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SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
$ B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR,
$ LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK,
$ IWORK, LIWORK, BWORK, INFO )
*
* -- LAPACK driver routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
$ SDIM
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
$ WORK( * )
* ..
* .. Function Arguments ..
LOGICAL SELCTG
EXTERNAL SELCTG
* ..
*
* Purpose
* =======
*
* ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
* (A,B), the generalized eigenvalues, the complex Schur form (S,T),
* and, optionally, the left and/or right matrices of Schur vectors (VSL
* and VSR). This gives the generalized Schur factorization
*
* (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
*
* where (VSR)**H is the conjugate-transpose of VSR.
*
* Optionally, it also orders the eigenvalues so that a selected cluster
* of eigenvalues appears in the leading diagonal blocks of the upper
* triangular matrix S and the upper triangular matrix T; computes
* a reciprocal condition number for the average of the selected
* eigenvalues (RCONDE); and computes a reciprocal condition number for
* the right and left deflating subspaces corresponding to the selected
* eigenvalues (RCONDV). The leading columns of VSL and VSR then form
* an orthonormal basis for the corresponding left and right eigenspaces
* (deflating subspaces).
*
* 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 for both being zero.
*
* A pair of matrices (S,T) is in generalized complex Schur form if T is
* upper triangular with non-negative diagonal and S is upper
* triangular.
*
* Arguments
* =========
*
* JOBVSL (input) CHARACTER*1
* = 'N': do not compute the left Schur vectors;
* = 'V': compute the left Schur vectors.
*
* JOBVSR (input) CHARACTER*1
* = 'N': do not compute the right Schur vectors;
* = 'V': compute the right Schur vectors.
*
* SORT (input) 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).
*
* SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX*16 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.
* Note that a selected complex eigenvalue may no longer satisfy
* SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
* ordering may change the value of complex eigenvalues
* (especially if the eigenvalue is ill-conditioned), in this
* case INFO is set to N+3 see INFO below).
*
* SENSE (input) CHARACTER*1
* Determines which reciprocal condition numbers are computed.
* = 'N' : None are computed;
* = 'E' : Computed for average of selected eigenvalues only;
* = 'V' : Computed for selected deflating subspaces only;
* = 'B' : Computed for both.
* If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
*
* N (input) INTEGER
* The order of the matrices A, B, VSL, and VSR. N >= 0.
*
* A (input/output) COMPLEX*16 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.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) COMPLEX*16 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.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* SDIM (output) INTEGER
* If SORT = 'N', SDIM = 0.
* If SORT = 'S', SDIM = number of eigenvalues (after sorting)
* for which SELCTG is true.
*
* ALPHA (output) COMPLEX*16 array, dimension (N)
* BETA (output) COMPLEX*16 array, dimension (N)
* On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
* generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are
* the diagonals of the complex Schur form (S,T). BETA(j) will
* be non-negative real.
*
* 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).
*
* VSL (output) COMPLEX*16 array, dimension (LDVSL,N)
* If JOBVSL = 'V', VSL will contain the left Schur vectors.
* Not referenced if JOBVSL = 'N'.
*
* LDVSL (input) INTEGER
* The leading dimension of the matrix VSL. LDVSL >=1, and
* if JOBVSL = 'V', LDVSL >= N.
*
* VSR (output) COMPLEX*16 array, dimension (LDVSR,N)
* If JOBVSR = 'V', VSR will contain the right Schur vectors.
* Not referenced if JOBVSR = 'N'.
*
* LDVSR (input) INTEGER
* The leading dimension of the matrix VSR. LDVSR >= 1, and
* if JOBVSR = 'V', LDVSR >= N.
*
* RCONDE (output) DOUBLE PRECISION array, dimension ( 2 )
* If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
* reciprocal condition numbers for the average of the selected
* eigenvalues.
* Not referenced if SENSE = 'N' or 'V'.
*
* RCONDV (output) DOUBLE PRECISION array, dimension ( 2 )
* If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
* reciprocal condition number for the selected deflating
* subspaces.
* Not referenced if SENSE = 'N' or 'E'.
*
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
* LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
* LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.
* Note also that an error is only returned if
* LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
* not be large enough.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the bound on the optimal size of the WORK
* array and the minimum size of the IWORK array, returns these
* values as the first entries of the WORK and IWORK arrays, and
* no error message related to LWORK or LIWORK is issued by
* XERBLA.
*
* RWORK (workspace) DOUBLE PRECISION array, dimension ( 8*N )
* Real workspace.
*
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
* On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK.
* If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
* LIWORK >= N+2.
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the bound on the optimal size of the
* WORK array and the minimum size of the IWORK array, returns
* these values as the first entries of the WORK and IWORK
* arrays, and no error message related to LWORK or LIWORK is
* issued by XERBLA.
*
* BWORK (workspace) LOGICAL array, dimension (N)
* Not referenced if SORT = 'N'.
*
* INFO (output) 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 ALPHA(j) and BETA(j) should be correct for
* j=INFO+1,...,N.
* > N: =N+1: other than QZ iteration failed in ZHGEQZ
* =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 ZTGSEN.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
$ LQUERY, WANTSB, WANTSE, WANTSN, WANTST, WANTSV
INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
$ ILEFT, ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK,
$ LIWMIN, LWRK, MAXWRK, MINWRK
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
$ PR, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DIF( 2 )
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD,
$ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGSEN, ZUNGQR,
$ ZUNMQR
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC 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' )
WANTSN = LSAME( SENSE, 'N' )
WANTSE = LSAME( SENSE, 'E' )
WANTSV = LSAME( SENSE, 'V' )
WANTSB = LSAME( SENSE, 'B' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
IF( WANTSN ) THEN
IJOB = 0
ELSE IF( WANTSE ) THEN
IJOB = 1
ELSE IF( WANTSV ) THEN
IJOB = 2
ELSE IF( WANTSB ) THEN
IJOB = 4
END IF
*
* Test the input arguments
*
INFO = 0
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( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
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
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.)
*
IF( INFO.EQ.0 ) THEN
IF( N.GT.0) THEN
MINWRK = 2*N
MAXWRK = N*(1 + ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) )
MAXWRK = MAX( MAXWRK, N*( 1 +
$ ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, -1 ) ) )
IF( ILVSL ) THEN
MAXWRK = MAX( MAXWRK, N*( 1 +
$ ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) ) )
END IF
LWRK = MAXWRK
IF( IJOB.GE.1 )
$ LWRK = MAX( LWRK, N*N/2 )
ELSE
MINWRK = 1
MAXWRK = 1
LWRK = 1
END IF
WORK( 1 ) = LWRK
IF( WANTSN .OR. N.EQ.0 ) THEN
LIWMIN = 1
ELSE
LIWMIN = N + 2
END IF
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -21
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY) THEN
INFO = -24
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGGESX', -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' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = ZLANGE( '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 ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = ZLANGE( '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 ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
* Permute the matrix to make it more nearly triangular
* (Real Workspace: need 6*N)
*
ILEFT = 1
IRIGHT = N + 1
IRWRK = IRIGHT + N
CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWRK ), IERR )
*
* Reduce B to triangular form (QR decomposition of B)
* (Complex Workspace: need N, prefer N*NB)
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = 1
IWRK = ITAU + IROWS
CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
*
* Apply the unitary transformation to matrix A
* (Complex Workspace: need N, prefer N*NB)
*
CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
$ LWORK+1-IWRK, IERR )
*
* Initialize VSL
* (Complex Workspace: need N, prefer N*NB)
*
IF( ILVSL ) THEN
CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
IF( IROWS.GT.1 ) THEN
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
END IF
CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VSR
*
IF( ILVSR )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
* (Workspace: none needed)
*
CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IERR )
*
SDIM = 0
*
* Perform QZ algorithm, computing Schur vectors if desired
* (Complex Workspace: need N)
* (Real Workspace: need N)
*
IWRK = ITAU
CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWRK ),
$ LWORK+1-IWRK, RWORK( IRWRK ), 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 40
END IF
*
* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
* condition number(s)
*
IF( WANTST ) THEN
*
* Undo scaling on eigenvalues before SELCTGing
*
IF( ILASCL )
$ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
IF( ILBSCL )
$ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
*
* Select eigenvalues
*
DO 10 I = 1, N
BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) )
10 CONTINUE
*
* Reorder eigenvalues, transform Generalized Schur vectors, and
* compute reciprocal condition numbers
* (Complex Workspace: If IJOB >= 1, need MAX(1, 2*SDIM*(N-SDIM))
* otherwise, need 1 )
*
CALL ZTGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PL, PR,
$ DIF, WORK( IWRK ), LWORK-IWRK+1, IWORK, LIWORK,
$ IERR )
*
IF( IJOB.GE.1 )
$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
IF( IERR.EQ.-21 ) THEN
*
* not enough complex workspace
*
INFO = -21
ELSE
IF( IJOB.EQ.1 .OR. IJOB.EQ.4 ) THEN
RCONDE( 1 ) = PL
RCONDE( 2 ) = PR
END IF
IF( IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
RCONDV( 1 ) = DIF( 1 )
RCONDV( 2 ) = DIF( 2 )
END IF
IF( IERR.EQ.1 )
$ INFO = N + 3
END IF
*
END IF
*
* Apply permutation to VSL and VSR
* (Workspace: none needed)
*
IF( ILVSL )
$ CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSL, LDVSL, IERR )
*
IF( ILVSR )
$ CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL ZLASCL( 'U', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
END IF
*
IF( ILBSCL ) THEN
CALL ZLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
END IF
*
IF( WANTST ) THEN
*
* Check if reordering is correct
*
LASTSL = .TRUE.
SDIM = 0
DO 30 I = 1, N
CURSL = SELCTG( ALPHA( I ), BETA( I ) )
IF( CURSL )
$ SDIM = SDIM + 1
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
LASTSL = CURSL
30 CONTINUE
*
END IF
*
40 CONTINUE
*
WORK( 1 ) = MAXWRK
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of ZGGESX
*
END
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