SUBROUTINE PDSYTRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK,
$ LWORK, INFO )
*
* -- ScaLAPACK routine (version 1.5) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. 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, PTOPGET,
$ PTOPSET, 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 PTOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' )
CALL PTOPSET( 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 PTOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
*
WORK( 1 ) = DBLE( LWMIN )
*
RETURN
*
* End of PDSYTRD
*
END