SUBROUTINE PZPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX,
$ INFO )
*
* -- ScaLAPACK routine (version 1.5) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
INTEGER IA, INFO, JA, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
DOUBLE PRECISION SC( * ), SR( * )
COMPLEX*16 A( * )
* ..
*
* Purpose
* =======
*
* PZPOEQU computes row and column scalings intended to
* equilibrate a distributed Hermitian positive definite matrix
* sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number
* (with respect to the two-norm). SR and SC contain the scale
* factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri-
* buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on
* the diagonal. This choice of SR and SC puts the condition number
* of B within a factor N of the smallest possible condition number
* over all possible diagonal scalings.
*
* The scaling factor are stored along process rows in SR and along
* process columns in SC. The duplication of information simplifies
* greatly the application of the factors.
*
* 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
* =========
*
* N (global input) INTEGER
* The number of rows and columns to be operated on i.e the
* order of the distributed submatrix sub( A ). N >= 0.
*
* A (local input) COMPLEX*16 pointer into the local memory to an
* array of local dimension ( LLD_A, LOCc(JA+N-1) ), the
* N-by-N Hermitian positive definite distributed matrix
* sub( A ) whose scaling factors are to be computed. Only the
* diagonal elements of sub( A ) are referenced.
*
* 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.
*
* SR (local output) DOUBLE PRECISION array, dimension LOCr(M_A)
* If INFO = 0, SR(IA:IA+N-1) contains the row scale factors
* for sub( A ). SR is aligned with the distributed matrix A,
* and replicated across every process column. SR is tied to the
* distributed matrix A.
*
* SC (local output) DOUBLE PRECISION array, dimension LOCc(N_A)
* If INFO = 0, SC(JA:JA+N-1) contains the column scale factors
* for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu-
* ted matrix A, and replicated down every process row. SC is
* tied to the distributed matrix A.
*
* SCOND (global output) DOUBLE PRECISION
* If INFO = 0, SCOND contains the ratio of the smallest SR(i)
* (or SC(j)) to the largest SR(i) (or SC(j)), with
* IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1
* and AMAX is neither too large nor too small, it is not worth
* scaling by SR (or SC).
*
* AMAX (global output) DOUBLE PRECISION
* Absolute value of largest 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 = K, the K-th diagonal entry of sub( A ) is
* nonpositive.
*
* =====================================================================
*
* .. 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 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
CHARACTER ALLCTOP, COLCTOP, ROWCTOP
INTEGER IACOL, IAROW, ICOFF, ICTXT, ICURCOL, ICURROW,
$ IDUMM, II, IIA, IOFFA, IOFFD, IROFF, J, JB, JJ,
$ JJA, JN, LDA, LL, MYCOL, MYROW, NP, NPCOL,
$ NPROW, NQ
DOUBLE PRECISION AII, SMIN
* ..
* .. Local Arrays ..
INTEGER DESCSC( DLEN_ ), DESCSR( DLEN_ )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, DGAMN2D,
$ DGAMX2D, DGSUM2D, IGAMN2D, INFOG2L,
$ PCHK1MAT, PTOPGET, PXERBLA
* ..
* .. External Functions ..
INTEGER ICEIL, NUMROC
DOUBLE PRECISION PDLAMCH
EXTERNAL ICEIL, NUMROC, PDLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN, MOD, SQRT
* ..
* .. 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 = -(500+CTXT_)
ELSE
CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 5, INFO )
CALL PCHK1MAT( N, 1, N, 1, IA, JA, DESCA, 5, 0, IDUMM, IDUMM,
$ INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PZPOEQU', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
*
CALL PTOPGET( ICTXT, 'Combine', 'All', ALLCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
*
* Compute some local indexes
*
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_ ) )
NP = NUMROC( N+IROFF, DESCA( MB_ ), MYROW, IAROW, NPROW )
NQ = NUMROC( N+ICOFF, DESCA( NB_ ), MYCOL, IACOL, NPCOL )
IF( MYROW.EQ.IAROW )
$ NP = NP - IROFF
IF( MYCOL.EQ.IACOL )
$ NQ = NQ - ICOFF
JN = MIN( ICEIL( JA, DESCA( NB_ ) ) * DESCA( NB_ ), JA+N-1 )
LDA = DESCA( LLD_ )
*
* Assign descriptors for SR and SC arrays
*
CALL DESCSET( DESCSR, N, 1, DESCA( MB_ ), 1, 0, 0, ICTXT,
$ MAX( 1, NP ) )
CALL DESCSET( DESCSC, 1, N, 1, DESCA( NB_ ), 0, 0, ICTXT, 1 )
*
* Initialize the scaling factors to zero.
*
DO 10 II = IIA, IIA+NP-1
SR( II ) = ZERO
10 CONTINUE
*
DO 20 JJ = JJA, JJA+NQ-1
SC( JJ ) = ZERO
20 CONTINUE
*
* Find the minimum and maximum diagonal elements.
* Handle first block separately.
*
II = IIA
JJ = JJA
JB = JN-JA+1
SMIN = ONE / PDLAMCH( ICTXT, 'S' )
AMAX = ZERO
*
IOFFA = II+(JJ-1)*LDA
IF( MYROW.EQ.IAROW .AND. MYCOL.EQ.IACOL ) THEN
IOFFD = IOFFA
DO 30 LL = 0, JB-1
AII = DBLE( A( IOFFD ) )
SR( II+LL ) = AII
SC( JJ+LL ) = AII
SMIN = MIN( SMIN, AII )
AMAX = MAX( AMAX, AII )
IF( AII.LE.ZERO .AND. INFO.EQ.0 )
$ INFO = LL + 1
IOFFD = IOFFD + LDA + 1
30 CONTINUE
END IF
*
IF( MYROW.EQ.IAROW ) THEN
II = II + JB
IOFFA = IOFFA + JB
END IF
IF( MYCOL.EQ.IACOL ) THEN
JJ = JJ + JB
IOFFA = IOFFA + JB*LDA
END IF
ICURROW = MOD( IAROW+1, NPROW )
ICURCOL = MOD( IACOL+1, NPCOL )
*
* Loop over remaining blocks of columns
*
DO 50 J = JN+1, JA+N-1, DESCA( NB_ )
JB = MIN( N-J+JA, DESCA( NB_ ) )
*
IF( MYROW.EQ.ICURROW .AND. MYCOL.EQ.ICURCOL ) THEN
IOFFD = IOFFA
DO 40 LL = 0, JB-1
AII = DBLE( A( IOFFD ) )
SR( II+LL ) = AII
SC( JJ+LL ) = AII
SMIN = MIN( SMIN, AII )
AMAX = MAX( AMAX, AII )
IF( AII.LE.ZERO .AND. INFO.EQ.0 )
$ INFO = J + LL - JA + 1
IOFFD = IOFFD + LDA + 1
40 CONTINUE
END IF
*
IF( MYROW.EQ.ICURROW ) THEN
II = II + JB
IOFFA = IOFFA + JB
END IF
IF( MYCOL.EQ.ICURCOL ) THEN
JJ = JJ + JB
IOFFA = IOFFA + JB*LDA
END IF
ICURROW = MOD( ICURROW+1, NPROW )
ICURCOL = MOD( ICURCOL+1, NPCOL )
*
50 CONTINUE
*
* Compute scaling factors
*
CALL DGSUM2D( ICTXT, 'Columnwise', COLCTOP, 1, NQ, SC( JJA ),
$ 1, -1, MYCOL )
CALL DGSUM2D( ICTXT, 'Rowwise', ROWCTOP, NP, 1, SR( IIA ),
$ MAX( 1, NP ), -1, MYCOL )
*
CALL DGAMX2D( ICTXT, 'All', ALLCTOP, 1, 1, AMAX, 1, IDUMM, IDUMM,
$ -1, -1, MYCOL )
CALL DGAMN2D( ICTXT, 'All', ALLCTOP, 1, 1, SMIN, 1, IDUMM, IDUMM,
$ -1, -1, MYCOL )
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
CALL IGAMN2D( ICTXT, 'All', ALLCTOP, 1, 1, INFO, 1, II, JJ, -1,
$ -1, MYCOL )
RETURN
*
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 60 II = IIA, IIA+NP-1
SR( II ) = ONE / SQRT( SR( II ) )
60 CONTINUE
*
DO 70 JJ = JJA, JJA+NQ-1
SC( JJ ) = ONE / SQRT( SC( JJ ) )
70 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I))
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
*
END IF
*
RETURN
*
* End of PZPOEQU
*
END