Contents


NAME

     cbelsm - block Ellpack format triangular solve

SYNOPSIS

       SUBROUTINE CBELSM( TRANSA, MB, N, UNITD, DV, ALPHA, DESCRA,
      *           VAL, BINDX, BLDA, MAXBNZ, LB,
      *           B, LDB, BETA, C, LDC, WORK, LWORK )
       INTEGER    TRANSA, MB, N, UNITD, DESCRA(5), BLDA, MAXBNZ, LB,
      *           LDB, LDC, LWORK
       INTEGER    BINDX(BLDA,MAXBNZ)
       COMPLEX    ALPHA, BETA
       COMPLEX    DV(MB*LB*LB), VAL(LB*LB*BLDA*MAXBNZ), B(LDB,*), C(LDC,*),
      *           WORK(LWORK)

       SUBROUTINE CBELSM_64( TRANSA, MB, N, UNITD, DV, ALPHA, DESCRA,
      *           VAL, BINDX, BLDA, MAXBNZ, LB,
      *           B, LDB, BETA, C, LDC, WORK, LWORK )
       INTEGER*8  TRANSA, MB, N, UNITD, DESCRA(5), BLDA, MAXBNZ, LB,
      *           LDB, LDC, LWORK
       INTEGER*8  BINDX(BLDA,MAXBNZ)
       COMPLEX    ALPHA, BETA
       COMPLEX    DV(MB*LB*LB), VAL(LB*LB*BLDA*MAXBNZ), B(LDB,*), C(LDC,*),
      *           WORK(LWORK)

     F95 INTERFACE

       SUBROUTINE BELSM( TRANSA, MB, [N], UNITD, DV, ALPHA, DESCRA, VAL, BINDX,
      *   BLDA, MAXBNZ, LB, B, [LDB], BETA, C, [LDC], [WORK], [LWORK])
       INTEGER    TRANSA, MB, UNITD,  BLDA, MAXBNZ, LB
       INTEGER, DIMENSION(:) ::    DESCRA,  BINDX
       COMPLEX    ALPHA, BETA
       COMPLEX, DIMENSION(:) :: VAL, DV
       COMPLEX, DIMENSION(:, :) ::   B, C

       SUBROUTINE BELSM_64( TRANSA, MB, [N], UNITD, DV, ALPHA, DESCRA, VAL, BINDX,
      *   BLDA, MAXBNZ, LB, B, [LDB], BETA, C, [LDC], [WORK], [LWORK])
       INTEGER*8  TRANSA, MB, UNITD,  BLDA, MAXBNZ, LB
       INTEGER*8, DIMENSION(:) ::    DESCRA,  BINDX
       COMPLEX    ALPHA, BETA
       COMPLEX, DIMENSION(:) :: VAL, DV
       COMPLEX, DIMENSION(:, :) ::   B, C

     C INTERFACE

     #include <sunperf.h>

     void cbelsm(int transa, int mb, int n, int unitd,
      complex *dv, complex *alpha, int *descra, complex *val,
      int *bindx, int blda, int maxbnz, int lb, complex *b,
      int ldb, complex *beta, complex *c, int ldc);
     void  cbelsm_64(long transa, long mb, long n, long unitd,
     complex *dv, complex *alpha, long *descra, complex *val,
     long *bindx, long blda, long maxbnz, long lb, complex *b,
     long ldb, complex *beta, complex *c, long ldc);

DESCRIPTION

      cbelsm 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 mb*lb by n dense matrices,
      D is a block  diagonal matrix,  A is a sparse mb*lb by mb*lb unit, or
      non-unit, upper or lower triangular matrix represented in the block ellpack
      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)   Integer TRANSA specifies the form of op( A ) to be
                      used in the sparse matrix inverse as follows:
                        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.

      MB(input)       On entry,  MB  specifies the number of block 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, integer  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)
                      Unchanged on exit.

      DV(input)       On entry, DV is an array of length MB*LB*LB consisting
                      of the elements of the diagonal blocks of the matrix D.
                      The size of each square block is LB-by-LB and each
                      block is stored in standard column-major form.
                      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-identity blocks on the main diagonal
                         1 : identity diagonal blocks
                         2 : diagonal blocks are dense matrices
                      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 two-dimensional LB*LB*BLDA-by-MAXBNZ
                      array consisting of the non-zero blocks, stored
                      column-major within each dense block. Unchanged on exit.

      BINDX(input)    On entry, BINDX is an integer two-dimensional BLDA-MAXBNZ
                      array such BINDX(i,:) consists of the block column indices
                      of the nonzero blocks in block row i, padded by the integer
                      value i if the number of nonzero blocks is less than MAXBNZ.
                      The block column indices MUST be sorted in increasing order
                      for each block row. Unchanged on exit.

      BLDA(input)     On entry, BLDA specifies the leading dimension of BINDX(:,:).
                      Unchanged on exit.

      MAXBNZ (input)  On entry, MAXBNZ specifies the max number of nonzeros
                      blocks per row. Unchanged on exit.

      LB (input)      On entry, LB specifies the dimension of dense blocks
                      composing A.  Unchanged on exit.

      B (input)       Array of DIMENSION ( LDB, N ).
                      On entry, the leading mb*lb 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 mb*lb 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 MB*LB.

                      For good performance, LWORK should generally be larger.
                      For optimum performance on multiple processors, LWORK
                      >=MB*LB*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 parallelized with the help of OPENMP and it is
      fully  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 DESCRA(3)=0 , the lower or upper triangular part of
     each diagonal block is used by the routine depending on
     DESCRA(2) .

     3. If DESCRA(3)=1 , the diagonal blocks in the block ellpack
     representation of A  don't need to be the identity matrices
     because these block entries are not used by the routine in
     this case.

     4. If DESCRA(3)=2 , diagonal blocks are considered as dense
     matrices and the LU factorization with partial pivoting is
     used by the routine.

     WORK(1)=0 on return if the factorization for all diagonal
     blocks has been completed successfully, otherwise WORK(1) =
     - i where i is the block number for which the LU
     factorization could not be computed.

     5. The routine is designed so that it checks the validity of
     each sparse block entry given in the sparse blas
     representation. Block 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 block 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 block lower triangle of A, U is the
     strictly block upper triangle of A, D is the block 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