*> \brief <b> SGGEVX 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 SGGEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sggevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
* ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
* IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
* RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER BALANC, JOBVL, JOBVR, SENSE
* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* REAL ABNRM, BBNRM
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* INTEGER IWORK( * )
* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), LSCALE( * ),
* $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
* $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
*> right eigenvectors (RCONDV).
*>
*> 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 eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*> A * v(j) = lambda(j) * B * v(j) .
*>
*> The left eigenvector u(j) corresponding to the eigenvalue 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] BALANC
*> \verbatim
*> BALANC is CHARACTER*1
*> Specifies the balance option to be performed.
*> = 'N': do not diagonally scale or permute;
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> Computed reciprocal condition numbers will be for the
*> matrices after permuting and/or balancing. Permuting does
*> not change condition numbers (in exact arithmetic), but
*> balancing does.
*> \endverbatim
*>
*> \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] SENSE
*> \verbatim
*> SENSE is CHARACTER*1
*> Determines which reciprocal condition numbers are computed.
*> = 'N': none are computed;
*> = 'E': computed for eigenvalues only;
*> = 'V': computed for eigenvectors only;
*> = 'B': computed for eigenvalues and 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 REAL array, dimension (LDA, N)
*> On entry, the matrix A in the pair (A,B).
*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
*> or both, then A contains the first part of the real Schur
*> form of the "balanced" versions of the input A and B.
*> \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 REAL array, dimension (LDB, N)
*> On entry, the matrix B in the pair (A,B).
*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
*> or both, then B contains the second part of the real Schur
*> form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is REAL array, dimension (N)
*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*> be the generalized eigenvalues. 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
*> ALPHA/BETA. 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] VL
*> \verbatim
*> VL is REAL array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
*> after another in the columns of VL, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> u(j) = VL(:,j), the j-th column of VL. If the j-th and
*> (j+1)-th 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 will be scaled so the largest component have
*> 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 REAL array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
*> after another in the columns of VR, in the same order as
*> their eigenvalues. If the j-th eigenvalue is real, then
*> v(j) = VR(:,j), the j-th column of VR. If the j-th and
*> (j+1)-th eigenvalues form a complex conjugate pair, then
*> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*> Each eigenvector will be scaled so the largest component have
*> 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] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are integer values such that on exit
*> A(i,j) = 0 and B(i,j) = 0 if i > j and
*> j = 1,...,ILO-1 or i = IHI+1,...,N.
*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*> LSCALE is REAL array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the left side of A and B. If PL(j) is the index of the
*> row interchanged with row j, and DL(j) is the scaling
*> factor applied to row j, then
*> LSCALE(j) = PL(j) for j = 1,...,ILO-1
*> = DL(j) for j = ILO,...,IHI
*> = PL(j) for j = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*> RSCALE is REAL array, dimension (N)
*> Details of the permutations and scaling factors applied
*> to the right side of A and B. If PR(j) is the index of the
*> column interchanged with column j, and DR(j) is the scaling
*> factor applied to column j, then
*> RSCALE(j) = PR(j) for j = 1,...,ILO-1
*> = DR(j) for j = ILO,...,IHI
*> = PR(j) for j = IHI+1,...,N
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*> ABNRM is REAL
*> The one-norm of the balanced matrix A.
*> \endverbatim
*>
*> \param[out] BBNRM
*> \verbatim
*> BBNRM is REAL
*> The one-norm of the balanced matrix B.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is REAL array, dimension (N)
*> If SENSE = 'E' or 'B', the reciprocal condition numbers of
*> the eigenvalues, stored in consecutive elements of the array.
*> For a complex conjugate pair of eigenvalues two consecutive
*> elements of RCONDE are set to the same value. Thus RCONDE(j),
*> RCONDV(j), and the j-th columns of VL and VR all correspond
*> to the j-th eigenpair.
*> If SENSE = 'N' or 'V', RCONDE is not referenced.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is REAL array, dimension (N)
*> If SENSE = 'V' or 'B', the estimated reciprocal condition
*> numbers of the eigenvectors, stored in consecutive elements
*> of the array. For a complex eigenvector two consecutive
*> elements of RCONDV are set to the same value. If the
*> eigenvalues cannot be reordered to compute RCONDV(j),
*> RCONDV(j) is set to 0; this can only occur when the true
*> value would be very small anyway.
*> If SENSE = 'N' or 'E', RCONDV is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL 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).
*> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
*> LWORK >= max(1,6*N).
*> If SENSE = 'E', LWORK >= max(1,10*N).
*> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
*>
*> 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] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N+6)
*> If SENSE = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> If SENSE = 'N', BWORK is not referenced.
*> \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: =N+1: other than QZ iteration failed in SHGEQZ.
*> =N+2: error return from STGEVC.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date April 2012
*
*> \ingroup realGEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing a matrix pair (A,B) includes, first, permuting rows and
*> columns to isolate eigenvalues, second, applying diagonal similarity
*> transformation to the rows and columns to make the rows and columns
*> as close in norm as possible. The computed reciprocal condition
*> numbers correspond to the balanced matrix. Permuting rows and columns
*> will not change the condition numbers (in exact arithmetic) but
*> diagonal scaling will. For further explanation of balancing, see
*> section 4.11.1.2 of LAPACK Users' Guide.
*>
*> An approximate error bound on the chordal distance between the i-th
*> computed generalized eigenvalue w and the corresponding exact
*> eigenvalue lambda is
*>
*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
*>
*> An approximate error bound for the angle between the i-th computed
*> eigenvector VL(i) or VR(i) is given by
*>
*> EPS * norm(ABNRM, BBNRM) / DIF(i).
*>
*> For further explanation of the reciprocal condition numbers RCONDE
*> and RCONDV, see section 4.11 of LAPACK User's Guide.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
$ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. Scalar Arguments ..
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
REAL ABNRM, BBNRM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), LSCALE( * ),
$ RCONDE( * ), RCONDV( * ), RSCALE( * ),
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
$ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
CHARACTER CHTEMP
INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
$ MINWRK, MM
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD,
$ SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGEVC,
$ STGSNA, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. 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
*
NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
WANTSN = LSAME( SENSE, 'N' )
WANTSE = LSAME( SENSE, 'E' )
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.( NOSCL .OR. LSAME( BALANC, 'S' ) .OR.
$ LSAME( BALANC, 'B' ) ) ) THEN
INFO = -1
ELSE IF( IJOBVL.LE.0 ) THEN
INFO = -2
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -3
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
$ THEN
INFO = -4
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( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -14
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -16
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV. The workspace is
* computed assuming ILO = 1 and IHI = N, the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
IF( NOSCL .AND. .NOT.ILV ) THEN
MINWRK = 2*N
ELSE
MINWRK = 6*N
END IF
IF( WANTSE ) THEN
MINWRK = 10*N
ELSE IF( WANTSV .OR. WANTSB ) THEN
MINWRK = 2*N*( N + 4 ) + 16
END IF
MAXWRK = MINWRK
MAXWRK = MAX( MAXWRK,
$ N + N*ILAENV( 1, 'SGEQRF', ' ', N, 1, N, 0 ) )
MAXWRK = MAX( MAXWRK,
$ N + N*ILAENV( 1, 'SORMQR', ' ', N, 1, N, 0 ) )
IF( ILVL ) THEN
MAXWRK = MAX( MAXWRK, N +
$ N*ILAENV( 1, 'SORGQR', ' ', N, 1, N, 0 ) )
END IF
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -26
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
*
* Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( '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 SLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = SLANGE( '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 SLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute and/or balance the matrix pair (A,B)
* (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
*
CALL SGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
$ WORK, IERR )
*
* Compute ABNRM and BBNRM
*
ABNRM = SLANGE( '1', N, N, A, LDA, WORK( 1 ) )
IF( ILASCL ) THEN
WORK( 1 ) = ABNRM
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
$ IERR )
ABNRM = WORK( 1 )
END IF
*
BBNRM = SLANGE( '1', N, N, B, LDB, WORK( 1 ) )
IF( ILBSCL ) THEN
WORK( 1 ) = BBNRM
CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
$ IERR )
BBNRM = WORK( 1 )
END IF
*
* Reduce B to triangular form (QR decomposition of B)
* (Workspace: need N, prefer N*NB )
*
IROWS = IHI + 1 - ILO
IF( ILV .OR. .NOT.WANTSN ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = 1
IWRK = ITAU + IROWS
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the orthogonal transformation to A
* (Workspace: need N, prefer N*NB)
*
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VL and/or VR
* (Workspace: need N, prefer N*NB)
*
IF( ILVL ) THEN
CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
IF( ILVR )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
IF( ILV .OR. .NOT.WANTSN ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IERR )
ELSE
CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
END IF
*
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
* Schur forms and Schur vectors)
* (Workspace: need N)
*
IF( ILV .OR. .NOT.WANTSN ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
*
CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
$ LWORK, 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 130
END IF
*
* Compute Eigenvectors and estimate condition numbers if desired
* (Workspace: STGEVC: need 6*N
* STGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
* need N otherwise )
*
IF( ILV .OR. .NOT.WANTSN ) THEN
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, N, IN, WORK, IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 130
END IF
END IF
*
IF( .NOT.WANTSN ) THEN
*
* compute eigenvectors (STGEVC) and estimate condition
* numbers (STGSNA). Note that the definition of the condition
* number is not invariant under transformation (u,v) to
* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
* Schur form (S,T), Q and Z are orthogonal matrices. In order
* to avoid using extra 2*N*N workspace, we have to recalculate
* eigenvectors and estimate one condition numbers at a time.
*
PAIR = .FALSE.
DO 20 I = 1, N
*
IF( PAIR ) THEN
PAIR = .FALSE.
GO TO 20
END IF
MM = 1
IF( I.LT.N ) THEN
IF( A( I+1, I ).NE.ZERO ) THEN
PAIR = .TRUE.
MM = 2
END IF
END IF
*
DO 10 J = 1, N
BWORK( J ) = .FALSE.
10 CONTINUE
IF( MM.EQ.1 ) THEN
BWORK( I ) = .TRUE.
ELSE IF( MM.EQ.2 ) THEN
BWORK( I ) = .TRUE.
BWORK( I+1 ) = .TRUE.
END IF
*
IWRK = MM*N + 1
IWRK1 = IWRK + MM*N
*
* Compute a pair of left and right eigenvectors.
* (compute workspace: need up to 4*N + 6*N)
*
IF( WANTSE .OR. WANTSB ) THEN
CALL STGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
$ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
$ WORK( IWRK1 ), IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 130
END IF
END IF
*
CALL STGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
$ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
$ RCONDV( I ), MM, M, WORK( IWRK1 ),
$ LWORK-IWRK1+1, IWORK, IERR )
*
20 CONTINUE
END IF
END IF
*
* Undo balancing on VL and VR and normalization
* (Workspace: none needed)
*
IF( ILVL ) THEN
CALL SGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
$ LDVL, IERR )
*
DO 70 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 70
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
30 CONTINUE
ELSE
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
40 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 70
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 50 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
50 CONTINUE
ELSE
DO 60 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
60 CONTINUE
END IF
70 CONTINUE
END IF
IF( ILVR ) THEN
CALL SGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
$ LDVR, IERR )
DO 120 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 120
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
80 CONTINUE
ELSE
DO 90 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
90 CONTINUE
END IF
IF( TEMP.LT.SMLNUM )
$ GO TO 120
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 100 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
100 CONTINUE
ELSE
DO 110 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
*
* Undo scaling if necessary
*
130 CONTINUE
*
IF( ILASCL ) THEN
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
END IF
*
IF( ILBSCL ) THEN
CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of SGGEVX
*
END