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dsytrd_sb2st.F
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*> \brief \b DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYTRD_SB2ST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrd_sb2st.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrd_sb2st.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd_sb2st.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYTRD_SB2ST( STAGE1, VECT, UPLO, N, KD, AB, LDAB,
* D, E, HOUS, LHOUS, WORK, LWORK, INFO )
*
* #if defined(_OPENMP)
* use omp_lib
* #endif
*
* IMPLICIT NONE
*
* .. Scalar Arguments ..
* CHARACTER STAGE1, UPLO, VECT
* INTEGER N, KD, IB, LDAB, LHOUS, LWORK, INFO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* DOUBLE PRECISION AB( LDAB, * ), HOUS( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
*> tridiagonal form T by a orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] STAGE1
*> \verbatim
*> STAGE1 is CHARACTER*1
*> = 'N': "No": to mention that the stage 1 of the reduction
*> from dense to band using the dsytrd_sy2sb routine
*> was not called before this routine to reproduce AB.
*> In other term this routine is called as standalone.
*> = 'Y': "Yes": to mention that the stage 1 of the
*> reduction from dense to band using the dsytrd_sy2sb
*> routine has been called to produce AB (e.g., AB is
*> the output of dsytrd_sy2sb.
*> \endverbatim
*>
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'N': No need for the Housholder representation,
*> and thus LHOUS is of size max(1, 4*N);
*> = 'V': the Householder representation is needed to
*> either generate or to apply Q later on,
*> then LHOUS is to be queried and computed.
*> (NOT AVAILABLE IN THIS RELEASE).
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> On exit, the diagonal elements of AB are overwritten by the
*> diagonal elements of the tridiagonal matrix T; if KD > 0, the
*> elements on the first superdiagonal (if UPLO = 'U') or the
*> first subdiagonal (if UPLO = 'L') are overwritten by the
*> off-diagonal elements of T; the rest of AB is overwritten by
*> values generated during the reduction.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] HOUS
*> \verbatim
*> HOUS is DOUBLE PRECISION array, dimension LHOUS, that
*> store the Householder representation.
*> \endverbatim
*>
*> \param[in] LHOUS
*> \verbatim
*> LHOUS is INTEGER
*> The dimension of the array HOUS. LHOUS = MAX(1, dimension)
*> If LWORK = -1, or LHOUS=-1,
*> then a query is assumed; the routine
*> only calculates the optimal size of the HOUS array, returns
*> this value as the first entry of the HOUS array, and no error
*> message related to LHOUS is issued by XERBLA.
*> LHOUS = MAX(1, dimension) where
*> dimension = 4*N if VECT='N'
*> not available now if VECT='H'
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK = MAX(1, dimension)
*> If LWORK = -1, or LHOUS=-1,
*> then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> LWORK = MAX(1, dimension) where
*> dimension = (2KD+1)*N + KD*NTHREADS
*> where KD is the blocking size of the reduction,
*> FACTOPTNB is the blocking used by the QR or LQ
*> algorithm, usually FACTOPTNB=128 is a good choice
*> NTHREADS is the number of threads used when
*> openMP compilation is enabled, otherwise =1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2017
*
*> \ingroup real16OTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Implemented by Azzam Haidar.
*>
*> All details are available on technical report, SC11, SC13 papers.
*>
*> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
*> Parallel reduction to condensed forms for symmetric eigenvalue problems
*> using aggregated fine-grained and memory-aware kernels. In Proceedings
*> of 2011 International Conference for High Performance Computing,
*> Networking, Storage and Analysis (SC '11), New York, NY, USA,
*> Article 8 , 11 pages.
*> http://doi.acm.org/10.1145/2063384.2063394
*>
*> A. Haidar, J. Kurzak, P. Luszczek, 2013.
*> An improved parallel singular value algorithm and its implementation
*> for multicore hardware, In Proceedings of 2013 International Conference
*> for High Performance Computing, Networking, Storage and Analysis (SC '13).
*> Denver, Colorado, USA, 2013.
*> Article 90, 12 pages.
*> http://doi.acm.org/10.1145/2503210.2503292
*>
*> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
*> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
*> calculations based on fine-grained memory aware tasks.
*> International Journal of High Performance Computing Applications.
*> Volume 28 Issue 2, Pages 196-209, May 2014.
*> http://hpc.sagepub.com/content/28/2/196
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSYTRD_SB2ST( STAGE1, VECT, UPLO, N, KD, AB, LDAB,
$ D, E, HOUS, LHOUS, WORK, LWORK, INFO )
*
#if defined(_OPENMP)
use omp_lib
#endif
*
IMPLICIT NONE
*
* -- LAPACK computational routine (version 3.8.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2017
*
* .. Scalar Arguments ..
CHARACTER STAGE1, UPLO, VECT
INTEGER N, KD, LDAB, LHOUS, LWORK, INFO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
DOUBLE PRECISION AB( LDAB, * ), HOUS( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION RZERO
DOUBLE PRECISION ZERO, ONE
PARAMETER ( RZERO = 0.0D+0,
$ ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WANTQ, UPPER, AFTERS1
INTEGER I, M, K, IB, SWEEPID, MYID, SHIFT, STT, ST,
$ ED, STIND, EDIND, BLKLASTIND, COLPT, THED,
$ STEPERCOL, GRSIZ, THGRSIZ, THGRNB, THGRID,
$ NBTILES, TTYPE, TID, NTHREADS, DEBUG,
$ ABDPOS, ABOFDPOS, DPOS, OFDPOS, AWPOS,
$ INDA, INDW, APOS, SIZEA, LDA, INDV, INDTAU,
$ SIDEV, SIZETAU, LDV, LHMIN, LWMIN
* ..
* .. External Subroutines ..
EXTERNAL DSB2ST_KERNELS, DLACPY, DLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN, MAX, CEILING, REAL
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV2STAGE
EXTERNAL LSAME, ILAENV2STAGE
* ..
* .. Executable Statements ..
*
* Determine the minimal workspace size required.
* Test the input parameters
*
DEBUG = 0
INFO = 0
AFTERS1 = LSAME( STAGE1, 'Y' )
WANTQ = LSAME( VECT, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 ) .OR. ( LHOUS.EQ.-1 )
*
* Determine the block size, the workspace size and the hous size.
*
IB = ILAENV2STAGE( 2, 'DSYTRD_SB2ST', VECT, N, KD, -1, -1 )
LHMIN = ILAENV2STAGE( 3, 'DSYTRD_SB2ST', VECT, N, KD, IB, -1 )
LWMIN = ILAENV2STAGE( 4, 'DSYTRD_SB2ST', VECT, N, KD, IB, -1 )
*
IF( .NOT.AFTERS1 .AND. .NOT.LSAME( STAGE1, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( VECT, 'N' ) ) THEN
INFO = -2
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KD.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.(KD+1) ) THEN
INFO = -7
ELSE IF( LHOUS.LT.LHMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.EQ.0 ) THEN
HOUS( 1 ) = LHMIN
WORK( 1 ) = LWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRD_SB2ST', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
HOUS( 1 ) = 1
WORK( 1 ) = 1
RETURN
END IF
*
* Determine pointer position
*
LDV = KD + IB
SIZETAU = 2 * N
SIDEV = 2 * N
INDTAU = 1
INDV = INDTAU + SIZETAU
LDA = 2 * KD + 1
SIZEA = LDA * N
INDA = 1
INDW = INDA + SIZEA
NTHREADS = 1
TID = 0
*
IF( UPPER ) THEN
APOS = INDA + KD
AWPOS = INDA
DPOS = APOS + KD
OFDPOS = DPOS - 1
ABDPOS = KD + 1
ABOFDPOS = KD
ELSE
APOS = INDA
AWPOS = INDA + KD + 1
DPOS = APOS
OFDPOS = DPOS + 1
ABDPOS = 1
ABOFDPOS = 2
ENDIF
*
* Case KD=0:
* The matrix is diagonal. We just copy it (convert to "real" for
* real because D is double and the imaginary part should be 0)
* and store it in D. A sequential code here is better or
* in a parallel environment it might need two cores for D and E
*
IF( KD.EQ.0 ) THEN
DO 30 I = 1, N
D( I ) = ( AB( ABDPOS, I ) )
30 CONTINUE
DO 40 I = 1, N-1
E( I ) = RZERO
40 CONTINUE
*
HOUS( 1 ) = 1
WORK( 1 ) = 1
RETURN
END IF
*
* Case KD=1:
* The matrix is already Tridiagonal. We have to make diagonal
* and offdiagonal elements real, and store them in D and E.
* For that, for real precision just copy the diag and offdiag
* to D and E while for the COMPLEX case the bulge chasing is
* performed to convert the hermetian tridiagonal to symmetric
* tridiagonal. A simpler coversion formula might be used, but then
* updating the Q matrix will be required and based if Q is generated
* or not this might complicate the story.
*
IF( KD.EQ.1 ) THEN
DO 50 I = 1, N
D( I ) = ( AB( ABDPOS, I ) )
50 CONTINUE
*
IF( UPPER ) THEN
DO 60 I = 1, N-1
E( I ) = ( AB( ABOFDPOS, I+1 ) )
60 CONTINUE
ELSE
DO 70 I = 1, N-1
E( I ) = ( AB( ABOFDPOS, I ) )
70 CONTINUE
ENDIF
*
HOUS( 1 ) = 1
WORK( 1 ) = 1
RETURN
END IF
*
* Main code start here.
* Reduce the symmetric band of A to a tridiagonal matrix.
*
THGRSIZ = N
GRSIZ = 1
SHIFT = 3
NBTILES = CEILING( REAL(N)/REAL(KD) )
STEPERCOL = CEILING( REAL(SHIFT)/REAL(GRSIZ) )
THGRNB = CEILING( REAL(N-1)/REAL(THGRSIZ) )
*
CALL DLACPY( "A", KD+1, N, AB, LDAB, WORK( APOS ), LDA )
CALL DLASET( "A", KD, N, ZERO, ZERO, WORK( AWPOS ), LDA )
*
*
* openMP parallelisation start here
*
#if defined(_OPENMP)
!$OMP PARALLEL PRIVATE( TID, THGRID, BLKLASTIND )
!$OMP$ PRIVATE( THED, I, M, K, ST, ED, STT, SWEEPID )
!$OMP$ PRIVATE( MYID, TTYPE, COLPT, STIND, EDIND )
!$OMP$ SHARED ( UPLO, WANTQ, INDV, INDTAU, HOUS, WORK)
!$OMP$ SHARED ( N, KD, IB, NBTILES, LDA, LDV, INDA )
!$OMP$ SHARED ( STEPERCOL, THGRNB, THGRSIZ, GRSIZ, SHIFT )
!$OMP MASTER
#endif
*
* main bulge chasing loop
*
DO 100 THGRID = 1, THGRNB
STT = (THGRID-1)*THGRSIZ+1
THED = MIN( (STT + THGRSIZ -1), (N-1))
DO 110 I = STT, N-1
ED = MIN( I, THED )
IF( STT.GT.ED ) EXIT
DO 120 M = 1, STEPERCOL
ST = STT
DO 130 SWEEPID = ST, ED
DO 140 K = 1, GRSIZ
MYID = (I-SWEEPID)*(STEPERCOL*GRSIZ)
$ + (M-1)*GRSIZ + K
IF ( MYID.EQ.1 ) THEN
TTYPE = 1
ELSE
TTYPE = MOD( MYID, 2 ) + 2
ENDIF
IF( TTYPE.EQ.2 ) THEN
COLPT = (MYID/2)*KD + SWEEPID
STIND = COLPT-KD+1
EDIND = MIN(COLPT,N)
BLKLASTIND = COLPT
ELSE
COLPT = ((MYID+1)/2)*KD + SWEEPID
STIND = COLPT-KD+1
EDIND = MIN(COLPT,N)
IF( ( STIND.GE.EDIND-1 ).AND.
$ ( EDIND.EQ.N ) ) THEN
BLKLASTIND = N
ELSE
BLKLASTIND = 0
ENDIF
ENDIF
*
* Call the kernel
*
#if defined(_OPENMP)
IF( TTYPE.NE.1 ) THEN
!$OMP TASK DEPEND(in:WORK(MYID+SHIFT-1))
!$OMP$ DEPEND(in:WORK(MYID-1))
!$OMP$ DEPEND(out:WORK(MYID))
TID = OMP_GET_THREAD_NUM()
CALL DSB2ST_KERNELS( UPLO, WANTQ, TTYPE,
$ STIND, EDIND, SWEEPID, N, KD, IB,
$ WORK ( INDA ), LDA,
$ HOUS( INDV ), HOUS( INDTAU ), LDV,
$ WORK( INDW + TID*KD ) )
!$OMP END TASK
ELSE
!$OMP TASK DEPEND(in:WORK(MYID+SHIFT-1))
!$OMP$ DEPEND(out:WORK(MYID))
TID = OMP_GET_THREAD_NUM()
CALL DSB2ST_KERNELS( UPLO, WANTQ, TTYPE,
$ STIND, EDIND, SWEEPID, N, KD, IB,
$ WORK ( INDA ), LDA,
$ HOUS( INDV ), HOUS( INDTAU ), LDV,
$ WORK( INDW + TID*KD ) )
!$OMP END TASK
ENDIF
#else
CALL DSB2ST_KERNELS( UPLO, WANTQ, TTYPE,
$ STIND, EDIND, SWEEPID, N, KD, IB,
$ WORK ( INDA ), LDA,
$ HOUS( INDV ), HOUS( INDTAU ), LDV,
$ WORK( INDW + TID*KD ) )
#endif
IF ( BLKLASTIND.GE.(N-1) ) THEN
STT = STT + 1
EXIT
ENDIF
140 CONTINUE
130 CONTINUE
120 CONTINUE
110 CONTINUE
100 CONTINUE
*
#if defined(_OPENMP)
!$OMP END MASTER
!$OMP END PARALLEL
#endif
*
* Copy the diagonal from A to D. Note that D is REAL thus only
* the Real part is needed, the imaginary part should be zero.
*
DO 150 I = 1, N
D( I ) = ( WORK( DPOS+(I-1)*LDA ) )
150 CONTINUE
*
* Copy the off diagonal from A to E. Note that E is REAL thus only
* the Real part is needed, the imaginary part should be zero.
*
IF( UPPER ) THEN
DO 160 I = 1, N-1
E( I ) = ( WORK( OFDPOS+I*LDA ) )
160 CONTINUE
ELSE
DO 170 I = 1, N-1
E( I ) = ( WORK( OFDPOS+(I-1)*LDA ) )
170 CONTINUE
ENDIF
*
HOUS( 1 ) = LHMIN
WORK( 1 ) = LWMIN
RETURN
*
* End of DSYTRD_SB2ST
*
END