SUBROUTINE PSGEHD2( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, $ LWORK, INFO ) * * -- 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, IHI, ILO, INFO, JA, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ) REAL A( * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * PSGEHD2 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+JLO-2 * and JA+JHI: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 + MAX( NpA0, NB ) * * where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), * IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), * NpA0 = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), * * 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 (local 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 follo- * wing 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 PARAMETER ( ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, IAROW, ICOFFA, ICTXT, IROFFA, J, K, LWMIN, $ MYCOL, MYROW, NPA0, NPCOL, NPROW REAL AII * .. * .. External Subroutines .. EXTERNAL BLACS_ABORT, BLACS_GRIDINFO, CHK1MAT, PSELSET, $ PSLARF, PSLARFG, 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 IROFFA = MOD( IA-1, DESCA( MB_ ) ) ICOFFA = MOD( JA-1, DESCA( NB_ ) ) IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ), $ NPROW ) NPA0 = NUMROC( IHI+IROFFA, DESCA( MB_ ), MYROW, IAROW, $ NPROW ) LWMIN = DESCA( NB_ ) + MAX( NPA0, DESCA( NB_ ) ) * 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 ) 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 END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PSGEHD2', -INFO ) CALL BLACS_ABORT( ICTXT, 1 ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * DO 10 K = ILO, IHI-1 I = IA + K - 1 J = JA + K - 1 * * Compute elementary reflector H(j) to annihilate * A(i+2:ihi+ia-1,j) * CALL PSLARFG( IHI-K, AII, I+1, J, A, MIN( I+2, N+IA-1 ), J, $ DESCA, 1, TAU ) CALL PSELSET( A, I+1, J, DESCA, ONE ) * * Apply H(k) to A(ia:ihi+ia-1,j+1:ihi+ja-1) from the right * CALL PSLARF( 'Right', IHI, IHI-K, A, I+1, J, DESCA, 1, TAU, A, $ IA, J+1, DESCA, WORK ) * * Apply H(j) to A(i+1:ia+ihi-1,j+1:ja+n-1) from the left * CALL PSLARF( 'Left', IHI-K, N-K, A, I+1, J, DESCA, 1, TAU, A, $ I+1, J+1, DESCA, WORK ) * CALL PSELSET( A, I+1, J, DESCA, AII ) 10 CONTINUE * WORK( 1 ) = REAL( LWMIN ) * RETURN * * End of PSGEHD2 * END