pdgbtrf.f

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      SUBROUTINE pdgbtrf( N, BWL, BWU, A, JA, DESCA, IPIV, AF, LAF,
     $                    WORK, LWORK, INFO )
*
*  -- ScaLAPACK routine (version 2.0.2) --
*     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver
*     May 1 2012
*
*     .. Scalar Arguments ..
      INTEGER            BWL, BWU, INFO, JA, LAF, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            DESCA( * ), IPIV( * )
      DOUBLE PRECISION   A( * ), AF( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  PDGBTRF computes a LU factorization
*  of an N-by-N real banded
*  distributed matrix
*  with bandwidth BWL, BWU: A(1:N, JA:JA+N-1).
*  Reordering is used to increase parallelism in the factorization.
*  This reordering results in factors that are DIFFERENT from those
*  produced by equivalent sequential codes. These factors cannot
*  be used directly by users; however, they can be used in
*  subsequent calls to PDGBTRS to solve linear systems.
*
*  The factorization has the form
*
*          P A(1:N, JA:JA+N-1) Q = L U
*
*  where U is a banded upper triangular matrix and L is banded
*  lower triangular, and P and Q are permutation matrices.
*  The matrix Q represents reordering of columns
*  for parallelism's sake, while P represents
*  reordering of rows for numerical stability using
*  classic partial pivoting.
*
*  =====================================================================
*
*  Arguments
*  =========
*
*
*  N       (global input) INTEGER
*          The number of rows and columns to be operated on, i.e. the
*          order of the distributed submatrix A(1:N, JA:JA+N-1). N >= 0.
*
*  BWL     (global input) INTEGER
*          Number of subdiagonals. 0 <= BWL <= N-1
*
*  BWU     (global input) INTEGER
*          Number of superdiagonals. 0 <= BWU <= N-1
*
*  A       (local input/local output) DOUBLE PRECISION pointer into
*          local memory to an array with first dimension
*          LLD_A >=(2*bwl+2*bwu+1) (stored in DESCA).
*          On entry, this array contains the local pieces of the
*          N-by-N unsymmetric banded distributed matrix
*          A(1:N, JA:JA+N-1) to be factored.
*          This local portion is stored in the packed banded format
*            used in LAPACK. Please see the Notes below and the
*            ScaLAPACK manual for more detail on the format of
*            distributed matrices.
*          On exit, this array contains information containing details
*            of the factorization.
*          Note that permutations are performed on the matrix, so that
*            the factors returned are different from those returned
*            by LAPACK.
*
*  JA      (global input) INTEGER
*          The index in the global array A that points to the start of
*          the matrix to be operated on (which may be either all of A
*          or a submatrix of A).
*
*  DESCA   (global and local input) INTEGER array of dimension DLEN.
*          if 1D type (DTYPE_A=501), DLEN >= 7;
*          if 2D type (DTYPE_A=1), DLEN >= 9 .
*          The array descriptor for the distributed matrix A.
*          Contains information of mapping of A to memory. Please
*          see NOTES below for full description and options.
*
*  IPIV    (local output) INTEGER array, dimension >= DESCA( NB ).
*          Pivot indices for local factorizations.
*          Users *should not* alter the contents between
*          factorization and solve.
*
*  AF      (local output) DOUBLE PRECISION array, dimension LAF.
*          Auxiliary Fillin Space.
*          Fillin is created during the factorization routine
*          PDGBTRF and this is stored in AF. If a linear system
*          is to be solved using PDGBTRS after the factorization
*          routine, AF *must not be altered* after the factorization.
*
*  LAF     (local input) INTEGER
*          Size of user-input Auxiliary Fillin space AF. Must be >=
*          (NB+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu)
*          If LAF is not large enough, an error code will be returned
*          and the minimum acceptable size will be returned in AF( 1 )
*
*  WORK    (local workspace/local output)
*          DOUBLE PRECISION temporary workspace. This space may
*          be overwritten in between calls to routines. WORK must be
*          the size given in LWORK.
*          On exit, WORK( 1 ) contains the minimal LWORK.
*
*  LWORK   (local input or global input) INTEGER
*          Size of user-input workspace WORK.
*          If LWORK is too small, the minimal acceptable size will be
*          returned in WORK(1) and an error code is returned. LWORK>=
*          1
*
*  INFO    (global output) INTEGER
*          = 0:  successful exit
*          < 0:  If the i-th argument is an array and the j-entry had
*                an illegal value, then INFO = -(i*100+j), if the i-th
*                argument is a scalar and had an illegal value, then
*                INFO = -i.
*          > 0:  If INFO = K<=NPROCS, the submatrix stored on processor
*                INFO and factored locally was not
*                nonsingular,  and
*                the factorization was not completed.
*                If INFO = K>NPROCS, the submatrix stored on processor
*                INFO-NPROCS representing interactions with other
*                processors was not
*                nonsingular,
*                and the factorization was not completed.
*
*  =====================================================================
*
*
*  Restrictions
*  ============
*
*  The following are restrictions on the input parameters. Some of these
*    are temporary and will be removed in future releases, while others
*    may reflect fundamental technical limitations.
*
*    Non-cyclic restriction: VERY IMPORTANT!
*      P*NB>= mod(JA-1,NB)+N.
*      The mapping for matrices must be blocked, reflecting the nature
*      of the divide and conquer algorithm as a task-parallel algorithm.
*      This formula in words is: no processor may have more than one
*      chunk of the matrix.
*
*    Blocksize cannot be too small:
*      If the matrix spans more than one processor, the following
*      restriction on NB, the size of each block on each processor,
*      must hold:
*      NB >= (BWL+BWU)+1
*      The bulk of parallel computation is done on the matrix of size
*      O(NB) on each processor. If this is too small, divide and conquer
*      is a poor choice of algorithm.
*
*    Submatrix reference:
*      JA = IB
*      Alignment restriction that prevents unnecessary communication.
*
*
*  =====================================================================
*
*
*  Notes
*  =====
*
*  If the factorization routine and the solve routine are to be called
*    separately (to solve various sets of righthand sides using the same
*    coefficient matrix), the auxiliary space AF *must not be altered*
*    between calls to the factorization routine and the solve routine.
*
*  The best algorithm for solving banded and tridiagonal linear systems
*    depends on a variety of parameters, especially the bandwidth.
*    Currently, only algorithms designed for the case N/P >> bw are
*    implemented. These go by many names, including Divide and Conquer,
*    Partitioning, domain decomposition-type, etc.
*
*  Algorithm description: Divide and Conquer
*
*    The Divide and Conqer algorithm assumes the matrix is narrowly
*      banded compared with the number of equations. In this situation,
*      it is best to distribute the input matrix A one-dimensionally,
*      with columns atomic and rows divided amongst the processes.
*      The basic algorithm divides the banded matrix up into
*      P pieces with one stored on each processor,
*      and then proceeds in 2 phases for the factorization or 3 for the
*      solution of a linear system.
*      1) Local Phase:
*         The individual pieces are factored independently and in
*         parallel. These factors are applied to the matrix creating
*         fillin, which is stored in a non-inspectable way in auxiliary
*         space AF. Mathematically, this is equivalent to reordering
*         the matrix A as P A P^T and then factoring the principal
*         leading submatrix of size equal to the sum of the sizes of
*         the matrices factored on each processor. The factors of
*         these submatrices overwrite the corresponding parts of A
*         in memory.
*      2) Reduced System Phase:
*         A small (max(bwl,bwu)* (P-1)) system is formed representing
*         interaction of the larger blocks, and is stored (as are its
*         factors) in the space AF. A parallel Block Cyclic Reduction
*         algorithm is used. For a linear system, a parallel front solve
*         followed by an analagous backsolve, both using the structure
*         of the factored matrix, are performed.
*      3) Backsubsitution Phase:
*         For a linear system, a local backsubstitution is performed on
*         each processor in parallel.
*
*
*  Descriptors
*  ===========
*
*  Descriptors now have *types* and differ from ScaLAPACK 1.0.
*
*  Note: banded codes can use either the old two dimensional
*    or new one-dimensional descriptors, though the processor grid in
*    both cases *must be one-dimensional*. We describe both types below.
*
*  Each global data object is described by an associated description
*  vector.  This vector stores the information required to establish
*  the mapping between an object element and its corresponding process
*  and memory location.
*
*  Let A be a generic term for any 2D block cyclicly distributed array.
*  Such a global array has an associated description vector DESCA.
*  In the following comments, the character _ should be read as
*  "of the global array".
*
*  NOTATION        STORED IN      EXPLANATION
*  --------------- -------------- --------------------------------------
*  DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
*                                 DTYPE_A = 1.
*  CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
*                                 the BLACS process grid A is distribu-
*                                 ted over. The context itself is glo-
*                                 bal, but the handle (the integer
*                                 value) may vary.
*  M_A    (global) DESCA( M_ )    The number of rows in the global
*                                 array A.
*  N_A    (global) DESCA( N_ )    The number of columns in the global
*                                 array A.
*  MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
*                                 the rows of the array.
*  NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
*                                 the columns of the array.
*  RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
*                                 row of the array A is distributed.
*  CSRC_A (global) DESCA( CSRC_ ) The process column over which the
*                                 first column of the array A is
*                                 distributed.
*  LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
*                                 array.  LLD_A >= MAX(1,LOCr(M_A)).
*
*  Let K be the number of rows or columns of a distributed matrix,
*  and assume that its process grid has dimension p x q.
*  LOCr( K ) denotes the number of elements of K that a process
*  would receive if K were distributed over the p processes of its
*  process column.
*  Similarly, LOCc( K ) denotes the number of elements of K that a
*  process would receive if K were distributed over the q processes of
*  its process row.
*  The values of LOCr() and LOCc() may be determined via a call to the
*  ScaLAPACK tool function, NUMROC:
*          LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
*          LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
*  An upper bound for these quantities may be computed by:
*          LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
*          LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
*
*
*  One-dimensional descriptors:
*
*  One-dimensional descriptors are a new addition to ScaLAPACK since
*    version 1.0. They simplify and shorten the descriptor for 1D
*    arrays.
*
*  Since ScaLAPACK supports two-dimensional arrays as the fundamental
*    object, we allow 1D arrays to be distributed either over the
*    first dimension of the array (as if the grid were P-by-1) or the
*    2nd dimension (as if the grid were 1-by-P). This choice is
*    indicated by the descriptor type (501 or 502)
*    as described below.
*
*    IMPORTANT NOTE: the actual BLACS grid represented by the
*    CTXT entry in the descriptor may be *either*  P-by-1 or 1-by-P
*    irrespective of which one-dimensional descriptor type
*    (501 or 502) is input.
*    This routine will interpret the grid properly either way.
*    ScaLAPACK routines *do not support intercontext operations* so that
*    the grid passed to a single ScaLAPACK routine *must be the same*
*    for all array descriptors passed to that routine.
*
*    NOTE: In all cases where 1D descriptors are used, 2D descriptors
*    may also be used, since a one-dimensional array is a special case
*    of a two-dimensional array with one dimension of size unity.
*    The two-dimensional array used in this case *must* be of the
*    proper orientation:
*      If the appropriate one-dimensional descriptor is DTYPEA=501
*      (1 by P type), then the two dimensional descriptor must
*      have a CTXT value that refers to a 1 by P BLACS grid;
*      If the appropriate one-dimensional descriptor is DTYPEA=502
*      (P by 1 type), then the two dimensional descriptor must
*      have a CTXT value that refers to a P by 1 BLACS grid.
*
*
*  Summary of allowed descriptors, types, and BLACS grids:
*  DTYPE           501         502         1         1
*  BLACS grid      1xP or Px1  1xP or Px1  1xP       Px1
*  -----------------------------------------------------
*  A               OK          NO          OK        NO
*  B               NO          OK          NO        OK
*
*  Let A be a generic term for any 1D block cyclicly distributed array.
*  Such a global array has an associated description vector DESCA.
*  In the following comments, the character _ should be read as
*  "of the global array".
*
*  NOTATION        STORED IN  EXPLANATION
*  --------------- ---------- ------------------------------------------
*  DTYPE_A(global) DESCA( 1 ) The descriptor type. For 1D grids,
*                                TYPE_A = 501: 1-by-P grid.
*                                TYPE_A = 502: P-by-1 grid.
*  CTXT_A (global) DESCA( 2 ) The BLACS context handle, indicating
*                                the BLACS process grid A is distribu-
*                                ted over. The context itself is glo-
*                                bal, but the handle (the integer
*                                value) may vary.
*  N_A    (global) DESCA( 3 ) The size of the array dimension being
*                                distributed.
*  NB_A   (global) DESCA( 4 ) The blocking factor used to distribute
*                                the distributed dimension of the array.
*  SRC_A  (global) DESCA( 5 ) The process row or column over which the
*                                first row or column of the array
*                                is distributed.
*  LLD_A  (local)  DESCA( 6 ) The leading dimension of the local array
*                                storing the local blocks of the distri-
*                                buted array A. Minimum value of LLD_A
*                                depends on TYPE_A.
*                                TYPE_A = 501: LLD_A >=
*                                   size of undistributed dimension, 1.
*                                TYPE_A = 502: LLD_A >=NB_A, 1.
*  Reserved        DESCA( 7 ) Reserved for future use.
*
*  =====================================================================
*
*  Implemented for ScaLAPACK by:
*     Andrew J. Cleary, Livermore National Lab and University of Tenn.,
*     and Markus Hegland, Australian National University. Feb., 1997.
*  Based on code written by    : Peter Arbenz, ETH Zurich, 1996.
*  Last modified by:  Peter Arbenz, Institute of Scientific Computing,
*    ETH, Zurich.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d+0 )
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
      INTEGER            INT_ONE
      parameter( int_one = 1 )
      INTEGER            DESCMULT, BIGNUM
      parameter( descmult = 100, bignum = descmult*descmult )
      INTEGER            BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
     $                   lld_, mb_, m_, nb_, n_, rsrc_
      parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
     $                     ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
     $                     rsrc_ = 7, csrc_ = 8, lld_ = 9 )
*     ..
*     .. Local Scalars ..
      INTEGER            APTR, BBPTR, BIPTR, BM, BM1, BM2, BMN, BN, BW,
     $                   csrc, dbptr, first_proc, i, i1, i2, ictxt,
     $                   ictxt_new, ictxt_save, idum3, j, ja_new, jptr,
     $                   l, laf_min, lbwl, lbwu, ldb, ldbb, llda, lm,
     $                   lmj, ln, lnj, lptr, mycol, myrow, my_num_cols,
     $                   nb, neicol, np, npact, npcol, nprow, npstr,
     $                   np_save, nrhs, odd_n, odd_size, odptr, ofst,
     $                   part_offset, part_size, return_code, store_n_a,
     $                   work_size_min
*     ..
*     .. Local Arrays ..
      INTEGER            DESCA_1XP( 7 ), PARAM_CHECK( 9, 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           blacs_gridexit, blacs_gridinfo, desc_convert,
     $                   dgbtrf, dgemm, dger, dgerv2d, dgesd2d, dgetrf,
     $                   dlamov, dlaswp, dlatcpy, dswap, dtrrv2d,
     $                   dtrsd2d, dtrsm, globchk, igamx2d, igebr2d,
     $                   igebs2d, pxerbla, reshape
*     ..
*     .. External Functions ..
      INTEGER            NUMROC
      EXTERNAL           numroc
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, mod
*     ..
*     .. Executable Statements ..
*
*
*     Test the input parameters
*
      info = 0
*
*     Convert descriptor into standard form for easy access to
*        parameters, check that grid is of right shape.
*
      desca_1xp( 1 ) = 501
*
      CALL desc_convert( desca, desca_1xp, return_code )
*
      IF( return_code.NE.0 ) THEN
         info = -( 6*100+2 )
      END IF
*
*     Get values out of descriptor for use in code.
*
      ictxt = desca_1xp( 2 )
      csrc = desca_1xp( 5 )
      nb = desca_1xp( 4 )
      llda = desca_1xp( 6 )
      store_n_a = desca_1xp( 3 )
*
*     Get grid parameters
*
*
      CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
      np = nprow*npcol
*
*
*
      IF( lwork.LT.-1 ) THEN
         info = -11
      ELSE IF( lwork.EQ.-1 ) THEN
         idum3 = -1
      ELSE
         idum3 = 1
      END IF
*
      IF( n.LT.0 ) THEN
         info = -1
      END IF
*
      IF( n+ja-1.GT.store_n_a ) THEN
         info = -( 6*100+6 )
      END IF
*
      IF( ( bwl.GT.n-1 ) .OR. ( bwl.LT.0 ) ) THEN
         info = -2
      END IF
*
      IF( ( bwu.GT.n-1 ) .OR. ( bwu.LT.0 ) ) THEN
         info = -3
      END IF
*
      IF( llda.LT.( 2*bwl+2*bwu+1 ) ) THEN
         info = -( 6*100+6 )
      END IF
*
      IF( nb.LE.0 ) THEN
         info = -( 6*100+4 )
      END IF
*
      bw = bwu + bwl
*
*     Argument checking that is specific to Divide & Conquer routine
*
      IF( nprow.NE.1 ) THEN
         info = -( 6*100+2 )
      END IF
*
      IF( n.GT.np*nb-mod( ja-1, nb ) ) THEN
         info = -( 1 )
         CALL pxerbla( ictxt, 'PDGBTRF, D&C alg.: only 1 block per proc'
     $                 , -info )
         RETURN
      END IF
*
      IF( ( ja+n-1.GT.nb ) .AND. ( nb.LT.( bwl+bwu+1 ) ) ) THEN
         info = -( 6*100+4 )
         CALL pxerbla( ictxt, 'PDGBTRF, D&C alg.: NB too small', -info )
         RETURN
      END IF
*
*
*     Check auxiliary storage size
*
      laf_min = ( nb+bwu )*( bwl+bwu ) + 6*( bwl+bwu )*( bwl+2*bwu )
*
      IF( laf.LT.laf_min ) THEN
         info = -9
*        put minimum value of laf into AF( 1 )
         af( 1 ) = laf_min
         CALL pxerbla( ictxt, 'PDGBTRF: auxiliary storage error ',
     $                 -info )
         RETURN
      END IF
*
*     Check worksize
*
      work_size_min = 1
*
      work( 1 ) = work_size_min
*
      IF( lwork.LT.work_size_min ) THEN
         IF( lwork.NE.-1 ) THEN
            info = -11
*        put minimum value of work into work( 1 )
            work( 1 ) = work_size_min
            CALL pxerbla( ictxt, 'PDGBTRF: worksize error ', -info )
         END IF
         RETURN
      END IF
*
*     Pack params and positions into arrays for global consistency check
*
      param_check( 9, 1 ) = desca( 5 )
      param_check( 8, 1 ) = desca( 4 )
      param_check( 7, 1 ) = desca( 3 )
      param_check( 6, 1 ) = desca( 1 )
      param_check( 5, 1 ) = ja
      param_check( 4, 1 ) = bwu
      param_check( 3, 1 ) = bwl
      param_check( 2, 1 ) = n
      param_check( 1, 1 ) = idum3
*
      param_check( 9, 2 ) = 605
      param_check( 8, 2 ) = 604
      param_check( 7, 2 ) = 603
      param_check( 6, 2 ) = 601
      param_check( 5, 2 ) = 5
      param_check( 4, 2 ) = 3
      param_check( 3, 2 ) = 2
      param_check( 2, 2 ) = 1
      param_check( 1, 2 ) = 11
*
*     Want to find errors with MIN( ), so if no error, set it to a big
*     number. If there already is an error, multiply by the the
*     descriptor multiplier.
*
      IF( info.GE.0 ) THEN
         info = bignum
      ELSE IF( info.LT.-descmult ) THEN
         info = -info
      ELSE
         info = -info*descmult
      END IF
*
*     Check consistency across processors
*
      CALL globchk( ictxt, 9, param_check, 9, param_check( 1, 3 ),
     $              info )
*
*     Prepare output: set info = 0 if no error, and divide by DESCMULT
*     if error is not in a descriptor entry.
*
      IF( info.EQ.bignum ) THEN
         info = 0
      ELSE IF( mod( info, descmult ).EQ.0 ) THEN
         info = -info / descmult
      ELSE
         info = -info
      END IF
*
      IF( info.LT.0 ) THEN
         CALL pxerbla( ictxt, 'PDGBTRF', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*
*     Adjust addressing into matrix space to properly get into
*        the beginning part of the relevant data
*
      part_offset = nb*( ( ja-1 ) / ( npcol*nb ) )
*
      IF( ( mycol-csrc ).LT.( ja-part_offset-1 ) / nb ) THEN
         part_offset = part_offset + nb
      END IF
*
      IF( mycol.LT.csrc ) THEN
         part_offset = part_offset - nb
      END IF
*
*     Form a new BLACS grid (the "standard form" grid) with only procs
*        holding part of the matrix, of size 1xNP where NP is adjusted,
*        starting at csrc=0, with JA modified to reflect dropped procs.
*
*     First processor to hold part of the matrix:
*
      first_proc = mod( ( ja-1 ) / nb+csrc, npcol )
*
*     Calculate new JA one while dropping off unused processors.
*
      ja_new = mod( ja-1, nb ) + 1
*
*     Save and compute new value of NP
*
      np_save = np
      np = ( ja_new+n-2 ) / nb + 1
*
*     Call utility routine that forms "standard-form" grid
*
      CALL reshape( ictxt, int_one, ictxt_new, int_one, first_proc,
     $              int_one, np )
*
*     Use new context from standard grid as context.
*
      ictxt_save = ictxt
      ictxt = ictxt_new
      desca_1xp( 2 ) = ictxt_new
*
*     Get information about new grid.
*
      CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
*
*     Drop out processors that do not have part of the matrix.
*
      IF( myrow.LT.0 ) THEN
         GO TO 210
      END IF
*
*     ********************************
*     Values reused throughout routine
*
*     User-input value of partition size
*
      part_size = nb
*
*     Number of columns in each processor
*
      my_num_cols = numroc( n, part_size, mycol, 0, npcol )
*
*     Offset in columns to beginning of main partition in each proc
*
      IF( mycol.EQ.0 ) THEN
         part_offset = part_offset + mod( ja_new-1, part_size )
         my_num_cols = my_num_cols - mod( ja_new-1, part_size )
      END IF
*
*     Offset in elements
*
      ofst = part_offset*llda
*
*     Size of main (or odd) partition in each processor
*
      odd_size = numroc( n, part_size, mycol, 0, npcol )
*
*
*     Zero out space for fillin
*
      DO 10 i = 1, laf_min
         af( i ) = zero
   10 CONTINUE
*
      DO 30 j = 1, odd_size
         DO 20 i = 1, bw
            a( i+( j-1 )*llda ) = zero
   20    CONTINUE
   30 CONTINUE
*
*     Begin main code
*
********************************************************************
*     PHASE 1: Local computation phase.
********************************************************************
*
*
*     Transfer triangle B_i of local matrix to next processor
*     for fillin. Overlap the send with the factorization of A_i.
*
      IF( mycol.LE.npcol-2 ) THEN
*
*     The last processor does not need to send anything.
*     BIPTR = location of triangle B_i in memory
         biptr = ( nb-bw )*llda + 2*bw + 1
*
         CALL dtrsd2d( ictxt, 'u', 'n',
     $                 MIN( BW, BWU+NUMROC( N, NB, MYCOL+1, 0,
     $                 NPCOL ) ), BW, A( BIPTR ), LLDA-1, 0, MYCOL+1 )
*
      END IF
*
*     Factor main partition P_i A_i = L_i U_i on each processor
*
*     LBWL, LBWU: lower and upper bandwidth of local solver
*     Note that for MYCOL > 0 one has lower triangular blocks!
*     LM is the number of rows which is usually NB except for
*     MYCOL = 0 where it is BWU less and MYCOL=NPCOL-1 where it
*     is NR+BWU where NR is the number of columns on the last processor
*     Finally APTR is the pointer to the first element of A. As LAPACK
*     has a slightly different matrix format than Scalapack the pointer
*     has to be adjusted on processor MYCOL=0.
*
.NE.      IF( MYCOL0 ) THEN
         LBWL = BW
         LBWU = 0
         APTR = 1
      ELSE
         LBWL = BWL
         LBWU = BWU
         APTR = 1 + BWU
      END IF
*
.NE.      IF( MYCOLNPCOL-1 ) THEN
         LM = NB - LBWU
         LN = NB - BW
.NE.      ELSE IF( MYCOL0 ) THEN
         LM = ODD_SIZE + BWU
         LN = MAX( ODD_SIZE-BW, 0 )
      ELSE
         LM = N
         LN = MAX( N-BW, 0 )
      END IF
*
.GT.      IF( LN0 ) THEN
*
         CALL DGBTRF( LM, LN, LBWL, LBWU, A( APTR ), LLDA, IPIV, INFO )
*
.NE.         IF( INFO0 ) THEN
            INFO = INFO + NB*MYCOL
            GO TO 80
         END IF
*
         NRHS = BW
         LDB = LLDA - 1
*
*     Update the last BW columns of A_i (code modified from DGBTRS)
*
*     Only the eliminations of unknowns > LN-BW have an effect on
*     the last BW columns. Loop over them...
*
         DO 40 J = MAX( LN-BW+1, 1 ), LN
*
            LMJ = MIN( LBWL, LM-J )
            LNJ = MIN( BW, J+BW-LN+APTR-1 )
*
            L = IPIV( J )
*
            JPTR = J - ( LN+1 ) + 2*BW + 1 - LBWL + LN*LLDA
*
.NE.            IF( LJ ) THEN
*
*        Element (L,LN+1) is swapped with element (J,LN+1) etc
*        Furthermore, the elements in the same row are LDB=LLDA-1 apart
*        The complicated formulas are to cope with the banded
*          data format:
*
               LPTR = L - ( LN+1 ) + 2*BW + 1 - LBWL + LN*LLDA
*
               CALL DSWAP( LNJ, A( LPTR ), LDB, A( JPTR ), LDB )
*
            END IF
*
*              LPTR is the pointer to the beginning of the
*                 coefficients of L
*
            LPTR = BW + 1 + APTR + ( J-1 )*LLDA
*
            CALL DGER( LMJ, LNJ, -ONE, A( LPTR ), 1, A( JPTR ), LDB,
     $                 A( JPTR+1 ), LDB )
   40    CONTINUE
*
      END IF
*
*      Compute spike fill-in, L_i F_i = P_i B_{i-1}
*
*      Receive triangle B_{i-1} from previous processor
*
.GT.      IF( MYCOL0 ) THEN
         CALL DTRRV2D( ICTXT, 'u', 'n', MIN( BW, LM ), BW, AF( 1 ), BW,
     $                 0, MYCOL-1 )
*
*        Transpose transmitted upper triangular (trapezoidal) matrix
*
         DO 60 I2 = 1, MIN( BW, LM )
            DO 50 I1 = I2 + 1, BW
               AF( I1+( I2-1 )*BW ) = AF( I2+( I1-1 )*BW )
               AF( I2+( I1-1 )*BW ) = ZERO
   50       CONTINUE
   60    CONTINUE
*
*      Permutation and forward elimination (triang. solve)
*
         DO 70 J = 1, LN
*
            LMJ = MIN( LBWL, LM-J )
            L = IPIV( J )
*
.NE.            IF( LJ ) THEN
               CALL DSWAP( BW, AF( ( L-1 )*BW+1 ), 1,
     $                     AF( ( J-1 )*BW+1 ), 1 )
            END IF
*
            LPTR = BW + 1 + APTR + ( J-1 )*LLDA
*
            CALL DGER( NRHS, LMJ, -ONE, AF( ( J-1 )*BW+1 ), 1,
     $                 A( LPTR ), 1, AF( J*BW+1 ), BW )
*
   70    CONTINUE
*
      END IF
*
   80 CONTINUE
*
********************************************************************
*     PHASE 2: Formation and factorization of Reduced System.
********************************************************************
*
*     Define the initial dimensions of the diagonal blocks
*     The offdiagonal blocks (for MYCOL > 0) are of size BM by BW
*
.NE.      IF( MYCOLNPCOL-1 ) THEN
         BM = BW - LBWU
         BN = BW
      ELSE
         BM = MIN( BW, ODD_SIZE ) + BWU
         BN = MIN( BW, ODD_SIZE )
      END IF
*
*     Pointer to first element of block bidiagonal matrix in AF
*     Leading dimension of block bidiagonal system
*
      BBPTR = ( NB+BWU )*BW + 1
      LDBB = 2*BW + BWU
*
*     Copy from A and AF into block bidiagonal matrix (tail of AF)
*
*     DBPTR = Pointer to diagonal blocks in A
      DBPTR = BW + 1 + LBWU + LN*LLDA
*
      CALL DLAMOV( 'g', BM, BN, A( DBPTR ), LLDA-1, AF( BBPTR+BW*LDBB ),
     $             LDBB )
*
*     Zero out any junk entries that were copied
*
      DO 100 J = 1, BM
         DO 90 I = J + LBWL, BM - 1
            AF( BBPTR+BW*LDBB+( J-1 )*LDBB+I ) = ZERO
   90    CONTINUE
  100 CONTINUE
*
.NE.      IF( MYCOL0 ) THEN
*
*        ODPTR = Pointer to offdiagonal blocks in A
*
         ODPTR = ( LM-BM )*BW + 1
         CALL DLATCPY( 'g', BW, BM, AF( ODPTR ), BW,
     $                 AF( BBPTR+2*BW*LDBB ), LDBB )
      END IF
*
.EQ.      IF( NPCOL1 ) THEN
*
*        In this case the loop over the levels will not be
*        performed.
         CALL DGETRF( N-LN, N-LN, AF( BBPTR+BW*LDBB ), LDBB,
     $                IPIV( LN+1 ), INFO )
*
      END IF
*
*     Loop over levels ... only occurs if npcol > 1
*
*     The two integers NPACT (nu. of active processors) and NPSTR
*     (stride between active processors) are used to control the
*     loop.
*
      NPACT = NPCOL
      NPSTR = 1
*
*       Begin loop over levels
*
  110 CONTINUE
.LE.      IF( NPACT1 )
     $   GO TO 190
*
*         Test if processor is active
*
.EQ.      IF( MOD( MYCOL, NPSTR )0 ) THEN
*
*   Send/Receive blocks
*
*
.EQ.         IF( MOD( MYCOL, 2*NPSTR )0 ) THEN
*
*            This node will potentially do more work later
*
            NEICOL = MYCOL + NPSTR
*
.LT.            IF( NEICOL / NPSTRNPACT-1 ) THEN
               BMN = BW
.EQ.            ELSE IF( NEICOL / NPSTRNPACT-1 ) THEN
               ODD_N = NUMROC( N, NB, NPCOL-1, 0, NPCOL )
               BMN = MIN( BW, ODD_N ) + BWU
            ELSE
*
*                  Last processor skips to next level
               GO TO 180
            END IF
*
*               BM1 = M for 1st block on proc pair, BM2 2nd block
*
            BM1 = BM
            BM2 = BMN
*
.LE.            IF( NEICOL / NPSTRNPACT-1 ) THEN
*
               CALL DGESD2D( ICTXT, BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB,
     $                       0, NEICOL )
*
               CALL DGERV2D( ICTXT, BMN, 2*BW, AF( BBPTR+BM ), LDBB, 0,
     $                       NEICOL )
*
.EQ.               IF( NPACT2 ) THEN
*
*                     Copy diagonal block to align whole system
*
                  CALL DLAMOV( 'g', BMN, BW, AF( BBPTR+BM ), LDBB,
     $                         AF( BBPTR+2*BW*LDBB+BM ), LDBB )
               END IF
*
            END IF
*
         ELSE
*
*               This node stops work after this stage -- an extra copy
*               is required to make the odd and even frontal matrices
*               look identical
*
            NEICOL = MYCOL - NPSTR
*
.EQ.            IF( NEICOL0 ) THEN
               BMN = BW - BWU
            ELSE
               BMN = BW
            END IF
*
            BM1 = BMN
            BM2 = BM
*
            CALL DGESD2D( ICTXT, BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB, 0,
     $                    NEICOL )
*
            CALL DLAMOV( 'g', BM, 2*BW, AF( BBPTR+BW*LDBB ), LDBB,
     $                   AF( BBPTR+BMN ), LDBB )
*
            DO 130 J = BBPTR + 2*BW*LDBB, BBPTR + 3*BW*LDBB - 1, LDBB
               DO 120 I = 0, LDBB - 1
                  AF( I+J ) = ZERO
  120          CONTINUE
  130       CONTINUE
*
            CALL DGERV2D( ICTXT, BMN, 2*BW, AF( BBPTR+BW*LDBB ), LDBB,
     $                    0, NEICOL )
*
.EQ.            IF( NPACT2 ) THEN
*
*                  Copy diagonal block to align whole system
*
               CALL DLAMOV( 'g', BM, BW, AF( BBPTR+BMN ), LDBB,
     $                      AF( BBPTR+2*BW*LDBB+BMN ), LDBB )
            END IF
*
         END IF
*
*            LU factorization with partial pivoting
*
.NE.         IF( NPACT2 ) THEN
*
            CALL DGETRF( BM+BMN, BW, AF( BBPTR+BW*LDBB ), LDBB,
     $                   IPIV( LN+1 ), INFO )
*
*   Backsolve left side
*
            DO 150 J = BBPTR, BBPTR + BW*LDBB - 1, LDBB
               DO 140 I = 0, BM1 - 1
                  AF( I+J ) = ZERO
  140          CONTINUE
  150       CONTINUE
*
            CALL DLASWP( BW, AF( BBPTR ), LDBB, 1, BW, IPIV( LN+1 ), 1 )
*
            CALL DTRSM( 'l', 'l', 'n', 'u', BW, BW, ONE,
     $                  AF( BBPTR+BW*LDBB ), LDBB, AF( BBPTR ), LDBB )
*
*               Use partial factors to update remainder
*
            CALL DGEMM( 'n', 'n', BM+BMN-BW, BW, BW, -ONE,
     $                  AF( BBPTR+BW*LDBB+BW ), LDBB, AF( BBPTR ), LDBB,
     $                  ONE, AF( BBPTR+BW ), LDBB )
*
*   Backsolve right side
*
            NRHS = BW
*
            CALL DLASWP( NRHS, AF( BBPTR+2*BW*LDBB ), LDBB, 1, BW,
     $                   IPIV( LN+1 ), 1 )
*
            CALL DTRSM( 'l', 'l', 'n', 'u', BW, NRHS, ONE,
     $                  AF( BBPTR+BW*LDBB ), LDBB,
     $                  AF( BBPTR+2*BW*LDBB ), LDBB )
*
*               Use partial factors to update remainder
*
            CALL DGEMM( 'n', 'n', bm+bmn-bw, nrhs, bw, -one,
     $                  af( bbptr+bw*ldbb+bw ), ldbb,
     $                  af( bbptr+2*bw*ldbb ), ldbb, one,
     $                  af( bbptr+2*bw*ldbb+bw ), ldbb )
*
*
*     Test if processor is active in next round
*
            IF( mod( mycol, 2*npstr ).EQ.0 ) THEN
*
*                  Reset BM
*
               bm = bm1 + bm2 - bw
*
*                  Local copying in the block bidiagonal area
*
*
               CALL dlamov( 'G', bm, bw, af( bbptr+bw ), ldbb,
     $                      af( bbptr+bw*ldbb ), ldbb )
               CALL dlamov( 'G', bm, bw, af( bbptr+2*bw*ldbb+bw ), ldbb,
     $                      af( bbptr+2*bw*ldbb ), ldbb )
*
*                  Zero out space that held original copy
*
               DO 170 j = 0, bw - 1
                  DO 160 i = 0, bm - 1
                     af( bbptr+2*bw*ldbb+bw+j*ldbb+i ) = zero
  160             CONTINUE
  170          CONTINUE
*
            END IF
*
         ELSE
*
*               Factor the final 2 by 2 block matrix
*
            CALL dgetrf( bm+bmn, bm+bmn, af( bbptr+bw*ldbb ), ldbb,
     $                   ipiv( ln+1 ), info )
         END IF
*
      END IF
*
*        Last processor in an odd-sized NPACT skips to here
*
  180 CONTINUE
*
      npact = ( npact+1 ) / 2
      npstr = npstr*2
      GO TO 110
*
  190 CONTINUE
*     End loop over levels
*
  200 CONTINUE
*     If error was found in Phase 1, processors jump here.
*
*     Free BLACS space used to hold standard-form grid.
*
      ictxt = ictxt_save
      IF( ictxt.NE.ictxt_new ) THEN
         CALL blacs_gridexit( ictxt_new )
      END IF
*
  210 CONTINUE
*     If this processor did not hold part of the grid it
*        jumps here.
*
*     Restore saved input parameters
*
      ictxt = ictxt_save
      np = np_save
*
*     Output worksize
*
      work( 1 ) = work_size_min
*
*         Make INFO consistent across processors
*
      CALL igamx2d( ictxt, 'A', ' ', 1, 1, info, 1, info, info, -1, 0,
     $              0 )
*
      IF( mycol.EQ.0 ) THEN
         CALL igebs2d( ictxt, 'A', ' ', 1, 1, info, 1 )
      ELSE
         CALL igebr2d( ictxt, 'A', ' ', 1, 1, info, 1, 0, 0 )
      END IF
*
*
      RETURN
*
*     End of PDGBTRF
*
      END