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Werner Saar
2015-11-20 09:45:46 +01:00
parent 64db4576e6
commit 33e37d01b3
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*> \brief <b> DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGES3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgges3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgges3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgges3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
* SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
* LDVSR, WORK, LWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR, SORT
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
* .. Function Arguments ..
* LOGICAL SELCTG
* EXTERNAL SELCTG
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
*> the generalized eigenvalues, the generalized real Schur form (S,T),
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR). This gives the generalized Schur factorization
*>
*> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T.The
*> leading columns of VSL and VSR then form an orthonormal basis for the
*> corresponding left and right eigenspaces (deflating subspaces).
*>
*> (If only the generalized eigenvalues are needed, use the driver
*> DGGEV instead, which is faster.)
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*> [ a 0 ]
*> [ 0 b ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is CHARACTER*1
*> Specifies whether or not to order the eigenvalues on the
*> diagonal of the generalized Schur form.
*> = 'N': Eigenvalues are not ordered;
*> = 'S': Eigenvalues are ordered (see SELCTG);
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*> SELCTG must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'N', SELCTG is not referenced.
*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
*> one of a complex conjugate pair of eigenvalues is selected,
*> then both complex eigenvalues are selected.
*>
*> Note that in the ill-conditioned case, a selected complex
*> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
*> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
*> in this case.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the first of the pair of matrices.
*> On exit, A has been overwritten by its generalized Schur
*> form S.
*> \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 second of the pair of matrices.
*> On exit, B has been overwritten by its generalized Schur
*> form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*> for which SELCTG is true. (Complex conjugate pairs for which
*> SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
*> form (S,T) that would result if the 2-by-2 diagonal blocks of
*> the real Schur form of (A,B) were further reduced to
*> triangular form using 2-by-2 complex unitary transformations.
*> 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) negative.
*>
*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*> may easily over- or underflow, and BETA(j) may even be zero.
*> Thus, the user should avoid naively computing the ratio.
*> However, ALPHAR and ALPHAI 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] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= 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.
*>
*> 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] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = '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. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: =N+1: other than QZ iteration failed in DHGEQZ.
*> =N+2: after reordering, roundoff changed values of
*> some complex eigenvalues so that leading
*> eigenvalues in the Generalized Schur form no
*> longer satisfy SELCTG=.TRUE. This could also
*> be caused due to scaling.
*> =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date January 2015
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
$ LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
$ VSR, LDVSR, WORK, LWORK, BWORK, INFO )
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* January 2015
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
* .. Function Arguments ..
LOGICAL SELCTG
EXTERNAL SELCTG
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
$ LQUERY, LST2SL, WANTST
INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
$ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, LWKOPT
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
$ PVSR, SAFMAX, SAFMIN, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DIF( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHD3, DHGEQZ, DLABAD,
$ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
WANTST = LSAME( SORT, 'S' )
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -15
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -17
ELSE IF( LWORK.LT.6*N+16 .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
*
* Compute workspace
*
IF( INFO.EQ.0 ) THEN
CALL DGEQRF( N, N, B, LDB, WORK, WORK, -1, IERR )
LWKOPT = MAX( 6*N+16, 3*N+INT( WORK ( 1 ) ) )
CALL DORMQR( 'L', 'T', N, N, N, B, LDB, WORK, A, LDA, WORK,
$ -1, IERR )
LWKOPT = MAX( LWKOPT, 3*N+INT( WORK ( 1 ) ) )
IF( ILVSL ) THEN
CALL DORGQR( N, N, N, VSL, LDVSL, WORK, WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, 3*N+INT( WORK ( 1 ) ) )
END IF
CALL DGGHD3( JOBVSL, JOBVSR, N, 1, N, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, 3*N+INT( WORK ( 1 ) ) )
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, 1, N, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK, -1, IERR )
LWKOPT = MAX( LWKOPT, 2*N+INT( WORK ( 1 ) ) )
IF( WANTST ) THEN
CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ SDIM, PVSL, PVSR, DIF, WORK, -1, IDUM, 1,
$ IERR )
LWKOPT = MAX( LWKOPT, 2*N+INT( WORK ( 1 ) ) )
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGES3 ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
SMLNUM = SQRT( SAFMIN ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
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 DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
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 DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrix to make it more nearly triangular
*
ILEFT = 1
IRIGHT = N + 1
IWRK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWRK
IWRK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to matrix A
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VSL
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
IF( IROWS.GT.1 ) THEN
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
END IF
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VSR
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL DGGHD3( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, WORK( IWRK ), LWORK+1-IWRK,
$ IERR )
*
* Perform QZ algorithm, computing Schur vectors if desired
*
IWRK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWRK ), LWORK+1-IWRK, 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 50
END IF
*
* Sort eigenvalues ALPHA/BETA if desired
*
SDIM = 0
IF( WANTST ) THEN
*
* Undo scaling on eigenvalues before SELCTGing
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IERR )
END IF
IF( ILBSCL )
$ CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
*
* Select eigenvalues
*
DO 10 I = 1, N
BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
10 CONTINUE
*
CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL,
$ PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
$ IERR )
IF( IERR.EQ.1 )
$ INFO = N + 3
*
END IF
*
* Apply back-permutation to VSL and VSR
*
IF( ILVSL )
$ CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
IF( ILVSR )
$ CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
* Check if unscaling would cause over/underflow, if so, rescale
* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
IF( ILASCL ) THEN
DO 20 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN
WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.
$ ( ANRMTO / ANRM ) .OR.
$ ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
$ THEN
WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
20 CONTINUE
END IF
*
IF( ILBSCL ) THEN
DO 30 I = 1, N
IF( ALPHAI( I ).NE.ZERO ) THEN
IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
$ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
BETA( I ) = BETA( I )*WORK( 1 )
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
END IF
END IF
30 CONTINUE
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
IF( WANTST ) THEN
*
* Check if reordering is correct
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 40 I = 1, N
CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
IF( ALPHAI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
40 CONTINUE
*
END IF
*
50 CONTINUE
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGGES3
*
END