SUBROUTINE PZLARFG( N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, $ TAU ) * * -- ScaLAPACK auxiliary routine (version 1.5) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 1, 1997 * * .. Scalar Arguments .. INTEGER IAX, INCX, IX, JAX, JX, N COMPLEX*16 ALPHA * .. * .. Array Arguments .. INTEGER DESCX( * ) COMPLEX*16 TAU( * ), X( * ) * .. * * Purpose * ======= * * PZLARFG generates a complex elementary reflector H of order n, such * that * * H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I. * ( x ) ( 0 ) * * where alpha is a real scalar, and sub( X ) is an (N-1)-element * complex distributed vector X(IX:IX+N-2,JX) if INCX = 1 and * X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form * * H = I - tau * ( 1 ) * ( 1 v' ) , * ( v ) * * where tau is a complex scalar and v is a complex (N-1)-element * vector. Note that H is not Hermitian. * * If the elements of sub( X ) are all zero and X(IAX,JAX) is real, * then tau = 0 and H is taken to be the unit matrix. * * Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1. * * 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 * * Because vectors may be viewed as a subclass of matrices, a * distributed vector is considered to be a distributed matrix. * * Arguments * ========= * * N (global input) INTEGER * The global order of the elementary reflector. N >= 0. * * ALPHA (local output) COMPLEX*16 * On exit, alpha is computed in the process scope having the * vector sub( X ). * * IAX (global input) INTEGER * The global row index in X of X(IAX,JAX). * * JAX (global input) INTEGER * The global column index in X of X(IAX,JAX). * * X (local input/local output) COMPLEX*16, pointer into the * local memory to an array of dimension (LLD_X,*). This array * contains the local pieces of the distributed vector sub( X ). * Before entry, the incremented array sub( X ) must contain * the vector x. On exit, it is overwritten with the vector v. * * IX (global input) INTEGER * The row index in the global array X indicating the first * row of sub( X ). * * JX (global input) INTEGER * The column index in the global array X indicating the * first column of sub( X ). * * DESCX (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix X. * * INCX (global input) INTEGER * The global increment for the elements of X. Only two values * of INCX are supported in this version, namely 1 and M_X. * INCX must not be zero. * * TAU (local output) COMPLEX*16, array, dimension LOCc(JX) * if INCX = 1, and LOCr(IX) otherwise. This array contains the * Householder scalars related to the Householder vectors. * TAU is tied to the distributed matrix X. * * ===================================================================== * * .. 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 .. INTEGER ICTXT, IIAX, INDXTAU, IXCOL, IXROW, J, JJAX, $ KNT, MYCOL, MYROW, NPCOL, NPROW DOUBLE PRECISION ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, INFOG2L, PDZNRM2, $ ZGEBR2D, ZGEBS2D, PZSCAL, $ PZDSCAL * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY3 COMPLEX*16 ZLADIV EXTERNAL DLAMCH, DLAPY3, ZLADIV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DIMAG, SIGN * .. * .. Executable Statements .. * * Get grid parameters. * ICTXT = DESCX( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) * IF( INCX.EQ.DESCX( M_ ) ) THEN * * sub( X ) is distributed across a process row. * CALL INFOG2L( IX, JAX, DESCX, NPROW, NPCOL, MYROW, MYCOL, $ IIAX, JJAX, IXROW, IXCOL ) * IF( MYROW.NE.IXROW ) $ RETURN * * Broadcast X(IAX,JAX) across the process row. * IF( MYCOL.EQ.IXCOL ) THEN J = IIAX+(JJAX-1)*DESCX( LLD_ ) CALL ZGEBS2D( ICTXT, 'Rowwise', ' ', 1, 1, X( J ), 1 ) ALPHA = X( J ) ELSE CALL ZGEBR2D( ICTXT, 'Rowwise', ' ', 1, 1, ALPHA, 1, $ MYROW, IXCOL ) END IF * INDXTAU = IIAX * ELSE * * sub( X ) is distributed across a process column. * CALL INFOG2L( IAX, JX, DESCX, NPROW, NPCOL, MYROW, MYCOL, $ IIAX, JJAX, IXROW, IXCOL ) * IF( MYCOL.NE.IXCOL ) $ RETURN * * Broadcast X(IAX,JAX) across the process column. * IF( MYROW.EQ.IXROW ) THEN J = IIAX+(JJAX-1)*DESCX( LLD_ ) CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 1, X( J ), 1 ) ALPHA = X( J ) ELSE CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 1, ALPHA, 1, $ IXROW, MYCOL ) END IF * INDXTAU = JJAX * END IF * IF( N.LE.0 ) THEN TAU( INDXTAU ) = ZERO RETURN END IF * CALL PDZNRM2( N-1, XNORM, X, IX, JX, DESCX, INCX ) ALPHR = DBLE( ALPHA ) ALPHI = DIMAG( ALPHA ) * IF( XNORM.EQ.ZERO .AND. ALPHI.EQ.ZERO ) THEN * * H = I * TAU( INDXTAU ) = ZERO * ELSE * * General case * BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR ) SAFMIN = DLAMCH( 'S' ) RSAFMN = ONE / SAFMIN IF( ABS( BETA ).LT.SAFMIN ) THEN * * XNORM, BETA may be inaccurate; scale X and recompute them * KNT = 0 10 CONTINUE KNT = KNT + 1 CALL PZDSCAL( N-1, RSAFMN, X, IX, JX, DESCX, INCX ) BETA = BETA*RSAFMN ALPHI = ALPHI*RSAFMN ALPHR = ALPHR*RSAFMN IF( ABS( BETA ).LT.SAFMIN ) $ GO TO 10 * * New BETA is at most 1, at least SAFMIN * CALL PDZNRM2( N-1, XNORM, X, IX, JX, DESCX, INCX ) ALPHA = DCMPLX( ALPHR, ALPHI ) BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR ) TAU( INDXTAU ) = DCMPLX( ( BETA-ALPHR ) / BETA, $ -ALPHI / BETA ) ALPHA = ZLADIV( DCMPLX( ONE ), ALPHA-BETA ) CALL PZSCAL( N-1, ALPHA, X, IX, JX, DESCX, INCX ) * * If ALPHA is subnormal, it may lose relative accuracy * ALPHA = BETA DO 20 J = 1, KNT ALPHA = ALPHA*SAFMIN 20 CONTINUE ELSE TAU( INDXTAU ) = DCMPLX( ( BETA-ALPHR ) / BETA, $ -ALPHI / BETA ) ALPHA = ZLADIV( DCMPLX( ONE ), ALPHA-BETA ) CALL PZSCAL( N-1, ALPHA, X, IX, JX, DESCX, INCX ) ALPHA = BETA END IF END IF * RETURN * * End of PZLARFG * END