SUBROUTINE PDLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, $ X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) * * -- ScaLAPACK auxiliary routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 1, 1997 * * .. Scalar Arguments .. INTEGER IA, IX, IY, JA, JX, JY, M, N, NB * .. * .. Array Arguments .. INTEGER DESCA( * ), DESCX( * ), DESCY( * ) DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), $ TAUQ( * ), X( * ), Y( * ), WORK( * ) * .. * * Purpose * ======= * * PDLABRD reduces the first NB rows and columns of 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 by an orthogonal transformation Q' * A * P, * and returns the matrices X and Y which are needed to apply the * transformation to the unreduced part of sub( A ). * * If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to * lower bidiagonal form. * * This is an auxiliary routine called by PDGEBRD. * * 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. * * NB (global input) INTEGER * The number of leading rows and columns of sub( A ) to be * reduced. * * 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 ) to be reduced. On exit, * the first NB rows and columns of the matrix are overwritten; * the rest of the distributed matrix sub( A ) is unchanged. * If m >= n, elements on and below the diagonal in the first NB * columns, with the array TAUQ, represent the orthogonal * matrix Q as a product of elementary reflectors; and * elements above the diagonal in the first NB rows, with the * array TAUP, represent the orthogonal matrix P as a product * of elementary reflectors. * If m < n, elements below the diagonal in the first NB * columns, with the array TAUQ, represent the orthogonal * matrix Q as a product of elementary reflectors, and * elements on and above the diagonal in the first NB rows, * 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 * LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. * The distributed diagonal elements of the bidiagonal matrix * B: D(i) = A(ia+i-1,ja+i-1). 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(ia+i-1,ja+i) for i = 1,2,...,n-1; * if m < n, E(i) = A(ia+i,ja+i-1) 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. * * X (local output) DOUBLE PRECISION pointer into the local memory * to an array of dimension (LLD_X,NB). On exit, the local * pieces of the distributed M-by-NB matrix * X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced * part of sub( A ). * * IX (global input) INTEGER * The row index in the global array X indicating the first * row of sub( X ). * * JX (global input) INTEGER * The column index in the global array X indicating the * first column of sub( X ). * * DESCX (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix X. * * Y (local output) DOUBLE PRECISION pointer into the local memory * to an array of dimension (LLD_Y,NB). On exit, the local * pieces of the distributed N-by-NB matrix * Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced * part of sub( A ). * * IY (global input) INTEGER * The row index in the global array Y indicating the first * row of sub( Y ). * * JY (global input) INTEGER * The column index in the global array Y indicating the * first column of sub( Y ). * * DESCY (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix Y. * * WORK (local workspace) DOUBLE PRECISION array, dimension (LWORK) * LWORK >= NB_A + NQ, with * * NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) * IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) * * INDXG2P and NUMROC are ScaLAPACK tool functions; * MYROW, MYCOL, NPROW and NPCOL can be determined by calling * the subroutine BLACS_GRIDINFO. * * Further Details * =============== * * The matrices Q and P are represented as products of elementary * reflectors: * * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) * * 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. * * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in * A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 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). * * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1: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: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 elements of the vectors v and u together form the m-by-nb matrix * V and the nb-by-n matrix U' which are needed, with X and Y, to apply * the transformation to the unreduced part of the matrix, using a block * update of the form: sub( A ) := sub( A ) - V*Y' - X*U'. * * The contents of sub( A ) on exit are illustrated by the following * examples with nb = 2: * * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): * * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) * ( v1 v2 a a a ) ( v1 1 a a a a ) * ( v1 v2 a a a ) ( v1 v2 a a a a ) * ( v1 v2 a a a ) ( v1 v2 a a a a ) * ( v1 v2 a a a ) * * where a denotes an element of the original matrix which is unchanged, * vi denotes an element of the vector defining H(i), and ui an element * of the vector defining G(i). * * ===================================================================== * * .. 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 .. INTEGER I, IACOL, IAROW, ICTXT, II, IPY, IW, J, JJ, $ JWY, K, MYCOL, MYROW, NPCOL, NPROW DOUBLE PRECISION ALPHA, TAU INTEGER DESCD( DLEN_ ), DESCE( DLEN_ ), $ DESCTP( DLEN_ ), DESCTQ( DLEN_ ), $ DESCW( DLEN_ ), DESCWY( DLEN_ ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, DESCSET, INFOG2L, PDCOPY, $ PDELGET, PDELSET, PDGEMV, PDLARFG, $ PDSCAL * .. * .. Intrinsic Functions .. INTRINSIC MIN, MOD * .. * .. Executable Statements .. * * Quick return if possible * IF( M.LE.0 .OR. N.LE.0 ) $ RETURN * ICTXT = DESCA( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, II, JJ, $ IAROW, IACOL ) IPY = DESCA( MB_ ) + 1 IW = MOD( IA-1, DESCA( NB_ ) ) + 1 ALPHA = ZERO * CALL DESCSET( DESCWY, 1, N+MOD( IA-1, DESCY( NB_ ) ), 1, $ DESCA( NB_ ), IAROW, IACOL, ICTXT, 1 ) CALL DESCSET( DESCW, DESCA( MB_ ), 1, DESCA( MB_ ), 1, IAROW, $ IACOL, ICTXT, DESCA( MB_ ) ) CALL DESCSET( DESCTQ, 1, JA+MIN(M,N)-1, 1, DESCA( NB_ ), IAROW, $ DESCA( CSRC_ ), DESCA( CTXT_ ), 1 ) CALL DESCSET( DESCTP, IA+MIN(M,N)-1, 1, DESCA( MB_ ), 1, $ DESCA( RSRC_ ), IACOL, DESCA( CTXT_ ), $ DESCA( LLD_ ) ) * IF( M.GE.N ) THEN * * Reduce to upper bidiagonal form * CALL DESCSET( DESCD, 1, JA+MIN(M,N)-1, 1, DESCA( NB_ ), MYROW, $ DESCA( CSRC_ ), DESCA( CTXT_ ), 1 ) CALL DESCSET( DESCE, IA+MIN(M,N)-1, 1, DESCA( MB_ ), 1, $ DESCA( RSRC_ ), MYCOL, DESCA( CTXT_ ), $ DESCA( LLD_ ) ) DO 10 K = 1, NB I = IA + K - 1 J = JA + K - 1 JWY = IW + K * * Update A(i:ia+m-1,j) * IF( K.GT.1 ) THEN CALL PDGEMV( 'No transpose', M-K+1, K-1, -ONE, A, I, JA, $ DESCA, Y, IY, JY+K-1, DESCY, 1, ONE, A, I, $ J, DESCA, 1 ) CALL PDGEMV( 'No transpose', M-K+1, K-1, -ONE, X, IX+K-1, $ JX, DESCX, A, IA, J, DESCA, 1, ONE, A, I, J, $ DESCA, 1 ) CALL PDELSET( A, I-1, J, DESCA, ALPHA ) END IF * * Generate reflection Q(i) to annihilate A(i+1:ia+m-1,j) * CALL PDLARFG( M-K+1, ALPHA, I, J, A, I+1, J, DESCA, 1, $ TAUQ ) CALL PDELSET( D, 1, J, DESCD, ALPHA ) CALL PDELSET( A, I, J, DESCA, ONE ) * * Compute Y(IA+I:IA+N-1,J) * CALL PDGEMV( 'Transpose', M-K+1, N-K, ONE, A, I, J+1, DESCA, $ A, I, J, DESCA, 1, ZERO, WORK( IPY ), 1, JWY, $ DESCWY, DESCWY( M_ ) ) CALL PDGEMV( 'Transpose', M-K+1, K-1, ONE, A, I, JA, DESCA, $ A, I, J, DESCA, 1, ZERO, WORK, IW, 1, DESCW, $ 1 ) CALL PDGEMV( 'Transpose', K-1, N-K, -ONE, Y, IY, JY+K, $ DESCY, WORK, IW, 1, DESCW, 1, ONE, WORK( IPY ), $ 1, JWY, DESCWY, DESCWY( M_ ) ) CALL PDGEMV( 'Transpose', M-K+1, K-1, ONE, X, IX+K-1, JX, $ DESCX, A, I, J, DESCA, 1, ZERO, WORK, IW, 1, $ DESCW, 1 ) CALL PDGEMV( 'Transpose', K-1, N-K, -ONE, A, IA, J+1, DESCA, $ WORK, IW, 1, DESCW, 1, ONE, WORK( IPY ), 1, $ JWY, DESCWY, DESCWY( M_ ) ) * CALL PDELGET( 'Rowwise', ' ', TAU, TAUQ, 1, J, DESCTQ ) CALL PDSCAL( N-K, TAU, WORK( IPY ), 1, JWY, DESCWY, $ DESCWY( M_ ) ) CALL PDCOPY( N-K, WORK( IPY ), 1, JWY, DESCWY, DESCWY( M_ ), $ Y, IY+K-1, JY+K, DESCY, DESCY( M_ ) ) * * Update A(i,j+1:ja+n-1) * CALL PDGEMV( 'Transpose', K, N-K, -ONE, Y, IY, JY+K, DESCY, $ A, I, JA, DESCA, DESCA( M_ ), ONE, A, I, J+1, $ DESCA, DESCA( M_ ) ) CALL PDGEMV( 'Transpose', K-1, N-K, -ONE, A, IA, J+1, DESCA, $ X, IX+K-1, JX, DESCX, DESCX( M_ ), ONE, A, I, $ J+1, DESCA, DESCA( M_ ) ) CALL PDELSET( A, I, J, DESCA, ALPHA ) * * Generate reflection P(i) to annihilate A(i,j+2:ja+n-1) * CALL PDLARFG( N-K, ALPHA, I, J+1, A, I, $ MIN( J+2, N+JA-1 ), DESCA, DESCA( M_ ), TAUP ) CALL PDELSET( E, I, 1, DESCE, ALPHA ) CALL PDELSET( A, I, J+1, DESCA, ONE ) * * Compute X(I+1:IA+M-1,J) * CALL PDGEMV( 'No transpose', M-K, N-K, ONE, A, I+1, J+1, $ DESCA, A, I, J+1, DESCA, DESCA( M_ ), ZERO, X, $ IX+K, JX+K-1, DESCX, 1 ) CALL PDGEMV( 'No transpose', K, N-K, ONE, Y, IY, JY+K, $ DESCY, A, I, J+1, DESCA, DESCA( M_ ), ZERO, $ WORK, IW, 1, DESCW, 1 ) CALL PDGEMV( 'No transpose', M-K, K, -ONE, A, I+1, JA, $ DESCA, WORK, IW, 1, DESCW, 1, ONE, X, IX+K, $ JX+K-1, DESCX, 1 ) CALL PDGEMV( 'No transpose', K-1, N-K, ONE, A, IA, J+1, $ DESCA, A, I, J+1, DESCA, DESCA( M_ ), ZERO, $ WORK, IW, 1, DESCW, 1 ) CALL PDGEMV( 'No transpose', M-K, K-1, -ONE, X, IX+K, JX, $ DESCX, WORK, IW, 1, DESCW, 1, ONE, X, IX+K, $ JX+K-1, DESCX, 1 ) * CALL PDELGET( 'Columnwise', ' ', TAU, TAUP, I, 1, DESCTP ) CALL PDSCAL( M-K, TAU, X, IX+K, JX+K-1, DESCX, 1 ) 10 CONTINUE * ELSE * * Reduce to lower bidiagonal form * CALL DESCSET( DESCD, IA+MIN(M,N)-1, 1, DESCA( MB_ ), 1, $ DESCA( RSRC_ ), MYCOL, DESCA( CTXT_ ), $ DESCA( LLD_ ) ) CALL DESCSET( DESCE, 1, JA+MIN(M,N)-1, 1, DESCA( NB_ ), MYROW, $ DESCA( CSRC_ ), DESCA( CTXT_ ), 1 ) DO 20 K = 1, NB I = IA + K - 1 J = JA + K - 1 JWY = IW + K * * Update A(i,j:ja+n-1) * IF( K.GT.1 ) THEN CALL PDGEMV( 'Transpose', K-1, N-K+1, -ONE, Y, IY, $ JY+K-1, DESCY, A, I, JA, DESCA, DESCA( M_ ), $ ONE, A, I, J, DESCA, DESCA( M_ ) ) CALL PDGEMV( 'Transpose', K-1, N-K+1, -ONE, A, IA, J, $ DESCA, X, IX+K-1, JX, DESCX, DESCX( M_ ), $ ONE, A, I, J, DESCA, DESCA( M_ ) ) CALL PDELSET( A, I, J-1, DESCA, ALPHA ) END IF * * Generate reflection P(i) to annihilate A(i,j+1:ja+n-1) * CALL PDLARFG( N-K+1, ALPHA, I, J, A, I, J+1, DESCA, $ DESCA( M_ ), TAUP ) CALL PDELSET( D, I, 1, DESCD, ALPHA ) CALL PDELSET( A, I, J, DESCA, ONE ) * * Compute X(i+1:ia+m-1,j) * CALL PDGEMV( 'No transpose', M-K, N-K+1, ONE, A, I+1, J, $ DESCA, A, I, J, DESCA, DESCA( M_ ), ZERO, X, $ IX+K, JX+K-1, DESCX, 1 ) CALL PDGEMV( 'No transpose', K-1, N-K+1, ONE, Y, IY, JY+K-1, $ DESCY, A, I, J, DESCA, DESCA( M_ ), ZERO, $ WORK, IW, 1, DESCW, 1 ) CALL PDGEMV( 'No transpose', M-K, K-1, -ONE, A, I+1, JA, $ DESCA, WORK, IW, 1, DESCW, 1, ONE, X, IX+K, $ JX+K-1, DESCX, 1 ) CALL PDGEMV( 'No transpose', K-1, N-K+1, ONE, A, IA, J, $ DESCA, A, I, J, DESCA, DESCA( M_ ), ZERO, $ WORK, IW, 1, DESCW, 1 ) CALL PDGEMV( 'No transpose', M-K, K-1, -ONE, X, IX+K, JX, $ DESCX, WORK, IW, 1, DESCW, 1, ONE, X, IX+K, $ JX+K-1, DESCX, 1 ) * CALL PDELGET( 'Columnwise', ' ', TAU, TAUP, I, 1, DESCTP ) CALL PDSCAL( M-K, TAU, X, IX+K, JX+K-1, DESCX, 1 ) * * Update A(i+1:ia+m-1,j) * CALL PDGEMV( 'No transpose', M-K, K-1, -ONE, A, I+1, JA, $ DESCA, Y, IY, JY+K-1, DESCY, 1, ONE, A, I+1, J, $ DESCA, 1 ) CALL PDGEMV( 'No transpose', M-K, K, -ONE, X, IX+K, JX, $ DESCX, A, IA, J, DESCA, 1, ONE, A, I+1, J, $ DESCA, 1 ) CALL PDELSET( A, I, J, DESCA, ALPHA ) * * Generate reflection Q(i) to annihilate A(i+2:ia+m-1,j) * CALL PDLARFG( M-K, ALPHA, I+1, J, A, MIN( I+2, M+IA-1 ), $ J, DESCA, 1, TAUQ ) CALL PDELSET( E, 1, J, DESCE, ALPHA ) CALL PDELSET( A, I+1, J, DESCA, ONE ) * * Compute Y(ia+i:ia+n-1,j) * CALL PDGEMV( 'Transpose', M-K, N-K, ONE, A, I+1, J+1, DESCA, $ A, I+1, J, DESCA, 1, ZERO, WORK( IPY ), 1, $ JWY, DESCWY, DESCWY( M_ ) ) CALL PDGEMV( 'Transpose', M-K, K-1, ONE, A, I+1, JA, DESCA, $ A, I+1, J, DESCA, 1, ZERO, WORK, IW, 1, DESCW, $ 1 ) CALL PDGEMV( 'Transpose', K-1, N-K, -ONE, Y, IY, JY+K, $ DESCY, WORK, IW, 1, DESCW, 1, ONE, WORK( IPY ), $ 1, JWY, DESCWY, DESCWY( M_ ) ) CALL PDGEMV( 'Transpose', M-K, K, ONE, X, IX+K, JX, DESCX, $ A, I+1, J, DESCA, 1, ZERO, WORK, IW, 1, DESCW, $ 1 ) CALL PDGEMV( 'Transpose', K, N-K, -ONE, A, IA, J+1, DESCA, $ WORK, IW, 1, DESCW, 1, ONE, WORK( IPY ), 1, $ JWY, DESCWY, DESCWY( M_ ) ) * CALL PDELGET( 'Rowwise', ' ', TAU, TAUQ, 1, J, DESCTQ ) CALL PDSCAL( N-K, TAU, WORK( IPY ), 1, JWY, DESCWY, $ DESCWY( M_ ) ) CALL PDCOPY( N-K, WORK( IPY ), 1, JWY, DESCWY, DESCWY( M_ ), $ Y, IY+K-1, JY+K, DESCY, DESCY( M_ ) ) 20 CONTINUE END IF * RETURN * * End of PDLABRD * END