You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
766 lines
25 KiB
766 lines
25 KiB
*> \brief <b> DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download DGEGV + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
|
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER JOBVL, JOBVR
|
|
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
|
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
|
* $ VR( LDVR, * ), WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> This routine is deprecated and has been replaced by routine DGGEV.
|
|
*>
|
|
*> DGEGV computes the eigenvalues and, optionally, the left and/or right
|
|
*> eigenvectors of a real matrix pair (A,B).
|
|
*> Given two square matrices A and B,
|
|
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
|
|
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
|
|
*> that
|
|
*>
|
|
*> A*x = lambda*B*x.
|
|
*>
|
|
*> An alternate form is to find the eigenvalues mu and corresponding
|
|
*> eigenvectors y such that
|
|
*>
|
|
*> mu*A*y = B*y.
|
|
*>
|
|
*> These two forms are equivalent with mu = 1/lambda and x = y if
|
|
*> neither lambda nor mu is zero. In order to deal with the case that
|
|
*> lambda or mu is zero or small, two values alpha and beta are returned
|
|
*> for each eigenvalue, such that lambda = alpha/beta and
|
|
*> mu = beta/alpha.
|
|
*>
|
|
*> The vectors x and y in the above equations are right eigenvectors of
|
|
*> the matrix pair (A,B). Vectors u and v satisfying
|
|
*>
|
|
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
|
|
*>
|
|
*> are left eigenvectors of (A,B).
|
|
*>
|
|
*> Note: this routine performs "full balancing" on A and B
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] JOBVL
|
|
*> \verbatim
|
|
*> JOBVL is CHARACTER*1
|
|
*> = 'N': do not compute the left generalized eigenvectors;
|
|
*> = 'V': compute the left generalized eigenvectors (returned
|
|
*> in VL).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] JOBVR
|
|
*> \verbatim
|
|
*> JOBVR is CHARACTER*1
|
|
*> = 'N': do not compute the right generalized eigenvectors;
|
|
*> = 'V': compute the right generalized eigenvectors (returned
|
|
*> in VR).
|
|
*> \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 DOUBLE PRECISION array, dimension (LDA, N)
|
|
*> On entry, the matrix A.
|
|
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
|
|
*> contains the real Schur form of A from the generalized Schur
|
|
*> factorization of the pair (A,B) after balancing.
|
|
*> If no eigenvectors were computed, then only the diagonal
|
|
*> blocks from the Schur form will be correct. See DGGHRD and
|
|
*> DHGEQZ for details.
|
|
*> \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 DOUBLE PRECISION array, dimension (LDB, N)
|
|
*> On entry, the matrix B.
|
|
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
|
|
*> upper triangular matrix obtained from B in the generalized
|
|
*> Schur factorization of the pair (A,B) after balancing.
|
|
*> If no eigenvectors were computed, then only those elements of
|
|
*> B corresponding to the diagonal blocks from the Schur form of
|
|
*> A will be correct. See DGGHRD and DHGEQZ for details.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDB
|
|
*> \verbatim
|
|
*> LDB is INTEGER
|
|
*> The leading dimension of B. LDB >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] ALPHAR
|
|
*> \verbatim
|
|
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
|
|
*> The real parts of each scalar alpha defining an eigenvalue of
|
|
*> GNEP.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] ALPHAI
|
|
*> \verbatim
|
|
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
|
|
*> The imaginary parts of each scalar alpha defining an
|
|
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
|
|
*> eigenvalue is real; if positive, then the j-th and
|
|
*> (j+1)-st eigenvalues are a complex conjugate pair, with
|
|
*> ALPHAI(j+1) = -ALPHAI(j).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] BETA
|
|
*> \verbatim
|
|
*> BETA is DOUBLE PRECISION array, dimension (N)
|
|
*> The scalars beta that define the eigenvalues of GNEP.
|
|
*>
|
|
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
|
|
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
|
|
*> pair (A,B), in one of the forms lambda = alpha/beta or
|
|
*> mu = beta/alpha. Since either lambda or mu may overflow,
|
|
*> they should not, in general, be computed.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] VL
|
|
*> \verbatim
|
|
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
|
|
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
|
|
*> in the columns of VL, in the same order as their eigenvalues.
|
|
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
|
|
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
|
*> pair, then
|
|
*> u(j) = VL(:,j) + i*VL(:,j+1)
|
|
*> and
|
|
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
|
|
*>
|
|
*> Each eigenvector is scaled so that its largest component has
|
|
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
|
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
|
*> are set to zero.
|
|
*> 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 DOUBLE PRECISION array, dimension (LDVR,N)
|
|
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
|
|
*> in the columns of VR, in the same order as their eigenvalues.
|
|
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
|
|
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
|
*> pair, then
|
|
*> x(j) = VR(:,j) + i*VR(:,j+1)
|
|
*> and
|
|
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
|
|
*>
|
|
*> Each eigenvector is scaled so that its largest component has
|
|
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
|
|
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
|
*> are set to zero.
|
|
*> 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 DOUBLE PRECISION 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(1,8*N).
|
|
*> For good performance, LWORK must generally be larger.
|
|
*> To compute the optimal value of LWORK, call ILAENV to get
|
|
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
|
|
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
|
|
*> The optimal LWORK is:
|
|
*> 2*N + MAX( 6*N, N*(NB+1) ).
|
|
*>
|
|
*> 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.
|
|
*> = 1,...,N:
|
|
*> The QZ iteration failed. No eigenvectors have been
|
|
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
|
|
*> should be correct for j=INFO+1,...,N.
|
|
*> > N: errors that usually indicate LAPACK problems:
|
|
*> =N+1: error return from DGGBAL
|
|
*> =N+2: error return from DGEQRF
|
|
*> =N+3: error return from DORMQR
|
|
*> =N+4: error return from DORGQR
|
|
*> =N+5: error return from DGGHRD
|
|
*> =N+6: error return from DHGEQZ (other than failed
|
|
*> iteration)
|
|
*> =N+7: error return from DTGEVC
|
|
*> =N+8: error return from DGGBAK (computing VL)
|
|
*> =N+9: error return from DGGBAK (computing VR)
|
|
*> =N+10: error return from DLASCL (various calls)
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \ingroup doubleGEeigen
|
|
*
|
|
*> \par Further Details:
|
|
* =====================
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> Balancing
|
|
*> ---------
|
|
*>
|
|
*> This driver calls DGGBAL to both permute and scale rows and columns
|
|
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
|
|
*> and PL*B*R will be upper triangular except for the diagonal blocks
|
|
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
|
|
*> possible. The diagonal scaling matrices DL and DR are chosen so
|
|
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
|
|
*> one (except for the elements that start out zero.)
|
|
*>
|
|
*> After the eigenvalues and eigenvectors of the balanced matrices
|
|
*> have been computed, DGGBAK transforms the eigenvectors back to what
|
|
*> they would have been (in perfect arithmetic) if they had not been
|
|
*> balanced.
|
|
*>
|
|
*> Contents of A and B on Exit
|
|
*> -------- -- - --- - -- ----
|
|
*>
|
|
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
|
|
*> both), then on exit the arrays A and B will contain the real Schur
|
|
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
|
|
*> are computed, then only the diagonal blocks will be correct.
|
|
*>
|
|
*> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
|
|
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
|
|
*> \endverbatim
|
|
*>
|
|
* =====================================================================
|
|
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
|
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, 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 ..
|
|
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
|
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
|
$ VR( LDVR, * ), WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
|
|
CHARACTER CHTEMP
|
|
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
|
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
|
|
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
|
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
|
|
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
|
|
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
|
|
* ..
|
|
* .. Local Arrays ..
|
|
LOGICAL LDUMMA( 1 )
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
|
|
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
INTEGER ILAENV
|
|
DOUBLE PRECISION DLAMCH, DLANGE
|
|
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, INT, MAX
|
|
* ..
|
|
* .. 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
|
|
*
|
|
LWKMIN = MAX( 8*N, 1 )
|
|
LWKOPT = LWKMIN
|
|
WORK( 1 ) = LWKOPT
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
INFO = 0
|
|
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 = -12
|
|
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
|
INFO = -14
|
|
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
|
INFO = -16
|
|
END IF
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
|
|
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
|
|
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
|
|
NB = MAX( NB1, NB2, NB3 )
|
|
LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
|
|
WORK( 1 ) = LOPT
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DGEGV ', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
* Get machine constants
|
|
*
|
|
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
|
|
SAFMIN = DLAMCH( 'S' )
|
|
SAFMIN = SAFMIN + SAFMIN
|
|
SAFMAX = ONE / SAFMIN
|
|
ONEPLS = ONE + ( 4*EPS )
|
|
*
|
|
* Scale A
|
|
*
|
|
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
|
|
ANRM1 = ANRM
|
|
ANRM2 = ONE
|
|
IF( ANRM.LT.ONE ) THEN
|
|
IF( SAFMAX*ANRM.LT.ONE ) THEN
|
|
ANRM1 = SAFMIN
|
|
ANRM2 = SAFMAX*ANRM
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( ANRM.GT.ZERO ) THEN
|
|
CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 10
|
|
RETURN
|
|
END IF
|
|
END IF
|
|
*
|
|
* Scale B
|
|
*
|
|
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
|
|
BNRM1 = BNRM
|
|
BNRM2 = ONE
|
|
IF( BNRM.LT.ONE ) THEN
|
|
IF( SAFMAX*BNRM.LT.ONE ) THEN
|
|
BNRM1 = SAFMIN
|
|
BNRM2 = SAFMAX*BNRM
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( BNRM.GT.ZERO ) THEN
|
|
CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 10
|
|
RETURN
|
|
END IF
|
|
END IF
|
|
*
|
|
* Permute the matrix to make it more nearly triangular
|
|
* Workspace layout: (8*N words -- "work" requires 6*N words)
|
|
* left_permutation, right_permutation, work...
|
|
*
|
|
ILEFT = 1
|
|
IRIGHT = N + 1
|
|
IWORK = IRIGHT + N
|
|
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 1
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Reduce B to triangular form, and initialize VL and/or VR
|
|
* Workspace layout: ("work..." must have at least N words)
|
|
* left_permutation, right_permutation, tau, work...
|
|
*
|
|
IROWS = IHI + 1 - ILO
|
|
IF( ILV ) THEN
|
|
ICOLS = N + 1 - ILO
|
|
ELSE
|
|
ICOLS = IROWS
|
|
END IF
|
|
ITAU = IWORK
|
|
IWORK = ITAU + IROWS
|
|
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
|
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
|
$ LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 3
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
|
|
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
|
$ VL( ILO+1, ILO ), LDVL )
|
|
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
|
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
|
$ IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 4
|
|
GO TO 120
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( ILVR )
|
|
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
|
|
*
|
|
* Reduce to generalized Hessenberg form
|
|
*
|
|
IF( ILV ) THEN
|
|
*
|
|
* Eigenvectors requested -- work on whole matrix.
|
|
*
|
|
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, IINFO )
|
|
ELSE
|
|
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
|
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
|
END IF
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 5
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Perform QZ algorithm
|
|
* Workspace layout: ("work..." must have at least 1 word)
|
|
* left_permutation, right_permutation, work...
|
|
*
|
|
IWORK = ITAU
|
|
IF( ILV ) THEN
|
|
CHTEMP = 'S'
|
|
ELSE
|
|
CHTEMP = 'E'
|
|
END IF
|
|
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
|
|
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
|
INFO = IINFO
|
|
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
|
INFO = IINFO - N
|
|
ELSE
|
|
INFO = N + 6
|
|
END IF
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
IF( ILV ) THEN
|
|
*
|
|
* Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
|
|
*
|
|
IF( ILVL ) THEN
|
|
IF( ILVR ) THEN
|
|
CHTEMP = 'B'
|
|
ELSE
|
|
CHTEMP = 'L'
|
|
END IF
|
|
ELSE
|
|
CHTEMP = 'R'
|
|
END IF
|
|
*
|
|
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
|
|
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 7
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Undo balancing on VL and VR, rescale
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 8
|
|
GO TO 120
|
|
END IF
|
|
DO 50 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 50
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 10 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
|
|
10 CONTINUE
|
|
ELSE
|
|
DO 20 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
|
|
$ ABS( VL( JR, JC+1 ) ) )
|
|
20 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SAFMIN )
|
|
$ GO TO 50
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 30 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
30 CONTINUE
|
|
ELSE
|
|
DO 40 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
|
|
40 CONTINUE
|
|
END IF
|
|
50 CONTINUE
|
|
END IF
|
|
IF( ILVR ) THEN
|
|
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 9
|
|
GO TO 120
|
|
END IF
|
|
DO 100 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 100
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 60 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
|
|
60 CONTINUE
|
|
ELSE
|
|
DO 70 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
|
|
$ ABS( VR( JR, JC+1 ) ) )
|
|
70 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SAFMIN )
|
|
$ GO TO 100
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 80 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
80 CONTINUE
|
|
ELSE
|
|
DO 90 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
|
|
90 CONTINUE
|
|
END IF
|
|
100 CONTINUE
|
|
END IF
|
|
*
|
|
* End of eigenvector calculation
|
|
*
|
|
END IF
|
|
*
|
|
* Undo scaling in alpha, beta
|
|
*
|
|
* Note: this does not give the alpha and beta for the unscaled
|
|
* problem.
|
|
*
|
|
* Un-scaling is limited to avoid underflow in alpha and beta
|
|
* if they are significant.
|
|
*
|
|
DO 110 JC = 1, N
|
|
ABSAR = ABS( ALPHAR( JC ) )
|
|
ABSAI = ABS( ALPHAI( JC ) )
|
|
ABSB = ABS( BETA( JC ) )
|
|
SALFAR = ANRM*ALPHAR( JC )
|
|
SALFAI = ANRM*ALPHAI( JC )
|
|
SBETA = BNRM*BETA( JC )
|
|
ILIMIT = .FALSE.
|
|
SCALE = ONE
|
|
*
|
|
* Check for significant underflow in ALPHAI
|
|
*
|
|
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
|
|
*
|
|
ELSE IF( SALFAI.EQ.ZERO ) THEN
|
|
*
|
|
* If insignificant underflow in ALPHAI, then make the
|
|
* conjugate eigenvalue real.
|
|
*
|
|
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
|
|
ALPHAI( JC-1 ) = ZERO
|
|
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
|
|
ALPHAI( JC+1 ) = ZERO
|
|
END IF
|
|
END IF
|
|
*
|
|
* Check for significant underflow in ALPHAR
|
|
*
|
|
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
|
|
END IF
|
|
*
|
|
* Check for significant underflow in BETA
|
|
*
|
|
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
|
|
END IF
|
|
*
|
|
* Check for possible overflow when limiting scaling
|
|
*
|
|
IF( ILIMIT ) THEN
|
|
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
|
$ ABS( SBETA ) )
|
|
IF( TEMP.GT.ONE )
|
|
$ SCALE = SCALE / TEMP
|
|
IF( SCALE.LT.ONE )
|
|
$ ILIMIT = .FALSE.
|
|
END IF
|
|
*
|
|
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
|
|
*
|
|
IF( ILIMIT ) THEN
|
|
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
|
|
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
|
|
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
|
END IF
|
|
ALPHAR( JC ) = SALFAR
|
|
ALPHAI( JC ) = SALFAI
|
|
BETA( JC ) = SBETA
|
|
110 CONTINUE
|
|
*
|
|
120 CONTINUE
|
|
WORK( 1 ) = LWKOPT
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DGEGV
|
|
*
|
|
END
|
|
|