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OpenBLAS/lapack-netlib/SRC/dgegs.f

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*> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGES.
*>
*> DGEGS computes the eigenvalues, real Schur form, and, optionally,
*> left and or/right Schur vectors of a real matrix pair (A,B).
*> Given two square matrices A and B, the generalized real Schur
*> factorization has the form
*>
*> A = Q*S*Z**T, B = Q*T*Z**T
*>
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> DGEGV should be used instead. See DGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \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 matrix A.
*> On exit, the upper quasi-triangular matrix S from the
*> generalized real Schur factorization.
*> \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.
*> On exit, the upper triangular matrix T from the generalized
*> real Schur factorization.
*> \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] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> 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', the matrix of right Schur vectors Z.
*> 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. LWORK >= max(1,4*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 + 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. (A,B) are not in Schur
*> form, 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 DGGBAK (computing VSL)
*> =N+8: error return from DGGBAK (computing VSR)
*> =N+9: error return from DLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver 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 ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
$ LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. 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
*
* Test the input arguments
*
LWKMIN = MAX( 4*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( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.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 + N*( NB+1 )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGS ', -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' )
SMLNUM = N*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 ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* 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 ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (2*N words -- "work..." not actually used)
* 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 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
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 10
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 10
END IF
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ 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 10
END IF
END IF
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ 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 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGS
*
END
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