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