SUBROUTINE PDSYTRD( UPLO, N, A, IA, JA, DESCA, D, E, 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 .. CHARACTER UPLO INTEGER IA, INFO, JA, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ) DOUBLE PRECISION A( * ), D( * ), E( * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * PDSYTRD reduces a real symmetric matrix sub( A ) to symmetric * tridiagonal form T by an orthogonal similarity transformation: * Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA: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 * ========= * * UPLO (global input) CHARACTER * Specifies whether the upper or lower triangular part of the * symmetric matrix sub( A ) is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * 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. * * 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 * symmetric distributed matrix sub( A ). If UPLO = 'U', the * leading N-by-N upper triangular part of sub( A ) contains * the upper triangular part of the matrix, and its strictly * lower triangular part is not referenced. If UPLO = 'L', the * leading N-by-N lower triangular part of sub( A ) contains the * lower triangular part of the matrix, and its strictly upper * triangular part is not referenced. On exit, if UPLO = 'U', * the diagonal and first superdiagonal of sub( A ) are over- * written by the corresponding elements of the tridiagonal * matrix T, and the elements above the first superdiagonal, * with the array TAU, represent the orthogonal matrix Q as a * product of elementary reflectors; if UPLO = 'L', the diagonal * and first subdiagonal of sub( A ) are overwritten by the * corresponding elements of the tridiagonal matrix T, and the * elements below the first subdiagonal, with the array TAU, * represent 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. * * D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) * The diagonal elements of the tridiagonal matrix T: * D(i) = A(i,i). D is tied to the distributed matrix A. * * E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) * if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal * elements of the tridiagonal matrix T: E(i) = A(i,i+1) if * UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the * distributed matrix A. * * TAU (local output) DOUBLE PRECISION array, dimension * LOCc(JA+N-1). This array contains the scalar factors TAU of * the elementary reflectors. TAU is tied to the distributed * matrix A. * * 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 * ( NP +1 ), 3 * NB ) * * where NB = MB_A = NB_A, * NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), * IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, 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 (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 * =============== * * If UPLO = 'U', the matrix Q is represented as a product of elementary * reflectors * * Q = H(n-1) . . . H(2) H(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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in * A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). * * If UPLO = 'L', the matrix Q is represented as a product of elementary * reflectors * * Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in * A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). * * The contents of sub( A ) on exit are illustrated by the following * examples with n = 5: * * if UPLO = 'U': if UPLO = 'L': * * ( d e v2 v3 v4 ) ( d ) * ( d e v3 v4 ) ( e d ) * ( d e v4 ) ( v1 e d ) * ( d e ) ( v1 v2 e d ) * ( d ) ( v1 v2 v3 e d ) * * where d and e denote diagonal and off-diagonal elements of T, and vi * denotes an element of the vector defining H(i). * * 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 .AND. IROFFA.EQ.0 ) with * IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). * * ===================================================================== * * .. 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, UPPER CHARACTER COLCTOP, ROWCTOP INTEGER I, IACOL, IAROW, ICOFFA, ICTXT, IINFO, IPW, $ IROFFA, J, JB, JX, K, KK, LWMIN, MYCOL, MYROW, $ NB, NP, NPCOL, NPROW, NQ * .. * .. Local Arrays .. INTEGER DESCW( DLEN_ ), IDUM1( 2 ), IDUM2( 2 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, PCHK1MAT, $ PDLATRD, PDSYR2K, PDSYTD2, PB_TOPGET, $ PB_TOPSET, PXERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER INDXG2L, INDXG2P, NUMROC EXTERNAL LSAME, INDXG2L, INDXG2P, NUMROC * .. * .. Intrinsic Functions .. INTRINSIC DBLE, ICHAR, 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( N, 2, N, 2, IA, JA, DESCA, 6, INFO ) UPPER = LSAME( UPLO, 'U' ) IF( INFO.EQ.0 ) THEN NB = DESCA( NB_ ) IROFFA = MOD( IA-1, DESCA( MB_ ) ) ICOFFA = MOD( JA-1, DESCA( NB_ ) ) IAROW = INDXG2P( IA, NB, MYROW, DESCA( RSRC_ ), NPROW ) IACOL = INDXG2P( JA, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) NP = NUMROC( N, NB, MYROW, IAROW, NPROW ) NQ = MAX( 1, NUMROC( N+JA-1, NB, MYCOL, DESCA( CSRC_ ), $ NPCOL ) ) LWMIN = MAX( (NP+1)*NB, 3*NB ) * WORK( 1 ) = DBLE( LWMIN ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( IROFFA.NE.ICOFFA .OR. ICOFFA.NE.0 ) THEN INFO = -5 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN INFO = -(600+NB_) ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 END IF END IF IF( UPPER ) THEN IDUM1( 1 ) = ICHAR( 'U' ) ELSE IDUM1( 1 ) = ICHAR( 'L' ) END IF IDUM2( 1 ) = 1 IF( LWORK.EQ.-1 ) THEN IDUM1( 2 ) = -1 ELSE IDUM1( 2 ) = 1 END IF IDUM2( 2 ) = 11 CALL PCHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, 2, IDUM1, IDUM2, $ INFO ) END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PDSYTRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.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' ) * IPW = NP * NB + 1 * IF( UPPER ) THEN * * Reduce the upper triangle of sub( A ). * KK = MOD( JA+N-1, NB ) IF( KK.EQ.0 ) $ KK = NB CALL DESCSET( DESCW, N, NB, NB, NB, IAROW, INDXG2P( JA+N-KK, $ NB, MYCOL, DESCA( CSRC_ ), NPCOL ), ICTXT, $ MAX( 1, NP ) ) * DO 10 K = N-KK+1, NB+1, -NB JB = MIN( N-K+1, NB ) I = IA + K - 1 J = JA + K - 1 * * Reduce columns I:I+NB-1 to tridiagonal form and form * the matrix W which is needed to update the unreduced part of * the matrix * CALL PDLATRD( UPLO, K+JB-1, JB, A, IA, JA, DESCA, D, E, TAU, $ WORK, 1, 1, DESCW, WORK( IPW ) ) * * Update the unreduced submatrix A(IA:I-1,JA:J-1), using an * update of the form: * A(IA:I-1,JA:J-1) := A(IA:I-1,JA:J-1) - V*W' - W*V' * CALL PDSYR2K( UPLO, 'No transpose', K-1, JB, -ONE, A, IA, J, $ DESCA, WORK, 1, 1, DESCW, ONE, A, IA, JA, $ DESCA ) * * Copy last superdiagonal element back into sub( A ) * JX = MIN( INDXG2L( J, NB, 0, IACOL, NPCOL ), NQ ) CALL PDELSET( A, I-1, J, DESCA, E( JX ) ) * DESCW( CSRC_ ) = MOD( DESCW( CSRC_ ) + NPCOL - 1, NPCOL ) * 10 CONTINUE * * Use unblocked code to reduce the last or only block * CALL PDSYTD2( UPLO, MIN( N, NB ), A, IA, JA, DESCA, D, E, $ TAU, WORK, LWORK, IINFO ) * ELSE * * Reduce the lower triangle of sub( A ) * KK = MOD( JA+N-1, NB ) IF( KK.EQ.0 ) $ KK = NB CALL DESCSET( DESCW, N, NB, NB, NB, IAROW, IACOL, ICTXT, $ MAX( 1, NP ) ) * DO 20 K = 1, N-NB, NB I = IA + K - 1 J = JA + K - 1 * * Reduce columns I:I+NB-1 to tridiagonal form and form * the matrix W which is needed to update the unreduced part * of the matrix * CALL PDLATRD( UPLO, N-K+1, NB, A, I, J, DESCA, D, E, TAU, $ WORK, K, 1, DESCW, WORK( IPW ) ) * * Update the unreduced submatrix A(I+NB:IA+N-1,I+NB:IA+N-1), * using an update of the form: A(I+NB:IA+N-1,I+NB:IA+N-1) := * A(I+NB:IA+N-1,I+NB:IA+N-1) - V*W' - W*V' * CALL PDSYR2K( UPLO, 'No transpose', N-K-NB+1, NB, -ONE, A, $ I+NB, J, DESCA, WORK, K+NB, 1, DESCW, ONE, A, $ I+NB, J+NB, DESCA ) * * Copy last subdiagonal element back into sub( A ) * JX = MIN( INDXG2L( J+NB-1, NB, 0, IACOL, NPCOL ), NQ ) CALL PDELSET( A, I+NB, J+NB-1, DESCA, E( JX ) ) * DESCW( CSRC_ ) = MOD( DESCW( CSRC_ ) + 1, NPCOL ) * 20 CONTINUE * * Use unblocked code to reduce the last or only block * CALL PDSYTD2( UPLO, KK, A, IA+K-1, JA+K-1, DESCA, D, E, $ TAU, WORK, LWORK, IINFO ) END IF * CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) * WORK( 1 ) = DBLE( LWMIN ) * RETURN * * End of PDSYTRD * END