SUBROUTINE PCGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB, \$ DESCB, TAUB, WORK, LWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * 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( * ) COMPLEX A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) * .. * * Purpose * ======= * * PCGGRQF 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 unitary matrix, Z is a P-by-P unitary 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 conjugate 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) 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, 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 unitary 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) COMPLEX, array, dimension LOCr(IA+M-1) * This array contains the scalar factors of the elementary * reflectors which represent the unitary matrix Q. TAUA is * tied to the distributed matrix A (see Further Details). * * B (local input/local output) COMPLEX 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 unitary 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) COMPLEX, array, dimension * LOCc(JB+MIN(P,N)-1). This array contains the scalar factors * TAUB of the elementary reflectors which represent the unitary * matrix Z. TAUB is tied to the distributed matrix B (see * Further Details). * * WORK (local workspace/local output) COMPLEX 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 complex scalar, and v is a complex vector with * v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(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 PCUNGRQ. * To use Q to update another matrix, use ScaLAPACK subroutine PCUNMRQ. * * 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 complex scalar, and v is a complex 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 PCUNGQR. * To use Z to update another matrix, use ScaLAPACK subroutine PCUNMQR. * * 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, PCGEQRF, PCGERQF, \$ PCHK2MAT, PCUNMRQ, PXERBLA * .. * .. Local Arrays .. INTEGER IDUM1( 1 ), IDUM2( 1 ) * .. * .. External Functions .. INTEGER INDXG2P, NUMROC EXTERNAL INDXG2P, NUMROC * .. * .. Intrinsic Functions .. INTRINSIC CMPLX, INT, MAX, MIN, MOD, REAL * .. * .. 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 ) = CMPLX( REAL( 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, 'PCGGRQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * RQ factorization of M-by-N matrix sub( A ): sub( A ) = R*Q * CALL PCGERQF( M, N, A, IA, JA, DESCA, TAUA, WORK, LWORK, INFO ) LWMIN = INT( WORK( 1 ) ) * * Update sub( B ) := sub( B )*Q' * CALL PCUNMRQ( 'Right', 'Conjugate 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 PCGEQRF( P, N, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = CMPLX( REAL( MAX( LWMIN, INT( WORK( 1 ) ) ) ) ) * RETURN * * End of PCGGRQF * END