SUBROUTINE PDGGQRF( N, M, P, 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( * ) DOUBLE PRECISION A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * ) * .. * * Purpose * ======= * * PDGGQRF computes a generalized QR factorization of * an N-by-M matrix sub( A ) = A(IA:IA+N-1,JA:JA+M-1) and * an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1): * * sub( A ) = Q*R, sub( B ) = Q*T*Z, * * 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 N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, * ( 0 ) N-M N M-N * M * * where R11 is upper triangular, and * * if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, * P-N N ( T21 ) P * P * * where T12 or T21 is upper triangular. * * In particular, if sub( B ) is square and nonsingular, the GQR * factorization of sub( A ) and sub( B ) implicitly gives the QR * factorization of inv( sub( B ) )* sub( A ): * * inv( sub( B ) )*sub( A )= Z'*(inv(T)*R) * * 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 * ========= * * N (global input) INTEGER * The number of rows to be operated on i.e the number of rows * of the distributed submatrices sub( A ) and sub( B ). N >= 0. * * M (global input) INTEGER * The number of columns to be operated on i.e the number of * columns of the distributed submatrix sub( A ). M >= 0. * * P (global input) INTEGER * The number of columns to be operated on i.e the number of * columns of the distributed submatrix sub( B ). P >= 0. * * A (local input/local output) DOUBLE PRECISION pointer into the * local memory to an array of dimension (LLD_A, LOCc(JA+M-1)). * On entry, the local pieces of the N-by-M distributed matrix * sub( A ) which is to be factored. On exit, the elements on * and above the diagonal of sub( A ) contain the min(N,M) by M * upper trapezoidal matrix R (R is upper triangular if N >= M); * the elements below the diagonal, with the array TAUA, * represent the orthogonal matrix Q as a product of min(N,M) * 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 * LOCc(JA+MIN(N,M)-1). This array contains the scalar factors * TAUA of the elementary reflectors which represent the * orthogonal 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+P-1)). * On entry, the local pieces of the N-by-P distributed matrix * sub( B ) which is to be factored. On exit, if N <= P, the * upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1) contains the * N by N upper triangular matrix T; if N > P, the elements on * and above the (N-P)-th subdiagonal contain the N by P upper * trapezoidal matrix T; the remaining elements, 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 LOCr(IB+N-1) * This array contains the scalar factors of the elementary * reflectors which represent the orthogonal unitary 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( NB_A * ( NpA0 + MqA0 + NB_A ), * MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + * NB_A * NB_A, * MB_B * ( NpB0 + PqB0 + MB_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 ), * NpA0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), * MqA0 = NUMROC( M+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 ), * NpB0 = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), * PqB0 = NUMROC( P+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(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m). * * 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(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in * A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1). * To form Q explicitly, use ScaLAPACK subroutine PDORGQR. * To use Q to update another matrix, use ScaLAPACK subroutine PDORMQR. * * The matrix Z is represented as a product of elementary reflectors * * Z = H(ib) H(ib+1) . . . H(ib+k-1), where k = min(n,p). * * 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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in * B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in TAUB(ib+n-k+i-1). * To form Z explicitly, use ScaLAPACK subroutine PDORGRQ. * To use Z to update another matrix, use ScaLAPACK subroutine PDORMRQ. * * Alignment requirements * ====================== * * The distributed submatrices sub( A ) and sub( B ) must verify some * alignment properties, namely the following expression should be true: * * ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW ) * * ===================================================================== * * .. 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, MQA0, MYCOL, \$ MYROW, NPA0, NPB0, NPCOL, NPROW, PQB0 * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, PCHK2MAT, PDGEQRF, \$ PDGERQF, PDORMQR, 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( N, 1, M, 2, IA, JA, DESCA, 7, INFO ) CALL CHK1MAT( N, 1, P, 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 ) NPA0 = NUMROC( N+IROFFA, DESCA( MB_ ), MYROW, IAROW, NPROW ) MQA0 = NUMROC( M+ICOFFA, DESCA( NB_ ), MYCOL, IACOL, NPCOL ) NPB0 = NUMROC( N+IROFFB, DESCB( MB_ ), MYROW, IBROW, NPROW ) PQB0 = NUMROC( P+ICOFFB, DESCB( NB_ ), MYCOL, IBCOL, NPCOL ) LWMIN = MAX( DESCA( NB_ ) * ( NPA0 + MQA0 + DESCA( NB_ ) ), \$ MAX( MAX( ( DESCA( NB_ )*( DESCA( NB_ ) - 1 ) ) / 2, \$ ( PQB0 + NPB0 ) * DESCA( NB_ ) ) + \$ DESCA( NB_ ) * DESCA( NB_ ), \$ DESCB( MB_ ) * ( NPB0 + PQB0 + DESCB( MB_ ) ) ) ) * WORK( 1 ) = DBLE( LWMIN ) LQUERY = ( LWORK.EQ.-1 ) IF( IAROW.NE.IBROW .OR. IROFFA.NE.IROFFB ) THEN INFO = -10 ELSE IF( DESCA( MB_ ).NE.DESCB( MB_ ) ) THEN INFO = -1203 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( LQUERY ) THEN IDUM1( 1 ) = -1 ELSE IDUM1( 1 ) = 1 END IF IDUM2( 1 ) = 15 CALL PCHK2MAT( N, 1, M, 2, IA, JA, DESCA, 7, N, 1, P, 3, IB, \$ JB, DESCB, 12, 1, IDUM1, IDUM2, INFO ) END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PDGGQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * QR factorization of N-by-M matrix sub( A ): sub( A ) = Q*R * CALL PDGEQRF( N, M, A, IA, JA, DESCA, TAUA, WORK, LWORK, INFO ) LWMIN = INT( WORK( 1 ) ) * * Update sub( B ) := Q'*sub( B ). * CALL PDORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, IA, JA, \$ DESCA, TAUA, B, IB, JB, DESCB, WORK, LWORK, INFO ) LWMIN = MIN( LWMIN, INT( WORK( 1 ) ) ) * * RQ factorization of N-by-P matrix sub( B ): sub( B ) = T*Z. * CALL PDGERQF( N, P, B, IB, JB, DESCB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = DBLE( MAX( LWMIN, INT( WORK( 1 ) ) ) ) * RETURN * * End of PDGGQRF * END