SUBROUTINE PSGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, 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, IHI, ILO, INFO, JA, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ) REAL A( * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * PSGEHRD reduces a real general distributed matrix sub( A ) * to upper Hessenberg form H by an orthogonal similarity transforma- * tion: Q' * sub( A ) * Q = H, where * sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1). * * 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 and columns to be operated on, i.e. the * order of the distributed submatrix sub( A ). N >= 0. * * ILO (global input) INTEGER * IHI (global input) INTEGER * It is assumed that sub( A ) is already upper triangular in * rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2 * and JA+IHI:JA+N-1. See Further Details. If N > 0, * 1 <= ILO <= IHI <= N; otherwise set ILO = 1, IHI = N. * * A (local input/local output) REAL 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 N-by-N * general distributed matrix sub( A ) to be reduced. On exit, * the upper triangle and the first subdiagonal of sub( A ) are * overwritten with the upper Hessenberg matrix H, and the ele- * ments below the first subdiagonal, with the array TAU, repre- * sent 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. * * TAU (local output) REAL array, dimension LOCc(JA+N-2) * The scalar factors of the elementary reflectors (see Further * Details). Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are * set to zero. TAU is tied to the distributed matrix A. * * WORK (local workspace/local output) REAL 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*NB + NB*MAX( IHIP+1, IHLP+INLQ ) * * where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), * ICOFFA = MOD( JA-1, NB ), IOFF = MOD( IA+ILO-2, NB ), * IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), * IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), * ILROW = INDXG2P( IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), * IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ), * ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL ), * INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, 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 matrix Q is represented as a product of (ihi-ilo) elementary * reflectors * * Q = H(ilo) H(ilo+1) . . . H(ihi-1). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a real scalar, and v is a real vector with * v(1:I) = 0, v(I+1) = 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on * exit in A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2). * * The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- * ing example, with N = 7, ILO = 2 and IHI = 6: * * on entry on exit * * ( a a a a a a a ) ( a a h h h h a ) * ( a a a a a a ) ( a h h h h a ) * ( a a a a a a ) ( h h h h h h ) * ( a a a a a a ) ( v2 h h h h h ) * ( a a a a a a ) ( v2 v3 h h h h ) * ( a a a a a a ) ( v2 v3 v4 h h h ) * ( a ) ( a ) * * where a denotes an element of the original matrix sub( A ), H denotes * a modified element of the upper Hessenberg matrix H, and vi denotes * an element of the vector defining H(JA+ILO+I-2). * * Alignment requirements * ====================== * * The distributed submatrix sub( A ) must verify some alignment proper- * ties, namely the following expression 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 ) REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY CHARACTER COLCTOP, ROWCTOP INTEGER I, IACOL, IAROW, IB, ICOFFA, ICTXT, IHIP, $ IHLP, IIA, IINFO, ILCOL, ILROW, IMCOL, INLQ, $ IOFF, IPT, IPW, IPY, IROFFA, J, JJ, JJA, JY, $ K, L, LWMIN, MYCOL, MYROW, NB, NPCOL, NPROW, $ NQ REAL EI * .. * .. Local Arrays .. INTEGER DESCY( DLEN_ ), IDUM1( 3 ), IDUM2( 3 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, INFOG1L, $ INFOG2L, PCHK1MAT, PSGEMM, PSGEHD2, $ PSLAHRD, PSLARFB, PB_TOPGET, PB_TOPSET, PXERBLA * .. * .. External Functions .. INTEGER INDXG2P, NUMROC EXTERNAL INDXG2P, NUMROC * .. * .. Intrinsic Functions .. INTRINSIC 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 = -(700+CTXT_) ELSE CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, INFO ) IF( INFO.EQ.0 ) THEN NB = DESCA( NB_ ) IROFFA = MOD( IA-1, NB ) ICOFFA = MOD( JA-1, NB ) CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, $ IIA, JJA, IAROW, IACOL ) IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ) IOFF = MOD( IA+ILO-2, NB ) ILROW = INDXG2P( IA+ILO-1, NB, MYROW, DESCA( RSRC_ ), $ NPROW ) IHLP = NUMROC( IHI-ILO+IOFF+1, NB, MYROW, ILROW, NPROW ) ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, DESCA( CSRC_ ), $ NPCOL ) INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, NPCOL ) LWMIN = NB*( NB + MAX( IHIP+1, IHLP+INLQ ) ) * WORK( 1 ) = REAL( LWMIN ) LQUERY = ( LWORK.EQ.-1 ) IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( IROFFA.NE.ICOFFA .OR. IROFFA.NE.0 ) THEN INFO = -6 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN INFO = -(700+NB_) ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 END IF END IF IDUM1( 1 ) = ILO IDUM2( 1 ) = 2 IDUM1( 2 ) = IHI IDUM2( 2 ) = 3 IF( LWORK.EQ.-1 ) THEN IDUM1( 3 ) = -1 ELSE IDUM1( 3 ) = 1 END IF IDUM2( 3 ) = 10 CALL PCHK1MAT( N, 1, N, 1, IA, JA, DESCA, 7, 3, IDUM1, IDUM2, $ INFO ) END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PSGEHRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Set elements JA:JA+ILO-2 and JA+JHI-1:JA+N-2 of TAU to zero. * NQ = NUMROC( JA+N-2, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) CALL INFOG1L( JA+ILO-2, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ, $ IMCOL ) DO 10 J = JJA, MIN( JJ, NQ ) TAU( J ) = ZERO 10 CONTINUE * CALL INFOG1L( JA+IHI-1, NB, NPCOL, MYCOL, DESCA( CSRC_ ), JJ, $ IMCOL ) DO 20 J = JJ, NQ TAU( J ) = ZERO 20 CONTINUE * * Quick return if possible * IF( IHI-ILO.LE.0 ) $ RETURN * 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' ) * IPT = 1 IPY = IPT + NB * NB IPW = IPY + IHIP * NB CALL DESCSET( DESCY, IHI+IROFFA, NB, NB, NB, IAROW, ILCOL, ICTXT, $ MAX( 1, IHIP ) ) * K = ILO IB = NB - IOFF JY = IOFF + 1 * * Loop over remaining block of columns * DO 30 L = 1, IHI-ILO+IOFF-NB, NB I = IA + K - 1 J = JA + K - 1 * * Reduce columns j:j+ib-1 to Hessenberg form, returning the * matrices V and T of the block reflector H = I - V*T*V' * which performs the reduction, and also the matrix Y = A*V*T * CALL PSLAHRD( IHI, K, IB, A, IA, J, DESCA, TAU, WORK( IPT ), $ WORK( IPY ), 1, JY, DESCY, WORK( IPW ) ) * * Apply the block reflector H to A(ia:ia+ihi-1,j+ib:ja+ihi-1) * from the right, computing A := A - Y * V'. * V(i+ib,ib-1) must be set to 1. * CALL PSELSET2( EI, A, I+IB, J+IB-1, DESCA, ONE ) CALL PSGEMM( 'No transpose', 'Transpose', IHI, IHI-K-IB+1, IB, $ -ONE, WORK( IPY ), 1, JY, DESCY, A, I+IB, J, $ DESCA, ONE, A, IA, J+IB, DESCA ) CALL PSELSET( A, I+IB, J+IB-1, DESCA, EI ) * * Apply the block reflector H to A(i+1:ia+ihi-1,j+ib:ja+n-1) from * the left * CALL PSLARFB( 'Left', 'Transpose', 'Forward', 'Columnwise', $ IHI-K, N-K-IB+1, IB, A, I+1, J, DESCA, $ WORK( IPT ), A, I+1, J+IB, DESCA, WORK( IPY ) ) * K = K + IB IB = NB JY = 1 DESCY( CSRC_ ) = MOD( DESCY( CSRC_ ) + 1, NPCOL ) * 30 CONTINUE * * Use unblocked code to reduce the rest of the matrix * CALL PSGEHD2( N, K, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, $ IINFO ) * CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) * WORK( 1 ) = REAL( LWMIN ) * RETURN * * End of PSGEHRD * END