SUBROUTINE PDGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB,
$ DESCB, TAUB, 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, IB, INFO, JA, JB, LWORK, M, N, P
* ..
* .. Array Arguments ..
INTEGER DESCA( * ), DESCB( * )
DOUBLE PRECISION A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PDGGRQF computes a generalized RQ factorization of
* an M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1)
* and a P-by-N matrix sub( B ) = B(IB:IB+P-1,JB:JB+N-1):
*
* sub( A ) = R*Q, sub( B ) = Z*T*Q,
*
* where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
* matrix, and R and T assume one of the forms:
*
* if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
* N-M M ( R21 ) N
* N
*
* where R12 or R21 is upper triangular, and
*
* if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
* ( 0 ) P-N P N-P
* N
*
* where T11 is upper triangular.
*
* In particular, if sub( B ) is square and nonsingular, the GRQ
* factorization of sub( A ) and sub( B ) implicitly gives the RQ
* factorization of sub( A )*inv( sub( B ) ):
*
* sub( A )*inv( sub( B ) ) = (R*inv(T))*Z'
*
* where inv( sub( B ) ) denotes the inverse of the matrix sub( B ),
* and Z' denotes the transpose of matrix Z.
*
* 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.
*
* P (global input) INTEGER
* The number of rows to be operated on i.e the number of
* rows of the distributed submatrix sub( B ). P >= 0.
*
* N (global input) INTEGER
* The number of columns to be operated on i.e the number of
* columns of the distributed submatrices sub( A ) and sub( B ).
* N >= 0.
*
* 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, the local pieces of the M-by-N distributed matrix
* sub( A ) which is to be factored. On exit, if M <= N, the
* upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) contains the
* M by M upper triangular matrix R; if M >= N, the elements on
* and above the (M-N)-th subdiagonal contain the M by N upper
* trapezoidal matrix R; the remaining elements, with the array
* TAUA, represent 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.
*
* TAUA (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1)
* This array contains the scalar factors of the elementary
* reflectors which represent the orthogonal unitary matrix Q.
* TAUA is tied to the distributed matrix A (see Further
* Details).
*
* B (local input/local output) DOUBLE PRECISION pointer into the
* local memory to an array of dimension (LLD_B, LOCc(JB+N-1)).
* On entry, the local pieces of the P-by-N distributed matrix
* sub( B ) which is to be factored. On exit, the elements on
* and above the diagonal of sub( B ) contain the min(P,N) by N
* upper trapezoidal matrix T (T is upper triangular if P >= N);
* the elements below the diagonal, with the array TAUB,
* represent the orthogonal matrix Z as a product of elementary
* reflectors (see Further Details).
*
* IB (global input) INTEGER
* The row index in the global array B indicating the first
* row of sub( B ).
*
* JB (global input) INTEGER
* The column index in the global array B indicating the
* first column of sub( B ).
*
* DESCB (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix B.
*
* TAUB (local output) DOUBLE PRECISION, array, dimension
* LOCc(JB+MIN(P,N)-1). This array contains the scalar factors
* TAUB of the elementary reflectors which represent the
* orthogonal matrix Z. TAUB is tied to the distributed matrix
* B (see Further Details).
*
* 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 >= MAX( MB_A * ( MpA0 + NqA0 + MB_A ),
* MAX( (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) +
* MB_A * MB_A,
* NB_B * ( PpB0 + NqB0 + NB_B ) ), where
*
* IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ),
* IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
* IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
* MpA0 = NUMROC( M+IROFFA, MB_A, MYROW, IAROW, NPROW ),
* NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ),
*
* IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ),
* IBROW = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ),
* IBCOL = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL ),
* PpB0 = NUMROC( P+IROFFB, MB_B, MYROW, IBROW, NPROW ),
* NqB0 = NUMROC( N+ICOFFB, NB_B, MYCOL, IBCOL, NPCOL ),
*
* and NUMROC, INDXG2P 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 elementary reflectors
*
* Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - taua * v * v'
*
* where taua is a real scalar, and v is a real vector with
* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
* A(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1).
* To form Q explicitly, use ScaLAPACK subroutine PDORGRQ.
* To use Q to update another matrix, use ScaLAPACK subroutine PDORMRQ.
*
* The matrix Z is represented as a product of elementary reflectors
*
* Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n).
*
* Each H(i) has the form
*
* H(i) = I - taub * v * v'
*
* where taub is a real scalar, and v is a real vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in
* B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1).
* To form Z explicitly, use ScaLAPACK subroutine PDORGQR.
* To use Z to update another matrix, use ScaLAPACK subroutine PDORMQR.
*
* Alignment requirements
* ======================
*
* The distributed submatrices sub( A ) and sub( B ) must verify some
* alignment properties, namely the following expression should be true:
*
* ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL )
*
* =====================================================================
*
* .. 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 )
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB,
$ ICTXT, IROFFA, IROFFB, LWMIN, MPA0, MYCOL,
$ MYROW, NPCOL, NPROW, NQA0, NQB0, PPB0
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, PCHK2MAT, PDGEQRF,
$ PDGERQF, PDORMRQ, PXERBLA
* ..
* .. Local Arrays ..
INTEGER IDUM1( 1 ), IDUM2( 1 )
* ..
* .. External Functions ..
INTEGER INDXG2P, NUMROC
EXTERNAL INDXG2P, NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, INT, 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 = -707
ELSE
CALL CHK1MAT( M, 1, N, 3, IA, JA, DESCA, 7, INFO )
CALL CHK1MAT( P, 2, N, 3, IB, JB, DESCB, 12, INFO )
IF( INFO.EQ.0 ) THEN
IROFFA = MOD( IA-1, DESCA( MB_ ) )
ICOFFA = MOD( JA-1, DESCA( NB_ ) )
IROFFB = MOD( IB-1, DESCB( MB_ ) )
ICOFFB = MOD( JB-1, DESCB( NB_ ) )
IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
$ NPROW )
IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ),
$ NPCOL )
IBROW = INDXG2P( IB, DESCB( MB_ ), MYROW, DESCB( RSRC_ ),
$ NPROW )
IBCOL = INDXG2P( JB, DESCB( NB_ ), MYCOL, DESCB( CSRC_ ),
$ NPCOL )
MPA0 = NUMROC( M+IROFFA, DESCA( MB_ ), MYROW, IAROW, NPROW )
NQA0 = NUMROC( N+ICOFFA, DESCA( NB_ ), MYCOL, IACOL, NPCOL )
PPB0 = NUMROC( P+IROFFB, DESCB( MB_ ), MYROW, IBROW, NPROW )
NQB0 = NUMROC( N+ICOFFB, DESCB( NB_ ), MYCOL, IBCOL, NPCOL )
LWMIN = MAX( DESCA( MB_ ) * ( MPA0 + NQA0 + DESCA( MB_ ) ),
$ MAX( MAX( ( DESCA( MB_ )*( DESCA( MB_ ) - 1 ) ) / 2,
$ ( PPB0 + NQB0 ) * DESCA( MB_ ) ) +
$ DESCA( MB_ ) * DESCA( MB_ ),
$ DESCB( NB_ ) * ( PPB0 + NQB0 + DESCB( NB_ ) ) ) )
*
WORK( 1 ) = DBLE( LWMIN )
LQUERY = ( LWORK.EQ.-1 )
IF( IACOL.NE.IBCOL .OR. ICOFFA.NE.ICOFFB ) THEN
INFO = -11
ELSE IF( DESCA( NB_ ).NE.DESCB( NB_ ) ) THEN
INFO = -1204
ELSE IF( ICTXT.NE.DESCB( CTXT_ ) ) THEN
INFO = -1207
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
END IF
IF( LWORK.EQ.-1 ) THEN
IDUM1( 1 ) = -1
ELSE
IDUM1( 1 ) = 1
END IF
IDUM2( 1 ) = 15
CALL PCHK2MAT( M, 1, N, 3, IA, JA, DESCA, 7, P, 2, N, 3, IB,
$ JB, DESCB, 12, 1, IDUM1, IDUM2, INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PDGGRQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* RQ factorization of M-by-N matrix sub( A ): sub( A ) = R*Q
*
CALL PDGERQF( M, N, A, IA, JA, DESCA, TAUA, WORK, LWORK, INFO )
LWMIN = INT( WORK( 1 ) )
*
* Update sub( B ) := sub( B )*Q'
*
CALL PDORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ), A,
$ MAX( IA, IA+M-N ), JA, DESCA, TAUA, B, IB, JB,
$ DESCB, WORK, LWORK, INFO )
LWMIN = MAX( LWMIN, INT( WORK( 1 ) ) )
*
* QR factorization of P-by-N matrix sub( B ): sub( B ) = Z*T
*
CALL PDGEQRF( P, N, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = DBLE( MAX( LWMIN, INT( WORK( 1 ) ) ) )
*
RETURN
*
* End of PDGGRQF
*
END