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+ SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
+ $ SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
+ $ LWORK, RWORK, BWORK, INFO )
+*
+* -- LAPACK driver routine (version 3.1) --
+* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+* November 2006
+*
+* .. Scalar Arguments ..
+ CHARACTER JOBVSL, JOBVSR, SORT
+ INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
+* ..
+* .. Array Arguments ..
+ LOGICAL BWORK( * )
+ DOUBLE PRECISION RWORK( * )
+ COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
+ $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
+ $ WORK( * )
+* ..
+* .. Function Arguments ..
+ LOGICAL SELCTG
+ EXTERNAL SELCTG
+* ..
+*
+* Purpose
+* =======
+*
+* ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
+* (A,B), the generalized eigenvalues, the generalized complex Schur
+* form (S, T), and optionally left and/or right 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. The leading
+* columns of VSL and VSR then form an unitary basis for the
+* corresponding left and right eigenspaces (deflating subspaces).
+*
+* (If only the generalized eigenvalues are needed, use the driver
+* ZGGEV 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, and even for both being zero.
+*
+* A pair of matrices (S,T) is in generalized complex Schur form if S
+* and T are upper triangular and, in addition, the diagonal elements
+* of T are non-negative real numbers.
+*
+* 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.
+* An eigenvalue ALPHA(j)/BETA(j) is selected if
+* SELCTG(ALPHA(j),BETA(j)) is true.
+*
+* 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+2 (See INFO below).
+*
+* 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), j=1,...,N and BETA(j),
+* j=1,...,N are the diagonals of the complex Schur form (A,B)
+* output by ZGGES. The 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.
+*
+* 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. LWORK >= max(1,2*N).
+* For good performance, LWORK must generally be larger.
+*
+* 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.
+*
+* RWORK (workspace) DOUBLE PRECISION array, dimension (8*N)
+*
+* 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 falied in ZTGSEN.
+*
+* =====================================================================
+*
+* .. Parameters ..
+ DOUBLE PRECISION ZERO, ONE
+ PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
+ COMPLEX*16 CZERO, CONE
+ PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
+ $ CONE = ( 1.0D0, 0.0D0 ) )
+* ..
+* .. Local Scalars ..
+ LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
+ $ LQUERY, WANTST
+ INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
+ $ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKMIN,
+ $ LWKOPT
+ DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
+ $ PVSR, SMLNUM
+* ..
+* .. Local Arrays ..
+ INTEGER IDUM( 1 )
+ 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' )
+*
+* 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 = -14
+ ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.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.)
+*
+ IF( INFO.EQ.0 ) THEN
+ LWKMIN = MAX( 1, 2*N )
+ LWKOPT = MAX( 1, N + N*ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) )
+ LWKOPT = MAX( LWKOPT, N +
+ $ N*ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, -1 ) )
+ IF( ILVSL ) THEN
+ LWKOPT = MAX( LWKOPT, N +
+ $ N*ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) )
+ END IF
+ WORK( 1 ) = LWKOPT
+*
+ IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
+ $ INFO = -18
+ END IF
+*
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'ZGGES ', -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 orthogonal 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 30
+ END IF
+*
+* Sort eigenvalues ALPHA/BETA if desired
+* (Workspace: none needed)
+*
+ IF( WANTST ) THEN
+*
+* Undo scaling on eigenvalues before selecting
+*
+ IF( ILASCL )
+ $ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, 1, ALPHA, N, IERR )
+ IF( ILBSCL )
+ $ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, 1, BETA, N, IERR )
+*
+* Select eigenvalues
+*
+ DO 10 I = 1, N
+ BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) )
+ 10 CONTINUE
+*
+ CALL ZTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHA,
+ $ 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
+* (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 20 I = 1, N
+ CURSL = SELCTG( ALPHA( I ), BETA( I ) )
+ IF( CURSL )
+ $ SDIM = SDIM + 1
+ IF( CURSL .AND. .NOT.LASTSL )
+ $ INFO = N + 2
+ LASTSL = CURSL
+ 20 CONTINUE
+*
+ END IF
+*
+ 30 CONTINUE
+*
+ WORK( 1 ) = LWKOPT
+*
+ RETURN
+*
+* End of ZGGES
+*
+ END
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