#include "ESMF_LapackBlas.inc" !> \brief \b DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DGEHD2 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehd2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgehd2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgehd2.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER IHI, ILO, INFO, LDA, N ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DGEHD2 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 <= max(1,N). !> \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). !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension (N) !> \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). !> \endverbatim !> ! ===================================================================== SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, 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, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 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 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEHD2', -INFO ) RETURN END IF ! DO 10 I = ILO, IHI - 1 ! ! Compute elementary reflector H(i) to annihilate A(i+2:ihi,i) ! CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAU( I ) ) AII = A( I+1, I ) A( I+1, I ) = ONE ! ! Apply H(i) to A(1:ihi,i+1:ihi) from the right ! CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), & A( 1, I+1 ), LDA, WORK ) ! ! Apply H(i) to A(i+1:ihi,i+1:n) from the left ! CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ), & A( I+1, I+1 ), LDA, WORK ) ! A( I+1, I ) = AII 10 CONTINUE ! RETURN ! ! End of DGEHD2 ! END