SUBROUTINE PSPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, 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, N REAL AMAX, SCOND * .. * .. Array Arguments .. INTEGER DESCA( * ) REAL A( * ), SC( * ), SR( * ) * .. * * Purpose * ======= * * PSPOEQU computes row and column scalings intended to * equilibrate a distributed symmetric 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) REAL pointer into the local memory to an * array of local dimension ( LLD_A, LOCc(JA+N-1) ), the * N-by-N symmetric 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) REAL 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) REAL 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) REAL * 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) REAL * 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 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+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 REAL AII, SMIN * .. * .. Local Arrays .. INTEGER DESCSC( DLEN_ ), DESCSR( DLEN_ ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, IGAMN2D, \$ INFOG2L, PCHK1MAT, PB_TOPGET, PXERBLA, \$ SGAMN2D, SGAMX2D, SGSUM2D * .. * .. External Functions .. INTEGER ICEIL, NUMROC REAL PSLAMCH EXTERNAL ICEIL, NUMROC, PSLAMCH * .. * .. Intrinsic Functions .. INTRINSIC 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, 'PSPOEQU', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF * CALL PB_TOPGET( ICTXT, 'Combine', 'All', ALLCTOP ) CALL PB_TOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) CALL PB_TOPGET( 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 / PSLAMCH( 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 = 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 = 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 SGSUM2D( ICTXT, 'Columnwise', COLCTOP, 1, NQ, SC( JJA ), \$ 1, -1, MYCOL ) CALL SGSUM2D( ICTXT, 'Rowwise', ROWCTOP, NP, 1, SR( IIA ), \$ MAX( 1, NP ), -1, MYCOL ) * CALL SGAMX2D( ICTXT, 'All', ALLCTOP, 1, 1, AMAX, 1, IDUMM, IDUMM, \$ -1, -1, MYCOL ) CALL SGAMN2D( 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 PSPOEQU * END