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SUBROUTINE PZGEEQU( 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
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
DOUBLE PRECISION C( * ), R( * )
COMPLEX*16 A( * )
* ..
*
* Purpose
* =======
*
* PZGEEQU 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*16 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
* 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) DOUBLE PRECISION
* 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) DOUBLE PRECISION
* 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 )
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+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
DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
COMPLEX*16 ZDUM
* ..
* .. Local Arrays ..
INTEGER DESCC( DLEN_ ), DESCR( DLEN_ )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, DGAMN2D,
$ DGAMX2D, IGAMX2D, INFOG2L, PCHK1MAT, PB_TOPGET,
$ PXERBLA
* ..
* .. External Functions ..
INTEGER INDXL2G, NUMROC
DOUBLE PRECISION PDLAMCH
EXTERNAL INDXL2G, NUMROC, PDLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, MOD
* ..
* .. Statement Functions ..
DOUBLE PRECISION ZABS1
* ..
* .. Statement Function definitions ..
ZABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( 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, 'PZGEEQU', -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 = PDLAMCH( 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 ), ZABS1( A( IOFFA + I ) ) )
20 CONTINUE
IOFFA = IOFFA + LDA
30 CONTINUE
CALL DGAMX2D( 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 DGAMX2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, RCMAX, 1, IDUMM,
$ IDUMM, -1, -1, MYCOL )
CALL DGAMN2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, RCMIN, 1, IDUMM,
$ IDUMM, -1, -1, MYCOL )
AMAX = RCMAX
*
IF( RCMIN.EQ.ZERO ) THEN
*
* Find the first zero scale factor and return an error code.
*
DO 50 I = IIA, IIA+MP-1
IF( R( I ).EQ.ZERO .AND. INFO.EQ.0 )
$ 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 )
IF( INFO.NE.0 )
$ 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 ), ZABS1( A( IOFFA + I ) )*R( I ) )
80 CONTINUE
IOFFA = IOFFA + LDA
90 CONTINUE
CALL DGAMX2D( 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 DGAMX2D( ICTXT, 'Columnwise', COLCTOP, 1, 1, RCMAX, 1, IDUMM,
$ IDUMM, -1, -1, MYCOL )
CALL DGAMN2D( 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 PZGEEQU
*
END
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