*> \brief <b> CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGGEV3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggev3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggev3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggev3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
* $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* REAL RWORK( * )
* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGGEV3 computes for a pair of N-by-N complex nonsymmetric matrices
*> (A,B), the generalized eigenvalues, and optionally, the left and/or
*> right generalized eigenvectors.
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right generalized eigenvector v(j) corresponding to the
*> generalized eigenvalue lambda(j) of (A,B) satisfies
*>
*> A * v(j) = lambda(j) * B * v(j).
*>
*> The left generalized eigenvector u(j) corresponding to the
*> generalized eigenvalues lambda(j) of (A,B) satisfies
*>
*> u(j)**H * A = lambda(j) * u(j)**H * B
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> On entry, the matrix A in the pair (A,B).
*> On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB, N)
*> On entry, the matrix B in the pair (A,B).
*> On exit, B has been overwritten.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX array, dimension (N)
*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
*> generalized eigenvalues.
*>
*> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
*> underflow, and BETA(j) may even be zero. Thus, the user
*> should avoid naively computing the ratio alpha/beta.
*> However, ALPHA will be always less than and usually
*> comparable with norm(A) in magnitude, and BETA always less
*> than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is COMPLEX array, dimension (LDVL,N)
*> If JOBVL = 'V', the left generalized eigenvectors u(j) are
*> stored one after another in the columns of VL, in the same
*> order as their eigenvalues.
*> Each eigenvector is scaled so the largest component has
*> abs(real part) + abs(imag. part) = 1.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is COMPLEX array, dimension (LDVR,N)
*> If JOBVR = 'V', the right generalized eigenvectors v(j) are
*> stored one after another in the columns of VR, in the same
*> order as their eigenvalues.
*> Each eigenvector is scaled so the largest component has
*> abs(real part) + abs(imag. part) = 1.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= MAX(1,2*N).
*> For good performance, LWORK must 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] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> =1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHA(j) and BETA(j) should be
*> correct for j=INFO+1,...,N.
*> > N: =N+1: other then QZ iteration failed in CHGEQZ,
*> =N+2: error return from CTGEVC.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggev3
*
* =====================================================================
SUBROUTINE CGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA,
$ BETA,
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR,
$ LWKOPT, LWKMIN
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
COMPLEX X
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHD3, CLAQZ0,
$ CLACPY,
$ CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANGE, SLAMCH, SROUNDUP_LWORK
EXTERNAL LSAME, CLANGE, SLAMCH,
$ SROUNDUP_LWORK
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, REAL, SQRT
* ..
* .. Statement Functions ..
REAL ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
LWKMIN = MAX( 1, 2*N )
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
*
* Compute workspace
*
IF( INFO.EQ.0 ) THEN
CALL CGEQRF( N, N, B, LDB, WORK, WORK, -1, IERR )
LWKOPT = MAX( LWKMIN, N+INT( WORK( 1 ) ) )
CALL CUNMQR( 'L', 'C', N, N, N, B, LDB, WORK, A, LDA, WORK,
$ -1, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
IF( ILVL ) THEN
CALL CUNGQR( N, N, N, VL, LDVL, WORK, WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
END IF
IF( ILV ) THEN
CALL CGGHD3( JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
CALL CLAQZ0( 'S', JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, -1,
$ RWORK, 0, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
ELSE
CALL CGGHD3( 'N', 'N', N, 1, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
CALL CLAQZ0( 'E', JOBVL, JOBVR, N, 1, N, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, -1,
$ RWORK, 0, IERR )
LWKOPT = MAX( LWKOPT, N+INT( WORK( 1 ) ) )
END IF
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
ELSE
WORK( 1 ) = SROUNDUP_LWORK( LWKOPT )
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGGEV3 ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SMLNUM = SLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
IF( ILASCL )
$ CALL CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
IF( ILBSCL )
$ CALL CLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrices A, B to isolate eigenvalues if possible
*
ILEFT = 1
IRIGHT = N + 1
IRWRK = IRIGHT + N
CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = 1
IWRK = ITAU + IROWS
CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
*
CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VL
*
IF( ILVL ) THEN
CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VR
*
IF( ILVR )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL CGGHD3( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, WORK( IWRK ), LWORK+1-IWRK,
$ IERR )
ELSE
CALL CGGHD3( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR,
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
* Schur form and Schur vectors)
*
IWRK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL CLAQZ0( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ),
$ LWORK+1-IWRK, RWORK( IRWRK ), 0, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
INFO = IERR
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
INFO = IERR - N
ELSE
INFO = N + 1
END IF
GO TO 70
END IF
*
* Compute Eigenvectors
*
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL CTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
$ LDVL,
$ VR, LDVR, N, IN, WORK( IWRK ), RWORK( IRWRK ),
$ IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 70
END IF
*
* Undo balancing on VL and VR and normalization
*
IF( ILVL ) THEN
CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VL, LDVL, IERR )
DO 30 JC = 1, N
TEMP = ZERO
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
10 CONTINUE
IF( TEMP.LT.SMLNUM )
$ GO TO 30
TEMP = ONE / TEMP
DO 20 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
20 CONTINUE
30 CONTINUE
END IF
IF( ILVR ) THEN
CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VR, LDVR, IERR )
DO 60 JC = 1, N
TEMP = ZERO
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
40 CONTINUE
IF( TEMP.LT.SMLNUM )
$ GO TO 60
TEMP = ONE / TEMP
DO 50 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
50 CONTINUE
60 CONTINUE
END IF
END IF
*
* Undo scaling if necessary
*
70 CONTINUE
*
IF( ILASCL )
$ CALL CLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
*
IF( ILBSCL )
$ CALL CLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
*
WORK( 1 ) = SROUNDUP_LWORK( LWKOPT )
RETURN
*
* End of CGGEV3
*
END