                      Unchanged on exit.

      B (input)       Array of DIMENSION ( LDB, N ).
                      On entry, the leading m by n part of the array B
                      must contain the matrix B. Unchanged on exit.
      LDB (input)     On entry, LDB specifies the first dimension of B as declared
                      in the calling (sub) program. Unchanged on exit.

      BETA (input)    On entry, BETA specifies the scalar beta. Unchanged on exit.

      C(input/output) Array of DIMENSION ( LDC, N ).
                      On entry, the leading m by n part of the array C
                      must contain the matrix C. On exit, the array C is
                      overwritten.

      LDC (input)     On entry, LDC specifies the first dimension of C as declared
                      in the calling (sub) program. Unchanged on exit.

      WORK(workspace)   Scratch array of length LWORK.
                      On exit, if LWORK= -1, WORK(1) returns the optimum  size
                      of LWORK.

      LWORK (input)   On entry, LWORK specifies the length of WORK array. LWORK
                      should be at least M.

                      For good performance, LWORK should generally be larger.
                      For optimum performance on multiple processors, LWORK
                      >=M*N_CPUS where N_CPUS is the maximum number of
                      processors available to the program.

                      If LWORK=0, the routine is to allocate workspace needed.

                      If LWORK = -1, then a workspace query is assumed; the
                      routine only calculates the optimum size of the WORK array,
                      returns this value as the first entry of the WORK array,
                      and no error message related to LWORK is issued by XERBLA.

SEE ALSO

     Libsunperf  SPARSE BLAS is fully parallel and compatible
     with NIST FORTRAN Sparse Blas but the sources are different.
     Libsunperf SPARSE BLAS is free of bugs found in NIST FORTRAN
     Sparse Blas.  Besides several new features and routines are
     implemented.

     NIST FORTRAN Sparse Blas User's Guide available at:

     http://math.nist.gov/mcsd/Staff/KRemington/fspblas/

     Based on the standard proposed in

     "Document for the Basic Linear Algebra Subprograms (BLAS)
     Standard", University of Tennessee, Knoxville, Tennessee,
     1996:

     http://www.netlib.org/utk/papers/sparse.ps
NOTES/BUGS
     1. No test for singularity or near-singularity is included
     in this routine. Such tests must be performed before calling
     this routine.

     2. If UNITD =4, the routine scales the columns of A such
     that their 2-norms are one. The scaling may improve the
     accuracy of the computed solution. Corresponding entries of
     VAL are changed only in the particular case. On return DV
     matrix stored as a vector contains the diagonal matrix by
     which the columns have been scaled. UNITD=3 should be used
     for the next calls to the routine with overwritten VAL and
     DV.

     WORK(1)=0 on return if the scaling has been completed
     successfully, otherwise WORK(1) = - i where i is the column
     number which 2-norm is exactly zero.

     3. If DESCRA(3)=1 and  UNITD < 4, the diagonal entries are
     each used with the mathematical value 1. The entries of the
     main diagonal in the CSC representation of a sparse matrix
     do not need to be 1.0 in this usage. They are not used by
     the routine in these cases. But if UNITD=4, the unit
     diagonal elements MUST be referenced in the CSC
     representation.

     4. The routine is designed so that it checks the validity of
     each sparse entry given in the sparse blas representation.
     Entries with incorrect indices are not used and no error
     message related to the entries is issued.

     The feature also provides a possibility to use the sparse
     matrix representation of a general matrix A for solving
     triangular systems with the upper or lower triangle of A.
     But DESCRA(1) MUST be equal to 3 even in this case.

     Assume that there is the sparse matrix representation a
     general matrix A decomposed in the form

                          A = L + D + U

     where L is the strictly lower triangle of A, U is the
     strictly upper triangle of A, D is the diagonal matrix.
     Let's I denotes the identity matrix.

     Then the correspondence between the first three values of
     DESCRA and the result matrix for the sparse representation
     of A is

       DESCRA(1)  DESCRA(2)   DESCRA(3)     RESULT

          3          1           1      alpha*op(L+I)*B+beta*C
          3          1           0      alpha*op(L+D)*B+beta*C

          3          2           1      alpha*op(U+I)*B+beta*C

          3          2           0      alpha*op(U+D)*B+beta*C

     5. It is known that there exists another representation of
     the compressed sparse column format (see for example Y.Saad,
     "Iterative Methods for Sparse Linear Systems", WPS, 1996).
     Its data structure consists of three array instead of the
     four used in the current implementation.  The main
     difference is that only one array, IA, containing the
     pointers to the beginning of each column  in the arrays VAL
     and INDX is used instead of two arrays PNTRB and PNTRE. To
     use the routine with this kind of sparse column format the
     following calling sequence should be used

       SUBROUTINE ZCSCSM( TRANSA, M, N, UNITD, DV, ALPHA, DESCRA,
      *           VAL, INDX, IA, IA(2), B, LDB, BETA,
      *           C, LDC, WORK, LWORK )