353 lines
11 KiB
Fortran
353 lines
11 KiB
Fortran
*> \brief \b CGEHRD
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download CGEHRD + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgehrd.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgehrd.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgehrd.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER IHI, ILO, INFO, LDA, LWORK, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* COMPLEX A( LDA, * ), TAU( * ), WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
|
|
*> an unitary similarity transformation: Q**H * 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 CGEBAL; 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 COMPLEX 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 unitary 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 COMPLEX 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 COMPLEX 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 optimum performance LWORK >= N*NB, where NB is the
|
|
*> optimal blocksize.
|
|
*>
|
|
*> 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 November 2011
|
|
*
|
|
*> \ingroup complexGEcomputational
|
|
*
|
|
*> \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**H
|
|
*>
|
|
*> where tau is a complex scalar, and v is a complex 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 CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
|
|
*
|
|
* -- LAPACK computational routine (version 3.4.0) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* November 2011
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER IHI, ILO, INFO, LDA, LWORK, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
INTEGER NBMAX, LDT
|
|
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
|
|
COMPLEX ZERO, ONE
|
|
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ),
|
|
$ ONE = ( 1.0E+0, 0.0E+0 ) )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY
|
|
INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
|
|
$ NBMIN, NH, NX
|
|
COMPLEX EI
|
|
* ..
|
|
* .. Local Arrays ..
|
|
COMPLEX T( LDT, NBMAX )
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL CAXPY, CGEHD2, CGEMM, CLAHR2, CLARFB, CTRMM,
|
|
$ XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER ILAENV
|
|
EXTERNAL ILAENV
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input parameters
|
|
*
|
|
INFO = 0
|
|
NB = MIN( NBMAX, ILAENV( 1, 'CGEHRD', ' ', N, ILO, IHI, -1 ) )
|
|
LWKOPT = N*NB
|
|
WORK( 1 ) = LWKOPT
|
|
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.NE.0 ) THEN
|
|
CALL XERBLA( 'CGEHRD', -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, 'CGEHRD', ' ', N, ILO, IHI, -1 ) )
|
|
NBMIN = 2
|
|
IWS = 1
|
|
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, 'CGEHRD', ' ', N, ILO, IHI, -1 ) )
|
|
IF( NX.LT.NH ) THEN
|
|
*
|
|
* Determine if workspace is large enough for blocked code
|
|
*
|
|
IWS = N*NB
|
|
IF( LWORK.LT.IWS ) 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, 'CGEHRD', ' ', N, ILO, IHI,
|
|
$ -1 ) )
|
|
IF( LWORK.GE.N*NBMIN ) THEN
|
|
NB = LWORK / 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
|
|
*
|
|
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**H
|
|
* which performs the reduction, and also the matrix Y = A*V*T
|
|
*
|
|
CALL CLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
|
|
$ WORK, LDWORK )
|
|
*
|
|
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
|
|
* right, computing A := A - Y * V**H. 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 CGEMM( 'No transpose', 'Conjugate 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 CTRMM( 'Right', 'Lower', 'Conjugate transpose',
|
|
$ 'Unit', I, IB-1,
|
|
$ ONE, A( I+1, I ), LDA, WORK, LDWORK )
|
|
DO 30 J = 0, IB-2
|
|
CALL CAXPY( 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 CLARFB( 'Left', 'Conjugate transpose', 'Forward',
|
|
$ 'Columnwise',
|
|
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
|
|
$ A( I+1, I+IB ), LDA, WORK, LDWORK )
|
|
40 CONTINUE
|
|
END IF
|
|
*
|
|
* Use unblocked code to reduce the rest of the matrix
|
|
*
|
|
CALL CGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
|
|
WORK( 1 ) = IWS
|
|
*
|
|
RETURN
|
|
*
|
|
* End of CGEHRD
|
|
*
|
|
END
|