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pcdttrf.f

      SUBROUTINE PCDTTRF( N, DL, D, DU, JA, DESCA, AF, LAF, WORK, LWORK,
     $                    INFO )
*
*
*
*  -- ScaLAPACK routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 25, 2001 
*
*     .. Scalar Arguments ..
      INTEGER            INFO, JA, LAF, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            DESCA( * )
      COMPLEX            AF( * ), D( * ), DL( * ), DU( * ), WORK( * )
*     ..
*
*
*  Purpose
*  =======
*
*  PCDTTRF computes a LU factorization
*  of an N-by-N complex tridiagonal
*  diagonally dominant-like distributed matrix
*  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 PCDTTRS to solve linear systems.
*
*  The factorization has the form
*
*          P A(1:N, JA:JA+N-1) P^T = L U
*
*  where U is a tridiagonal upper triangular matrix and L is tridiagonal
*  lower triangular, and P is a permutation matrix.
*
*  =====================================================================
*
*  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.
*
*  DL      (local input/local output) COMPLEX pointer to local
*          part of global vector storing the lower diagonal of the
*          matrix. Globally, DL(1) is not referenced, and DL must be
*          aligned with D.
*          Must be of size >= DESCA( NB_ ).
*          On exit, this array contains information containing the
*            factors of the matrix.
*
*  D       (local input/local output) COMPLEX pointer to local
*          part of global vector storing the main diagonal of the
*          matrix.
*          On exit, this array contains information containing the
*            factors of the matrix.
*          Must be of size >= DESCA( NB_ ).
*
*  DU       (local input/local output) COMPLEX pointer to local
*          part of global vector storing the upper diagonal of the
*          matrix. Globally, DU(n) is not referenced, and DU must be
*          aligned with D.
*          On exit, this array contains information containing the
*            factors of the matrix.
*          Must be of size >= DESCA( NB_ ).
*
*  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 or 502), 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.
*
*  AF      (local output) COMPLEX array, dimension LAF.
*          Auxiliary Fillin Space.
*          Fillin is created during the factorization routine
*          PCDTTRF and this is stored in AF. If a linear system
*          is to be solved using PCDTTRS 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 >=
*          2*(NB+2)
*          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)
*          COMPLEX 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>=
*          8*NPCOL
*
*  INFO    (local 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
*                diagonally dominant-like,  and
*                the factorization was not completed.
*                If INFO = K>NPROCS, the submatrix stored on processor
*                INFO-NPROCS representing interactions with other
*                processors was not
*                stably factorable wo/interchanges,
*                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 >= 2
*      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.
*    For tridiagonal matrices, it is obvious: N/P >> bw(=1), and so D&C
*    algorithms are the appropriate choice.
*
*  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 tridiagonal 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 ((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: tridiagonal 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.
*    However, for tridiagonal matrices, since the objects being
*    distributed are the individual vectors storing the diagonals, we
*    have adopted the convention that both the P-by-1 descriptor and
*    the 1-by-P descriptor are allowed and are equivalent for
*    tridiagonal matrices. Thus, for tridiagonal matrices,
*    DTYPE_A = 501 or 502 can be used interchangeably
*    without any other change.
*  We require that the distributed vectors storing the diagonals of a
*    tridiagonal matrix be aligned with each other. Because of this, a
*    single descriptor, DESCA, serves to describe the distribution of
*    of all diagonals simultaneously.
*
*    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          OK          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.
*  Ignored         DESCA( 6 ) Ignored for tridiagonal matrices.
*  Reserved        DESCA( 7 ) Reserved for future use.
*
*
*
*  =====================================================================
*
*  Code Developer: Andrew J. Cleary, University of Tennessee.
*    Current address: Lawrence Livermore National Labs.
*  This version released: August, 2001.
*
*  =====================================================================
*
*     ..
*     .. Parameters ..
      REAL               ONE, ZERO
      PARAMETER          ( ONE = 1.0E+0 )
      PARAMETER          ( ZERO = 0.0E+0 )
      COMPLEX            CONE, CZERO
      PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
      PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+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            COMM_PROC, CSRC, FIRST_PROC, I, ICTXT,
     $                   ICTXT_NEW, ICTXT_SAVE, IDUM3, JA_NEW, LAF_MIN,
     $                   LEVEL_DIST, LLDA, MYCOL, MYROW, MY_NUM_COLS,
     $                   NB, NP, NPCOL, NPROW, NP_SAVE, ODD_SIZE,
     $                   PART_OFFSET, PART_SIZE, RETURN_CODE, STORE_N_A,
     $                   TEMP, WORK_SIZE_MIN, WORK_U
*     ..
*     .. Local Arrays ..
      INTEGER            DESCA_1XP( 7 ), PARAM_CHECK( 7, 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           BLACS_GET, BLACS_GRIDEXIT, BLACS_GRIDINFO,
     $                   CAXPY, CGEMM, CGERV2D, CGESD2D, CLACPY,
     $                   CLATCPY, CPBTRF, CPOTRF, CSYRK, CTBTRS, CTRMM,
     $                   CTRRV2D, CTRSD2D, CTRSM, CTRTRS, DESC_CONVERT,
     $                   GLOBCHK, PXERBLA, RESHAPE
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            NUMROC
      COMPLEX            CDOTC
      EXTERNAL           CDOTC, LSAME, NUMROC
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ICHAR, 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
*
      TEMP = DESCA( DTYPE_ )
      IF( TEMP .EQ. 502 ) THEN
*        Temporarily set the descriptor type to 1xP type
         DESCA( DTYPE_ ) = 501
      ENDIF
*
      CALL DESC_CONVERT( DESCA, DESCA_1XP, RETURN_CODE )
*
      DESCA( DTYPE_ ) = TEMP
*
      IF( RETURN_CODE .NE. 0) THEN
         INFO = -( 6*100 + 2 )
      ENDIF
*
*     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 = -10
      ELSE IF ( LWORK .EQ. -1 ) THEN
         IDUM3 = -1
      ELSE
         IDUM3 = 1
      ENDIF
*
      IF( N .LT. 0 ) THEN
         INFO = -1
      ENDIF
*
      IF( N+JA-1 .GT. STORE_N_A ) THEN
         INFO = -( 6*100 + 6 )
      ENDIF
*
*     Argument checking that is specific to Divide & Conquer routine
*
      IF( NPROW .NE. 1 ) THEN
         INFO = -( 6*100+2 )
      ENDIF
*
      IF( N .GT. NP*NB-MOD( JA-1, NB )) THEN
         INFO = -( 1 )
         CALL PXERBLA( ICTXT,
     $      'PCDTTRF, D&C alg.: only 1 block per proc',
     $      -INFO )
         RETURN
      ENDIF
*
      IF((JA+N-1.GT.NB) .AND. ( NB.LT.2*INT_ONE )) THEN
         INFO = -( 6*100+4 )
         CALL PXERBLA( ICTXT,
     $      'PCDTTRF, D&C alg.: NB too small',
     $      -INFO )
         RETURN
      ENDIF
*
*
*     Check auxiliary storage size
*
      LAF_MIN = (12*NPCOL+3*NB)
*
      IF( LAF .LT. LAF_MIN ) THEN
         INFO = -8
*        put minimum value of laf into AF( 1 )
         AF( 1 ) = LAF_MIN
         CALL PXERBLA( ICTXT,
     $      'PCDTTRF: auxiliary storage error ',
     $      -INFO )
         RETURN
      ENDIF
*
*     Check worksize
*
      WORK_SIZE_MIN = 8*NPCOL
*
      WORK( 1 ) = WORK_SIZE_MIN
*
      IF( LWORK .LT. WORK_SIZE_MIN ) THEN
         IF( LWORK .NE. -1 ) THEN
           INFO = -10
           CALL PXERBLA( ICTXT,
     $      'PCDTTRF: worksize error ',
     $      -INFO )
         ENDIF
         RETURN
      ENDIF
*
*     Pack params and positions into arrays for global consistency check
*
      PARAM_CHECK(  7, 1 ) = DESCA(5)
      PARAM_CHECK(  6, 1 ) = DESCA(4)
      PARAM_CHECK(  5, 1 ) = DESCA(3)
      PARAM_CHECK(  4, 1 ) = DESCA(1)
      PARAM_CHECK(  3, 1 ) = JA
      PARAM_CHECK(  2, 1 ) = N
      PARAM_CHECK(  1, 1 ) = IDUM3
*
      PARAM_CHECK(  7, 2 ) = 605
      PARAM_CHECK(  6, 2 ) = 604
      PARAM_CHECK(  5, 2 ) = 603
      PARAM_CHECK(  4, 2 ) = 601
      PARAM_CHECK(  3, 2 ) = 5
      PARAM_CHECK(  2, 2 ) = 1
      PARAM_CHECK(  1, 2 ) = 10
*
*     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, 7, PARAM_CHECK, 7,
     $              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, 'PCDTTRF', -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
      ENDIF
*
      IF ( MYCOL .LT. CSRC ) THEN
         PART_OFFSET = PART_OFFSET - NB
      ENDIF
*
*     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
         GOTO 1234
      ENDIF
*
*     ********************************
*     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 )
      ENDIF
*
*     Size of main (or odd) partition in each processor
*
      ODD_SIZE = MY_NUM_COLS
      IF ( MYCOL .LT. NP-1 ) THEN
         ODD_SIZE = ODD_SIZE - INT_ONE
      ENDIF
*
*     Offset to workspace for Upper triangular factor
*
      WORK_U = INT_ONE*ODD_SIZE + 3
*
*
*       Zero out space for fillin
*
        DO 10  I=1, LAF_MIN
          AF( I ) = CZERO
   10   CONTINUE
*
*     Begin main code
*
*
********************************************************************
*       PHASE 1: Local computation phase.
********************************************************************
*
*
        IF ( MYCOL .LT. NP-1 ) THEN
*         Transfer last triangle D_i of local matrix to next processor
*         which needs it to calculate fillin due to factorization of
*         its main (odd) block A_i.
*         Overlap the send with the factorization of A_i.
*
          CALL CTRSD2D( ICTXT, 'U', 'N', 1, 1,
     $                  DU( PART_OFFSET+ODD_SIZE+1 ), LLDA-1, 0,
     $                  MYCOL+1 )
*
        ENDIF
*
*
*       Factor main partition A_i = L_i {U_i} in each processor
*
        CALL CDTTRF( ODD_SIZE, DL( PART_OFFSET+2 ), D( PART_OFFSET+1 ),
     $                    DU( PART_OFFSET+1 ), INFO )
*
        IF( INFO.NE.0 ) THEN
           INFO = MYCOL+1
           GOTO 1500
        ENDIF
*
*
        IF ( MYCOL .LT. NP-1 ) THEN
*
*         Apply factorization to lower connection block BL_i
*         Apply factorization to upper connection block BU_i
*
*
*         Perform the triangular solve {U_i}^C{BL'}_i^C = {BL_i}^C
*
*
          DL( PART_OFFSET+ODD_SIZE+1 ) =
     $            ( DL( PART_OFFSET+ODD_SIZE+1 ) )
     $          / ( D( PART_OFFSET+ODD_SIZE ) )
*
*
*         Compute contribution to diagonal block(s) of reduced system.
*          {C'}_i = {C_i}-{{BL'}_i}{{BU'}_i}
*
*
          D( PART_OFFSET+ODD_SIZE+1 ) = D( PART_OFFSET+ODD_SIZE+1 )-
     $       DL( PART_OFFSET+ODD_SIZE+1 )*DU( PART_OFFSET+ODD_SIZE )
*
        ENDIF
*       End of "if ( MYCOL .lt. NP-1 )..." loop
*
*
 1500   CONTINUE
*       If the processor could not locally factor, it jumps here.
*
        IF ( MYCOL .NE. 0 ) THEN
*
*         Move entry that causes spike to auxiliary storage
*
          AF( WORK_U+1 ) = ( DL( PART_OFFSET+1 ) )
*
          IF (INFO.EQ.0) THEN
*
*         Calculate the "spike" fillin, ${L_i} {{GU}_i} = {DL_i}$ .
*
          CALL CDTTRSV( 'L', 'N', ODD_SIZE, INT_ONE,
     $                  DL( PART_OFFSET+2 ), D( PART_OFFSET+1 ),
     $                  DU( PART_OFFSET+1 ), AF( WORK_U+1 ), ODD_SIZE,
     $                  INFO )
*
*
*         Calculate the "spike" fillin, ${U_i}^C {{GL}_i}^C = {DU_i}^C$
*
          CALL CTRRV2D( ICTXT, 'U', 'N', 1, 1, AF( 1 ), ODD_SIZE, 0,
     $                  MYCOL-1 )
*
          AF( 1 ) = CONJG( AF( 1 ) )
*
          CALL CDTTRSV( 'U', 'C', ODD_SIZE, INT_ONE,
     $            DL( PART_OFFSET+2 ), D( PART_OFFSET+1 ),
     $            DU( PART_OFFSET+1 ),
     $            AF( 1 ), ODD_SIZE, INFO )
*
*
*         Calculate the update block for previous proc, E_i = GL_i{GU_i}
*
          AF( ODD_SIZE+3 ) = -CONE *
     $        CDOTC( ODD_SIZE, AF( 1 ), 1, AF( WORK_U+1 ), 1 )
*
*
*         Initiate send of E_i to previous processor to overlap
*           with next computation.
*
          CALL CGESD2D( ICTXT, INT_ONE, INT_ONE, AF( ODD_SIZE+3 ),
     $                  INT_ONE, 0, MYCOL-1 )
*
*
          IF ( MYCOL .LT. NP-1 ) THEN
*
*           Calculate off-diagonal block(s) of reduced system.
*           Note: for ease of use in solution of reduced system, store
*           L's off-diagonal block in conjugate transpose form.
*
            AF( ODD_SIZE+1 ) = -CONE
     $                         * CONJG( DL( PART_OFFSET+ODD_SIZE+1 )
     $                         * AF( WORK_U+ODD_SIZE ) )
*
*
            AF(WORK_U+(ODD_SIZE)+1 ) = -CONE
     $                   * DU( PART_OFFSET+ODD_SIZE )
     $                   * CONJG( AF( ODD_SIZE ) )
*
          ENDIF
*
        ENDIF
*       End of "if ( MYCOL .ne. 0 )..."
*
        ENDIF
*       End of "if (info.eq.0) then"
*
*
*       Check to make sure no processors have found errors
*
        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 )
        ENDIF
*
        IF ( INFO.NE.0 ) THEN
           GOTO 1000
        ENDIF
*       No errors found, continue
*
*
********************************************************************
*       PHASE 2: Formation and factorization of Reduced System.
********************************************************************
*
*       Gather up local sections of reduced system
*
*
*     The last processor does not participate in the factorization of
*       the reduced system, having sent its E_i already.
      IF( MYCOL .EQ. NPCOL-1 ) THEN
        GOTO 14
      ENDIF
*
*       Initiate send of off-diag block(s) to overlap with next part.
*       Off-diagonal block needed on neighboring processor to start
*       algorithm.
*
        IF( (MOD( MYCOL+1, 2 ) .EQ. 0) .AND. ( MYCOL .GT. 0 ) ) THEN
*
          CALL CGESD2D( ICTXT, INT_ONE, INT_ONE,
     $                       AF( ODD_SIZE+1 ),
     $                       INT_ONE, 0, MYCOL-1 )
*
          CALL CGESD2D( ICTXT, INT_ONE, INT_ONE,
     $                       AF( WORK_U+ODD_SIZE+1 ),
     $                       INT_ONE, 0, MYCOL-1 )
*
        ENDIF
*
*       Copy last diagonal block into AF storage for subsequent
*         operations.
*
        AF( ODD_SIZE+2 ) =
     $        CMPLX( D( PART_OFFSET+ODD_SIZE+1 ) )
*
*       Receive cont. to diagonal block that is stored on this proc.
*
        IF( MYCOL.LT. NPCOL-1 ) THEN
*
           CALL CGERV2D( ICTXT, INT_ONE, INT_ONE,
     $               AF( ODD_SIZE+2+1 ),
     $               INT_ONE, 0,
     $               MYCOL+1 )
*
*          Add contribution to diagonal block
*
           AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+AF( ODD_SIZE+3 )
*
        ENDIF
*
*
*       *************************************
*       Modification Loop
*
*       The distance for sending and receiving for each level starts
*         at 1 for the first level.
        LEVEL_DIST = 1
*
*       Do until this proc is needed to modify other procs' equations
*
   12   CONTINUE
        IF( MOD( (MYCOL+1)/LEVEL_DIST, 2) .NE. 0 ) GOTO 11
*
*         Receive and add contribution to diagonal block from the left
*
          IF( MYCOL-LEVEL_DIST .GE. 0 ) THEN
            CALL CGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ),
     $                     INT_ONE, 0, MYCOL-LEVEL_DIST )
*
            AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+WORK( 1 )
*
          ENDIF
*
*         Receive and add contribution to diagonal block from the right
*
          IF( MYCOL+LEVEL_DIST .LT. NPCOL-1 ) THEN
            CALL CGERV2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ),
     $                        INT_ONE, 0, MYCOL+LEVEL_DIST )
*
            AF( ODD_SIZE+2 ) = AF( ODD_SIZE+2 )+WORK( 1 )
*
          ENDIF
*
          LEVEL_DIST = LEVEL_DIST*2
*
        GOTO 12
   11   CONTINUE
*       [End of GOTO Loop]
*
*
*       *********************************
*       Calculate and use this proc's blocks to modify other procs'...
        IF( AF( ODD_SIZE+2 ) .EQ. CZERO ) THEN
               INFO = NPCOL + MYCOL
        ENDIF
*
*       ****************************************************************
*       Receive offdiagonal block from processor to right.
*         If this is the first group of processors, the receive comes
*         from a different processor than otherwise.
*
        IF( LEVEL_DIST .EQ. 1 )THEN
          COMM_PROC = MYCOL + 1
*
*           Move block into place that it will be expected to be for
*             calcs.
*
          AF( WORK_U+ODD_SIZE+3 ) = AF( ODD_SIZE+1 )
*
          AF( ODD_SIZE+3 ) = AF( WORK_U+ODD_SIZE+1 )
*
        ELSE
          COMM_PROC = MYCOL + LEVEL_DIST/2
        ENDIF
*
        IF( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-2 )THEN
*
          CALL CGERV2D( ICTXT, INT_ONE, INT_ONE,
     $        AF( ODD_SIZE+1 ),
     $                       INT_ONE, 0, COMM_PROC )
*
          CALL CGERV2D( ICTXT, INT_ONE, INT_ONE,
     $        AF( WORK_U+ODD_SIZE+1 ),
     $                       INT_ONE, 0, COMM_PROC )
*
          IF( INFO .EQ. 0 ) THEN
*
*
*         Modify lower off_diagonal block with diagonal block
*
*
          AF( ODD_SIZE+1 ) = AF( ODD_SIZE+1 )
     $                 / CONJG( AF( ODD_SIZE+2 ) )
*
          ENDIF
*         End of "if ( info.eq.0 ) then"
*
*         Calculate contribution from this block to next diagonal block
*
          WORK( 1 ) = -ONE*CONJG( AF( ODD_SIZE+1 ) )*
     $                AF( WORK_U+(ODD_SIZE)+1 )
*
*         Send contribution to diagonal block's owning processor.
*
          CALL CGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
     $                     0, MYCOL+LEVEL_DIST )
*
        ENDIF
*       End of "if( mycol/level_dist .le. (npcol-1)/level_dist-2 )..."
*
*
*       ****************************************************************
*       Receive off_diagonal block from left and use to finish with this
*         processor.
*
        IF( (MYCOL/LEVEL_DIST .GT. 0 ).AND.
     $      ( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-1 ) ) THEN
*
          IF( LEVEL_DIST .GT. 1)THEN
*
*           Receive offdiagonal block(s) from proc level_dist/2 to the
*           left
*
           CALL CGERV2D( ICTXT, INT_ONE, INT_ONE,
     $            AF( WORK_U+ODD_SIZE+2+1 ),
     $            INT_ONE, 0, MYCOL-LEVEL_DIST/2 )
*
*           Receive offdiagonal block(s) from proc level_dist/2 to the
*           left
*
           CALL CGERV2D( ICTXT, INT_ONE, INT_ONE,
     $            AF( ODD_SIZE+2+1 ),
     $            INT_ONE, 0, MYCOL-LEVEL_DIST/2 )
*
          ENDIF
*
*
          IF( INFO.EQ.0 ) THEN
*
*         Use diagonal block(s) to modify this offdiagonal block
*
          AF( ODD_SIZE+3 ) = AF( ODD_SIZE+3 )
     $         / ( AF( ODD_SIZE+2 ) )
*
*
          ENDIF
*         End of "if( info.eq.0 ) then"
*
*         Use offdiag block(s) to calculate modification to diag block
*           of processor to the left
*
          WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )
     $              *CONJG( AF( WORK_U+ODD_SIZE+3 ) )
*
*         Send contribution to diagonal block's owning processor.
*
          CALL CGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
     $                     0, MYCOL-LEVEL_DIST )
*
*         *******************************************************
*
          IF( MYCOL/LEVEL_DIST .LE. (NPCOL-1)/LEVEL_DIST-2 ) THEN
*
*           Decide which processor offdiagonal block(s) goes to
*
            IF( ( MOD( MYCOL/( 2*LEVEL_DIST ),2 )) .EQ.0 ) THEN
              COMM_PROC = MYCOL + LEVEL_DIST
            ELSE
              COMM_PROC = MYCOL - LEVEL_DIST
            ENDIF
*
*           Use offdiagonal blocks to calculate offdiag
*             block to send to neighboring processor. Depending
*             on circumstances, may need to transpose the matrix.
*
              WORK( 1 ) = -ONE*AF( WORK_U+ODD_SIZE+3 )
     $                  * AF( ODD_SIZE+1 )
*
*           Send contribution to offdiagonal block's owning processor.
*
            CALL CGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
     $                         0, COMM_PROC )
*
              WORK( 1 ) = -ONE*AF( ODD_SIZE+3 )
     $                  * AF( WORK_U+ODD_SIZE+1 )
*
*           Send contribution to offdiagonal block's owning processor.
*
            CALL CGESD2D( ICTXT, INT_ONE, INT_ONE, WORK( 1 ), INT_ONE,
     $                         0, COMM_PROC )
*
          ENDIF
*
        ENDIF
*       End of "if( mycol/level_dist.le. (npcol-1)/level_dist -1 )..."
*
   14   CONTINUE
*
*
 1000 CONTINUE
*
*
*     Free BLACS space used to hold standard-form grid.
*
      IF( ICTXT_SAVE .NE. ICTXT_NEW ) THEN
         CALL BLACS_GRIDEXIT( ICTXT_NEW )
      ENDIF
*
 1234 CONTINUE
*
*     Restore saved input parameters
*
      ICTXT = ICTXT_SAVE
      NP = NP_SAVE
*
*     Output minimum 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 )
          ENDIF
*
*
      RETURN
*
*     End of PCDTTRF
*
      END

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