SUBROUTINE PDGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK,
$ LWORK, INFO )
*
* -- ScaLAPACK routine (version 1.5) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
INTEGER IA, IHI, ILO, INFO, JA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
DOUBLE PRECISION A( * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PDGEHRD reduces a real general distributed matrix sub( A )
* to upper Hessenberg form H by an orthogonal similarity transforma-
* tion: Q' * sub( A ) * Q = H, where
* sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-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
*
* 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.
*
* ILO (global input) INTEGER
* IHI (global input) INTEGER
* It is assumed that sub( A ) is already upper triangular in
* rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2
* and JA+IHI:JA+N-1. See Further Details. If N > 0,
* 1 <= ILO <= IHI <= N; otherwise set ILO = 1, IHI = N.
*
* A (local input/local output) DOUBLE PRECISION pointer into the
* local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).
* On entry, this array contains the local pieces of the N-by-N
* general distributed matrix sub( A ) to be reduced. On exit,
* the upper triangle and the first subdiagonal of sub( A ) are
* overwritten with the upper Hessenberg matrix H, and the ele-
* ments below the first subdiagonal, with the array TAU, repre-
* sent the orthogonal matrix Q as a product of elementary
* reflectors. See Further Details.
*
* 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) DOUBLE PRECISION array, dimension LOCc(JA+N-2)
* The scalar factors of the elementary reflectors (see Further
* Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are
* set to zero. TAU is tied to the distributed matrix A.
*
* WORK (local workspace/local output) DOUBLE PRECISION array,
* dimension (LWORK)
* On exit, WORK( 1 ) returns the minimal and optimal LWORK.
*
* LWORK (local or global input) INTEGER
* The dimension of the array WORK.
* LWORK is local input and must be at least
* LWORK >= NB*NB + NB*MAX( IHIP+1, IHLP+INLQ )
*
* where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ),
* ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ),
* IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
* IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ),
* ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ),
* IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ),
* ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ),
* INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ),
*
* INDXG2P and NUMROC are ScaLAPACK tool functions;
* MYROW, MYCOL, NPROW and NPCOL can be determined by calling
* the subroutine BLACS_GRIDINFO.
*
* If LWORK = -1, then LWORK is global input and a workspace
* query is assumed; the routine only calculates the minimum
* and optimal size for all work arrays. Each of these
* values is returned in the first entry of the corresponding
* work array, and no error message is issued by PXERBLA.
*
* 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.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on
* exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2).
*
* The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow-
* ing example, with N = 7, ILO = 2 and IHI = 6:
*
* on entry on exit
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( a ) ( a )
*
* where a denotes an element of the original matrix sub( A ), H denotes
* a modified element of the upper Hessenberg matrix H, and vi denotes
* an element of the vector defining H(JA+ILO+I-2).
*
* Alignment requirements
* ======================
*
* The distributed submatrix sub( A ) must verify some alignment proper-
* ties, namely the following expression should be true:
* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )
*
* =====================================================================
*
* .. 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 ..
LOGICAL LQUERY
CHARACTER COLCTOP, ROWCTOP
INTEGER I, IACOL, IAROW, IB, ICOFFA, ICTXT, IHIP,
$ IHLP, IIA, IINFO, ILCOL, ILROW, IMCOL, INLQ,
$ IOFF, IPT, IPW, IPY, IROFFA, J, JJ, JJA, JY,
$ K, L, LWMIN, MYCOL, MYROW, NB, NPCOL, NPROW,
$ NQ
DOUBLE PRECISION EI
* ..
* .. Local Arrays ..
INTEGER DESCY( DLEN_ ), IDUM1( 3 ), IDUM2( 3 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, INFOG1L,
$ INFOG2L, PCHK1MAT, PDGEMM, PDGEHD2,
$ PDLAHRD, PDLARFB, PTOPGET, PTOPSET, PXERBLA
* ..
* .. External Functions ..
INTEGER INDXG2P, NUMROC
EXTERNAL INDXG2P, NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, 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 = -(700+CTXT_)
ELSE
CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, INFO )
IF( INFO.EQ.0 ) THEN
NB = DESCA( NB_ )
IROFFA = MOD( IA-1, NB )
ICOFFA = MOD( JA-1, NB )
CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL,
$ IIA, JJA, IAROW, IACOL )
IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW )
IOFF = MOD( IA+ILO-2, NB )
ILROW = INDXG2P( IA+ILO-1, NB, MYROW, DESCA( RSRC_ ),
$ NPROW )
IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW )
ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, DESCA( CSRC_ ),
$ NPCOL )
INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL )
LWMIN = NB*( NB + MAX( IHIP+1, IHLP+INLQ ) )
*
WORK( 1 ) = DBLE( LWMIN )
LQUERY = ( LWORK.EQ.-1 )
IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( IROFFA.NE.ICOFFA .OR. IROFFA.NE.0 ) THEN
INFO = -6
ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN
INFO = -(700+NB_)
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
END IF
IDUM1( 1 ) = ILO
IDUM2( 1 ) = 2
IDUM1( 2 ) = IHI
IDUM2( 2 ) = 3
IF( LWORK.EQ.-1 ) THEN
IDUM1( 3 ) = -1
ELSE
IDUM1( 3 ) = 1
END IF
IDUM2( 3 ) = 10
CALL PCHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, 3, IDUM1, IDUM2,
$ INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PDGEHRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Set elements JA:JA+ILO-2 and JA+JHI-1:JA+N-2 of TAU to zero.
*
NQ = NUMROC( JA+N-2, NB, MYCOL, DESCA( CSRC_ ), NPCOL )
CALL INFOG1L( JA+ILO-2, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ,
$ IMCOL )
DO 10 J = JJA, MIN( JJ, NQ )
TAU( J ) = ZERO
10 CONTINUE
*
CALL INFOG1L( JA+IHI-1, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ,
$ IMCOL )
DO 20 J = JJ, NQ
TAU( J ) = ZERO
20 CONTINUE
*
* Quick return if possible
*
IF( IHI-ILO.LE.0 )
$ RETURN
*
CALL PTOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' )
*
IPT = 1
IPY = IPT + NB * NB
IPW = IPY + IHIP * NB
CALL DESCSET( DESCY, IHI+IROFFA, NB, NB, NB, IAROW, ILCOL, ICTXT,
$ MAX( 1, IHIP ) )
*
K = ILO
IB = NB - IOFF
JY = IOFF + 1
*
* Loop over remaining block of columns
*
DO 30 L = 1, IHI-ILO+IOFF-NB, NB
I = IA + K - 1
J = JA + K - 1
*
* Reduce columns j:j+ib-1 to Hessenberg form, returning the
* matrices V and T of the block reflector H = I - V*T*V'
* which performs the reduction, and also the matrix Y = A*V*T
*
CALL PDLAHRD( IHI, K, IB, A, IA, J, DESCA, TAU, WORK( IPT ),
$ WORK( IPY ), 1, JY, DESCY, WORK( IPW ) )
*
* Apply the block reflector H to A(ia:ia+ihi-1,j+ib:ja+ihi-1)
* from the right, computing A := A - Y * V'.
* V(i+ib,ib-1) must be set to 1.
*
CALL PDELSET2( EI, A, I+IB, J+IB-1, DESCA, ONE )
CALL PDGEMM( 'No transpose', 'Transpose', IHI, IHI-K-IB+1, IB,
$ -ONE, WORK( IPY ), 1, JY, DESCY, A, I+IB, J,
$ DESCA, ONE, A, IA, J+IB, DESCA )
CALL PDELSET( A, I+IB, J+IB-1, DESCA, EI )
*
* Apply the block reflector H to A(i+1:ia+ihi-1,j+ib:ja+n-1) from
* the left
*
CALL PDLARFB( 'Left', 'Transpose', 'Forward', 'Columnwise',
$ IHI-K, N-K-IB+1, IB, A, I+1, J, DESCA,
$ WORK( IPT ), A, I+1, J+IB, DESCA, WORK( IPY ) )
*
K = K + IB
IB = NB
JY = 1
DESCY( CSRC_ ) = MOD( DESCY( CSRC_ ) + 1, NPCOL )
*
30 CONTINUE
*
* Use unblocked code to reduce the rest of the matrix
*
CALL PDGEHD2( N, K, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK,
$ IINFO )
*
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
*
WORK( 1 ) = DBLE( LWMIN )
*
RETURN
*
* End of PDGEHRD
*
END