SUBROUTINE PZHETD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK,
$ LWORK, INFO )
*
* -- ScaLAPACK auxiliary 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 D( * ), E( * )
COMPLEX*16 A( * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PZHETD2 reduces a complex Hermitian matrix sub( A ) to Hermitian
* tridiagonal form T by an unitary 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
* Hermitian 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) COMPLEX*16 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
* Hermitian 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 unitary 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 unitary 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) COMPLEX*16, 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) COMPLEX*16 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 >= 3*N.
*
* 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
* ===============
*
* 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 complex scalar, and v is a complex 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 complex scalar, and v is a complex 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 ) 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 )
COMPLEX*16 HALF, ONE, ZERO
PARAMETER ( HALF = ( 0.5D+0, 0.0D+0 ),
$ ONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER IACOL, IAROW, ICOFFA, ICTXT, II, IK, IROFFA, J,
$ JJ, JK, JN, LDA, LWMIN, MYCOL, MYROW, NPCOL,
$ NPROW
COMPLEX*16 ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL BLACS_ABORT, BLACS_GRIDINFO, CHK1MAT, INFOG2L,
$ PXERBLA, ZAXPY, ZGEBR2D, ZGEBS2D,
$ ZHEMV, ZHER2, ZLARFG
* ..
* .. External Functions ..
LOGICAL LSAME
COMPLEX*16 ZDOTC
EXTERNAL LSAME, ZDOTC
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX
* ..
* .. 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
UPPER = LSAME( UPLO, 'U' )
CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, INFO )
LWMIN = 3 * N
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
LQUERY = ( LWORK.EQ.-1 )
IF( INFO.EQ.0 ) THEN
IROFFA = MOD( IA-1, DESCA( MB_ ) )
ICOFFA = MOD( JA-1, DESCA( NB_ ) )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( IROFFA.NE.ICOFFA ) 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
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PZHETD2', -INFO )
CALL BLACS_ABORT( ICTXT, 1 )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Compute local information
*
LDA = DESCA( LLD_ )
CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, II, JJ,
$ IAROW, IACOL )
*
IF( UPPER ) THEN
*
* Process(IAROW, IACOL) owns block to be reduced
*
IF( MYCOL.EQ.IACOL ) THEN
IF( MYROW.EQ.IAROW ) THEN
*
* Reduce the upper triangle of sub( A )
*
IK = II+N-1+(JJ+N-2)*LDA
A( IK ) = DBLE( A( IK ) )
DO 10 J = N-1, 1, -1
IK = II + J - 1
JK = JJ + J - 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(IA:IA+J-1,JA:JA+J-1)
*
ALPHA = A( IK+JK*LDA )
CALL ZLARFG( J, ALPHA, A( II+JK*LDA ), 1, TAUI )
E( JK+1 ) = DBLE( ALPHA )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to
* A(IA:IA+J-1,JA:JA+J-1)
*
A( IK+JK*LDA ) = ONE
*
* Compute x := tau * A * v storing x in TAU(1:i)
*
CALL ZHEMV( UPLO, J, TAUI, A( II+(JJ-1)*LDA ),
$ LDA, A( II+JK*LDA ), 1, ZERO,
$ TAU( JJ ), 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
ALPHA = -HALF*TAUI*ZDOTC( J, TAU( JJ ), 1,
$ A( II+JK*LDA ), 1 )
CALL ZAXPY( J, ALPHA, A( II+JK*LDA ), 1,
$ TAU( JJ ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL ZHER2( UPLO, J, -ONE, A( II+JK*LDA ), 1,
$ TAU( JJ ), 1, A( II+(JJ-1)*LDA ),
$ LDA )
END IF
*
* Copy D, E, TAU to broadcast them columnwise.
*
A( IK+JK*LDA ) = DCMPLX( E( JK+1 ) )
D( JK+1 ) = DBLE( A( IK+1+JK*LDA ) )
WORK( J+1 ) = DCMPLX( D( JK+1 ) )
WORK( N+J+1 ) = DCMPLX( E( JK+1 ) )
TAU( JK+1 ) = TAUI
WORK( 2*N+J+1 ) = TAU( JK+1 )
*
10 CONTINUE
D( JJ ) = DBLE( A( II+(JJ-1)*LDA ) )
WORK( 1 ) = DCMPLX( D( JJ ) )
WORK( N+1 ) = ZERO
WORK( 2*N+1 ) = ZERO
*
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1 )
*
ELSE
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1,
$ IAROW, IACOL )
DO 20 J = 2, N
JN = JJ + J - 1
D( JN ) = DBLE( WORK( J ) )
E( JN ) = DBLE( WORK( N+J ) )
TAU( JN ) = WORK( 2*N+J )
20 CONTINUE
D( JJ ) = DBLE( WORK( 1 ) )
END IF
END IF
*
ELSE
*
* Process (IAROW, IACOL) owns block to be factorized
*
IF( MYCOL.EQ.IACOL ) THEN
IF( MYROW.EQ.IAROW ) THEN
*
* Reduce the lower triangle of sub( A )
*
A( II+(JJ-1)*LDA ) = DBLE( A( II+(JJ-1)*LDA ) )
DO 30 J = 1, N - 1
IK = II + J - 1
JK = JJ + J - 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(IA+J-JA+2:IA+N-1,JA+J-1)
*
ALPHA = A( IK+1+(JK-1)*LDA )
CALL ZLARFG( N-J, ALPHA, A( IK+2+(JK-1)*LDA ), 1,
$ TAUI )
E( JK ) = DBLE( ALPHA )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to
* A(IA+J-JA+1:IA+N-1,JA+J+1:JA+N-1)
*
A( IK+1+(JK-1)*LDA ) = ONE
*
* Compute x := tau * A * v storing y in TAU(i:n-1)
*
CALL ZHEMV( UPLO, N-J, TAUI, A( IK+1+JK*LDA ),
$ LDA, A( IK+1+(JK-1)*LDA ), 1,
$ ZERO, TAU( JK ), 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
ALPHA = -HALF*TAUI*ZDOTC( N-J, TAU( JK ), 1,
$ A( IK+1+(JK-1)*LDA ), 1 )
CALL ZAXPY( N-J, ALPHA, A( IK+1+(JK-1)*LDA ),
$ 1, TAU( JK ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL ZHER2( UPLO, N-J, -ONE,
$ A( IK+1+(JK-1)*LDA ), 1,
$ TAU( JK ), 1, A( IK+1+JK*LDA ),
$ LDA )
END IF
*
* Copy D(JK), E(JK), TAU(JK) to broadcast them
* columnwise.
*
A( IK+1+(JK-1)*LDA ) = DCMPLX( E( JK ) )
D( JK ) = DBLE( A( IK+(JK-1)*LDA ) )
WORK( J ) = DCMPLX( D( JK ) )
WORK( N+J ) = DCMPLX( E( JK ) )
TAU( JK ) = TAUI
WORK( 2*N+J ) = TAU( JK )
30 CONTINUE
JN = JJ + N - 1
D( JN ) = DBLE( A( II+N-1+(JN-1)*LDA ) )
WORK( N ) = DCMPLX( D( JN ) )
TAU( JN ) = ZERO
WORK( 2*N ) = ZERO
*
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK,
$ 1 )
*
ELSE
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK,
$ 1, IAROW, IACOL )
DO 40 J = 1, N - 1
JN = JJ + J - 1
D( JN ) = DBLE( WORK( J ) )
E( JN ) = DBLE( WORK( N+J ) )
TAU( JN ) = WORK( 2*N+J )
40 CONTINUE
JN = JJ + N - 1
D( JN ) = DBLE( WORK( N ) )
TAU( JN ) = ZERO
END IF
END IF
END IF
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
*
RETURN
*
* End of PZHETD2
*
END