*> \brief \b DSYTRD_SY2SB * * @generated from zhetrd_he2hb.f, fortran z -> d, Wed Dec 7 08:22:39 2016 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSYTRD_SY2SB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSYTRD_SY2SB( UPLO, N, KD, A, LDA, AB, LDAB, TAU, * WORK, LWORK, INFO ) * * IMPLICIT NONE * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, LDAB, LWORK, N, KD * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), AB( LDAB, * ), * TAU( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric *> band-diagonal form AB by a orthogonal similarity transformation: *> Q**T * A * Q = AB. *> \endverbatim * * Arguments: * ========== * *> \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 reduced matrix if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. *> The reduced matrix is stored in the array AB. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA,N) *> On entry, the symmetric matrix A. If UPLO = 'U', the leading *> N-by-N upper triangular part of A contains the upper *> triangular part of the matrix A, and the strictly lower *> triangular part of A is not referenced. If UPLO = 'L', the *> leading N-by-N lower triangular part of A contains the lower *> triangular part of the matrix A, and the strictly upper *> triangular part of A is not referenced. *> On exit, if UPLO = 'U', the diagonal and first superdiagonal *> of A are overwritten 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 A are over- *> written 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] AB *> \verbatim *> AB is DOUBLE PRECISION array, dimension (LDAB,N) *> On exit, 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). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KD+1. *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is DOUBLE PRECISION array, dimension (N-KD) *> The scalar factors of the elementary reflectors (see Further *> Details). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (LWORK) *> On exit, if INFO = 0, or if LWORK=-1, *> WORK(1) returns the size of LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK which should be calculated *> by a workspace query. LWORK = MAX(1, LWORK_QUERY) *> If LWORK = -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_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD *> where FACTOPTNB is the blocking used by the QR or LQ *> algorithm, usually FACTOPTNB=128 is a good choice otherwise *> putting LWORK=-1 will provide the size of WORK. *> \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 June 2017 * *> \ingroup doubleSYcomputational * *> \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 *> *> \verbatim *> *> If UPLO = 'U', the matrix Q is represented as a product of elementary *> reflectors *> *> Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd. *> *> Each H(i) has the form *> *> H(i) = I - tau * v * v**T *> *> where tau is a real scalar, and v is a real vector with *> v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in *> A(i,i+kd+1:n), and tau in TAU(i). *> *> If UPLO = 'L', the matrix Q is represented as a product of elementary *> reflectors *> *> Q = H(1) H(2) . . . H(k), where k = n-kd. *> *> Each H(i) has the form *> *> H(i) = I - tau * v * v**T *> *> where tau is a real scalar, and v is a real vector with *> v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in *> A(i+kd+2:n,i), and tau in TAU(i). *> *> The contents of A on exit are illustrated by the following examples *> with n = 5: *> *> if UPLO = 'U': if UPLO = 'L': *> *> ( ab ab/v1 v1 v1 v1 ) ( ab ) *> ( ab ab/v2 v2 v2 ) ( ab/v1 ab ) *> ( ab ab/v3 v3 ) ( v1 ab/v2 ab ) *> ( ab ab/v4 ) ( v1 v2 ab/v3 ab ) *> ( ab ) ( v1 v2 v3 ab/v4 ab ) *> *> where d and e denote diagonal and off-diagonal elements of T, and vi *> denotes an element of the vector defining H(i). *> \endverbatim *> * ===================================================================== SUBROUTINE DSYTRD_SY2SB( UPLO, N, KD, A, LDA, AB, LDAB, TAU, \$ WORK, LWORK, INFO ) * IMPLICIT NONE * * -- LAPACK computational routine (version 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2017 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDAB, LWORK, N, KD * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), AB( LDAB, * ), \$ TAU( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION RONE DOUBLE PRECISION ZERO, ONE, HALF PARAMETER ( RONE = 1.0D+0, \$ ZERO = 0.0D+0, \$ ONE = 1.0D+0, \$ HALF = 0.5D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER I, J, IINFO, LWMIN, PN, PK, LK, \$ LDT, LDW, LDS2, LDS1, \$ LS2, LS1, LW, LT, \$ TPOS, WPOS, S2POS, S1POS * .. * .. External Subroutines .. EXTERNAL XERBLA, DSYR2K, DSYMM, DGEMM, \$ DLARFT, DGELQF, DGEQRF, DLASET * .. * .. Intrinsic Functions .. INTRINSIC MIN, MAX * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV * .. * .. Executable Statements .. * * Determine the minimal workspace size required * and test the input parameters * INFO = 0 UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) LWMIN = ILAENV( 20, 'DSYTRD_SY2SB', '', N, KD, -1, -1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDAB.LT.MAX( 1, KD+1 ) ) THEN INFO = -7 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRD_SY2SB', -INFO ) RETURN ELSE IF( LQUERY ) THEN WORK( 1 ) = LWMIN RETURN END IF * * Quick return if possible * Copy the upper/lower portion of A into AB * IF( N.LE.KD+1 ) THEN IF( UPPER ) THEN DO 100 I = 1, N LK = MIN( KD+1, I ) CALL DCOPY( LK, A( I-LK+1, I ), 1, \$ AB( KD+1-LK+1, I ), 1 ) 100 CONTINUE ELSE DO 110 I = 1, N LK = MIN( KD+1, N-I+1 ) CALL DCOPY( LK, A( I, I ), 1, AB( 1, I ), 1 ) 110 CONTINUE ENDIF WORK( 1 ) = 1 RETURN END IF * * Determine the pointer position for the workspace * LDT = KD LDS1 = KD LT = LDT*KD LW = N*KD LS1 = LDS1*KD LS2 = LWMIN - LT - LW - LS1 * LS2 = N*MAX(KD,FACTOPTNB) TPOS = 1 WPOS = TPOS + LT S1POS = WPOS + LW S2POS = S1POS + LS1 IF( UPPER ) THEN LDW = KD LDS2 = KD ELSE LDW = N LDS2 = N ENDIF * * * Set the workspace of the triangular matrix T to zero once such a * way everytime T is generated the upper/lower portion will be always zero * CALL DLASET( "A", LDT, KD, ZERO, ZERO, WORK( TPOS ), LDT ) * IF( UPPER ) THEN DO 10 I = 1, N - KD, KD PN = N-I-KD+1 PK = MIN( N-I-KD+1, KD ) * * Compute the LQ factorization of the current block * CALL DGELQF( KD, PN, A( I, I+KD ), LDA, \$ TAU( I ), WORK( S2POS ), LS2, IINFO ) * * Copy the upper portion of A into AB * DO 20 J = I, I+PK-1 LK = MIN( KD, N-J ) + 1 CALL DCOPY( LK, A( J, J ), LDA, AB( KD+1, J ), LDAB-1 ) 20 CONTINUE * CALL DLASET( 'Lower', PK, PK, ZERO, ONE, \$ A( I, I+KD ), LDA ) * * Form the matrix T * CALL DLARFT( 'Forward', 'Rowwise', PN, PK, \$ A( I, I+KD ), LDA, TAU( I ), \$ WORK( TPOS ), LDT ) * * Compute W: * CALL DGEMM( 'Conjugate', 'No transpose', PK, PN, PK, \$ ONE, WORK( TPOS ), LDT, \$ A( I, I+KD ), LDA, \$ ZERO, WORK( S2POS ), LDS2 ) * CALL DSYMM( 'Right', UPLO, PK, PN, \$ ONE, A( I+KD, I+KD ), LDA, \$ WORK( S2POS ), LDS2, \$ ZERO, WORK( WPOS ), LDW ) * CALL DGEMM( 'No transpose', 'Conjugate', PK, PK, PN, \$ ONE, WORK( WPOS ), LDW, \$ WORK( S2POS ), LDS2, \$ ZERO, WORK( S1POS ), LDS1 ) * CALL DGEMM( 'No transpose', 'No transpose', PK, PN, PK, \$ -HALF, WORK( S1POS ), LDS1, \$ A( I, I+KD ), LDA, \$ ONE, WORK( WPOS ), LDW ) * * * Update the unreduced submatrix A(i+kd:n,i+kd:n), using * an update of the form: A := A - V'*W - W'*V * CALL DSYR2K( UPLO, 'Conjugate', PN, PK, \$ -ONE, A( I, I+KD ), LDA, \$ WORK( WPOS ), LDW, \$ RONE, A( I+KD, I+KD ), LDA ) 10 CONTINUE * * Copy the upper band to AB which is the band storage matrix * DO 30 J = N-KD+1, N LK = MIN(KD, N-J) + 1 CALL DCOPY( LK, A( J, J ), LDA, AB( KD+1, J ), LDAB-1 ) 30 CONTINUE * ELSE * * Reduce the lower triangle of A to lower band matrix * DO 40 I = 1, N - KD, KD PN = N-I-KD+1 PK = MIN( N-I-KD+1, KD ) * * Compute the QR factorization of the current block * CALL DGEQRF( PN, KD, A( I+KD, I ), LDA, \$ TAU( I ), WORK( S2POS ), LS2, IINFO ) * * Copy the upper portion of A into AB * DO 50 J = I, I+PK-1 LK = MIN( KD, N-J ) + 1 CALL DCOPY( LK, A( J, J ), 1, AB( 1, J ), 1 ) 50 CONTINUE * CALL DLASET( 'Upper', PK, PK, ZERO, ONE, \$ A( I+KD, I ), LDA ) * * Form the matrix T * CALL DLARFT( 'Forward', 'Columnwise', PN, PK, \$ A( I+KD, I ), LDA, TAU( I ), \$ WORK( TPOS ), LDT ) * * Compute W: * CALL DGEMM( 'No transpose', 'No transpose', PN, PK, PK, \$ ONE, A( I+KD, I ), LDA, \$ WORK( TPOS ), LDT, \$ ZERO, WORK( S2POS ), LDS2 ) * CALL DSYMM( 'Left', UPLO, PN, PK, \$ ONE, A( I+KD, I+KD ), LDA, \$ WORK( S2POS ), LDS2, \$ ZERO, WORK( WPOS ), LDW ) * CALL DGEMM( 'Conjugate', 'No transpose', PK, PK, PN, \$ ONE, WORK( S2POS ), LDS2, \$ WORK( WPOS ), LDW, \$ ZERO, WORK( S1POS ), LDS1 ) * CALL DGEMM( 'No transpose', 'No transpose', PN, PK, PK, \$ -HALF, A( I+KD, I ), LDA, \$ WORK( S1POS ), LDS1, \$ ONE, WORK( WPOS ), LDW ) * * * Update the unreduced submatrix A(i+kd:n,i+kd:n), using * an update of the form: A := A - V*W' - W*V' * CALL DSYR2K( UPLO, 'No transpose', PN, PK, \$ -ONE, A( I+KD, I ), LDA, \$ WORK( WPOS ), LDW, \$ RONE, A( I+KD, I+KD ), LDA ) * ================================================================== * RESTORE A FOR COMPARISON AND CHECKING TO BE REMOVED * DO 45 J = I, I+PK-1 * LK = MIN( KD, N-J ) + 1 * CALL DCOPY( LK, AB( 1, J ), 1, A( J, J ), 1 ) * 45 CONTINUE * ================================================================== 40 CONTINUE * * Copy the lower band to AB which is the band storage matrix * DO 60 J = N-KD+1, N LK = MIN(KD, N-J) + 1 CALL DCOPY( LK, A( J, J ), 1, AB( 1, J ), 1 ) 60 CONTINUE END IF * WORK( 1 ) = LWMIN RETURN * * End of DSYTRD_SY2SB * END