C DGEHRD SOURCE BP208322 20/09/18 21:15:50 10718 *> \brief \b DGEHRD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGEHRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, ILO, INFO, LDA, LWORK, N * .. * .. Array Arguments .. * REAL*8 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 REAL*8 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 REAL*8 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 REAL*8 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 IMPLICIT INTEGER(I-N) IMPLICIT REAL*8(A-H,O-Z) * * .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N * .. * .. Array Arguments .. REAL*8 A( LDA, * ), TAU( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER NBMAX, LDT, TSIZE PARAMETER ( NBMAX = 64, LDT = NBMAX+1, \$ TSIZE = LDT*NBMAX ) REAL*8 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 REAL*8 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