SUBROUTINE PCLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK )
*
* -- ScaLAPACK routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* December 31, 1998
*
* .. Scalar Arguments ..
INTEGER IA, JA, L, M, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
COMPLEX A( * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PCLATRZ reduces the M-by-N ( M<=N ) complex upper trapezoidal
* matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)]
* to upper triangular form by means of unitary transformations.
*
* The upper trapezoidal matrix sub( A ) is factored as
*
* sub( A ) = ( R 0 ) * Z,
*
* where Z is an N-by-N unitary matrix and R is an M-by-M upper
* triangular matrix.
*
* 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.
*
* L (global input) INTEGER
* The columns of the distributed submatrix sub( A ) containing
* the meaningful part of the Householder reflectors. L > 0.
*
* A (local input/local output) COMPLEX pointer into the
* local memory to an array of dimension (LLD_A, LOCc(JA+N-1)).
* On entry, the local pieces of the M-by-N distributed matrix
* sub( A ) which is to be factored. On exit, the leading M-by-M
* upper triangular part of sub( A ) contains the upper trian-
* gular matrix R, and elements N-L+1 to N of the first M rows
* of sub( A ), with the array TAU, represent the unitary matrix
* Z as a product of M elementary reflectors.
*
* 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.
*
* TAU (local output) COMPLEX, array, dimension LOCr(IA+M-1)
* This array contains the scalar factors of the elementary
* reflectors. TAU is tied to the distributed matrix A.
*
* WORK (local workspace) COMPLEX array, dimension (LWORK)
* LWORK >= Nq0 + MAX( 1, Mp0 ), where
*
* IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ),
* IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
* IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
* Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ),
* Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ),
*
* and NUMROC, INDXG2P are ScaLAPACK tool functions;
* MYROW, MYCOL, NPROW and NPCOL can be determined by calling
* the subroutine BLACS_GRIDINFO.
*
* Further Details
* ===============
*
* The factorization is obtained by Householder's method. The kth
* transformation matrix, Z( k ), whose conjugate transpose is used to
* introduce zeros into the (m - k + 1)th row of sub( A ), is given in
* the form
*
* Z( k ) = ( I 0 ),
* ( 0 T( k ) )
*
* where
*
* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
* ( 0 )
* ( z( k ) )
*
* tau is a scalar and z( k ) is an ( n - m ) element vector.
* tau and z( k ) are chosen to annihilate the elements of the kth row
* of sub( A ).
*
* The scalar tau is returned in the kth element of TAU and the vector
* u( k ) in the kth row of sub( A ), such that the elements of z( k )
* are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned
* in the upper triangular part of sub( A ).
*
* Z is given by
*
* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*
* =====================================================================
*
* .. 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 )
COMPLEX ONE, ZERO
PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ),
$ ZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, IAROW, ICTXT, II, J, J1, MP, MYCOL, MYROW,
$ NPCOL, NPROW
COMPLEX AII
* ..
* .. Local Arrays ..
INTEGER DESCTAU( DLEN_ )
* ..
* .. External Subroutines ..
EXTERNAL DESCSET, INFOG1L, PCELSET, PCLACGV,
$ PCLARFG, PCLARZ
* ..
* .. External Functions ..
INTEGER NUMROC
EXTERNAL NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC CONJG, MAX
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
MP = NUMROC( IA+M-1, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
$ NPROW )
*
CALL DESCSET( DESCTAU, DESCA( M_ ), 1, DESCA( MB_ ), 1,
$ DESCA( RSRC_ ), MYCOL, ICTXT, MAX( 1, MP ) )
*
IF( M.EQ.N ) THEN
*
CALL INFOG1L( IA, DESCA( MB_ ), NPROW, MYROW, DESCA( RSRC_ ),
$ II, IAROW )
DO 10 I = II, MP
TAU( I ) = ZERO
10 CONTINUE
*
ELSE
*
AII = ZERO
*
J1 = JA + N - L
DO 20 I = IA+M-1, IA, -1
J = JA + I - IA
*
* Generate elementary reflector H(i) to annihilate
* [ A(i, j) A(i,j1:ja+n-1) ]
*
CALL PCLACGV( 1, A, I, J, DESCA, DESCA( M_ ) )
CALL PCLACGV( L, A, I, J1, DESCA, DESCA( M_ ) )
CALL PCLARFG( L+1, AII, I, J, A, I, J1, DESCA, DESCA( M_ ),
$ TAU )
*
* Apply H(i) to A(ia:i-1,j:ja+n-1) from the right
*
CALL PCLARZ( 'Right', I-IA, JA+N-J, L, A, I, J1, DESCA,
$ DESCA( M_ ), TAU, A, IA, J, DESCA, WORK )
CALL PCELSET( A, I, J, DESCA, CONJG( AII ) )
*
20 CONTINUE
*
CALL PCLACGV( M, TAU, IA, 1, DESCTAU, 1 )
*
END IF
*
RETURN
*
* End of PCLATRZ
*
END