pcgeequ.f

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      SUBROUTINE pcgeequ( M, N, A, IA, JA, DESCA, R, C, ROWCND, COLCND,
     $                    AMAX, INFO )
*
*  -- ScaLAPACK routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 1, 1997
*
*     .. Scalar Arguments ..
      INTEGER            IA, INFO, JA, M, N
      REAL               AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      INTEGER            DESCA( * )
      REAL               C( * ), R( * )
      COMPLEX            A( * )
*     ..
*
*  Purpose
*  =======
*
*  PCGEEQU computes row and column scalings intended to equilibrate an
*  M-by-N distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA:JA+N-1) and
*  reduce its condition number.  R returns the row scale factors and C
*  the column scale factors, chosen to try to make the largest entry in
*  each row and column of the distributed matrix B with elements
*  B(i,j) = R(i) * A(i,j) * C(j) have absolute value 1.
*
*  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
*  number and BIGNUM = largest safe number.  Use of these scaling
*  factors is not guaranteed to reduce the condition number of
*  sub( A ) but works well in practice.
*
*  Notes
*  =====
*
*  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
*
*  Arguments
*  =========
*
*  M       (global input) INTEGER
*          The number of rows to be operated on i.e the number of rows
*          of the distributed submatrix sub( A ). M >= 0.
*
*  N       (global input) INTEGER
*          The number of columns to be operated on i.e the number of
*          columns of the distributed submatrix sub( A ). N >= 0.
*
*  A       (local input) COMPLEX pointer into the local memory
*          to an array of dimension ( LLD_A, LOCc(JA+N-1) ), the
*          local pieces of the M-by-N distributed matrix whose
*          equilibration factors are to be computed.
*
*  IA      (global input) INTEGER
*          The row index in the global array A indicating the first
*          row of sub( A ).
*
*  JA      (global input) INTEGER
*          The column index in the global array A indicating the
*          first column of sub( A ).
*
*  DESCA   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix A.
*
*  R       (local output) REAL array, dimension LOCr(M_A)
*          If INFO = 0 or INFO > IA+M-1, R(IA:IA+M-1) contains the row
*          scale factors for sub( A ). R is aligned with the distributed
*          matrix A, and replicated across every process column. R is
*          tied to the distributed matrix A.
*
*  C       (local output) REAL array, dimension LOCc(N_A)
*          If INFO = 0,  C(JA:JA+N-1) contains the column scale factors
*          for sub( A ). C is aligned with the distributed matrix A, and
*          replicated down every process row. C is tied to the distri-
*          buted matrix A.
*
*  ROWCND  (global output) REAL
*          If INFO = 0 or INFO > IA+M-1, ROWCND contains the ratio of
*          the smallest R(i) to the largest R(i) (IA <= i <= IA+M-1).
*          If ROWCND >= 0.1 and AMAX is neither too large nor too small,
*          it is not worth scaling by R(IA:IA+M-1).
*
*  COLCND  (global output) REAL
*          If INFO = 0, COLCND contains the ratio of the smallest C(j)
*          to the largest C(j) (JA <= j <= JA+N-1). If COLCND >= 0.1, it
*          is not worth scaling by C(JA:JA+N-1).
*
*  AMAX    (global output) REAL
*          Absolute value of largest distributed matrix element.  If
*          AMAX is very close to overflow or very close to underflow,
*          the matrix should be scaled.
*
*  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 = i,  and i is
*                <= M:  the i-th row of the distributed matrix sub( A )
*                       is exactly zero,
*                >  M:  the (i-M)-th column of the distributed
*                       matrix sub( A ) is exactly zero.
*
*  =====================================================================
*
*     .. Parameters ..
      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 )
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      CHARACTER          COLCTOP, ROWCTOP
      INTEGER            I, IACOL, IAROW, ICOFF, ICTXT, IDUMM, IIA,
     $                   ioffa, iroff, j, jja, lda, mp, mycol, myrow,
     $                   npcol, nprow, nq
      REAL               BIGNUM, RCMAX, RCMIN, SMLNUM
      COMPLEX            ZDUM
*     ..
*     .. Local Arrays ..
      INTEGER            DESCC( DLEN_ ), DESCR( DLEN_ )
*     ..
*     .. External Subroutines ..
      EXTERNAL           blacs_gridinfo, chk1mat, descset, igamx2d,
     $                   infog2l, pchk1mat, pb_topget, pxerbla, sgamn2d,
     $                   sgamx2d
*     ..
*     .. External Functions ..
      INTEGER            INDXL2G, NUMROC
      REAL               PSLAMCH
      EXTERNAL           indxl2g, numroc, pslamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, aimag, max, min, mod, real
*     ..
*     .. Statement Functions ..
      REAL               CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
*     ..
*     .. Executable Statements ..
*
*     Get grid parameters
*
      ictxt = desca( ctxt_ )
      CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
*
*     Test the input parameters.
*
      info = 0
      IF( nprow.EQ.-1 ) THEN
         info = -(600+ctxt_)
      ELSE
         CALL chk1mat( m, 1, n, 2, ia, ja, desca, 6, info )
         CALL pchk1mat( m, 1, n, 2, ia, ja, desca, 6, 0, idumm, idumm,
     $                  info )
      END IF
*
      IF( info.NE.0 ) THEN
         CALL pxerbla( ictxt, 'PCGEEQU', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( m.EQ.0 .OR. n.EQ.0 ) THEN
         rowcnd = one
         colcnd = one
         amax = zero
         RETURN
      END IF
*
      CALL pb_topget( ictxt, 'Combine', 'Rowwise', rowctop )
      CALL pb_topget( ictxt, 'Combine', 'Columnwise', colctop )
*
*     Get machine constants and local indexes.
*
      smlnum = pslamch( ictxt, 'S' )
      bignum = one / smlnum
      CALL infog2l( ia, ja, desca, nprow, npcol, myrow, mycol, iia, jja,
     $              iarow, iacol )
      iroff = mod( ia-1, desca( mb_ ) )
      icoff = mod( ja-1, desca( nb_ ) )
      mp = numroc( m+iroff, desca( mb_ ), myrow, iarow, nprow )
      nq = numroc( n+icoff, desca( nb_ ), mycol, iacol, npcol )
      IF( myrow.EQ.iarow )
     $   mp = mp - iroff
      IF( mycol.EQ.iacol )
     $   nq = nq - icoff
      lda = desca( lld_ )
*
*     Assign descriptors for R and C arrays
*
      CALL descset( descr, m, 1, desca( mb_ ), 1, 0, 0, ictxt,
     $               max( 1, mp ) )
      CALL descset( descc, 1, n, 1, desca( nb_ ), 0, 0, ictxt, 1 )
*
*     Compute row scale factors.
*
      DO 10 i = iia, iia+mp-1
         r( i ) = zero
   10 CONTINUE
*
*     Find the maximum element in each row.
*
      ioffa = (jja-1)*lda
      DO 30 j = jja, jja+nq-1
         DO 20 i = iia, iia+mp-1
            r( i ) = max( r( i ), cabs1( a( ioffa + i ) ) )
   20    CONTINUE
         ioffa = ioffa + lda
   30 CONTINUE
      CALL sgamx2d( ictxt, 'rowwise', ROWCTOP, MP, 1, R( IIA ),
     $              MAX( 1, MP ), IDUMM, IDUMM, -1, -1, MYCOL )
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 40 I = IIA, IIA+MP-1
         RCMAX = MAX( RCMAX, R( I ) )
         RCMIN = MIN( RCMIN, R( I ) )
   40 CONTINUE
      CALL SGAMX2D( ICTXT, 'columnwise', COLCTOP, 1, 1, RCMAX, 1, IDUMM,
     $              IDUMM, -1, -1, MYCOL )
      CALL SGAMN2D( ICTXT, 'columnwise', COLCTOP, 1, 1, RCMIN, 1, IDUMM,
     $              IDUMM, -1, -1, MYCOL )
      AMAX = RCMAX
*
.EQ.      IF( RCMINZERO ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 50 I = IIA, IIA+MP-1
.EQ..AND..EQ.            IF( R( I )ZERO  INFO0 )
     $         INFO = INDXL2G( I, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
     $                NPROW ) - IA + 1
   50    CONTINUE
         CALL IGAMX2D( ICTXT, 'columnwise', COLCTOP, 1, 1, INFO, 1,
     $                 IDUMM, IDUMM, -1, -1, MYCOL )
.NE.         IF( INFO0 )
     $      RETURN
      ELSE
*
*        Invert the scale factors.
*
         DO 60 I = IIA, IIA+MP-1
            R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
   60    CONTINUE
*
*        Compute ROWCND = min(R(I)) / max(R(I))
*
         ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
*
      END IF
*
*     Compute column scale factors
*
      DO 70 J = JJA, JJA+NQ-1
         C( J ) = ZERO
   70 CONTINUE
*
*     Find the maximum element in each column,
*     assuming the row scaling computed above.
*
      IOFFA = (JJA-1)*LDA
      DO 90 J = JJA, JJA+NQ-1
         DO 80 I = IIA, IIA+MP-1
            C( J ) = MAX( C( J ), CABS1( A( IOFFA + I ) )*R( I ) )
   80    CONTINUE
         IOFFA = IOFFA + LDA
   90 CONTINUE
      CALL SGAMX2D( ICTXT, 'columnwise', COLCTOP, 1, NQ, C( JJA ),
     $              1, IDUMM, IDUMM, -1, -1, MYCOL )
*
*     Find the maximum and minimum scale factors.
*
      RCMIN = BIGNUM
      RCMAX = ZERO
      DO 100 J = JJA, JJA+NQ-1
         RCMIN = MIN( RCMIN, C( J ) )
         RCMAX = MAX( RCMAX, C( J ) )
  100 CONTINUE
      CALL SGAMX2D( ICTXT, 'columnwise', colctop, 1, 1, rcmax, 1, idumm,
     $              idumm, -1, -1, mycol )
      CALL sgamn2d( ictxt, 'Columnwise', colctop, 1, 1, rcmin, 1, idumm,
     $              idumm, -1, -1, mycol )
*
      IF( rcmin.EQ.zero ) THEN
*
*        Find the first zero scale factor and return an error code.
*
         DO 110 j = jja, jja+nq-1
            IF( c( j ).EQ.zero .AND. info.EQ.0 )
     $         info = m + indxl2g( j, desca( nb_ ), mycol,
     $                desca( csrc_ ), npcol ) - ja + 1
  110    CONTINUE
         CALL igamx2d( ictxt, 'Columnwise', colctop, 1, 1, info, 1,
     $                 idumm, idumm, -1, -1, mycol )
         IF( info.NE.0 )
     $      RETURN
      ELSE
*
*        Invert the scale factors.
*
         DO 120 j = jja, jja+nq-1
            c( j ) = one / min( max( c( j ), smlnum ), bignum )
  120    CONTINUE
*
*        Compute COLCND = min(C(J)) / max(C(J))
*
         colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
*
      END IF
*
      RETURN
*
*     End of PCGEEQU
*
      END