SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices
* (A,B), the generalized eigenvalues, and optionally, the left and/or
* right generalized eigenvectors.
*
* 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 generalized eigenvector v(j) corresponding to the
* generalized eigenvalue lambda(j) of (A,B) satisfies
*
* A * v(j) = lambda(j) * B * v(j).
*
* The left generalized eigenvector u(j) corresponding to the
* generalized eigenvalues 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).
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': do not compute the left generalized eigenvectors;
* = 'V': compute the left generalized eigenvectors.
*
* JOBVR (input) CHARACTER*1
* = 'N': do not compute the right generalized eigenvectors;
* = 'V': compute the right generalized eigenvectors.
*
* N (input) INTEGER
* The order of the matrices A, B, VL, and VR. N >= 0.
*
* A (input/output) COMPLEX*16 array, dimension (LDA, N)
* On entry, the matrix A in the pair (A,B).
* On exit, A has been overwritten.
*
* LDA (input) INTEGER
* The leading dimension of A. LDA >= max(1,N).
*
* B (input/output) COMPLEX*16 array, dimension (LDB, N)
* On entry, the matrix B in the pair (A,B).
* On exit, B has been overwritten.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB >= max(1,N).
*
* 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.
*
* 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).
*
* VL (output) COMPLEX*16 array, dimension (LDVL,N)
* If JOBVL = 'V', the left generalized eigenvectors u(j) are
* stored one after another in the columns of VL, in the same
* order as their eigenvalues.
* Each eigenvector is scaled so the largest component has
* abs(real part) + abs(imag. part) = 1.
* Not referenced if JOBVL = 'N'.
*
* LDVL (input) INTEGER
* The leading dimension of the matrix VL. LDVL >= 1, and
* if JOBVL = 'V', LDVL >= N.
*
* VR (output) COMPLEX*16 array, dimension (LDVR,N)
* If JOBVR = 'V', the right generalized eigenvectors v(j) are
* stored one after another in the columns of VR, in the same
* order as their eigenvalues.
* Each eigenvector is scaled so the largest component has
* abs(real part) + abs(imag. part) = 1.
* Not referenced if JOBVR = 'N'.
*
* LDVR (input) INTEGER
* The leading dimension of the matrix VR. LDVR >= 1, and
* if JOBVR = 'V', LDVR >= 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/output) DOUBLE PRECISION array, dimension (8*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. No eigenvectors have been
* calculated, but ALPHA(j) and BETA(j) should be
* correct for j=INFO+1,...,N.
* > N: =N+1: other then QZ iteration failed in DHGEQZ,
* =N+2: error return from DTGEVC.
*
* =====================================================================
*
* .. 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 ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR,
$ LWKMIN, LWKOPT
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SMLNUM, TEMP
COMPLEX*16 X
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD,
$ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR,
$ ZUNMQR
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
* ..
* .. Statement Functions ..
DOUBLE PRECISION ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
* ..
* .. 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
*
* 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( 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( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -13
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
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, 0 ) )
IF( ILVL ) 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 = -15
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGGEV ', -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' )
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 matrices A, B to isolate eigenvalues if possible
* (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
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
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 VL
* (Complex Workspace: need N, prefer N*NB)
*
IF( ILVL ) THEN
CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
IF( IROWS.GT.1 ) THEN
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
END IF
CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
END IF
*
* Initialize VR
*
IF( ILVR )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IERR )
ELSE
CALL ZGGHRD( '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 form and Schur vectors)
* (Complex Workspace: need N)
* (Real Workspace: need N)
*
IWRK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, 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 70
END IF
*
* Compute Eigenvectors
* (Real Workspace: need 2*N)
* (Complex Workspace: need 2*N)
*
IF( ILV ) THEN
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWRK ), RWORK( IRWRK ),
$ IERR )
IF( IERR.NE.0 ) THEN
INFO = N + 2
GO TO 70
END IF
*
* Undo balancing on VL and VR and normalization
* (Workspace: none needed)
*
IF( ILVL ) THEN
CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VL, LDVL, IERR )
DO 30 JC = 1, N
TEMP = ZERO
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
10 CONTINUE
IF( TEMP.LT.SMLNUM )
$ GO TO 30
TEMP = ONE / TEMP
DO 20 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
20 CONTINUE
30 CONTINUE
END IF
IF( ILVR ) THEN
CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VR, LDVR, IERR )
DO 60 JC = 1, N
TEMP = ZERO
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
40 CONTINUE
IF( TEMP.LT.SMLNUM )
$ GO TO 60
TEMP = ONE / TEMP
DO 50 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
50 CONTINUE
60 CONTINUE
END IF
END IF
*
* Undo scaling if necessary
*
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 )
*
70 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZGGEV
*
END