SUBROUTINE PDGEEQU( 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 A( * ), C( * ), R( * ) * .. * * Purpose * ======= * * PDGEEQU 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) DOUBLE PRECISION 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 * .. * .. 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, MAX, MIN, MOD * .. * .. 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, 'PDGEEQU', -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 ), ABS( 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 ), ABS( 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 PDGEEQU * END