#include "ESMF_LapackBlas.inc" !> \brief \b DGEHRD ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DGEHRD + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehrd.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehrd.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehrd.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DGEHRD reduces a real general matrix A to upper Hessenberg form H by !> an orthogonal similarity transformation: Q**T * A * Q = H . !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] N !> \verbatim !> N is INTEGER !> The order of the matrix A. N >= 0. !> \endverbatim !> !> \param[in] ILO !> \verbatim !> ILO is INTEGER !> \endverbatim !> !> \param[in] IHI !> \verbatim !> IHI is INTEGER !> !> It is assumed that A is already upper triangular in rows !> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally !> set by a previous call to DGEBAL; otherwise they should be !> set to 1 and N respectively. See Further Details. !> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. !> \endverbatim !> !> \param[in,out] A !> \verbatim !> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the N-by-N general matrix to be reduced. !> On exit, the upper triangle and the first subdiagonal of A !> are overwritten with the upper Hessenberg matrix H, 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] TAU !> \verbatim !> TAU is DOUBLE PRECISION array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to !> zero. !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension (LWORK) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> \endverbatim !> !> \param[in] LWORK !> \verbatim !> LWORK is INTEGER !> The length of the array WORK. LWORK >= max(1,N). !> For good performance, LWORK should generally be larger. !> !> 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. !> \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 December 2016 ! !> \ingroup doubleGEcomputational ! !> \par Further Details: ! ===================== !> !> \verbatim !> !> The matrix Q is represented as a product of (ihi-ilo) elementary !> reflectors !> !> Q = H(ilo) H(ilo+1) . . . H(ihi-1). !> !> 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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on !> exit in A(i+2:ihi,i), and tau in TAU(i). !> !> The contents of A are illustrated by the following example, with !> n = 7, ilo = 2 and ihi = 6: !> !> on entry, on exit, !> !> ( a a a a a a a ) ( a a h h h h a ) !> ( a a a a a a ) ( a h h h h a ) !> ( a a a a a a ) ( h h h h h h ) !> ( a a a a a a ) ( v2 h h h h h ) !> ( a a a a a a ) ( v2 v3 h h h h ) !> ( a a a a a a ) ( v2 v3 v4 h h h ) !> ( a ) ( a ) !> !> where a denotes an element of the original matrix A, h denotes a !> modified element of the upper Hessenberg matrix H, and vi denotes an !> element of the vector defining H(i). !> !> This file is a slight modification of LAPACK-3.0's DGEHRD !> subroutine incorporating improvements proposed by Quintana-Orti and !> Van de Geijn (2006). (See DLAHR2.) !> \endverbatim !> ! ===================================================================== SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK computational routine (version 3.7.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! December 2016 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT, TSIZE PARAMETER ( NBMAX = 64, LDT = NBMAX+1, & TSIZE = LDT*NBMAX ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, & ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB, & NBMIN, NH, NX DOUBLE PRECISION EI ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF ! IF( INFO.EQ.0 ) THEN ! ! Compute the workspace requirements ! NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) LWKOPT = N*NB + TSIZE WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEHRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Set elements 1:ILO-1 and IHI:N-1 of TAU to zero ! DO 10 I = 1, ILO - 1 TAU( I ) = ZERO 10 CONTINUE DO 20 I = MAX( 1, IHI ), N - 1 TAU( I ) = ZERO 20 CONTINUE ! ! Quick return if possible ! NH = IHI - ILO + 1 IF( NH.LE.1 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Determine the block size ! NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) NBMIN = 2 IF( NB.GT.1 .AND. NB.LT.NH ) THEN ! ! Determine when to cross over from blocked to unblocked code ! (last block is always handled by unblocked code) ! NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) IF( NX.LT.NH ) THEN ! ! Determine if workspace is large enough for blocked code ! IF( LWORK.LT.N*NB+TSIZE ) THEN ! ! Not enough workspace to use optimal NB: determine the ! minimum value of NB, and reduce NB or force use of ! unblocked code ! NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI, & -1 ) ) IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN NB = (LWORK-TSIZE) / N ELSE NB = 1 END IF END IF END IF END IF LDWORK = N ! IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN ! ! Use unblocked code below ! I = ILO ! ELSE ! ! Use blocked code ! IWT = 1 + N*NB DO 40 I = ILO, IHI - 1 - NX, NB IB = MIN( NB, IHI-I ) ! ! Reduce columns i:i+ib-1 to Hessenberg form, returning the ! matrices V and T of the block reflector H = I - V*T*V**T ! which performs the reduction, and also the matrix Y = A*V*T ! CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), & WORK( IWT ), LDT, WORK, LDWORK ) ! ! Apply the block reflector H to A(1:ihi,i+ib:ihi) from the ! right, computing A := A - Y * V**T. V(i+ib,ib-1) must be set ! to 1 ! EI = A( I+IB, I+IB-1 ) A( I+IB, I+IB-1 ) = ONE CALL DGEMM( 'No transpose', 'Transpose', & IHI, IHI-I-IB+1, & IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE, & A( 1, I+IB ), LDA ) A( I+IB, I+IB-1 ) = EI ! ! Apply the block reflector H to A(1:i,i+1:i+ib-1) from the ! right ! CALL DTRMM( 'Right', 'Lower', 'Transpose', & 'Unit', I, IB-1, & ONE, A( I+1, I ), LDA, WORK, LDWORK ) DO 30 J = 0, IB-2 CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1, & A( 1, I+J+1 ), 1 ) 30 CONTINUE ! ! Apply the block reflector H to A(i+1:ihi,i+ib:n) from the ! left ! CALL DLARFB( 'Left', 'Transpose', 'Forward', & 'Columnwise', & IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, & WORK( IWT ), LDT, A( I+1, I+IB ), LDA, & WORK, LDWORK ) 40 CONTINUE END IF ! ! Use unblocked code to reduce the rest of the matrix ! CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) WORK( 1 ) = LWKOPT ! RETURN ! ! End of DGEHRD ! END