SUBROUTINE PDGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, $ WORK, LWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 25, 2001 * * .. Scalar Arguments .. INTEGER IA, INFO, JA, LWORK, M, N * .. * .. Array Arguments .. INTEGER DESCA( * ) DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), $ WORK( * ) * .. * * Purpose * ======= * * PDGEBRD reduces a real general M-by-N distributed matrix * sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal * form B by an orthogonal transformation: Q' * sub( A ) * P = B. * * If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. * * 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. * * 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 * general distributed matrix sub( A ). On exit, if M >= N, * the diagonal and the first superdiagonal of sub( A ) are * overwritten with the upper bidiagonal matrix B; the elements * below the diagonal, with the array TAUQ, represent the * orthogonal matrix Q as a product of elementary reflectors, * and the elements above the first superdiagonal, with the * array TAUP, represent the orthogonal matrix P as a product * of elementary reflectors. If M < N, the diagonal and the * first subdiagonal are overwritten with the lower bidiagonal * matrix B; the elements below the first subdiagonal, with the * array TAUQ, represent the orthogonal matrix Q as a product of * elementary reflectors, and the elements above the diagonal, * with the array TAUP, represent the orthogonal matrix P 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. * * D (local output) DOUBLE PRECISION array, dimension * LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. * The distributed diagonal elements of the bidiagonal matrix * B: D(i) = A(i,i). D is tied to the distributed matrix A. * * E (local output) DOUBLE PRECISION array, dimension * LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. * The distributed off-diagonal elements of the bidiagonal * distributed matrix B: * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. * E is tied to the distributed matrix A. * * TAUQ (local output) DOUBLE PRECISION array dimension * LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary * reflectors which represent the orthogonal matrix Q. TAUQ * is tied to the distributed matrix A. See Further Details. * * TAUP (local output) DOUBLE PRECISION array, dimension * LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary * reflectors which represent the orthogonal matrix P. TAUP * is tied to the distributed matrix A. 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 >= NB*( MpA0 + NqA0 + 1 ) + NqA0 * * where NB = MB_A = NB_A, * IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), * IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), * IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ), * MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), * NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, 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 matrices Q and P are represented as products of elementary * reflectors: * * If m >= n, * * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) * * Each H(i) and G(i) has the form: * * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' * * where tauq and taup are real scalars, and v and u are real vectors; * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in * A(ia+i:ia+m-1,ja+i-1); * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in * A(ia+i-1,ja+i+1:ja+n-1); * tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). * * If m < n, * * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) * * Each H(i) and G(i) has the form: * * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' * * where tauq and taup are real scalars, and v and u are real vectors; * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in * A(ia+i+1:ia+m-1,ja+i-1); * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in * A(ia+i-1,ja+i:ja+n-1); * tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1). * * The contents of sub( A ) on exit are illustrated by the following * examples: * * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): * * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) * ( v1 v2 v3 v4 v5 ) * * where d and e denote diagonal and off-diagonal elements of B, vi * denotes an element of the vector defining H(i), and ui an element of * the vector defining G(i). * * Alignment requirements * ====================== * * The distributed submatrix sub( A ) must verify some alignment proper- * ties, namely the following expressions 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 PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY CHARACTER COLCTOP, ROWCTOP INTEGER I, IACOL, IAROW, ICTXT, IINFO, IOFF, IPW, IPY, $ IW, J, JB, JS, JW, K, L, LWMIN, MN, MP, MYCOL, $ MYROW, NB, NPCOL, NPROW, NQ * .. * .. Local Arrays .. INTEGER DESCWX( DLEN_ ), DESCWY( DLEN_ ), IDUM1( 1 ), $ IDUM2( 1 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, PCHK1MAT, $ PDELSET, PDGEBD2, PDGEMM, PDLABRD, $ PB_TOPGET, PB_TOPSET, PXERBLA * .. * .. External Functions .. INTEGER INDXG2L, INDXG2P, NUMROC EXTERNAL INDXG2L, 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 = -(600+CTXT_) ELSE CALL CHK1MAT( M, 1, N, 2, IA, JA, DESCA, 6, INFO ) IF( INFO.EQ.0 ) THEN NB = DESCA( MB_ ) IOFF = MOD( IA-1, DESCA( MB_ ) ) IAROW = INDXG2P( IA, NB, MYROW, DESCA( RSRC_ ), NPROW ) IACOL = INDXG2P( JA, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) MP = NUMROC( M+IOFF, NB, MYROW, IAROW, NPROW ) NQ = NUMROC( N+IOFF, NB, MYCOL, IACOL, NPCOL ) LWMIN = NB*( MP+NQ+1 ) + NQ * WORK( 1 ) = DBLE( LWMIN ) LQUERY = ( LWORK.EQ.-1 ) IF( IOFF.NE.MOD( JA-1, DESCA( NB_ ) ) ) THEN INFO = -5 ELSE IF( NB.NE.DESCA( NB_ ) ) THEN INFO = -(600+NB_) ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF IF( LQUERY ) THEN IDUM1( 1 ) = -1 ELSE IDUM1( 1 ) = 1 END IF IDUM2( 1 ) = 12 CALL PCHK1MAT( M, 1, N, 2, IA, JA, DESCA, 6, 1, IDUM1, IDUM2, $ INFO ) END IF * IF( INFO.LT.0 ) THEN CALL PXERBLA( ICTXT, 'PDGEBRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * MN = MIN( M, N ) IF( MN.EQ.0 ) $ RETURN * * Initialize parameters. * CALL PB_TOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' ) * IPY = MP * NB + 1 IPW = NQ * NB + IPY * CALL DESCSET( DESCWX, M+IOFF, NB, NB, NB, IAROW, IACOL, ICTXT, $ MAX( 1, MP ) ) CALL DESCSET( DESCWY, NB, N+IOFF, NB, NB, IAROW, IACOL, ICTXT, $ NB ) * MP = NUMROC( M+IA-1, NB, MYROW, DESCA( RSRC_ ), NPROW ) NQ = NUMROC( N+JA-1, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) K = 1 JB = NB - IOFF IW = IOFF + 1 JW = IOFF + 1 * DO 10 L = 1, MN+IOFF-NB, NB I = IA + K - 1 J = JA + K - 1 * * Reduce rows and columns i:i+nb-1 to bidiagonal form and return * the matrices X and Y which are needed to update the unreduced * part of the matrix. * CALL PDLABRD( M-K+1, N-K+1, JB, A, I, J, DESCA, D, E, TAUQ, $ TAUP, WORK, IW, JW, DESCWX, WORK( IPY ), IW, $ JW, DESCWY, WORK( IPW ) ) * * Update the trailing submatrix A(i+nb:ia+m-1,j+nb:ja+n-1), using * an update of the form A := A - V*Y' - X*U'. * CALL PDGEMM( 'No transpose', 'No transpose', M-K-JB+1, $ N-K-JB+1, JB, -ONE, A, I+JB, J, DESCA, $ WORK( IPY ), IW, JW+JB, DESCWY, ONE, A, I+JB, $ J+JB, DESCA ) CALL PDGEMM( 'No transpose', 'No transpose', M-K-JB+1, $ N-K-JB+1, JB, -ONE, WORK, IW+JB, JW, DESCWX, A, I, $ J+JB, DESCA, ONE, A, I+JB, J+JB, DESCA ) * * Copy last off-diagonal elements of B back into sub( A ). * IF( M.GE.N ) THEN JS = MIN( INDXG2L( I+JB-1, NB, 0, DESCA( RSRC_ ), NPROW ), $ MP ) IF( JS.GT.0 ) $ CALL PDELSET( A, I+JB-1, J+JB, DESCA, E( JS ) ) ELSE JS = MIN( INDXG2L( J+JB-1, NB, 0, DESCA( CSRC_ ), NPCOL ), $ NQ ) IF( JS.GT.0 ) $ CALL PDELSET( A, I+JB, J+JB-1, DESCA, E( JS ) ) END IF * K = K + JB JB = NB IW = 1 JW = 1 DESCWX( M_ ) = DESCWX( M_ ) - JB DESCWX( RSRC_ ) = MOD( DESCWX( RSRC_ ) + 1, NPROW ) DESCWX( CSRC_ ) = MOD( DESCWX( CSRC_ ) + 1, NPCOL ) DESCWY( N_ ) = DESCWY( N_ ) - JB DESCWY( RSRC_ ) = MOD( DESCWY( RSRC_ ) + 1, NPROW ) DESCWY( CSRC_ ) = MOD( DESCWY( CSRC_ ) + 1, NPCOL ) * 10 CONTINUE * * Use unblocked code to reduce the remainder of the matrix. * CALL PDGEBD2( M-K+1, N-K+1, A, IA+K-1, JA+K-1, DESCA, D, E, TAUQ, $ TAUP, WORK, LWORK, IINFO ) * CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) * WORK( 1 ) = DBLE( LWMIN ) * RETURN * * End of PDGEBRD * END